Von Lokalisierung zu Delokalisierung
in ungeordneten Systemen
Numerishe Untersuhungen zum Einu von
Unordnung und Wehselwirkung auf die elektronishen
Transporteigenshaften von Festkorpern
Habilitationsshrift von
Rudolf A. R
omer
18. August 1999
zur Erlangung des akademishen Grades
dotor rerum naturalium habilitatus
eingereiht an der
Fakultat fur Naturwissenshaften
der
Tehnishen Universitat Chemnitz
From Loalization to Deloalization
in Disordered Systems
Numerial studies of the inuene of
disorder and many-body interations on the
eletroni transport properties of solids
Habilitation thesis of
Rudolf A. R
omer
18 August 1999
for the obtainment of the aademi degree of
dotor rerum naturalium habilitatus
submitted to the
Department of Natural Siene
at the
Chemnitz University of Tehnology
Revision : 1:2, reprints are not inluded
fur Yoginie
Number 8 (detail) by Jakson Pollok [226℄:
A disordered system with fratal dimension 1:5 [289℄.
v
vi
Prefae
The present online thesis reviews my sienti works on disordered systems from 1995 until
today. They an be roughly ategorized into three main lasses: (1) non-interating disordered systems, (2) the two-interating partile problem, and (3) the interplay of disorder
and many-partile interation. A (4)th hapter is onerned with the implementation of
the numerial algorithms. The struture of the thesis reets this division.
The reprints have been deleted from the online version due to memory onstraints. For the
onveniene of the reader, I give the full ond-mat itations in addition to the standard
paper referenes. Furthermore, the relevant itations appear at the end of eah hapter
inluding a page number referene to the ond-mat listing. Citations whih do not refer to
my work are numbered and are ordered in the bibliography aording to the names of the
authors.
I do not intend for this thesis to mature into a review artile. Nevertheless, I have tried to
give a thorough representation of related works and hope that the extensive bibliography
turns out to be useful to the reader. For eah paper, I briey introdue the physis ontext
whih is relevant for its understanding. Whenever possible, I have given full itations
inluding additional eletroni addresses for the preprint versions of the papers.
For readers who are interested in learning more about the subjet, I an reommend the
review artiles ited throughout the text. I personally very muh appreiate the review
[165℄ by B. Kramer and A. MaKinnon. Other nie reviews of disordered systems are
given in Ref. [22℄ (saling and interation eets), Ref. [294℄ (pre-saling arguments), Ref.
[24℄ (experiments on weak loalization), Ref. [52℄ (theory of weak loalization), Refs. [132,
202℄ (amorphous alloys), Ref. [302℄ (perturbation theory) and Ref. [179℄ (saling theory
and its extensions). Furthermore, useful reviews on related topis are in Refs. [255, 251℄
(RG approah the interations), Ref. [250℄ (Luttinger liquids), Ref. [119℄ (random matrix
theory), Ref. [316℄ (density-renormalization group), and Ref. [113℄ (quasirystals).
I hope that reading this thesis is enjoyable and perhaps even eduational. Mistakes and
errors have been erradiated as muh as possible. Nevertheless, some errors might have
esaped my attention and I would appreiate if the kind reader would let me know.
Rudolf A. Romer
Institut f
ur Physik
Tehnishe Universit
at Chemnitz
Email: r.roemerphysik.tu-hemnitz.de
D-09107 Chemnitz
URL: http://www.tu-hemnitz.de/~arr
vii
viii
Contents
Prefae
vii
List of gures
xi
List of publiations relevant for this Habilitation thesis
1 Introdution
1.1 The saling theory of loalization . . . . . . . . . . . . .
1.2 Experimental evidene in favor of saling . . . . . . . . .
1.3 Saling and the Anderson model of loalization . . . . .
1.4 An MIT in 2D! . . . . . . . . . . . . . . . . . . . . . . .
1.5 Numerial methods for disordered systems . . . . . . . .
1.6 A brief survey of this work . . . . . . . . . . . . . . . . .
1.7 A very brief survey of what is not disussed in this work
xiii
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2 Disorder and the Anderson model of loalization
2.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Anderson model with anisotropi hopping . . . . . . . . . . . . .
2.3 Comparisons with the theory of random matries and the onept of
versality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 An Anderson model with random hopping . . . . . . . . . . . . . . .
2.5 The inuene of topologial disorder . . . . . . . . . . . . . . . . . .
2.6 Thermoeletri transport oeÆients in the Anderson model . . . . .
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1
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10
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14
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19
21
24
3 The interplay of interations and disorder I: two interating partiles 26
3.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Numerial results for the random-matrix model of TIP . . . . . . . . . . . 28
3.3 Failure of the RMM approah for toy models . . . . . . . . . . . . . . . . . 29
3.4 The transfer-matrix approahes . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Using deimation to study TIP in random environments . . . . . . . . . . 37
3.6 The TIP eet in a 2D random environment . . . . . . . . . . . . . . . . . 38
3.7 The TIP eet lose to an MIT . . . . . . . . . . . . . . . . . . . . . . . . 39
ix
4 The interplay of interations and disorder II: nite partile density
4.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Integrable impurities in 1D . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 RG approah for a mesosopi Hubbard model . . . . . . . . . . . . . .
4.4 A DMRG study for the 1D Hubbard model . . . . . . . . . . . . . . . .
5 Massively parallel algorithms for the eigenvalue problem
systems
5.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 A parallel implementation of the TMM . . . . . . . . . . . .
5.3 A parallel implementation of CWI . . . . . . . . . . . . . . .
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in disordered
51
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6 Conlusions and Outlook
57
Aknowledgments
60
Bibliography
61
Curriulum vitae
84
Erklarung gema Habilitationsordnung x5 (1) 5/6
86
x
List of Figures
1.1
1.2
1.3
1.4
Shemati drawing of a disordered system . . . . . . . . . . . . . . . . . .
Shemati view of an extended and a loalized state . . . . . . . . . . . . .
Shemati view of the funtion aording to the saling hypothesis . . .
The averaged DOS for a 3D Anderson model and a shemati view of the
ondutivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Extended and loalized wave funtions for the 3D Anderson model . . . . .
1.6 Critial wave funtion for the 3D Anderson model . . . . . . . . . . . . . .
1.7 Shemati diagram of the TMM method . . . . . . . . . . . . . . . . . . .
2.1 Shemati drawing of plane and hains in the 3D anisotropi Anderson model
2.2 Critial wave funtion for the 3D anisotropi Anderson model . . . . . . .
2.3 Energy dependene of the loalization length for a 1D random hopping
Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Phase diagram of the 3D random hopping Anderson model . . . . . . . . .
2.5 Mosai and triangulation ordered and after a single ip . . . . . . . . . . .
2.6 Ordered and maximally disordered mosai latties . . . . . . . . . . . . . .
2.7 Ordered and maximally disordered triangulation latties . . . . . . . . . .
2.8 Standard periodi approximant of the Ammann-Beenker tiling . . . . . . .
2.9 Saling plot of the temperature-dependent ondutivity (T ) with ritial
exponent = 1:3 for various eletron llings . . . . . . . . . . . . . . . . .
1
3
4
7
8
9
11
16
18
20
21
22
22
22
23
25
3.1 Shemati piture of the TIP arguments of Ref. [256℄ . . . . . . . . . . . .
3.2 Dependene of 1 (M ) on disorder W for the 2D Anderson model at E = 0
3.3 Unnormalized distribution for the diagonal and o-diagonal oupling matrix
elements u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Dependene of uabs and utyp on 1 (M ) for the perturbed 2D Anderson model
3.5 Dependene of uabs and utyp on 1 for the perturbed 1D Anderson model .
3.6 Unnormalized distribution for the diagonal and o-diagonal oupling matrix
elements u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Shemati diagram for the TMM approah to TIP . . . . . . . . . . . . . .
3.8 Shemati diagram for the bag TMM . . . . . . . . . . . . . . . . . . . . .
31
32
33
4.1 Shemati diagram of the innite-size DMRG algorithm . . . . . . . . . . .
4.2 Shemati diagram of the nite-size DMRG algorithm . . . . . . . . . . . .
44
46
xi
27
30
33
35
36
4.3 Distribution P (J ) of random oupling strength for the eetive RAF in the
half-lled Hubbard model at large U . . . . . . . . . . . . . . . . . . . . . 48
4.4 The RG senario for random ouplings in the disordered XXX antiferromagnet 49
5.1 Shemati diagram of the two possible shemes for storing vetors in the
P-TMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Shemati diagram of the orthogonalization proedure for the P-TMM based
on the distributed vetor approah . . . . . . . . . . . . . . . . . . . . . .
5.3 Performane data for the P-TMM . . . . . . . . . . . . . . . . . . . . . . .
5.4 Comparison of the best P-TMM implementation on the GCPP with the
TMM implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Shemati diagram of P-CWI . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 FSS plots of the TIP loalization lengths in 2D . . . . . . . . . . . . . . .
xii
52
53
54
55
56
58
List of publiations relevant for this
Habilitation thesis
Peer-reviewed publiations
[EGRS99℄
Two interating partiles at the metal-insulator transition, A. Eilmes,
[EMMRS99℄
The Anderson model of loalization: a hallenge for modern eigenvalue methods, U. Elsner, V. Mehrmann, F. Milde, R. A. Romer,
[EPR97℄
[ERS98a℄
[LRS99a℄
[MRS97℄
[RP95℄
[RS97a℄
[RS97b℄
U. Grimm, R. A. Romer, and M. Shreiber, Eur. Phys. J. B 8, 547{554
(1999), (ond-mat/9808192)
and M. Shreiber, SIAM, J. Si. Comp 20, 2089{2102 (1999),
(physis/9802009)
Absene of baksattering at integrable impurities in one-dimensional
quantum many-body systems, H.-P. Ekle, A. Punnoose, and R. A.
Romer, Europhys. Lett. 39, 293{298 (1997), (ond-mat/9512139)
The two-dimensional Anderson model of loalization with random hopping, A. Eilmes, R. A. Romer, and M. Shreiber, Eur. Phys. J. B 1,
29{38 (1998), (ond-mat/9706265)
Interation-dependent enhanement of the loalisation length for two
interating partiles in a one-dimensional random potential, M. Lead-
beater, R. A. Romer, and M. Shreiber, Eur. Phys. J. B 8, 643{652
(1999), (ond-mat/9806255)
Multifratal analysis of the metal-insulator transition in anisotropi
systems, F. Milde, R. A. Romer, and M. Shreiber, Phys. Rev. B 55,
9463{9469 (1997), (ond-mat/9609276)
Enhaned Charge and Spin Currents in the One-Dimensional Disordered Mesosopi Hubbard Ring, R. A. Romer, and A. Punnoose, Phys.
Rev. B 52, 14809{14816 (1995), (ond-mat/9512139)
No enhanement of the loalization length for two interating partiles
in a random potential, R. A. Romer, and M. Shreiber, Phys. Rev. Lett.
78, 515-518 (1997), (ond-mat/9612034)
The enhanement of the loalization length for two-interating partiles
is vanishingly small in transfer-matrix alulations, R. A. Romer, and
M. Shreiber, Phys. Rev. Lett. 78, 4890 (1997), (ond-mat/9702246)
xiii
[RSV99℄
Two-interating partiles in a random potential: Numerial alulations
of the interation matrix elements, R. A. Romer, M. Shreiber, and T.
Vojta, phys. stat. sol. (b) 211, 681{691 (1999), (ond-mat/9702241)
[UMRS99℄
Smoothed universal orrelations in the two-dimensional Anderson
model, V. Uski, B. Mehlig, R. A. Romer, and M. Shreiber, Phys. Rev.
[VRS99℄
Thermoeletri transport properties in disordered systems near the Anderson transition, C. Villagonzalo, R. A. Romer, and M. Shreiber, a-
[ZGRS98℄
Level Spaings Distributions of Planar Quasiperiodi Tight-Binding
Models, J. X. Zhong, U. Grimm, R. A. Romer, and M. Shreiber, Phys.
B 59, 4080{4090 (1999), (ond-mat/9811258)
epted for publiation in Eur. Phys. J. B, (1999), (ond-mat/9904362)
Rev. Lett. 80, 3996{3999 (1998), (ond-mat/9710006)
Peer-reviewed onferene proeedings
[ERS98b℄
[GGR98℄
Critial behavior in the two-dimensional Anderson model of loalization
with random hopping, A. Eilmes, R. A. Romer, and M. Shreiber, phys.
stat. sol. (b) 205, 229{232 (1998)
Lax pair formulation for a small-polaron hain with integrable boundaries, X.-W. Guan, U. Grimm, and R. A. Romer, Ann. Phys. (Leipzig)
7, 518{522 (1998), (ond-mat/9811089)
[GRS98℄
Eletroni states in topologially disordered systems, U. Grimm, R.
[LRS98℄
Formation of eletron-hole pairs in a one-dimensional random environment, M. Leadbeater, R. A. Romer, and M. Shreiber, in Exitoni
[MR98℄
[RLS99a℄
[RLS99b℄
[RS98℄
[SGRZ99a℄
A. Romer, and G. Shlieker, Ann. Phys. (Leipzig) 7, 389{393 (1998),
(ond-mat/ond-mat/9811360)
Proesses in Condensed Matter, R. T. Williams and W. M. Yen, Editors, PV 98-25, p. 349{354, The Eletrohemial Soiety, Pennington,
NJ (1998), (ond-mat/9806350)
Energy level statistis at the metal-insulator transition in the Anderson
model of loalization with anisotropi hopping, F. Milde, and R. A.
Romer, Ann. Phys. (Leipzig) 7, 452{456 (1998), (ond-mat/9811186)
Saling the loalisation lengths for two interating partiles in onedimensional random potentials, R. A. Romer, M. Leadbeater, and M.
Shreiber, Physia A 266, 481{485 (1999), (ond-mat/9809369)
Numerial results for two interating partiles in a random environment,
R. A. Romer, M. Leadbeater and M. Shreiber, aepted for publiation
in Ann. Phys. (Leipzig) 8, (1999), (ond-mat/9908256)
Weak deloalization due to long-range interation for two eletrons in
a random potential hain, R. A. Romer, and M. Shreiber, phys. stat.
sol. (b) 205, 275{279 (1998)
Energy Levels of Quasiperiodi Hamiltonians, Spetral Unfolding, and
Random Matrix Theory, M. Shreiber, U. Grimm, R. A. Romer, and
xiv
[SGRZ99b℄
[SMREM99℄
[UMR98℄
[VR98℄
J. X. Zhong, aepted for publiation in Comp. Phys. Comm., (1999),
(ond-mat9811359)
Appliation of random matrix theory to quasiperiodi systems, M.
Shreiber, U. Grimm, R. A. Romer, and J. X. Zhong, Physia A 266,
477{480 (1999), (ond-mat/9809370)
Eletroni states in the Anderson model of loalization: benhmarking
eigenvalue algorithms, M. Shreiber, F. Milde, R. A. Romer, U. Elsner,
and V. Mehrmann, aepted for publiation in Comp. Phys. Comm.,
(1999)
A numerial study of wave-funtion and matrix-element statistis in the
Anderson model of loalization, V. Uski, B. Mehlig, and R. A. Romer,
Ann. Phys. (Leipzig) 7, 437{441 (1998), (ond-mat/9811119)
Low temperature behavior of the thermopower in disordered systems
near the Anderson transition, C. Villagonzalo, and R. A. Romer, Ann.
Phys. (Leipzig) 7, 394{399 (1998), (ond-mat/9811092)
Submitted manusripts and preprints
[PPR97℄
The Mott-Anderson transition in the disordered one-dimensional Hubbard model, R. Pai, A. Punnoose, and R. A. Romer, preprint series of
the SFB 393, No. 97-12 (Chemnitz, 1997), ond-mat/9704027, submitted to Phys. Rev. Lett., (1997), (ond-mat/9704027)
xv
xvi
Chapter 1
Introdution
Traditionally, researh in solid state physis has onentrated on the properties and appliations of rystalline solids [10℄. Nevertheless true rystals represent a lear minority
of real materials. Mostly, solids will have distortions of the rystalline struture due to,
e.g., disloations, vaanies, the presene of impurity atoms, and isotope defets as shown
shematially in Fig. 1.1. The distortions an even beome so large that the assumption
of a rystalline struture is longer meaningful as in, e.g., amorphous materials [132℄ and
glasses. Furthermore, the deviation from order need not only eet the dynami (mass)
and eletroni (harge) degrees of freedom but an also manifest itself in, e.g., spin disorder
suh as in spin glasses. In a sense, all these materials an be subsumed as being disordered
[22, 165, 179℄. How disorder an quantitatively or qualitatively inuene the (mehaniVacancy
Impurity
Dislocation
Isotope defect
Figure 1.1: Shemati drawing of a disordered system. The dierently olored dots denote dierent types of atoms suh as, e.g., boron or phosphate impurities in silion rystals. The size of the
dots is used to indiate dierent isotopes. The red line pitorially indiates the sattered path of
an eletron through the solid.
1
2
Introdution
al, thermal, eletri, magneti) properties of the material, depends on fators suh as the
strength of the disorder, the spatial dimension of the system and also on the exat property
under onsideration. However, the existene of at least a quantitative inuene is lear by
the existene of the well-known Ohm's law [10℄, whih relates the urrent I and the voltage
U in a metalli wire as
U = RI = I=G;
(1.1)
with R denoting the eletrial resistane and G = 1=R the eletrial ondutane of the
wire. For perfet rystals, it an be shown from the Bloh theorem [10℄ that R = 0 and
G = 1. However, even good ondutors suh as Au and Cu have a nite resistane even at
temperature T = 0 and thus annot be desribed by a lean rystalline struture [10, 165℄.
Furthermore, at low T , an even more signiant dierene between the behavior of rystals on the one hand and disordered solids on the other is seen. As was pointed out by
P.W. Anderson for the rst time in 1958 [7℄, suÆiently strong disorder an give rise to a
transition of the transport properties from onduting behavior with R > 0 to insulating
behavior with R = 1 as T ! 0. This phenomena is alled the disorder-driven metalinsulator transition (MIT) [22, 165, 179℄ and it is harateristi to non-rystalline solids.
The mehanism underlying this MIT was attributed by Anderson not to be due to a nite
gap in the energy spetrum whih is responsible for an MIT in band gap or Mott insulators
[10℄. Rather, he argued that the disorder will lead to interferene of the eletroni wave
funtion (x) with itself suh that it is no longer extended over the whole solid but is instead onned to a small part of the solid. This loalization eet exludes the possibility
of diusion at T = 0 and the system is an insulator.
In Fig. 1.2, I show shematially, how one might visualize these onepts for a long wire.
Note that the envelope of the loalized wave funtion away from the loalization enter
may be desribed by an exponential deay suh that
(x) exp
jx x0 j :
(1.2)
The deay length desribes the spatial extent of the loalized wave funtion and is alled
the loalization length.
1.1 The saling theory of loalization
A highly suessful theoretial approah to this disorder-indued MIT was put forward
in 1979 by Abrahams et al. [1℄. This \saling hypothesis of loalization" details the
existene of an MIT for non-interating eletrons in 3D disordered systems at zero magneti
eld B and in the absene of spin-orbit oupling. The starting point for the approah is
the realization that the sample size dependene of the (extensive) ondutane should be
investigated [294, 295, 296℄. We rst note that the resistane R of a metalli ube of
length L with ross-setional area A = L2 is given as R = L=A, with the resistivity.
3
Introdution
0.4
Ψ(x)
0.2
0.0
−0.2
−0.4
0
50
100
150
200
x
Figure 1.2: Shemati view of an extended (dashed blue line) and a loalized (solid green line)
state for a 1D disordered system with N = 200 sites and periodi boundary onditions. The thin
solid line indiates an exponential envelope with loalization length 12 aording to Eq. (1.2).
Generalizing this to arbitrary dimensions d, we thus see that the ondutane G an be
written as
2
e
G = Ld 2 = g
~
(1.3)
with = 1= denoting the ondutivity and d the spatial dimension onsidered. Furthermore, g denes the dimensionless ondutane, whih we will use in the following. On the
other hand, for strong disorder Ohm's law is no longer valid. The wave funtions will be
exponentially loalized as in Eq. (1.2) and thus the ondutane in a nite system will be
g exp( L=):
(1.4)
Dening a funtion
d ln g
;
(1.5)
d ln L
we an then easily ompute the behavior of (g ) for the two limits of metalli and insulating
behavior. Note that we have impliitly assumed that depends on g only and details suh
as energy, disorder or size dependene do not enter expliitly into the funtion argument.
In order to obtain at least the qualitative behavior of , one now assumes [1℄ that it is
possible to interpolate between the two limiting ases by a ontinuous, monotoni funtion
as shown in Fig. 1.3. For > 0, g inreases with inreasing L until = d 2 as we
expet for Ohmi behavior. On the other hand for < 0, g dereases as L inreases and
(g ) = ln g in the loalized regime. A speial point is (g ) = 0, whih indiates that a
hange in L does not hange g . This size independene orresponds to the MIT.
From Fig. 1.3, we see that < 0 in 1D and 2D and thus an inrease in L will drive the
system towards the insulator. This indiates that there are no extended states and thus
no MIT. However, the urve for 3D has both positive and negative parts and depending
(g ) =
4
Introdution
β
MIT
d-1
g~L
1
3D
metallic
ln g
insulating
2D
1D
-1
no
MIT
Figure 1.3: Shemati view of the funtion aording to the saling hypothesis. For the 1D and
the 2D ase, g(L) dereases with inreasing system size L sine < 0. Only for 3D, one has a
region with > 0 orresponding to extended states. The MIT in 3D orresponds to = 0.
on the initial value of g , i.e., on the strength of disorder or value of energy, an inrease of
L will drive the system either to metalli or to insulating behavior. Note that the saling
argument an not prove that the urve for the 2D ase does not ross the ritial line
= 0 for some weak disorder. However, as I will show later in this thesis, there is a large
body of numerial work that supports the omplete loalization of eletron states in 2D
at arbitrary weak disorder in the absene of many-body interations and at zero magneti
eld [165℄. Furthermore, perturbative results for weak disorder [302℄ show that the initial
orretions of the ln g = 1 results towards smaller g are given by < 0.
The senario proposed by the saling hypothesis is that of a ontinuous seond order phase
transition [179℄. Then the DC ondutivity and the loalization length should behave
as
/ (E E)s E E;
/ (E E ) E E:
(1.6)
(1.7)
with s = (d 2) due to the saling relations [301, 311℄. Introduing a similarly dened
dynamial exponent z for the temperature saling as (T ) / T 1=z , we an write the full
nite-temperature saling form as
(; T ) / [( E )=E
℄s
F
T
;
[( E )=E ℄z
(1.8)
with the hemial potential, F the saling funtion and z = d aording to Ref. [311℄.
The speial energy E is alled the mobility edge [7℄ and separates loalized states with
jE j > E from extended states with jE j < E. States diretly at the transition with E = E
are alled ritial and will be examined later in muh detail. Thus the disorder-driven MIT
has been reformulated in terms of the theory of ritial phenomena [22℄. In priniple, if one
an desribe experimental or theoretial data in a disordered system by the saling form of
Eq. (1.8) with the appropriate saling exponents, then one has shown that the transition
is of the Anderson type.
Introdution
5
1.2 Experimental evidene in favor of saling
Muh further work has subsequently supported these saling arguments at B = 0 experimentally, analytially and numerially [142, 165, 179℄. For weakly disordered systems, the
experiments of Bergmann [24℄ and others [68, 70, 160℄ in thin metalli lms established
the importane of quantum interferene orretions in the lassial transport behavior.
These experimental results are in good agreement with preditions of the theory of \weak
loalization" [4, 52℄.
Upon further inreasing the disorder, the MIT an be observed by measuring the ondutivity on the metalli side and the dieletri suseptibility on the insulating side of the
transition [292, 293℄. For doped Si:P, many experiments have been performed following
the original work of Paalanen and Thomas [217℄. The one-parameter saling hypothesis
has been beautifully validated in these experiments by, e.g., onstruting saling urves for
the ondutivity [304℄. The reent experiments in Si:P [235, 284, 285, 304℄ are onerned
with the exat estimation of the ritial exponent as in Eq. (1.6). The urrent estimate
is 1. The interest in the exat value of arises sine ompensated semiondutors
apparently have 1 as do amorphous metals [132, 202, 292℄. On the other hand, for
unompensated semiondutors one has also found 0:5 [217, 291, 292, 293℄. From the
point of view of saling theory, these dierenes are unexpeted, sine the ompensation
should only hange the degree of disorder present in the system: An n-type semiondutor
with donor (aeptor) onentration ND (NA ) has a arrier onentration n = ND NA .
With K = NA =ND , a ompensated system orresponds to a sea of eletrons with density
n sattering among impurities with onentration ND + NA = n[(1 + K )=(1 K )℄ [137℄.
The reent estimations of 1 for unompensated Si:P [304℄, based on a areful saling
analysis aording to Eq. (1.8) and a onsideration of various temperature regimes, may
suggest at least a hint towards to resolution of the exponent puzzle [137, 290℄.
Other experiments in 3D have been performed, e.g., on Si:B [27, 28℄. Saling aording
to Eq. (1.8) yields = 1:6. The large value of | as ompared to the Si:P data | was
attributed to the presene of interation eets. Another ompliation always present in
the experiments is the possible inhomogeneity in the impurity distribution, i.e., impurities
an luster and thus the transition may be loser to the lassial perolation senario [278℄
than to the quantum interferene problem studied here. An experimentally onvenient way
to onstrut homogeneous but disordered samples is the transmutation doping tehnique
[148, 237, 308℄. The main idea of this tehnique is the use the homogeneous properties of
neutron rays. Under irradiation with a thermal neutron 70 Ge beomes 71 Ge whih deays
within a half-life period of approximately 11 days to a 70 Ga aeptor. Annealing of the
sample an be used to redue other irradiation damage [237℄. The results again suggest
0:5, but have been questioned in Ref. [304℄.
As the loalization phenomenon in disordered solids is intrinsially due to the wave nature
of the eletrons, it an also be observed in other systems exhibiting wave motion [165℄.
Loalization has been studied, e.g. for water waves [189℄ in shallow basins with random
obstales, for light waves [195, 317, 321℄ in the presene of a ne dust of semiondutor material, for mirowaves [18, 69, 173, 282, 283℄ in mirowave avities with random satterers,
6
Introdution
and also for surfae plasmon polariton waves [32, 33, 34, 35, 36℄ on rough semionduting
surfaes.
In summary, many experimental settings have been established in whih the disorder-driven
MIT an be studied. The relevane of quantum interferene on the transport properties of
eletrons and more generally for waves in random media is thus proven. In most systems,
the one-parameter saling hypothesis an also be shown to be valid [22, 165℄. Deviations
from saling and variations in the value of the ritial exponents, that are not due to
experimental diÆulties suh as sample inhomogeneities and ooling problems, have been
observed in some ases as, e.g., in Refs. [27, 148, 237, 308℄ and attributed to the possible
inux of many-body Coulomb interation between the harge arriers.
1.3 Saling and the Anderson model of loalization
In order to model a disordered system of the type desribed in the last setions, let us onsider the ontinuum one-partile Hamiltonian, the so-alled Anderson model of loalization
[7℄,
N
p2 X
Hont =
+ V (r Rj )
2m j =1 j
(1.9)
where p denotes the momentum, m the (eetive) mass of the partile, N the number of
sites, and Vj is the potential of the lattie ion at site j . Disorder an then be introdued
into the Hamiltonian, by hoosing this potential to be randomly distributed aording to
some appropriately hosen distribution. In the following, I will mostly be onerned with
the lattie version of this Hamiltonian,
H=
X
j;k
tj;k jjihk j +
X
j
j jjihjj:
(1.10)
The o-diagonal matrix tj;k denotes the hopping integrals between the states fg at sites
fj g with the states f g at sites fkg and represents the disretization of the kineti term
of Eq. (1.9). The disorder in inorporated into the diagonal matrix j , whose elements
are random numbers in analogy to the potential term of (1.9). For simpliity, one mostly
assumes that = = 1 suh that there is only one state per site. Furthermore, the
random numbers j are usually taken from a uniform distribution [ W=2; W=2℄ with W
parameterizing the strength of the disorder. Other distributions suh as Gaussian and
Lorentzian have also been used [44℄.
The Anderson model of loalization has been used extensively in onjuntion with powerful
numerial methods in order to study the loalization problem [165℄. Perhaps the most
signiant result of these studies is the numerial veriation of the saling hypothesis by
the so-alled transfer-matrix method (TMM) in Refs. [191, 192, 224℄ in the beginning of the
1980's. The authors of these papers showed that one an onstrut saling urves whih lead
to the funtions of Fig. 1.3. For 3D these saling urves have two branhes orresponding
7
Introdution
−Ec
Ec
0.08
σ(E) (a.u.)
ρ(E)
0.06
0.04
0.02
0.00
−10
−5
0
5
10
Energy
Figure 1.4: The averaged DOS for a 3D Anderson model at W = 12 (left axis) and a shemati
view of the orresponding ondutivity (E ) (right axis). The mobility edges at E = E
separate loalized states (shaded) from extended states (no shading).
to the metalli and insulating phases, whereas in 1D and 2D only the insulating branh
exists [191, 192℄. Muh further work has sine been done, in partiular regarding the
omputation of the ritial exponents. The most reent numerial results | some of whih
shall be presented in the oming hapters | indiate that = 1:58 0:02 [190, 270℄ whih
is somewhat larger than in the experiments as reviewed above.
One of the distinguishing features of the Anderson-type MIT is the absene of any gap in
the density of states (DOS). Additionally, there annot be any singularities in the DOS
similar to the van Hove singularities in perfet rystals as that would imply some kind of
long-range order [165℄. Also, there are no sharp band edges, but rather smooth Lifshitz
tails [101, 188, 330℄. As an example, I show in Fig. 1.4 the DOS for a Anderson model in
3D at intermediate disorder. In the gure, I have also shematially indiated the behavior
of . As one inreases the disorder, the mobility edges [165℄ move inwards until nally
at some ritial disorder W (= 16:5t for a uniform distribution of the j ) all states are
loalized [44, 117℄. In Figs. 1.5 and 1.6, I show representations of typial wave funtions
for extended (W < W ), loalized (W > W ), and ritial (W = W ) regimes in 3D,
respetively. The lear distintions between the forms of these wave funtions an be used
to haraterize them qualitatively and quantitatively based on the onept of multifratality
[37, 114, 115, 116, 117, 244, 245, 246, 247℄ as I will show in hapter 2.
1.4 An MIT in 2D!
The arguments and results reviewed in the last setions have been aumulated over a
period of 25 years and established the general belief that, exept for \details" suh as the
8
Introdution
Figure 1.5: Extended (left) and loalized (right) wave funtions for the 3D Anderson model at
E = 0 with N = 483 and W = 1 and 25, respetively. Every site with probability j pi j2 larger
than the average N is shown as a box with volume j i j2N 1=3 . Boxes with j ij2 N 1=3 > 1000 are
plotted with blak edges. The olor sale distinguishes between dierent slies of the system along
the axis into the page. For the extended state, only every seond box in eah spatial diretion is
shown.
exat value of , the saling theory orretly desribes the physis of disordered systems
[22, 165, 179℄. In partiular, all investigations in 2D showed loalized behavior, at least
after appropriately saling the system towards the thermodynami limit. Aording to
these results a disordered material in 2D annot be a metal, but is an insulator.
In 1994, this belief was shaken by S. V. Kravhenko, G. V. Kravhenko, J. E. Furneaux,
V. M. Pudalov, and M. D'Iorio who performed a series of transport measurements in
a 2D eletron gas [167, 168℄. They showed that their resistivity vs. temperature data
ould be saled onto a single saling urve with two branhes orresponding to an MIT
just as for the usual 3D MIT. Saling as a funtion of applied eletri eld and eletron
density has subsequently been shown in Ref. [171℄ with 1:5 and z 0:8. These
experiments have sine then been repeated many times and extended to other materials
and experimental setups [87, 147, 159, 228℄. Whereas no MIT has been reported in an
n-doped 2D GaAs/AlGaAs heterostruture [142, 158℄, the MIT has been observed in a
2D GaAs/AlGaAs hole gas [128, 129, 262, 263, 323℄. Furthermore, saling has also been
argued to exist in Si/SiGe quantum wells [176℄.
Today it seems well established that an MIT in 2D exists. However, the physial mehanisms underlying the transition are still not understood. For example, it has been
shown that the metalli state vanishes upon swithing on an external magneti eld B
[170, 264, 238, 231℄. Certainly the most striking feature of these experiments is the observation that the MIT happens at very low eletron onentration around n 1010 m 2 .
On an intuitive level, sreening of the eletrons will be less eetive at low density and the
long-range harater of the eletron-eletron interation should be quite prominent. Another way of saying this is by noting that in 2D the Fermi energy EF sales linearly with
Introdution
9
Figure 1.6: Critial wave funtion for the 3D Anderson model at E = 0 with N = 100 and
W = 16:5. Sizes of the boxes and olor oding are as in Fig. 1.5. Note that the spatial struture
of the wave funtion is lose to being extended throughout the system as well as being loalized
to ertain distint regions in spae, i.e., it is a multifratal entity [86, 194℄.
10
Introdution
p
the eletron density ne , whereas the typial interation energy Etyp 1=rtyp n. Thus
at low values of n, the interation energy beomes important. Following these arguments,
most reent attempts at a theoretial explanation of the 2D MIT have foused on the problem of whether interation an give rise to a metalli behavior when without interation
the system would be an insulator [67℄. One way in whih this might be envisaged is that
the observed insulator-metal transition might be an insulator-to-superondutor transition
[23, 220, 221, 222, 327℄. Furthermore at low enough arrier density, one expets to see the
Wigner rystal [51℄. Other possible explanations are onerned with the expliit inuene
of the MOSFET struture on the 2D eletron gas [5, 133, 169℄ and the importane of spinorbit interation and the spin of the eletron [54, 55, 230℄. Regarding the latter proposal, it
has been observed experimentally that it is irrelevant whether the B eld is perpendiular
or parallel to the 2D eletron gas [170, 231℄: the MIT vanishes in both ases. However,
even this is being questioned experimentally by even more reent data as in Ref. [209℄.
Thus many proposals for an explanation of this lear ontradition to the traditional saling
theory of disordered systems [1℄ exist, but at present none of them an onviningly provide
an entirely satisfatory solution. The resolution of this ontradition onstitutes perhaps
the most hallenging problem in the physis of disordered systems today.
1.5 Numerial methods for disordered systems
The numerial studies presented in this work are to a large extent based on methods whih
use algorithms for the numerial diagonalization of large sparse matries. Most of these
methods have been well doumented in the literature and I shall thus only briey review
them here. In hapter 5, I will look at some of these algorithms more arefully from the
point of view of parallelization.
The preferred numerial method for aurately omputing loalization lengths in disordered
quantum systems is the transfer-matrix method (TMM) [166, 191, 192, 224℄. The TMM
is based on a reursive reformulation of the Shrodinger equation suh that, e.g. in a 2D
strip of width M , length K M and uniform hopping tj;k = 1,
n+1;m
= (E
n;m )
n;m
n;m
1
n;m+1
n 1;m
(1.11)
where n;m is the wave funtion at site (n; m). Eq. (1.11) an be reformulated into a matrix
equation as
n+1
n
=
= Tn
(E
n
n
n
1
H? )
1
;
1
0
n
n
1
(1.12)
(1.13)
where n = ( n;1 ; : : : ; n;M )T denotes the wave funtion at all sites of the nth slie,
n = diag(n;1 ; : : : ; n;M ), and H? represents the hopping terms in the transverse diretion.
The evolution of the wave funtion is given by the produt of the transfer matries K =
11
1.4
1.2
1.0
Ψn,m
m+1
m
m-1
n-1
n+1
}
M
convergence
1/λ
Introdution
matrixmultiplication
n
orthonormalizations
Figure 1.7: Shemati diagram of the TMM method for the example of a 2D model. At slie n and
position m, one needs to know the (past) n 1;m value, the (present) values of n;m and n;m1
and an then ompute the (future) value of n+1;m. The red lines indiate the orthogonalizations.
The diagram at the top shows the behavior of the typial onvergene for a Lyapunov exponent
measured after eah reorthonormalization, the atual number of reorthonormalization used here
exeeds 104 . The olors of n;m are hosen suh that they indiate the eventual onvergene.
TK TK 1 : : : T2 T1 . Aording to Oselede's theorem [213℄ the eigenvalues exp[i (M )℄ of
= limK !1(Ky K )1=2K exist and the smallest Lyapunov exponent min > 0 determines
the largest loalization length (M ) = 1=min at energy E . The auray of the 's is
determined as outlined in Ref. [191, 192℄ from the variane of the hanges of the exponents
in the ourse of the iteration. Usually the method is performed with a omplete and
orthonormal set of initial vetors ( 1 ; 0 )T . In order to preserve this orthogonality, the
iterated vetors will have to be reorthogonalized during the iteration proess. In Fig. 1.7, I
illustrate this proedure graphially. As will be shown in setion 5, this orthogonalization
is very time onsuming and should be the starting point of any parallelization sheme.
For small disorders, the (M ) values are of the same order of magnitude as the strip width
M and thus subjet to nite-size modiations. In order to avoid simple extrapolation
shemes, a nite-size saling (FSS) tehnique had been developed in Refs. [191, 192, 224℄.
The FSS approah is based on real spae renormalization arguments for systems with nite
size M and intimately related to the original saling approah [1, 312℄. The onnetion to
the experimentally perhaps more relevant nite-temperature saling as in Eq. (1.8) is based
on the idea that a nite system size M may be assumed to be equivalent to a measurement
at nite temperature T sine a nite temperature indues an eetive length sale beyond
whih the eletrons will satter inelastially and thus lose the phase oherene neessary
for quantum interferene [22, 192℄. Thus saling the (M )=M data onto a saling urve,
i.e.,
(M )=M = f (=M ):
(1.14)
is the analogue of the saling approah as in Eq. (1.8). One determines the FSS funtion
f and the values of the saling parameter by a least-squares t and the absolute sale
12
Introdution
of an be obtained in the insulating regime by tting =M = =M + b(=M )2 for the
smallest loalization lengths [192℄. For diagonal disorder in 3D, this hypothesis has been
shown to be valid with very high auray, and two branhes of the saling urve f exists
whih orrespond to extended and loalized behavior [191, 224℄. Thus FSS allows for a
numerial proof of the one-parameter-saling hypothesis. Furthermore, the values of the
extended (loalized) branh are equal to the orrelation (loalization) length in the innite
system.
Perhaps the most diret numerial approah for studying transport in disordered systems
is provided by the Kubo formula for the d.. ondutivity [58, 112, 172℄. This formula
requires the knowledge of the omplete spetrum inluding all eigenvetors. As I show in
Ref. [MRS97℄, this severely limits the attainable system sizes to very small values suh
that in pratie no reliable extrapolations | nor FSS | to the thermodynami limit are
feasible. Thus a number of more indiret numerial approahes to transport phenomena
have been developed that either only require seleted parts of the spetrum or a few seleted
eigenvetors in the spetrum.
The most prominent of suh methods is based on the onnetion of Anderson loalization to random matrix theory (RMT) [73, 199, 318℄ as explained, e.g., in Refs. [119, 120℄.
Depending on the overall symmetries of the Hamiltonian suh as geometrial or timereversal invariane or Kramers degeneray, the spetrum of a metalli disordered system may be haraterized by utuations aording to the Gaussian orthogonal ensemble
(GOE) [6, 139, 261, 297, 328℄ or the Gaussian unitary ensemble (GUE) [122, 140, 141℄ or
the Gaussian sympleti ensemble (GSE) [156, 157℄, respetively. The nearest-neighbor
energy-level statistis (ELS) P (s), after suitable unfolding [120℄ of the overall systemdependent struture in the DOS, shows level repulsion with P (s ! 0) / s with = 1; 2,
and 4 for GOE, GUE and GSE, respetively. On the other hand, in the insulating regime,
P (s) / exp( s) [6, 139, 261℄, there is no level repulsion and RMT is no longer appliable.
At the MIT, yet another system-size independent P (s) has been found and argued to be
universal [39, 139, 140, 141, 252, 261, 229, 297, 328℄. Thus a areful analysis of P (s) and
related quantities allows for the numerial estimation of the MIT and also its ritial properties [140, 328℄. Corresponding to the ELS, there are also preditions for wave funtion
statistis (WFS) due to RMT whih an be used similarly [84, 85, 102, 120, 201, 207, 272℄.
Note that
P the seond moment of the WFS is related to the inverse partiipation number
PN 1 = Nj=1 j j j4 whih has also been investigated in many appliations [165℄.
Another onvenient way to haraterize the MIT is the multifratal analysis (MFA) [244℄.
From Fig. 1.6 a desription of the wave funtion at the MIT as a fratal appears justied
[9, 16, 194℄. Upon loser inspetion one nds that the wave funtion is a multifratal,
i.e., one needs not one but a whole set of exponents to desribe its saling properties
[114, 115, 116, 117, 244, 245, 246, 247℄. Suh a spetrum of exponents an be omputed
by the methods of Refs. [86, 114, 245℄. For a given disorder, one an then read o from
the system size dependene whether the spetrum tends towards the metalli, insulating
or truly ritial behavior. Thus we again have a means of studying the MIT. Note that
WFS and MFA are not independent of eah other, but one may be derived from the other
[201℄.
Introdution
13
For the studies of the Anderson model based on ELS, WFS and MFA, we employ the
Lanzos algorithm for the diagonalization of the Hamiltonian (1.10) using the CullumWilloughby implementation (CWI) [61, 62℄. This method is ideally suited for these tasks,
sine the matrix is real, symmetri, sparse and indenite. Furthermore, due to the disorder,
there are only very few if any degeneraies in the spetrum suh that the usual problem
of spurious or \ghost" solutions [61℄ onstruted by the Lanzos method is still important
but an be handled quite suessfully [61℄. With the CWI, we an diagonalize matries up
to 106 106 entries. The wave funtions in Figs. 1.5 and 1.6 have been omputed in this
way. In hapter 5, I will show what problems need to be onsidered upon parallelizing the
Lanzos sheme.
Other algorithms, suh as the deimation method of hapter 3 or the density-matrix renormalization group of hapter 4, I will omment on in their respetive setions.
1.6 A brief survey of this work
In this work, I present a series of papers whih represent extensive numerial investigations
regarding the loalization problem and the interplay of disorder and interations. The
papers of hapter 2 are onerned with the problem of transport in the presene of disorder
for purely non-interating eletrons. In setions 2.2 and 2.4, I study two models whih are
variations of the standard Anderson model of loalization as in Eq. (1.10). Namely, in
setion 2.4 the disorder is purely o-diagonal random hopping disorder, whereas in 2.2
the disorder is onsite as usual, but the hopping integrals are hosen to be anisotropi. In
setion 2.3, I present a study whih shows a detailed omparison of random matrix theory,
semilassial arguments and numerial results for the Anderson model as a funtion of an
external magneti eld. Setion 2.5 is onerned with the investigation of the inuene of
topologial disorder on the loalization properties by hoosing appropriate quasi-periodi
and random latties. Lastly, hapter 2.6 deals with a study of thermoeletri transport
properties at nite temperature, fousing on the alulation of the thermopower.
In hapter 3, I present results whih are onerned with the problem of two interating
partiles in random environments. Setion 3.2 deals with a numerial investigation of the
validity of the original approah by Shepelyansky. I show in setion 3.4 that TMM data
does not reprodue the proposed enhanement of the loalization lengths due to interation.
In setion 3.5, the deimation method is applied to the two-interating partiles (TIP)
problem and I argue that this method provides a numerially ontrolled approah whih
gives a two-partile loalization length 2 1 , with 1 the single partile loalization
length. Setion 3.6 then shows the existene of a loalizaton-deloalizaton transition for
TIP in 2D by the deimation method. Lastly, in setion 3.7, I study the TIP problem
for a quasi-periodi 1D system with an MIT and show that long-range interation is quite
dierent from onsite interation for this system.
Chapter 4 is onerned with the interplay of disorder and interation at nite eletron
densities in 1D. In setion 4.2 I will rst show that speial models an be onstruted
with loal defets that still retain omplete integrability and for whih the spetrum and
14
Introdution
some transport properties an be alulated exatly. The physis of these systems is,
however, quite dierent from the usual disordered systems studied in hapters 2 and 3.
For a mesosopi disordered Hubbard model, I apply renormalization group arguments in
setion 4.3 and use those to ompute the persistent urrents in the presene of an external
magneti ux. Lastly, setion 4.4 in onerned with the ompetition between Anderson
loalization and the Mott insulator mehanism in the disordered Hubbard model at halflling.
Most studies presented in hapters 2, 3, and 4 have extensively used numerial algorithms
for large sale numerial omputing. In hapter 5 I review some of the strategies used in
these algorithms and in partiular explain some of the parallel algorithms employed.
1.7 A very brief survey of what is not disussed in
this work
The present ompilation represents only a brief exerpt of the (numerial) physis of disordered systems and the seletion reets purely my own interests. In partiular, I will
not touh in any detail the physis of weak loalization [4, 52℄, nor the quantum Hall
eet present in disordered 2D systems in a strong magneti eld [122℄. I shall also not
omment further on the related transition in the Anderson model at nite magneti eld
[19, 165, 269, 271, 277℄ or in the presene of spin-orbit oupling [156, 157℄. I only briey
mention the analytial approahes based on eld theories, perturbative approahes [302℄
and the non-linear sigma model [77, 78℄. Lastly, I shall not omment on approahes whih
try to explain the disorder-indued MIT in terms of Mott's minimum metalli ondutivity
[202, 203, 205, 206℄.
Chapter 2
Disorder and the Anderson model of
loalization
2.1 Introdution
The Anderson Hamiltonian (1.10) models the disorder by hoosing a spatially varying random potential. For example, a ommon hoie is j 2 [ W=2; W=2℄ with W parameterizing
the disorder strength. However, as shown in Fig. 1.1, the mirosopi auses of disorder
may be quite dierent, ranging from impurity atoms to lattie defets. Fortunately, at
least lose to the transition, the validity of the saling hypothesis [1℄ in fat tells us that
mirosopi detail should not matter. Nevertheless, the Anderson model should be viewed
as an eetive model and one should expet it to reprodue only the qualitative or universal behavior, but to dier quantitatively when ompared to experimental results in real
materials [165℄. Studies of the Anderson model an then proeed along at least two lines,
namely, (i) explore the ranges of validity of universality, and (ii) explore the physis beyond
universality.
Regarding (i), I show in setions 2.2 and 2.3 studies in whih we have investigated what
happens, if we hange the mirosopi details of the model in order to test the universality
lasses by introduing anisotropy or a magneti eld. Regarding (ii), setions 2.4 and
2.5 are onerned with the inuene of randomness not via potential disorder, but rather
via kineti and topologial disorder. Lastly, in most studies of the Anderson model, the
fous is on the eletroni transport properties at low T . However, experiments have also
measured thermal transport quantities suh as, e.g., the thermoeletri power. In setion
2.6, I present results of a numerial study of these thermal transport quantities.
2.2 The Anderson model with anisotropi hopping
The results presented in hapter 1 for the Anderson model and the saling hypothesis
show the existene of the MIT in 3D, whereas there is no MIT in 2D in the absene
of many-body interations, magneti eld and spin-orbit interations. Furthermore, the
15
16
Disorder and the Anderson model of loalization
Figure 2.1: Shemati drawing of weakly oupled planes (left) and hains (right) in the 3D
anisotropi Anderson model. For oupled planes, we have tx = ty = 1 (solid lines) and tz 1
(dashed lines). For oupled hains, we have tx = 1 (solid lines) and ty = tz 1 (dashed lines).
2 + expansion within the non-linear model [311℄ and numerial studies based on TMM
data for bifratals [248℄ suggest that the ritial exponent in 2 + dimensions hanges
ontinuously as ! 1 for between 0 and 1. Thus one an ask the question if a similarly
ontinuous hange does also happen, if we vary the hopping elements anisotropially. E.g.,
we derease the hopping homogeneously in one or two diretions giving weakly oupled
planes or hains as indiated in Fig. 2.1. This then might model a transition from 3D to
2D or 1D, respetively. Of ourse from the onept of universality introdued above, one
expets that suh a hange should not hange the value of the ritial exponents.
This problem had already been studied in previous investigations [183, 184, 324, 325, 326℄
using TMM. However, the nite-size orretion to saling are quite large in these systems
and the auray of the results, espeially for the ritial exponent, is quite limited. We
have investigated this problem using MFA [MRS97℄, TMM and WFS [MR98℄. We nd
that the ritial disorder hanges ontinuously with dereasing hopping strength ta suh
that
W ta ;
(2.1)
where is lose to 41 for planes and 21 for hains. Here a represents z for oupled planes
and y and z for hains. Eq. (2.1) is in good agreement with results based on perturbative
alulations [325℄. The value of the ritial exponent is however not aeted by the
anisotropy [MR98℄ and retains its usual value 1:6 0:1 as in 3D [190, 270℄. Thus the 2D
and 1D ases are reahed only for ta = 0 and we observe the 3D MIT at any nite ta .
The ELS and the singularity spetrum of the MFA at the MIT are independent of the
system size and this size independene an be used to identify the MIT [139, 140, 141,
245, 246, 247, 328℄. However, both ELS and MFA are inuened by the anisotropy and
an be hanged onsiderably in omparison to the isotropi ase. Also, the eigenfuntions
are dierent from the isotropi ase as shown in Fig. 2.2. Therefore ELS and the MFA
singularity spetrum at the MIT are not stritly universal, i.e., not independent on mirosopi details of the system, as had been proposed previously [297, 328℄. This dependene
Disorder and the Anderson model of loalization
17
on mirosopi details is similar to the dependene on boundary onditions established
reently for the ELS at the MIT [39, 229, 252℄.
List of enlosed publiations relevant for this setion
[MR98/xiv℄
Energy level statistis at the metal-insulator transition in the Anderson model of
, F. Milde, and R. A. Romer, Ann. Phys.
loalization with anisotropi hopping
[MRS97/xiii℄
(Leipzig) 7, 452{456 (1998).
, F.
Milde, R. A. Romer, and M. Shreiber, Phys. Rev. B 55, 9463{9469 (1997).
Multifratal analysis of the metal-insulator transition in anisotropi systems
2.3 Comparisons with the theory of random matries
and the onept of universality
Random matrix theory [73, 120, 199, 318℄ provides the underlying universal theoretial
desription of disordered systems in the diusive regime suh that the mean free path
[10℄ is larger than the lattie spaing, but muh smaller than the lattie size L [119℄.
Furthermore, the loalization length must obey L. Additionally, RMT is espeially
powerful, when the system is subjeted to an external perturbation suh as an magneti
eld or an additional external potential [38, 265, 266, 288℄. In this ase detailed analytial
expressions exist whih predit the funtional form not only of the ELS, but also of various
orrelation funtions of the level energies [265, 266, 288℄. Also, semilassial methods an
be used independently to ompute the same quantities [25, 65, 74, 319, 320℄. Further
analytial preditions exist for the WFS in the diusive regime [83, 84, 85, 102, 201, 272℄.
In order to hek whether these analytial preditions hold true for the Anderson model,
one has to be areful in identifying the universal regime. A priori the only input parameter
to the model for a given L is the disorder strength W . For any nite system, the MIT is
smeared out and the mobility edges E as in Fig. 1.4 are onsequently not well dened.
Also, E will vary slightly for eah disorder onguration from its thermodynami value.
Thus one has to be areful to hoose an interval in the spetrum suh that there are
no loalized states inside the interval. Furthermore, for problems in whih the dierene
between disrete and ontinuum model might be important, one should abstain from using
energies lose to the band enter due to the possibly dierent dispersion relations.
In Ref. [UMRS99℄, we have studied along these lines (i) three orrelations funtions in the
transition regime from GOE to GUE upon inreasing the magneti ux, and (ii) three
parametri orrelations in the GUE regime. There are three parameters needed for the
t to the analytial expressions. One of these is the ondutane g , whih then allows
for an a posteriori hek whether one has indeed studied the right regime. Namely, we
need g 1 for the diusive regime. Note that we have omputed g by three independent
approahes. Thus Ref. [UMRS99℄ also shows how to reliably ompute the non-universal
value of g numerially. In Ref. [UMR98℄, we turn our attention to a omparison with the
analytial preditions [83, 201℄ for the WFS in the Anderson model. We show that the
18
Disorder and the Anderson model of loalization
y
x
z
Figure 2.2: Critial wave funtion for the 3D anisotropi Anderson model at E = 0 with tx =
ty = 1, tz = 0:01, W = 4:5 and N = 48. Sizes of the boxes and olor odings are as in Fig.
1.5. The wave funtion is lose to being extended over a few weakly oupled planes and highly
loalized in the z diretion. The thik solid line is the logarithm of the summed probability for
eah plane perpendiular to the z diretion.
Disorder and the Anderson model of loalization
19
rossover from zero to nite ux is aurately modeled by the formulas of Refs. [83, 201℄
for the rossover from GOE to GUE behavior. We are urrently extending this study to
inlude 1=g orretions to the WFS [83, 84, 85, 102, 201, 272℄.
List of enlosed publiations relevant for this setion
[UMR98/xv℄
A numerial study of wave-funtion and matrix-element statistis in the Anderson
, V. Uski, B. Mehlig, and R. A. Romer, Ann. Phys. (Leipzig)
model of loalization
[UMRS99/xiv℄
7, 437{441
(1998).
, V. Uski,
B. Mehlig, R. A. Romer, and M. Shreiber, Phys. Rev. B 59, 4080{4090 (1999).
Smoothed universal orrelations in the two-dimensional Anderson model
2.4 An Anderson model with random hopping
The saling hypothesis argues that in 1D all states are loalized. However, prior to the
advent of saling it was shown in a series of papers that a 1D model of purely random
hopping disorder behaves quite dierently from the usual 1D onsite potential disorder
[76, 80, 135℄. The model is dened suh that all t values an be hosen randomly. In what
follows, I assume that t 2 [ w=2; + w=2℄ with and w denoting the enter and width
of the distribution. In this parameterization, the ordered tight-binding model is reovered
in the limit ! 1 after a suitable resaling.
For this model, the DOS has a peak at energy E = 0 for any strength of hopping disorder whih is known as Dyson singularity [72℄ with (E ) D=jE j(ln jE=Dj)3, where D
inorporates the variane of the o-diagonal disorder [198℄. Correspondingly, the loalization length at E = 0 diverges even in 1D [135℄. Assuming that Eq. (1.2) holds, then
(E ) ln jD=E j. Using the TMM, this behavior an be reprodued as shown in Fig. 2.3.
There is urrently an argument in the literature whether the assumption of Eq. (1.2) is
justied or whether a power-law deay should be assumed [12, 144, 164, 281℄. I point out
that the model is also related to the random ux model [155℄ as outlined in Ref. [ERS98a℄.
We have studied the random hopping model in 2D [ERS98a, ERS98b℄, where no analytial
results exist to the best of my knowledge [185, 186, 280, 310℄, by TMM and Lanzos
diagonalization. We an show that the singularity in the DOS still exists for bipartite
square latties. Furthermore, the loalization length is also diverging at E = 0 up to the
system sizes onsidered. However, this time it is lear that the divergene is not as expeted
for extended states, but rather losely mimis the behavior of ritial states similarly to
the MIT in 3D. This TMM result is orroborated by further MFA data. We also show that
even a very small amount of additional diagonal disorder or energies away from 0 leads
immediately to omplete loalization of the wave funtion and the model again falls into
the generi saling piture. These results were obtained by onstruting the FSS urves. I
wish to emphasize that these results are in perfet agreement with the saling hypothesis,
whih states that 2 is the limiting dimension suh that extended states are exluded [1℄.
20
Disorder and the Anderson model of loalization
7.5
c: W:
0.50 0.01
0.00 0.01
0.25 0.01
0.50 0.10
0.00 0.10
0.25 0.10
6.5
λ
5.5
4.5
3.5
2.5
−0.10
−0.05
0.00
0.05
0.10
E
Figure 2.3: Energy dependene of the loalization length for a 1D random hopping Anderson
model with w = 1. Note that dereasing the admixture of additional diagonal disorder W leads
to the development of a divergene at E = 0.
In 3D, the hopping disorder is no longer suÆient to loalize all states [76℄ as happens for
(uniform) diagonal disorder at W = 16:5. In Fig. 2.4 I show the (E; ) phase diagram
for the 3D system. Results of FSS for the 3D system indiate that the ritial exponent
is the same regardless whether we study the MIT as a funtion of E or as a funtion
of additional diagonal disorder W . Taking into aount irrelevant saling terms, we nd
that = 1:59 0:05. Thus the results are again in agreement with the usual 3D ase and
the saling hypothesis [48℄. Reently, the random hopping model has been extended to
non-bipartite quasi-1D latties [43, 42℄ and it has been argued that the divergene in the
DOS is of dierent nature. We are urrently investigating whether this dierent behavior
also manifests itself in the loalization properties.
List of enlosed publiations relevant for this setion
[ERS98a/xiii℄
[ERS98b/xiv℄
, A.
The two-dimensional Anderson model of loalization with random hopping
Eilmes, R. A. Romer, and M. Shreiber, Eur. Phys. J. B 1, 29{38 (1998).
Critial behavior in the two-dimensional Anderson model of loalization with ran-
, A. Eilmes, R. A. Romer, and M. Shreiber, phys. stat. sol. (b) 205,
229{232 (1998).
dom hopping
21
Disorder and the Anderson model of loalization
88
Ec
EB
Emax
|E|
66
44
22
00
0.0
0.0
extended
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1.0
1.0
c
Figure 2.4: Phase diagram of the 3D random hopping Anderson model. The mobility edge E
() has been obtained by TMM. The averaged band edge EB (2) has been determined from the
spetrum of 10000 latties of size 103 . The light gray region indiates extended states whereas
the dark gray region marks loalized states. The straight dashed line denotes the largest possible
energy Emax = 6 + 3w available, i.e., the true band edge, in the thermodynami limit.
2.5 The inuene of topologial disorder
In the original artile of Anderson [7℄, latties dierent from regular strutures suh as
square and triangular were also inluded in the general arguments presented. Most studies
sine then have used the model (1.10) without onsidering suh dierent latties [249℄.
However, in most natural strutures disorder and random oordination are the rule rather
than an exeption [109, 234, 279, 309℄. A ommonly used prototype for the onstrution of
topologially disordered strutures in physis [59℄ is provided by the Voronoi onstrution.
It may be viewed as a generalization of the Wigner-Seitz onstrution and allows to generate
tessellations from sets of arbitrarily distributed points [208℄.
In this hapter, I present results of ELS and WFS studies for topologially disordered
systems whih are not of the Voronoi type. Still the the disorder is primarily due to the
random struture in the onnetedness of the lattie sites. As a rst example [GRS98℄,
we onsider strutures whih an be obtained by ipping loally the orientation of a given
bond as in Fig. 2.5. The ordered latties orrespond to a hexagonal brikwall mosai and
its dual, a triangulation. In Figs. 2.6 and 2.7 I show the original strutures and a sample
of the most disordered ases. Next, we use the Hamiltonian
H=
X
ij
tij jiihj j;
(2.2)
22
Disorder and the Anderson model of loalization
b)
a)
6
6
6
6
5
7
5
7
c)
d)
6
6
7
5
6
6
5
7
Figure 2.5: Ordered p = 0 mosai (a) and triangulation (). The numbers indiate the number
of edges of a losed loop (mosai), or, equivalently, the number of neighbors of eah vertex
(triangulation). b) and d): The same struture after a single ip.
Figure 2.6: Ordered (left, p = 0) and maximally disordered (right, p = 0:5) typial mosai latties
aording to Ref. [47℄.
Figure 2.7: Ordered (left, p = 0) and maximally disordered (right, p = 0:5) typial triangulation
latties aording to Ref. [47℄. These latties are the duals to the latties of Fig. 2.6.
Disorder and the Anderson model of loalization
23
Figure 2.8: Standard periodi approximant of the Ammann-Beenker tiling with 1393 verties,
displayed together with a opy of itself that has been rotated through 90 degrees. The \worms"
where mismathes between the two rotated opies our are shaded.
where tij = 1 along the bonds and zero otherwise. Sine the appliation of TMM is not
possible for these strutures, we study the ELS by exat and Lanzos diagonalization.
We nd for both the mosai and the triangular strutures that the ELS is given by the
Wigner-Dyson distribution for the GOE ase, similar to the diusive regime of setion 2.3.
At rst glane this seems to be in ontradition to the saling hypothesis. However, any
amount of additional diagonal onsite disorder will again lead to loalization of all states.
Furthermore, from the point of view of RMT, this result appears quite natural sine the
model is simply a matrix with 0's and 1's at nearly random plaes in the matrix, viz, a
random matrix.
A perhaps even more interesting model is studied in Refs. [SGRZ99a, SGRZ99b, ZGRS98℄
where we investigate the ELS for a quasiperiodi system [66, 136, 151, 197, 287℄. In suh
systems, the topologial struture is dened by a deterministi onstrution proedure and
no randomness is present [113℄. Previously, the ELS had been reported to be either similar
to the loalized ase [329℄ or to be lose to a log-normal distribution [225℄. As we show,
the rst assumption is due to not taking into aount appropriately the symmetries of the
models, whereas the seond assumption is due to using the standard periodi | and thus
not stritly-speaking quasiperiodi | approximants for these non-periodi strutures. In
Fig. 2.8 I show the standard periodi approximant of the Ammann-Beenker tiling onsidered in Ref. [225℄. Note that there is an approximate fourfold rotational symmetry in the
system giving rise to \worm"like mismathes after rotating the tiling by 90 degrees. This
almost symmetry gives rise to harateristi deviations of the ELS from GOE.
For the quasi-periodi strutures, and after removing any symmetries of the tilings, we nd
[SGRZ99a, SGRZ99b, ZGRS98℄ that the ELS is aurately desribed by the GOE up to
the system sizes onsidered. Furthermore, the auray of our data is good enough so that
we an distinguish between the approximative Wigner surmise and the exat GOE result
24
Disorder and the Anderson model of loalization
[120℄. I stress that suh a dierene is very hard to resolve in numerial studies for the
Anderson model [328℄.
Finally, let me stress that an additional onsite disorder in the 2D Hamiltonian (2.2) leads
to loalization just as for the 2D random hopping ase of setion 2.4. Thus the results
presented here do not ontradit the saling hypothesis for generi disordered systems.
List of enlosed publiations relevant for this setion
[GRS98/xiv℄
[SGRZ99a/xiv℄
, U. Grimm, R. A. Romer,
and G. Shlieker, Ann. Phys. (Leipzig) 7, 389{393 (1998).
Eletroni states in topologially disordered systems
Energy Levels of Quasiperiodi Hamiltonians, Spetral Unfolding, and Random
, M. Shreiber, U. Grimm, R. A. Romer, and J. X. Zhong, to be
published in Comp. Phys. Comm., (1999).
[SGRZ99b/xv℄
, M. Shreiber, U.
Grimm, R. A. Romer, and J. X. Zhong, Physia A 266, 477{480 (1999).
[ZGRS98/xiv℄
, J.
X. Zhong, U. Grimm, R. A. Romer, and M. Shreiber, Phys. Rev. Lett. 80,
3996{3999 (1998).
Matrix Theory
Appliation of random matrix theory to quasiperiodi systems
Level Spaings Distributions of Planar Quasiperiodi Tight-Binding Models
2.6 Thermoeletri transport oeÆients in the Anderson model
Up to now, most of my attention was foused on omputing parameters whih in one way
or the other an be related to the ondutivity . This was motivated by the fat that is
the quantity whih is most often studied in transport measurements of disordered systems
[27, 28, 137, 217, 235, 284, 285, 291, 292, 293, 304℄. However, other transport properties
suh as the thermopower S , the thermal ondutivity K and the Lorenz number L0 have
also been measured [175, 178, 260℄. Previous theoretial onsiderations for the thermopower
in the viinity of the MIT have been based on low- and high-T expansions [49, 267℄ or exat
alulations diretly at the MIT in 3D [81℄ assuming the validity of Eq. (1.6). The low-T
Sommerfeld expansion [49, 267℄ indiates that S=T diverges as 1=(EF E ) when the MIT
is approahed from the metalli side. On the other hand, the high-T expansion [267℄ and
the exat alulation at = E [81℄ show that S remains onstant diretly at the MIT
[81℄.
In Refs. [VR98, VRS99℄, we have studied the behavior of S , K and L0 by straightforwardly
alulating the integrals in the linear response formulation of Chester-Greenwood-KuboThellung [58, 112, 172℄. The only additional ingredients in our study were an averaged
DOS as in Fig. 1.4 and the assumption of (E ) as in Eq. (1.6) [44, 117℄. We an show that
the previous analytial onsiderations are orret but only in limited regimes of validity.
Thus S=T diverges when the MIT at T = 0 is approahed from the metalli side, but S
itself remains onstant at the MIT. Furthermore, we an show that all data for S (T; EF ),
K (T; EF ), (T; EF ), and L0 (T; EF ) an be saled onto saling urves when plotted as
25
Disorder and the Anderson model of loalization
5
(σ/σ0)|(µ−Ec)/Ec|
−1.3
10
3
10
1
10
−1
10
3
10
5
10
7
10
T/|(µ−Ec)/Ec|
Figure 2.9: Saling plot of the temperature-dependent ondutivity (T ) with, e.g., ritial exponent = 1:3 for various eletron llings ranging from 97:74% (Æ, metalli) via 97:768% (, at
the MIT) to 97:80% (, insulating) for a 3D Anderson model at disorder W = 12.
funtions of T=((T ) E ). E.g., for the ondutivity we have
T
(; T )
:
=
F
[( E )=E ℄
[( E )=E ℄
(2.3)
Note that this saling implies that the dynamial ritial exponent z = 1= . This is in
ontrast to the usual saling arguments [22, 131℄ outlined in hapter 1.1 whih would lead
to z = 3 in 3D. In Fig. 2.9 I show the saling urve for, e.g., the ondutivity. We are
urrently investigating if the agreement with the experiments beomes better when instead
of (E ) as in Eq. (1.6) we also inlude nite-temperature eets.
List of enlosed publiations relevant for this setion
[VR98/xv℄
Low temperature behavior of the thermopower in disordered systems near the An-
, C. Villagonzalo, and R. A. Romer, Ann. Phys. (Leipzig) 7, 394{
399 (1998).
[VRS99/xiv℄
, C. Villagonzalo, R. A. Romer, and M. Shreiber, to be published in Eur.
Phys. J. B, (1999).
derson transition
Thermoeletri transport properties in disordered systems near the Anderson tran-
sition
Chapter 3
The interplay of interations and
disorder I: two interating partiles
3.1 Introdution
The researh presented in the last hapter learly supports the saling hypothesis of loalization for non-interating eletrons. The models usually fall into the predited universality
lasses, and, if they don't, then they an be shown not to be generi as, e.g., in hapter
2.4. However, real eletrons of ourse interat [60℄, and their interation is of relevane for
the transport properties of disordered systems [79, 206℄. Espeially in 2D and 1D where
sreening [10℄ is muh less eÆient than in 3D. Renormalization group (RG) arguments
have been applied to the loalization problem, whih treat both disorder and interations
perturbatively [105, 106, 107, 108, 153, 253℄. In 1D this has lead to the predition that
all thermodynami states remain loalized in the presene of repulsive many-body interations.
Reently, these theoretial onsiderations reeived a lot of renewed attention due to the
persistent urrent problem [3, 45, 53, 182, 193, 242℄ and the experimental disovery of
the 2D MIT as outlined in hapter 1.4. In order to theoretially study the eets of the
interplay between disorder and interations, one should in priniple solve a problem with
exponentially growing number of states in the Hilbert spae with inreasing system size.
At present, this an be ahieved only for a few partiles in 1D (see hapter 4) and very
few partiles in 2D [259, 275, 300, 82℄. However, in 1994 Shepelyansky [256, 257℄ proposed
to simply look at two interating partiles (TIP) in a random environment. In partiular,
he suggested that the two partiles would form pairs even for repulsive interations suh
that the TIP pairs would have a larger loalization length than the two single partiles
(SP) separately. Thus the interation would lead to an enhaned possibility of transport
through the system [146℄. The perhaps even more surprising part of the predition is that
the TIP pairs will have a loalization length 2 suh that at pair energy E = 0
2 / U 2 1 2 ;
26
(3.1)
The interplay of interations and disorder I: two interating partiles
<U>
ϕkl ∼ ϕk ϕl
27
ϕnm∼ ϕn ϕm
λ1
λ1
Figure 3.1: Shemati piture of the TIP arguments of Ref. [256℄. The two-partile state kl
(dark blue and green urves) is loalized within a distane 1 from the two-partile state nm
(red and light blue urves). The resulting overlap-matrix element u = hU i hkl jU jnm i leads to
a longer deay length 2 for the TIP state as explained in the text. This eet an be visualized
as an eetive redution (yellow) of the original disorder potential (brown urve).
where U represents the onsite interation strength and 1 is the single partile loalization
length. Sine 1 / 105=W 2 in 1D, this implies rather large values of 2 for small disorders
W . The rst numerial studies devoted to the TIP problem used the TMM to investigate the proposed enhanement of the pair loalization length 2 [97, 98, 123, 125, 127,
RS97a, RS97b, RS98, 256℄. Other diret numerial approahes to the TIP problem have
been based on the time evolution of wave pakets [41, 256℄, exat diagonalization [313℄,
variants of ELS [2, 314℄ and MFA [305, 306℄, Green funtion approahes [93, 211, 273, 274℄,
perturbative methods [149, 150℄ and mappings to eetive models [94, 95, 96, 145, 227℄. In
these investigations usually an enhanement of 2 ompared to 1 has been found but the
quantitative results tend to dier both from the analytial predition in Eq. (3.1), and from
eah other. Furthermore, a hek of the funtional dependene of 2 on 1 is numerially
very expensive sine it requires very large system sizes L 2 1 . Extensions of the
original arguments have been proposed in Refs. [30, 145, 146℄ for higher dimensions, TIP
lose to a Fermi sea [210℄ and long-range interations in 1D [41, RS97a, RS98℄ and 2D
[259℄.
The basi idea leading to the predition of Shepelyansky is based on looking at the interation matrix element of a pair state kl = k l with another pair state nm = n m
[256, 257℄. Here k ; l ; n ; m denote SP states loalized with 1 around sites k; l; n; m.
Furthermore, we restrit the states suh that jk lj 1 , jn mj 1 and jk nj 1 ,
jl mj 1. In Fig. 3.1 I show these onditions graphially. The interation matrix
28
The interplay of interations and disorder I: two interating partiles
elements then are
u=h
kl
jU j
nm
i=U
N
X
j =1
y
y
k (j ) l (j ) n (j ) m (j );
(3.2)
where I have
PN used that the interation Hamiltonian is given by the Hubbard onsite term
[RP95℄ U j =1 nj # nj " with nj denoting the number operator at site j and spin . If we
now assume [145, 256℄ that the SP state is given as
k (j )
/ p1
1
exp [
jj kj + i(j )℄
1
(3.3)
with (j ) a random phase, then we nd [256℄ that the average interation matrix element
has a magnitude of
u / 1 3=2 :
(3.4)
Shepelyansky next alulates the deay rate of a non-interating eigenstate by means of
Fermi's golden rule U 2 =1 t [149, 256, 257℄. Sine the typial hopping distane is of the
order of 1 the diusion onstant is D U 2 1 =t. Within a time the partile pair visits
N U31=2 t 1=2 1=2 states. Diusion stops when the level spaing of the visited states is
of the order of the frequeny resolution 1= . This determines
ut-o time and the
p the
orresponding pair-loalization length is obtained as 2 D (U=t)2 21 in agreement
with Eq. (3.1). Appliability of Fermi's golden rule requires t=21 whih is equivalent
to U 2 1 =t2 1. This is exatly the ondition for an enhanement of 2 ompared to 1 .
Alternatively, the model may be mapped to a random-matrix model (RMM) with entries
hosen aording to Eq. (3.4) [95, 96, 256℄.
3.2 Numerial results for the random-matrix model
of TIP
The arguments presented in setion 3.1 are of qualitatively nature and Eq. (3.1) must
be heked for quantitative auray. Even before testing (3.1), it is already worthwhile
to hek the validity of (3.4) and the subsequent arguments or the RMM approah [95,
96, 256℄. In Ref. [97℄, it had already been shown that the assumption of a Gaussian
distribution of the matrix elements u | neessary for taking the arithmeti average | was
oversimplied. In Ref. [RSV99℄ we have paid speial attention to the exat dependene
of u on 1 and system size. To this end, we diagonalized the 1D Anderson model for a
given length M and disorder W and omputed u by averaging over all suitable states and
many disorder ongurations. We showed that due to the non-Gaussian distribution of u,
one should rather use the typial average than the arithmeti average for the omputation
of u. But whereas the arithmeti average [305℄ gives u / 1 1:5 , the typial average obeys
utyp / 1 1:95 . Following the arguments above, this would imply 2 / 11:1 , i.e., a very small
The interplay of interations and disorder I: two interating partiles
29
enhanement. I emphasize that this result does not mean that there is no enhanement of
the loalization length. Rather, the results of Ref. [RSV99℄ indiate that the arguments
of Ref. [256℄ do apture the physis in a somewhat oversimplied form. One step in this
diretion is to take into aount the energy denominators in the omputation of u, e.g.,
one only onsiders interation matrix elements for states whose energy spaings are of the
order of U or smaller [RLS99a℄. In this ase we nd that there is a slight derease in the
value of and subsequently a slight inrease in TIP deloalization yielding 2 / 11:40:2 .
This suggests that higher orders in perturbation theory than the rst order RMM approah
[256℄ are important. Furthermore, the exponent 1:4 0:2 is in reasonable agreement with
the result of hapter 3.5 and previous results in the literature [RS98, 2, 41, 93, 97, 124,
123, 125, 127, 145, 149, 150, 211, 227, 273, 274, 305, 306, 314℄.
List of enlosed publiations relevant for this setion
[RLS99b/xiv℄
, R. A.
Romer, M. Leadbeater, and M. Shreiber, aepted for publiation in Ann. Phys.
(Leipzig) 8, (1999).
[RSV99/xiii℄
, R. A. Romer, M. Shreiber, and T. Vojta, phys. stat.
sol. (b) 211, 681{691 (1999).
Numerial results for two interating partiles in a random environment
Two-interating partiles in a random potential: Numerial alulations of the
interation matrix elements
3.3 Failure of the RMM approah for toy models
In this setion I want to show that even an RMM whih ontains the orret dependene of
the oupling matrix elements on the SP loalization length may give qualitatively inorret
results. To this end I onsider two toy models [299℄, viz. Anderson models of loalization
with additional perturbing random potentials. By a proedure analogous to that at the end
of setion 3.1 or by the mapping onto RMMs, I nd that there is an erroneous enhanement
of the loalization length.
2D Anderson model with perturbation on a line
The rst example is set up to lead to the same RMM as the TIP problem. It onsists
of the usual 2D Anderson model of loalization perturbed by an additional weak random
potential of strength U at the diagonal x = y in real spae. Sine this inreases the width of
the disorder distribution at the diagonal we expet the loalization length to derease. We
now map onto an RMM as in Refs. [30, 256℄. As above, the eigenstates of the unperturbed
system are loalized with a loalization length 1 and approximately given by
1
j
r rn j
exp
+ in (r)
(3.5)
n (x; y ) 1
1
where r = (x; y )T is the oordinate vetor of the partile and is again a phase whih is
assumed to be random. The Hamiltonian of this model diers from the TIP Hamiltonian
30
The interplay of interations and disorder I: two interating partiles
λ1(M)
100
10
1
5
10
15
20
25
W
Figure 3.2: Dependene of 1(M ) on disorder W for the 2D Anderson model at E = 0 for M =
10; 25; 30; 35 and 50 indiated by inreasing symbol size. We use the M = 50 data, emphasized
by the solid line, as nite-size estimate of 1 .
in two points: (i) the diagonal elements are independent random numbers instead of being
partially orrelated as in the TIP problem and (ii) the interation potential U (x; x) 2
[ U; U ℄ at eah site of the diagonal is random instead of having a denite sign and modulus
U as in the TIP problem. However, none of these points enters the mapping proedure
outlined above. Thus, applying exatly the same arguments as for the TIP problem in
setion 3.2 we nd that the perturbation ouples eah state lose to the diagonal (jxn
yn j < 1 ) to O(21 ) other suh states. The interation matrix element is again a sum
of O(1 ) terms of magnitude U=21 and random phases giving a typial value of U1 3=2 .
Consequently, our toy model is mapped onto exatly the same RMM as TIP in a random
potential.
As for the TIP ase we now numerially hek the relation between the oupling matrix
element and the SP loalization length 1 . We rst note that the disorder dependene of
1 in the 2D Anderson model is no longer approximated by the simple power law ited in
setion 3.1 [165℄. In fat, 1 is usually muh larger in the 2D ase for the same value of W .
Thus we ompute estimates 1 (M ) as a funtion of W for quasi-1D strips of nite strip
width M with 1% auray by TMM. We remark that due to the self-averaging [165℄ of
1=1 (M ) this is equivalent to omputing 1 (M ) for many samples of M M disordered
squares. In Fig. 3.2, we show data of 1 (M ) as a funtion of W . We take 1 (50) to ompute
the oupling matrix elements. Sine 1 (50) is always larger than for smaller system size,
this hoie only means that we sum over a few additional but very small terms when
omputing u. Next, we alulate both uabs and utyp for dierent values of W and various
31
The interplay of interations and disorder I: two interating partiles
12
5
10
4
2
3
6
2
Po(u)/10
Pd(u)
8
4
1
2
0
0.00
0.05
u
0.10
−0.05
0.00
0
0.05
u
Figure 3.3: Unnormalized distribution for the diagonal (left panel) and o-diagonal (right panel)
oupling matrix elements u with bin width = 0:0015 for the perturbed 2D Anderson model
with 1 = 3:1 (W = 12), U = 1, and M = 25. The irles indiate the 20 smallest u (largest
Po (u)) for diagonal (o-diagonal) data.
M M squares. Disorder averaging is over 20 samples and we study uabs and utyp as
funtions of 1 (M ). We emphasize that instead of the well-known extrapolations of 1 (M )
to innite system size by means of FSS [165℄, we take the nite-size approximants 1 (M )
on purpose, sine we ompute 2 also for omparable nite sizes only.
In Fig. 3.3 we show the omputed distributions Pd/o (u) of diagonal and o-diagonal matrix
elements u for the present model. As for the TIP model [RSV99℄ the diagonal elements
are non-negative and the distribution Pd (u) of diagonal matrix elements has a large peak
at u = 0; the distribution Po (u) of o-diagonal elements is again [RSV99℄ strongly nonGaussian. The results for uabs and utyp (see [RSV99℄ for denitions) are presented in Fig.
3.4. The dependene of uabs on 1 (M ) for 2 1 (M ) 12 follows uabs / 1 (M ) 1:60:1 in
agreement with Ref. [RSV99℄. Furthermore, here we also have utyp / 1 (M ) 1:50:1 . We
note that the slopes of uabs and utyp beome smaller for 1 (M ) M=2 due to the nite
sample sizes [RSV99℄. This nite-size eet is just the same as for TIP and thus further
supports our use of the nite-size values 1 (M ). We remark that if instead of 1 (M ),
we use 1 (50) for plotting the uabs and utyp data, that is irrespetive of the system sizes
for whih they had been omputed, we obtain uabs / 1 1:540:10 and utyp / 1 1:470:10 .
Thus both hoies of 1 show that uabs and utyp vary as 1 1:5 within the auray of the
alulation. Sine our toy model is mapped onto the same RMM as the TIP problem the
resulting loalization length along the diagonal is also given by Eq. (3.1). We thus arrive
at the surprising onlusion, that adding a weak random potential at the diagonal of a 2D
Anderson model leads to an enormous enhanement of the loalization length along this
32
The interplay of interations and disorder I: two interating partiles
−1
10
−2
u
10
−3
10
5
λ1(M)
10
15 20 25
Figure 3.4: Dependene of uabs (squares) and utyp (irles) on 1 (M ) for the perturbed 2D
Anderson model with U = 1 and M = 10; 25; 301:6 and 35 indiated1:5 by inreasing symbol size. The
solid lines represent the power laws uabs 1 and utyp 1 .
diagonal, in ontradition to the expetation expressed above, viz. that inreasing disorder
leads to stronger loalization.
1D Anderson model with perturbation
An even more striking ontradition an be obtained for a 1D Anderson model of loalization. The eigenstates are again given by Eq. (3.3) with 1 known from seond order perturbation theory [75, 154, 296℄ and numerial alulations [63, 223℄ to vary as 1 t2 =W 2
for small disorder. We now add a weak random potential of strength U at all sites. Sine
the result is obviously a 1D Anderson model with a slightly higher disorder strength the
loalization length will be redued, 1 (U ) t2 =(W 2 + U 2 ). Now we map onto an RMM
aording to Refs. [256, 257℄. The additional potential leads to transitions between the
unperturbed eigenstates n . Eah suh state is now oupled to O(1 ) other states by
oupling matrix elements h n jU j n i with magnitude u U1 1=2 sine we sum over 1
ontributions with magnitude U=1 and supposedly random phases.
Again we numerially hek the relation between uabs and utyp as funtions of 1 . In Fig.
3.5, we show results obtained for hains with various lengths and 50 disorder ongurations
for eah W . 1 is omputed by TMM. In Fig. 3.6 we show the distributions Pd/o (u). We
note that Po (u) is non-Gaussian as for the TIP model and the perturbed 2D Anderson
0
33
The interplay of interations and disorder I: two interating partiles
−1
u
10
−2
10
10
λ1
100
1000
Figure 3.5: Dependene of uabs (squares) and utyp (irles) on 1 for the perturbed 1D Anderson
model with U = 1 and M = 200; 300; 5000:and
800 indiated by inreasing symbol size. The solid
lines represent the power laws uabs 1 48 and utyp 1 0:59.
250
1.00
200
150
Po(u)
Pd(u)
0.75
0.50
100
0.25
0.00
0.0
50
0.1
0.2
u
−0.1
0.0
0
0.1
u
Figure 3.6: Unnormalized distribution for the diagonal (left panel) and o-diagonal (right panel)
oupling matrix elements u with bin width = 0:0015 for the perturbed 1D Anderson model
with 1 = 26 (W = 2), U = 1, and M = 200. The irles indiate the 20 smallest u (largest
Po (u)) for the diagonal (o-diagonal) data.
34
The interplay of interations and disorder I: two interating partiles
model. Pd (u) is similar to the previous models, but the utuations are muh larger. For
10 1 250, uabs varies as 1 0:480:10 as we predited above. utyp varies as 1 0:590:10 .
Both variations are ompatible with = 1=2. Again we need at least 1 & M=2 in order
to suppress the eets of the nite hain lengths.
In analogy to setion 3.2 the appliation of Fermi's golden rule in this 1D ase leads to
a diusion onstant D U 2 21 =t. The number of states visited within a time is now
N U1 t 1=2 1=2 . Again, diusion stops at a time when the level spaing of the states
visited equals the frequeny resolution. p
This gives U 2 21 =t3 . The loalization length of the perturbed system thus reads D U 2 21 as in Eq. (3.1), in lear ontradition
to the orret result.
Appliation of the blok-saling piture to toy models
Let me now disuss the relation of our results to Imry's blok-saling piture (BSP) [145,
146℄ for the TIP problem. In this approah one onsiders bloks of linear size 1 and
alulates the dimensionless pair ondutane on that sale,
u2
(3.6)
g2 2 ;
where u represents the typial interation-indued oupling matrix element between states
in neighboring bloks and t=21 is the level spaing within the blok. If the typial
oupling matrix element depends on 1 as u U1 the pair ondutane obeys
g2 (U=t)2 41 2 :
(3.7)
For the 2D Anderson model with perturbation onsidered above, the BSP an be applied analogously. Again, we onsider bloks of linear size 1 and ompute the typial
perturbation-indued matrix elements between these bloks. We then nd that aording
to the BSP the ondutane of a 2D Anderson model with additional weak perturbing
potential along the diagonal is given by Eq. (3.6). Using = 1:5 0:1 as obtained above
from the numerial data for uabs and utyp , we then have g2 (U=t)2 1 . Thus the BSP
yields the same unphysial result as the RMM approah of setion 3.3. Let us also apply
the BSP to the 1D toy example. The level spaing in a 1D blok of size 1 is t=1 , and
the oupling matrix element between states in neighboring bloks is u U1 1=2 . Thus, the
ondutane of the perturbed system on a sale 1 is obtained as gp (U=t)2 1 . For large
1 this again ontradits the orret result, viz. a derease of the ondutane ompared
to the unperturbed system. Thus, the BSP applied to the two toy models introdued in
the present setion gives the same qualitatively inorret results for the loalization properties as the RMM. This is not surprising sine the only ingredients of the BSP are the
intra-blok level spaing t=21 and the inter-blok oupling matrix elements u whih
also enter the RMM.
In summary, I have shown that ounter examples for the RMM approah exist, whih give
inorret results. This does not neessarily imply that the RMM approah fails for the TIP
The interplay of interations and disorder I: two interating partiles
35
m
TMM
1
n
Figure 3.7: Shemati diagram for the TMM approah to TIP. The partile oordinates are
denoted by n and m and the interation U gives an eetive additional potential along the
diagonal as indiated by the green line. The red 1 denotes that lattie onstant. The TMM
proeeds along an SP oordinate. The reader should ompare this to the original TMM approah
as in Fig. 1.7.
problem. Rather, the ounter examples suggest that additional physial insight is needed
for a more satisfatory explanation. In partiular, we expet that taking into aount the
energies of the states as at the end of setion 3.2 for TIP[RLS99a℄ will lead to a redution
in the erroneous enhanement.
3.4 The transfer-matrix approahes
In order to test whether the enhanement (3.1) exists, Shepelyansky already in his original
artile [256℄ had employed a variant of the TMM and also studied the diusion of a TIP
pair. However, both investigations only showed 2 > 1 , but ould not learly validate
Eq. (3.1). The rst study intended to do exatly this was performed in Ref. [97℄, in whih
two dierent implementations of the TMM were employed. The appliability of the TMM
is based on the fat that the TIP problem in 1D an be mapped onto a SP problem in a
highly orrelated 2D random potential as shown in Fig. 3.7. Note that in this TMM the
antisymmetry of the fermioni TIP wave funtion k (n) l (m) = k (m) l (n) has to be
negleted. The results of Ref. [97℄ for this TMM support Eq. (3.1) but with an exponent
of 1:65 instead of 2. However, the loalization lengths shown in Ref. [97℄ are muh larger
than the system sizes onsidered and no FSS had been done. Thus in Ref. [RS97a℄, we used
a slightly dierent TMM proedure and found that (i) the enhanement 2 =1 dereases
with inreasing system size M , (ii) the behavior of 2 for U = 0 is equal to 1 in the
limit M ! 1 only, and (iii) for U 6= 0 the enhanement 2 =1 also vanishes ompletely
in this limit. Therefore we onluded [98, RS97a, RS97b℄ that the TMM applied to the
TIP problem in 1D measures an enhanement of the loalization length whih is due to
the niteness of the systems onsidered. The same onlusion also applies for a long-range
The interplay of interations and disorder I: two interating partiles
M
by
1/
Sq
rt(
2
)
sy
R
m
m
et
ry
r
ba
g
bo
TM
un
da
ry
36
Figure 3.8: Shemati diagram for the bag TMM. The TMM now proeeds along the diagonal
(green). The bag | equivalent to an additional strong attration | is indiated by the bold
green line. The symmetry of the wave funtion has been taken into aount as indiated by the
pink region. Furthermore, there is a resaling due to the dierent distane between suessive
TMM slies.
interation U=(jn mj + 1), with n, m denoting the positions of the TIP.
Another TMM proposed in Ref. [97℄ is apable of taking into aount the antisymmetry of
the wave funtion and also is more suited in piking out the TIP modes [125, 123, RS98℄.
However, these features neessitate the introdution of a nite maximal partile separation
whih is physially equivalent to introduing a bag. As the separation reahes the bag
boundary, the partiles feel an innitely attrative interation as shown in Fig. 3.8. Many
studies have sine then tried to isolate the eet of the bag interation from the physially
relevant TIP interation, but have not sueeded thus far [123, 125, 127, RS98℄. Partiularly
puzzling is the result that there is a dierene whether the TIP problem is being studied
in SP oordinates, or whether the problem is rst transformed into relative and enterof-mass oordinates and then disretized [123, 125℄. Even after FSS, the TMMs of Refs.
[125, 127℄ show an enhaned loalization length already for U = 0 [127℄. Note that a small
but nite enhanement 2 > 1 an by found when a longer-range interation is assumed
for the TMM with bag interation [RS98℄. However, the dependene of 2 on 1 does not
follow to proposed form (3.1).
List of enlosed publiations relevant for this setion
[RS97a/xiii℄
[RS97b/xiii℄
No enhanement of the loalization length for two interating partiles in a random
, R. A. Romer, and M. Shreiber, Phys. Rev. Lett. 78, 515{518 (1997).
potential
The enhanement of the loalization length for two-interating partiles is vanish-
, R. A. Romer, and M. Shreiber, Phys.
ingly small in transfer-matrix alulations
Rev. Lett. 78, 4890 (1997).
The interplay of interations and disorder I: two interating partiles
[RS98/xiv℄
37
Weak deloalization due to long-range interation for two eletrons in a random
, R. A. Romer, and M. Shreiber, phys. stat. sol. (b) 205, 275{279
potential hain
(1998).
3.5 Using deimation to study TIP in random environments
The obvious failure of the TMM approah to the TIP problem in a random potential
has lead us to searh for and apply another well tested method of omputing loalization
lengths for disordered system: the deimation method [177℄. Furthermore, instead of
simply onsidering loalization lengths 2 (U ) obtained for nite systems [97, 211, 273, 313℄,
or by simple extrapolations to large M [93, RS97a℄, we onstruted FSS urves and ompute
from these urves saling parameters whih are the innite-sample-size estimates [273℄ of
the loalization lengths 2 (U ) at interation strength U . We nd [LRS99a, LRS98, RLS99a℄
that onsite interation in 1D indeed leads to a TIP loalization length whih is larger than
the SP loalization length at E = 0. However, the atual funtional dependene is not
simply given by Eq. (3.1). In fat our data allow us to see 2 (U ) 2 (0) with an
exponent whih inreases with inreasing jU j at E = 0. The enhanement of the TIP
loalization length 2 (U ) is up to 75% due to the onsite interation. This enhanement
persists, unlike for TMM, in the limit of large system size and after onstruting innitesample-size estimates from the FSS urves. We tted our results to various suggested
models. The best t was obtained when the enhanement 2 (U )=2 (0) depends on an
exponent whih is a funtion of the interation strength U . Suh a U -dependent exponent
had been previously predited in Ref. [227℄ for interation strengths up to U = 1 with up to 2. However, we nd that reahes at most 1:5 for U = 1. We do not see a behavior
as in Eq. (3.1) with exponent 2 when using the t funtion of Ref. [227℄. On the other
hand, after saling the data onto a single saling urve and using a t funtion as proposed
with = 2 in Ref. [211℄, we nd indeed = 2 for not too small disorder strength, e.g.,
W 2:5 for U = 1, but observe a rossover to a behavior with = 3=2 for smaller W .
For values of U & 1:5 we observe that the enhanement dereases again; the position of
the maximum depends upon W . Thus the proposed duality [306℄ between the behavior at
small and large U is approximately valid. Similar results are obtained by plaing the two
partiles in dierent random potentials [LRS98, LRS99a℄. This latter situation is lose to
a setup of interating eletrons and holes that has been proposed for an experimental test
of the TIP eet [41, 111℄.
List of enlosed publiations relevant for this setion
[LRS98/xiv℄
, M.
Leadbeater, R. A. Romer, and M. Shreiber, in Exitoni Proesses in Condensed
Matter, R. T. Williams and W. M. Yen, Editors, PV 98-25, p. 349{354, The
Eletrohemial Soiety, Pennington, NJ (1998).
Formation of eletron-hole pairs in a one-dimensional random environment
38
[LRS99a/xiii℄
The interplay of interations and disorder I: two interating partiles
Interation-dependent enhanement of the loalization length for two interating
, M. Leadbeater, R. A. Romer,
and M. Shreiber, Eur. Phys. J. B 8, 643{652 (1999).
partiles in a one-dimensional random potential
[RLS99/xiv℄
Saling the loalisation lengths for two interating partiles in one-dimensional
, R. A. Romer, M. Leadbeater, and M. Shreiber, Physia A
266, 481{485 (1999).
random potentials
3.6 The TIP eet in a 2D random environment
In Ref. [RLS99b℄ we have employed the deimation method for the ase of TIP in quasi-1D
strips of xed length L and small rossetion M < L at E = 0 and for 36 various disorders
and 51 interations strengths. The analytial onsiderations for 2D [146, 145℄ are based on
similar RMM arguments as for 1D and it has been predited that the enhanement of 2
at E = 0 should be
2 / 1 exp
2 2
U t2
1 ;
(3.8)
with 1 / exp t2 =W 2 the SP loalization length in 2D [192℄. As shown in Ref. [RLS99b℄,
we nd that the enhanement is even stronger and leads to saling urves that have two
branhes indiating a transition of TIP states from loalized to deloalized behavior. This
result is in qualitative agreement with a previous study of Ortu~no and Cuevas [212℄ who
numerially nd a transition for U = 1 using the reursive Green funtion method previously employed for the 1D TIP ase [211℄. Our 51 interation strengths and 36 disorder
values allow us to map the (U , W ) phase diagram of the TIP deloalization-loalization
transition and we an study how the ritial exponent of the loalization length hanges
with hanging U . We nd that for all U 2 (0; 2℄, the exponent is systematially larger
than the ritial exponent of the usual Anderson transition for non-interating eletrons
in 3D. Note that although our system sizes are fairly small, we nevertheless nd that FSS
of the data works better when inluding data for the largest M values. Thus we see that
the transition apparently beomes even more robust for large system sizes. I emphasize
that this transition is not a metal-insulator transition in the standard sense sine only the
TIP states show the deloalization transition. The majority of non-paired states remains
loalized.
List of enlosed publiations relevant for this setion
[RLS99b/xiv℄
, R. A.
Romer, M. Leadbeater, and M. Shreiber, aepted for publiation in Ann. Phys.
(Leipzig) 8, (1999).
Numerial results for two interating partiles in a random environment
The interplay of interations and disorder I: two interating partiles
39
3.7 The TIP eet lose to an MIT
Thus far I have been mostly onerned with the TIP eet in 1D where without interation
all states are loalized. Of ourse it would be even more interesting to study the inuene of
interation diretly at the MIT in 3D. However, the numerial eort to do so is prohibitive
even for TIP, sine the problem is equivalent to a six-dimensional SP system with orrelated
disorder. Fortunately, there is a model whih exhibits an MIT whih is driven by inreasing
a loal site potential even in 1D. This model is known as the Aubry-Andre model [11, 130,
138℄ with Hamiltonian
M
X
M
X
H = (yn+1 n + h::) + n yn n
n=1
n=1
(3.9)
where n = 2 os(n + ) with =2p
an irrational number, whih we have hosen as the
inverse of the golden mean =2 = ( 5 1)=2, and is an arbitrary phase shift. The yn
and n are the reation and annihilation operators for an eletron at site n. For TIP, the
interation Hamiltonian is given as
M
X
n;m=1
Un;m yn# n# ym" m" ;
(3.10)
and we assume that the TIP have dierent spins. Un;m denotes the interation between
partiles: Un;m = UÆnm for Hubbard onsite interation or Un;m = U=(jn mj + 1) for
long-range interation. For < 1, all SP states in the model with U = 0 have been proven
rigorously to be extended, whereas for > 1 all SP states are loalized [11, 161, 162,
215, 214, 239, 298℄. Diretly at = = 1, the model is equivalent to the Harper model
[130, 138℄ and the SP states are ritial. Thus the MIT is similar to the MIT in the 3D
Anderson model, but there are no mobility edges as in Fig. 1.4.
The model has been previously onsidered at U > 0 from the TIP point of view in Refs.
[13, 14, 258℄. It has been shown that on the loalized side, the TIP eets persists, i.e., the
TIP loalization length 2 > 1 . On the extended side, it was argued that the interation
leads to a loalization of the TIP states. In Ref. [EGRS99℄, we have used the TMM
together with FSS to study the problem. Let me emphasize that ontrary to the problem
with TMM for the TIP situation in nite systems, together with FSS the TMM approah
an be used to give meaningful results. However, the omputed loalization lengths are
no longer diretly the loalization length of a TIP pair, but rather measure the inuene
of the presene of the seond partile on the transport properties of the rst. In addition
to investigating the onsite interating ase, we have also studied long-range interations
in Ref. [EGRS99℄ as in setion 3.4. The numerial data of our study suggest that the
qualitative piture put forward in Refs. [13, 14, 258℄ is orret. Additionally, we nd that
whereas onsite interation does not shift the MIT from = 1, long-range interation does
hange the MIT towards smaller values 0:92. I remark that these results an also be
obtained by the deimation method of hapter 3.5.
40
The interplay of interations and disorder I: two interating partiles
List of enlosed publiations relevant for this setion
[EGRS99/xiii℄
, A. Eilmes, U. Grimm,
R. A. Romer, and M. Shreiber, Eur. Phys. J. B 8, 547{554 (1999).
Two interating partiles at the metal-insulator transition
Chapter 4
The interplay of interations and
disorder II: nite partile density
4.1 Introdution
As shown in the last hapter, interation an lead to an enhaned transport for two partiles
in a random environment. Unfortunately, there is no straightforward way to extrapolate
this result to the physially relevant ase of nite partile density, although at present there
are attempts [126℄ to study the problem for three and four partiles along the lines indiated
in hapter 3. A more detailed understanding of the interplay of disorder and interations
at nite density appears neessary before the \exponent puzzle" [137, 290, 304℄ or the 2D
MIT [167, 168, 171℄ are to be explained. Furthermore, various experiments over the last few
years have measured the magneti response of quasi-1D rings [3, 45, 53, 182, 193℄. These
experiments onrm the existene of the previously predited Aharonov-Bohm persistent
urrents [45, 46℄. As it turns out, the experimental value for the persistent urrent is two
to three orders of magnitude larger than the theoretial preditions based on alulations
in a disordered but noninterating eletron gas [3℄. It is ommonly believed that eletroneletron interations might again be the key to resolving this disrepany.
If one were to study the full eletron problem as in the Hubbard model [103℄, the Hilbert
spae H for N" (N# ) eletrons with spin up (down) on a hain of length L would be
dimH =
L
N"
L
N#
:
(4.1)
Even for moderate numbers of partiles, say, N" = N# = 10 on a hain of length L = 20,
one has to take into aount 34,134,779,536 states. This should be related to the 5 weeks
it takes on an Hewlett-Pakard 9000/889 K460 (see hapter 5) to ompute the single state
shown in Fig. 1.6 for a omparably small Hilbert spae of 106 states.
In order to proeed, one has to employ appropriate approximation shemes. The rst
approximation that I will present is related to simply hoosing a somewhat less physial
model. The advantage of the model is its exat solvability. Another approah is taken
41
42
The interplay of interations and disorder II: nite partile density
in setion 4.3, where the eets of disorder and interations are studied by a perturbative
renormalization group (RG) treatment. Finally, I will demonstrate an ingenious sheme
whih reently has been devised [315, 316℄ and allows the approximate numerial treatment
of large 1D disordered interating models by reduing the Hilbert spae to a manageable
number of states.
4.2 Integrable impurities in 1D
Integrable quantum systems have a rih and diverse history. For example, the 1D Hubbard
model without disorder has been solved exatly in 1963 by Lieb and Wu [187℄ and this Bethe
ansatz [20, 26℄ solution until today remains one of the benhmarks for low-dimensional
orrelated eletron systems [196℄. The main feature of these systems is their integrability,
i.e., the presene of an innite number of onserved quantities [20, 163, 286℄. Due to
these, the system is severely restrited and one an show that the wave funtions are plane
waves but with momenta whih are renormalized due to the many-partile interations
[163, 196, 286℄. One of the onserved quantities of motion is usually the total momentum
of the system due to the translational invariane of the model.
In 1984, it was shown by Johannesson and Andrei [8℄ that a breaking of translational
invariane need not destroy the integrability. Their approah onsisted in plaing a spin
with s = 1 at an arbitrary site of a spin- 21 hain. This suess was nearly forgotten, when
in 1994 and 1995 two papers appeared that showed a onstrution of integrable many-body
models with peuliar impurities [15, 243℄. These models do not show any loalization of
the ground-state wave funtion and their energy spetrum is independent of the loation
of the impurities. Nevertheless, the Hamiltonians of the models inlude impurity terms
that are very similar to onsite-randomness as in the Anderson model of loalization of Eq.
(1.10).
In hapter 2, I argued that Anderson loalization is due to the interferene of the eletron
with itself beause of the reetion at the random sattering potentials. Sine there is no
loalization of the wave funtion at the integrable impurities, it suggests that there is also
no suh baksattering at the impurities. In Ref. [EPR97℄, we show that this is indeed
the ase: the impurities at purely as forward satterers, i.e., just hange the phase of the
wave funtion. This property remains unhanged even in the presene of the many-body
interation.
An analogue of these integrable impurities an be found in the ase of light waves. Consider
a long strip of glass, interspersed with piees of bifringent material (BM) of the same index
of refration as the glass. Then as irular polarized light enters the strip and reahes the
rst BM, its plane of polarization will be rotated by an angle / l1 where l1 represents
the length of the rst BM and is the material spei rotation angle per unit length
[134℄. There is no reetion at the ontat due to the idential indies of refration. After
the next BM, we have = l1 + l2 and so on. Thus the net eet of the impurity BMs
is a rotation of the plane of polarization, i.e., a hange in the overall phase of the wave
funtion of light just as for the eletroni wave funtion in Ref. [EPR97℄.
The interplay of interations and disorder II: nite partile density
43
In Ref. [GGR98℄, we study a 1D model of interating eletrons [118℄ in the presene of
general boundary onditions [174, 268℄ whih may be viewed as soures and sinks for
eletrons. This study is preursor to a longer study in whih we ombine the perfetly
reeting boundaries with forward-sattering impurities in order to see whether more a
realisti disorder an be ahieved. But as the model remains integrable [21, 57, 92, 143,
240, 307℄, the physis is not omparable to the physis presented in the previous hapters.
List of enlosed publiations relevant for this setion
[EPR97/xiii℄
Absene of baksattering at integrable impurities in one-dimensional quantum
, H.-P. Ekle, A. Punnoose, and R. A. Romer, Europhys. Lett.
(1997).
[GGR98/xiv℄
, X.-W.
Guan, U. Grimm, and R. A. Romer, Ann. Phys. (Leipzig) 7, 518{522 (1998).
many-body systems
39, 293{298
Lax pair formulation for a small-polaron hain with integrable boundaries
4.3 RG approah for a mesosopi Hubbard model
The inlusion of disorder in the 1D Hubbard model destroys the integrability and no exat
solution exists within the Bethe ansatz [20, 26℄. Nevertheless, at least a qualitative insight
an be ahieved by RG arguments whih treat both interation and disorder perturbatively
[105, 106, 107, 108, 153℄. For the persistent urrent problem mentioned in the last setion,
the approah had been developed by Shulz and Giamarhi [106, 107℄ and by Shankar
[253, 254, 255℄.
The RG usually proeeds by employing a uto in momentum spae whih orresponds to
the lattie spaing. In 1995, it was shown by Giamarhi and Shastry [108℄ that the niteness
of the uto an be also interpreted as the eetive length sale of the model and thus allows
for the study of interation and disorder eets at a mesosopi length sale [152, 233℄.
They found that the ground state persistent urrent at nite repulsive interation strength
U exeeds the urrent at U = 0 already for very small values of the disorder. However,
in this study they negleted the derease of the urrent due to the interation [90, 91℄. In
Ref. [RP95℄, we rst used the Bethe ansatz results at no disorder [90, 91, 187℄ to study the
derease in the harge stiness D | a quantity whih is proportional to the urrent [RP95℄
| for nite U . Then we used these values as starting points for the RG of Ref. [108℄ and
omputed the renormalization of D as the mesosopi length of the system | and thus
also the disorder | inreases. We nd that for given U , a nite amount of disorder is
needed until D(U ) > D(0), i.e., until there is enhanement of the ground state urrent for
the interating system ompared to its value at no interation. Furthermore, the values of
disorder needed for this rossover are already quite large so that the RG results an only be
trusted up to U 0:8. In this regime, we also nd that the spin urrents are not enhaned.
For larger U values, the spin urrents an be interpreted as indiating a transition to a
random antiferromagnet (RAF), however, it is doubtful whether the RG results are still
reliable for suh large disorder values [204℄. In fat, we have heked that even with U = 0,
44
The interplay of interations and disorder II: nite partile density
l
BL/2-1
xl
xr
r
BL/2-1
Figure 4.1: Shemati diagram of the innite-size DMRG algorithm
for a disordered system. xl;r
l;r
denote the two sites added to the system. The new bloks BL=2 are indiated by the dashed lines.
The solid line denotes the superblok of the system with L sites.
the RG ow for large disorder erroneously indiates the existene of the RAF phase. Thus
the existene of a RAF in the disordered 1D Hubbard model remains plausible [204℄ but
unproven.
List of enlosed publiations relevant for this setion
[RP95/xiii℄
Enhaned Charge and Spin Currents in the One-Dimensional Disordered Mesosopi
, R. A. Romer, and A. Punnoose, Phys. Rev. B 52, 14809{14816
Hubbard Ring
(1995).
4.4 A DMRG study for the 1D Hubbard model
The density-matrix renormalization group (DMRG) is a numerial diagonalization sheme
for interating quantum many-body systems whih eetively implements a weighting proedure for eigenstates based on the eigenvalues of the density matrix of parts of the system
[316℄. The method has been suggested by S.R. White [315, 316℄ and sine then it has been
widely used, espeially for the alulation of orrelation funtions of orrelated eletron
systems [104, 218, 240, 241℄. It is most reliable in 1D, but extensions to 2D exist, too
[316℄. From an analytial point of view, it an be seen as a numerial onstrution of a
variational wave funtion [216℄.
Innite- and nite-size DMRG
There are in priniple two versions of the DMRG algorithm [240, 315, 316℄. In the innitesize algorithm, one starts with a system of size L 2 and deomposes it into a left and a
l;r
right blok BL=
2 1 as shown in Fig. 4.1. Next, two new sites with their respetive partiles
are added to the bloks and the full system | also alled the superblok | is diagonalized.
The interplay of interations and disorder II: nite partile density
45
l;r
The target state of the full system is then given in terms of the bloks BL=
2 as
j i=
X
ij
i;j
jiiB jj iB ;
r
L=2
l
L=2
(4.2)
where jiiB 2 and jj iB 2 denote the states in the left and right blok, respetively. Note
that not only the ground state of a given system but also exited states an be used as
target states. Then the redued density matrix for eah new blok, i.e.,
X
(4.3)
lkm =
k;j m;j
l
L=
r
L=
rkm =
j
X
j
j;k j;m
(4.4)
is diagonalized and only the M most probable states, orresponding to the M largest eigenl;r
values of l;r , are kept in the bloks BL=
2 . The Hamiltonian and other relevant operators
are trunated onto the M states. This proess is then repeated by adding two more sites
while keeping the partile density xed. The algorithm stops when the desired number of
lattie sites is reahed. The advantage of the algorithm lies in the fat that the number
of states is kept xed throughout the iteration although the number of sites and partiles
inreases. I emphasize that the algorithm as reviewed here is slightly dierent from the
traditional approah [240, 315℄, where, instead of using dierent left and right bloks, the
mirror image of the rst blok is used as the seond blok. The dierenes are small, but
the present approah is better suited to disordered systems.
The innite-size algorithm allows to reah very large system sizes, but it has at least one
severe problem when being used for a disordered system: When adding additional sites
with random potentials, the wave funtions in the left and right bloks have no freedom
to relax into the additional random potential. E.g., suppose that up to system size L all
randomness was absent and we now add a new site with a deep potential well. Then in
priniple the wave funtion should have a high probability amplitude at this additional
site. However, this will not happen sine the ratio of the probability amplitudes of the
new site and the blok is kept xed at 1=L. We have heked that this problem results
in only small errors for expetation values of operators suh as the target state energy.
However, for orrelation funtions the problem is muh more severe and might give rise to
an erroneous funtional form of the orrelation.
In order to irumvent this, the nite-size DMRG algorithm an be used. This algorithm
is very similar to the innite-size algorithm. In Fig. 4.2 I show it shematially. The main
idea of the algorithm is that, after having reahed a given system size by the innitesize DMRG, one now keeps the system size L xed, but onseutively breaks the system
towards the right into bloks of sizes fL=2; L=2 2g; fL=2 + 1; L=2 3g; : : : ; fL 3; 1g
and then does the reverse towards the left and nally bak towards fL=2 1; L=2 1g. In
this way, the wave funtion an relax in response to the values of the random potentials
and estimates of orrelations funtions beome muh more reliable [240℄. The numerial
eort, unfortunately, an be more than doubled when ompared to a ode based on the
innite-size DMRG algorithm alone.
46
The interplay of interations and disorder II: nite partile density
l
BL/2
x
y
r
BL/2-2
Figure 4.2: Shemati diagram of the nite-size DMRG algorithm for a disordered system. x and
y denote the two sites being added. Note that left and right blok need no longer orrespond to
the same lattie size.
The half-lled, disordered Hubbard model at large U
In Ref. [PPR97℄, we studied the losing of the Mott gap in the 1D Hubbard model as
the onsite potential disorder is inreased. From simple potential energy arguments, one
expets that the gap loses at U = W=2, namely when the energy gain due to two partiles
sitting in a deep well with n W=2 is balaned by the repulsive interation strength U
on the same site. The innite-size DMRG used in Ref. [PPR97℄ orretly reprodues this
behavior. We also study the behavior of the spin-spin orrelation funtion in Ref. [PPR97℄
and nd that in the gapless phase the orrelation deays exponentially as expeted in this
high disorder phase. However, in the gapped phase with U > 0, our DMRG results indiate
that the orrelation an be desribed by a power-law deay just as for the lean ase at
W = 0. This result appears to be in ontradition to RG arguments for the disordered
XXX Heisenberg model [64, 88℄.
In order to understand this ontradition, let me rst explain how the Hubbard model at
half-lling is related to the disordered Heisenberg antiferromagnet [196, 232℄. It is wellknown [89℄ from degenerate perturbation theory that the ordered and half-lled Hubbard
model at large U redues to the XXX antiferromagneti Heisenberg hain with oupling
onstant J0 = 4t2 =U . In order to derive the orresponding result for the disordered system,
let me dene
H = HU;W + Ht ;
HU;W = U
Ht =
N
X
ni" ni# + i (ni" + ni# ) ;
i=1
N X X
t
yi+1 i + h.. ;
=";# i=1
(4.5)
(4.6)
(4.7)
with t, U , and W denoting the hopping, interation and disorder strength, respetively,
dened analogously in the previous hapters. At half-lling, we have one partile for eah
site for U W . Let ji denote any of the singly oupied degenerate eigenstates of HU;W
and let EU;W be the orresponding energy, i.e., HU;W ji = EU;W ji. Dene ji to be the
47
The interplay of interations and disorder II: nite partile density
eigenstate of the full system, i.e., H ji = E ji. Then we an write formally
ji = E 1H Ht ji:
U;W
P
Introduing the projetor P = 1
jihj, this an be rewritten as
X
ji = E PH Htji + ji EhjHEtji :
U;W
Let me dene ji =
P
a
(4.8)
(4.9)
U;W
ji and a = EhjHE ji . Then
t
U;W
ji = ji + E PH
U;W
Ht ji
(4.10)
and we have used that [P; HU;W ℄ = 0. Iterating Eq. (4.10) to rst order in powers of
P
Ht , and substituting the result bak into the expression for a , one has
E H
U;W
(E
EU;W )a =
X
a hjHt
P
H j i:
E HU;W t
(4.11)
Consequently, terms of the form
yi+1 i
P
yj +1 j
E HU;W
(4.12)
will enter the r.h.s. of Eq. (4.11). The projetion P asserts that only singly oupied states
survive. HU;W is diagonal for these states and (4.11) redues to
(E
EU;W )a =
X
a
X
i;;
0
!
h
jyi i+1 yi+1 i j i hjyi+1 i yi i+1 j i
2
:
t
+
U ( )
U + ( )
0
i
0
i+1
0
0
i
i+1
(4.13)
Note that the denominator in this expression simply reets the energy of the perturbed
state. The reation and annihilation operators an be expressed as spin operators S~ via
[89℄
yi+1 i yi i+1 = 2S~i S~i+1
0
0
(4.14)
and onsequently the Hamiltonian (4.5) is
N
X
i=1
Ji;i+1 S~i S~i+1 + h..
(4.15)
48
The interplay of interations and disorder II: nite partile density
W=0.1
W=0.4
W=0.7
W=1.0
W=1.3
W=1.6
W=1.9
2
10
1
P(J)
10
0
10
−1
10
−2
10
−3
−2
10
−1
10
0
10
10
J
Figure 4.3: Distribution P (J ) of random oupling strength for the eetive RAF in the half-lled
Hubbard model at large U for various values of W and t = 1, U = 2. Only every fth data point
is shown.
with
Ji;j =
1
J0
i j 2
U
J0 :
(4.16)
In the absene of disorder i = j = 0 and the half-lled Hubbard model maps onto the
XXX hain [196℄ for large U . On the other hand, at nite disorder W , the Ji;j are not
unorrelated but only if we make this assumption does the disordered half-lled Hubbard
model map diretly onto the RAF. Assuming that the random onsite potentials i are
uniformly distributed in [ W=2; W=2℄ and assuming further that the Ji;j are unorrelated,
one an show [232℄ that the distribution for the Ji;j 's is given as
U
P (J ) = W
1
Up
1 J0 =J
W
J0 =J 2
1 (W=U )2
p
(4.17)
J0
for J0 < J < 1 (W=U
)2 . In Fig. 4.3 I plot this distribution for various values of W . P
is learly not a uniform, Gaussian or Lorentzian distribution as is ommonly assumed in
studies of the random XXX antiferromagnet. It deays very fast suh that there are only
very few large values of J .
The interplay of interations and disorder II: nite partile density
Jl
σl
J
49
Jr
σ0 σ1 σr
J’
σl
σr
Figure 4.4: Left: The RG senario for random ouplings in the disordered XXX antiferromagnet. The irles denote spins , the lines mark the ouplings J . In one RG step, the oupling
J is taken out of the system and the resulting eetive J 0 ouples the neighboring spins l;r .
Right: Shemati view of short- and long-ranged spin singlets in the disordered and half-lled 1D
Hubbard model at large values of U .
Correlation funtions at large U and the onnetion to the XXX
RAF
Let me now try to explain the basi idea of Refs. [64, 88℄ regarding the development of
the spin-spin orrelation in the random XXX antiferromagnet. Suppose that in the hain
there exists a strong bond J whih ouples spins 1 and 2 as shown in the right panel
of Fig. 4.4. Thus 0 and 1 most likely will form a singlet with energy E . The exited
states are the three degenerate triplet states with E . These states will be mixed due
to the presene of the neighboring spins. The mixing leads to an eetive oupling of
spins l and r with oupling strength J 0 = Jl Jr =2J . If this eetive oupling is again
larger than the other neighboring ouplings, this will again lead to the formation of a
singlet. Continuation of this proess to more and more sites leads to an eetive oupling
of spins whih are spatially separated over large distanes. In the left panel of Fig. 4.4
I show this senario shematially. If this situation is indeed present, then for a typial
onguration the probability of getting very large bonds is very low and the spin-spin
orrelations will be dominated by the short-ranged spin singlets. These are distributed
randomly in the system aording to the distribution of J 's and thus the typial orrelation
will be exponentially deaying. However, if in the thermodynami limit there remains a
small but nite probability of having long bonds of the order of the system size, then the
average orrelation will be dominated by these. The average orrelation funtion should
thus be a power law sine these rare singlet pairs are strongly orrelated over large distanes
of the 1D lattie.
The most favorable ase for the formation of loal singlets in a given hain ours for
J = Jmax and Jr;l = J0 . For t = 1 and U = 2, one an ompute from Eq. (4.17) the values
of Jmax and J 0 . One nds that Jmax 2J0 for W 1:4. Thus for small disorders, loal
singlets are unfavorable and should not form. However, already for W < W 2 suh loal
singlets may dominate. Thus the statement of Ref. [PPR97℄ regarding the oinidene of
losing of the Mott gap and the rossover from power law to exponential behavior of the
spin-spin orrelation funtion | based on the innite-size DMRG algorithm | appears
50
The interplay of interations and disorder II: nite partile density
questionable.
The reader may probably wonder why I have given so muh spae to the disussion of these
arguments. As explained above, the main point is that if the RG arguments are valid, then
one should see this in a dierent behavior of the typial and average spin-spin orrelation.
Our results [PPR97℄ for the innite-size algorithm do not show this dierene. However,
use of the nite-size DMRG should allow the study of the eet. Thus at present the
losing of the interation-indued Mott gap due to inreasing disorder seems to be valid,
but the aompanying transition in the nature of the orrelation funtions remains on open
problem and hallenging problem. Work in this diretion using the nite-size DMRG is in
progress.
List of enlosed publiations relevant for this setion
[PPR97/xv℄
,
R. Pai, A. Punnoose, and R. A. Romer, preprint series of the SFB 393, No. 97-12
(Chemnitz, 1997), ond-mat/9704027, submitted to Phys. Rev. Lett., (1997).
The Mott-Anderson transition in the disordered one-dimensional Hubbard model
Chapter 5
Massively parallel algorithms for the
eigenvalue problem in disordered
systems
5.1 Introdution
Many of the studies presented in hapters 2, 3, and 4 rely on advaned numerial tehniques
and the presene of state of the art omputing failities. Suh knowledge of numerial
algorithms and muh omputer power are available at the TU Chemnitz and I have been
fortunate to be able to use both. Nevertheless, the quest for evermore preise data leads
to yet more demand for large system sizes and number of ongurations. For example, the
preise determination of the ritial exponents for the MIT in the 3D anisotropi or random
hopping Anderson model as in hapter 2 requires at least a redution in the statistial error
of the raw data of one order of magnitude. This then implies that the number of transfermatrix multipliations has to be inreased by 2 orders of magnitude [192℄. Another example
of the high demand for omputer power is shown in Fig. 1.6. In order to onstrut this
wave funtion, about 5 weeks of omputing time on an Hewlett-Pakard 9000/889 K460
are needed when using one proessor.
A promising approah that should help in meeting these demands is based on the onept
of massively parallel omputing [40, 99, 200℄. The idea behind this approah is that |
instead of inreasing the omputing power by onstruting evermore powerful proessors
| it might be advantageous to simultaneously use many, perhaps somewhat less powerful,
mahines. The omputational problem onsequently needs to be divided into tasks that
an be done on the individual mahines. Depending on the struture of the problem, the
individual proesses will need to exhange data at ertain stages in the omputation. Thus
the art of developing massively parallel algorithms onsists in the hoie of optimizing
the load on eah mahine while minimizing the time spent on ommuniating between
mahines [99℄. In the present hapter, I will detail our strategies for ahieving both for the
most important algorithms used in this thesis.
51
52
Massively parallel algorithms for disordered systems
1
3
4
1
3
6
0
1
0
0
0
2
7
6
5
4
3
5
648
7
5
85
7
4
4
5
43
7
8
7
3
2
2
8
7
8
Figure 5.1: Shemati diagram of the two possible shemes for storing vetors in the P-TMM.
Eah data olumn of the 6 6 matrix represents a single vetor. The solid lines indiate the
distributed elements sheme (DES), whereas the dashed lines show the distributed vetor sheme
(DVS).
5.2 A parallel implementation of the TMM
The TMM onsists of two main parts as explained in hapter 1.5: (a) the matrix multipliation as in Eq. (1.13), and (b) the orthonormalization of eah vetor after a ertain
number of multipliations. On a single-proessor mahine, both parts take up about 50%
of the omputation time and are thus equally important. I emphasize | independent
of issues related to parallelization | that for the standard Anderson model (1.10), it is
most advantageous not to store the Hamilton matrix into mahine memory, but rather
enode it into the algorithm. For other systems, e.g., the ones studied in hapter 2.5, this
is no longer feasible, resulting in a onsiderably slower performane. Let me denote the
parallelized version of the TMM by P-TMM in the following.
For the matrix multipliation of the TMM there are at least two possibilities of parallelization. The rst sheme is based on storing the elements of eah vetor ( n+1 ; n )T on
dierent mahines as shown in Fig. 5.1. This I all the distributed element sheme (DES).
Eah mahine performs only part of the matrix-vetor multipliation for eah vetor. Thus
the speedup of this part of the algorithm is proportional to the number of mahines used.
However, at the boundaries, the mahines need to ommuniate due to the hopping in the
diretion perpendiular to the TMM propagation, e.g., the n;m1 terms in Eq. (1.11) for
the 2D loalization problem. This dereases the speedup. On the other hand, the reorthogonalization an be done fairly fast in this DES sheme sine the relevant salar produts an
be omputed rst loally and only a simple addition over all mahines is needed. Thus the
DES dereases the amount of omputer time needed for the matrix-vetor multipliations
and allows for a simple parallelization of the reorthogonalization.
The seond parallelization sheme is based in storing the omplete vetors on individual
53
Massively parallel algorithms for disordered systems
orthonormalization
proc
#0
proc
#1
proc
#2
proc
#k
proc
#1
proc
#2
proc
#k
proc
#2
proc
#k
step 0
step 1
...
step 2
step k
proc
#k
Figure 5.2: Shemati diagram of the orthogonalization proedure for the P-TMM based on the
DVS. In step 0, the rst of n0 vetors on proessor 0 is orthonormalized with respet to all
other n0 1 vetors and then sent to proessor 1. Then the seond vetor on proessor 0 is
orthonormalized and so on until all n0 vetors are orthonormal. In the meantime, the other
proessors have already orthonormalized their vetors with respet to the ones sent to them from
proessor 0. The algorithm now ontinues in step 1 with the n1 vetors on proessor 1 until
nally in step k all vetors on all k mahines are orthonormal.
mahines as also shown in Fig. 5.1. I denote it as the distributed vetor sheme (DVS). No
ommuniation is needed for the matrix-vetor multipliation part of the algorithm for the
DVS and the speedup is proportional to the number of mahines used. Unfortunately, the
DVS needs a lot of ommuniation for the reorthonormalization sine it requires to send
eah vetor to every other vetor in order to ompute the neessary salar produts. In
Fig. 5.2 I show an implementation of the DVS sheme that is nevertheless rather eetive
for the orthogonalization.
At rst glane, the seond sheme looks muh less onvining than the rst, sine in the
reorthonormalization, mahines suh as pro.#0 are idle for a long time. Nevertheless,
a diret omparison between the distributed element (DES) and the distributed vetor
shemes (DVS) show that the latter is faster. The DVS an be additionally aelerated by
reduing the number of reorthonormalizations. This redution an be ahieved by adapting
the number of matrix multipliations between suessive reorthonormalizations aording
to the dierene in norms of smallest and largest Lyapunov exponent [123, 192℄ after eah
reorthonormalization. In Fig. 5.3 I show how these improvements lead to a redution in
54
Massively parallel algorithms for disordered systems
20
5
1 proc.
4 proc., fixed ortho. scheme
4 proc., adaptive ortho. scheme
4
speedup
comp. time in ksec
15
10
3
2
5
0
8
10
12
14
1
4
6
8
10
system width
12
14
1
8
16
32
# procs
Figure 5.3: Left: Performane data of the P-TMM for various implementations of the DVS
algorithm. The times for data indiated by squares and diamonds in the left panel orrespond
to xed and adaptive orthonormalization shemes as outlined in the text. Right: Speedup for
dierent system sizes from 8 8 to 14 14 as a funtion of the number of proessors used for
the DVS. Note that the speedup for small system sizes deays upon inreasing the number of
proessors already for a small number of proessors due to the inreased ommuniation.
the time needed for the P-TMM runs. For example, P-TMM runs for the 3D Anderson
model with random hopping as in hapter 2 with system size 14 14 need only about 20%
of omputation time for 32 proessors when ompared to 1 single proessor. I emphasize
that many of the papers presented in hapters 2 and 3 would not have reahed the desired
auray without the P-TMM. Most importantly, the large system sizes shown in Fig. 1 of
Ref. [RS97a℄ were omputed by P-TMM on a parallel omputer arhiteture.
The speedup urves of Fig. 5.3 show that for the given parallel arhiteture of the GC/PowerPlus-128 (Parsyte)-parallel omputer (GCPP), the implementation of the P-TMM does
in fat give a net redution in omputing time for large system sizes. However, the real test
of the usefulness of these odes omes when omparing to the performane on the fastest
available serial omputers. In Fig. 5.4 I show that at present due to the slowness of an
individual proessor on the GCPP, it is only for very large system sizes that the P-TMM
performs better then its single-proessor version on a Pentium II arhiteture.
5.3 A parallel implementation of CWI
As already mentioned in hapter 1.5, we have used the CWI of the Lanzos algorithm
[61, 62℄ for the diret diagonalization of the Anderson Hamiltonian. In Refs. [EMMRS99,
SMREM99℄, we had shown that this algorithm is still the fastest [56℄ and least memory
onsuming when ompared to more reent developments in iterative eigensystem [29, 50,
71, 180, 181, 219, 276℄ and linear equation solvers [17, 100, 121, 236, 303℄. In partiular,
the CWI is faster than Arnoldi-based eigensolver [181℄ even when the latter algorithm
is used together with polynomial aeleration or shift-and-invert tehniques for iterative
solvers [110℄. I emphasize, however, that the Arnoldi-based algorithm an also handle non-
55
Massively parallel algorithms for disordered systems
4
comp. time in sec
10
3
10
Linux (NAG)
Linux (PGI)
GCPP (8P)
GCPP (16P)
GCPP (32P)
2
10
8
10
12
14
system width
Figure 5.4: Comparison of the best 3D P-TMM implementation on the GCPP with up to 32
proessors with the 3D TMM implementation on a 400 MHz Pentium II (NAG, PGI) as a funtion
of system size M M . NAG and PGI distinguish dierent Fortran ompilers; the PGI ompiler
an optimize ode for the Pentium II arhiteture.
symmetri matries, whih the CWI annot. For the present problem of the Anderson-type
matries, CWI is nearly as fast as the sparse symmetri solver MA27 [322℄, and nearly two
orders of magnitude more memory eÆient. Thus the CWI provides the ideal starting
point for a parallized version.
At the heart of the Lanzos sheme is again a matrix-vetor multipliation as explained
in Eq. (3.1) of Ref. [EMMRS99℄. Thus both DES and DVS an be used. However, this
time we do not have to use a omplete set of vetors and the DES sheme an be applied
quite eetively. Furthermore, by making use of the speial struture of the Anderson
Hamiltonian (1.10), we an redue the ommuniations needs quite eetively by storing
also the neighboring elements of the vetor as shown in Fig. 5.5. The parallized version of
the CWI, alled P-CWI, has been used for physis appliations, most notably the ELS, in
Refs. [ERS98a, MR98, MRS97, UMRS99℄. We have tested the performane of the algorithm
in Refs. [EMMRS99, SMREM99℄ where we also show some speedup urves similar to Fig.
5.3.
The P-CWI works very eetively when used to onstrut a few eigenvetors in a given
energy interval, e.g., for the MFA of hapter 2. For the ELS, all or at least many eigenvalues
of many dierent disorder realizations are needed. In this ase it turns out to be most
eetive to naively parallize the bisetion part of CWI. Thus we simply divide the desired
part of the spetrum into many small intervals and then let the standard CWI run for
eah interval independently on eah proessor. This \naive parallelization" approah for
the bisetion part of the CWI has already been inluded into the P-CWI together with
56
Massively parallel algorithms for disordered systems
matrix
A1 1
1
0
0
1
proc #1
A2 1
0
A1 1
1
1
vector #1
0
A3 1
1
A4
1
1
x
1
111111111111111111111
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3
000000000000000000000
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111111111111111111111
4
x 1x4x
proc #4
x
A
x
1
2
4
proc #2
1
proc #3
1
x 1x2x 3
A2 1
x 4x3x 2
A3 1
Figure 5.5: Shemati diagram of P-CWI for, e.g., a 43 system with periodi boundary onditions. The 16 16 sub-matries Ai denote the Hamiltonian within a given 4 4 plane, unit
(1) and zero (0) matries represent the ouplings of planes by the orresponding unit and zero
matries. The full matrix will be distributed aording to the DES. However, instead of storing
matrix entries, we make use of the form of the Hamiltonian (1.10) and enode the matrix in the
algorithm. Further, the parts denoted by zero matries an be negleted. Thus on proessor 1,
we need to store only the orresponding part x1 of the vetor (x1 ; x2 ; x3 ; x4)T . We also store
the vetor parts orresponding to the neighboring slies, i.e., x2 and x4 (due to the boundary
onditions). After eah multipliation, the updated neighboring information is most onveniently
passed bakwards and forwards to the respetive proessors by employing a ring arhiteture for
the parallel onnetivity.
the DES and the ode is urrently being used for omputing the ELS of the anisotropi
Anderson model as outlined in hapter 2.2.
List of enlosed publiations relevant for this setion
[EMMRS99/xiii℄
The Anderson model of loalization: a hallenge for modern eigenvalue meth-
, U. Elsner, V. Mehrmann, F. Milde, R. A. Romer, and M. Shreiber,
SIAM, J. Si. Comp 20, 2089{2102 (1999).
[SMREM99/xv℄
, M. Shreiber, F. Milde, R. A. Romer, U. Elsner, and V.
Mehrmann, to be published in Comp. Phys. Comm., (1999).
ods
Eletroni states in the Anderson model of loalization: benhmarking eigenvalue algorithms
Chapter 6
Conlusions and Outlook
In the preeding hapters, I have presented results of transport properties in disordered
systems. In hapter 2, I was onerned with various modiations of the standard Anderson model of loalization in whih the mirosopi details were varied. For the ase of
anisotropi hopping of setion 2.2, the results based on ELS, MFA, and TMM show that
the 3D universality lass of the standard Anderson model remains unaeted: the MIT
exists as long as the hopping is non zero and the orresponding ritial exponent is unhanged. In setion 2.3, I show that numerial results for energy-level orrelation funtions
an reliably reprodue and thus support the eld theoreti, semilassial and RMT-based
analytial preditions. Furthermore, WFS is investigated as yet another tool for the haraterization of disordered systems. The random hopping ase of setion 2.4 shows that
even though there are no extended states in disordered and non-interating 2D systems,
there is evidene for ritial states at E = 0. In 3D, the MIT for the random hopping model
has been studied by TMM and the ritial exponent is estimated with very high auray
to be onsistent with the exponent of the standard Anderson model. The random hopping
ase an also be viewed as reeting a random distortion of the square lattie struture
whih gives rise to dierent distanes between the atomi wave funtions and thus loally
modied hopping integrals tij . In setion 2.5, we went one step further and studied the
loalization and transport properties of pure tight-binding Hamiltonians on planar random
and quasi-periodi graphs. Somewhat surprising, it seems that the ELS of these 2D models
is with very good auray given by the ELS of the GOE as on the metalli side of the
3D MIT. I emphasize that although this might be unexpeted, it is not in ontradition to
the general arguments of saling theory for generi systems: adding an arbitrary amount
of additional potential disorder, the models show again the usual 2D loalization behavior.
Chapter 2 ends with a phenomenologial investigation of the transport properties at nite
temperature. This is of partiular importane for an eventual omparison of non-universal
properties with experiments. Our results suggest that the simplied assumption of (E )
as in Eq. 1.6 is yet to simplied for a diret omparison with experimental results. In
summary, the results of hapter 2 provide a large set of numerial evidene in support of
the saling hypothesis for all generi non-interating systems.
In hapter 3, I studied the eets of a weak two-body interation on the loalization
57
58
Conlusions and Outlook
properties of 1D disordered hains. Although previous numerial studies based on TMM
failed to learly exhibit the proposed funtional form 2 / 21 , reent results based on
the deimation method, reursive Green funtion tehniques and the diusion of wave
pakets show onsistently that the TIP eet leads to the formation of pair states in
the spetrum even for repulsive partiles and these pairs have a large loalization length
2 . The quadrati behavior as a funtion of 1 is still being disussed, but the results
presented in setion 3.2 show that suh deviations are to be expeted due to the relevane
of higher order terms in the perturbation theory. The results of Ref. [145℄ suggest that the
enhanement should be even larger in 2D. Reent numerial results as in Ref. [212℄ and
setion 3.6 suggest that even a loalization-deloalization transition may exist. In Fig. 6.1 I
show FSS results of Ref. [RLS99a℄ for the loalization lengths of TIP states in the 2D system
omputed with the deimation method of hapter 3.5. Although the FSS urves are not
10
U=0
U=1
U=2
0.5
ξ2
log10λ2/M
2D TMM
U=0.00
U=1.04
U=1.56
2
1.0
1
10
0.0
−0.5
0
10
−1.0
0
2
4
6
0
2
4
log10ξ2/M
6
0
2
4
6
3
4
5
6
7
8
9
10
11
12
W
Figure 6.1: Left: FSS plots of the redued TIP loalization lengths for three values of the onsite
interation U in 2D. Right: Plots of the innite-size TIP loalization lengths as obtained from
the FSS plots for three U values. The dashed line indiates the loalization lengths for a noninterating 2D Anderson model as omputed by TMM. The dierene with respet to the U = 0
urve is due to a geometrial resaling fator.
very good | the reader should ompare them to the FSS urves in Fig. 10 of Ref. [LRS99a℄
| they nevertheless demonstrate the possibility of onstruting a single FSS urve with
two branhes as expeted at the Anderson MIT. The saling urves also show that the MIT
appears at dierent values of W depending on the strength of the interation U . Reent
studies [259, 275℄ suggest that the MIT is more pronouned for long-ranged interation
potentials in 2D. I emphasize that all these studies involve extensive omputational eort,
i.e., we used 10 Pentium II mahines with 400 MHz operating frequeny for nearly 9 months
to onstrut the graphs in Fig. 6.1.
The results of hapter 3 may suggest a possible mehanism for the existene of the metalli
state in the 2D semiondutors [167, 168, 171℄. However, I wish to stress that a straightforward extrapolation of the TIP results to nite partile densities is unwarranted. Firstly,
the TIP arguments do not apply to the ground state of a many-body system, but only to
some exited states for a nearly zero density ase. Furthermore, the results are appliable
Conlusions and Outlook
59
to the attrative interation regime as well, but from hapter 4.3 we know that in this ase
the ground state is always more loalized in the presene of interation. Thus there is no
reason to believe that the extrapolation of the TIP results to nite densities may work
in the repulsive ase when it fails in the attrative ase. Nevertheless, the importane of
the TIP results of hapter 3 lies in the fat that they give a new mehanism by whih
even repulsive interation an lead to the formation of bound pairs and a orrespondingly
enhaned transport. This is somewhat reminisent of BCS superondutivity, where the
eletron-phonon interation leads to the formation of eetive eletron pairs [10℄. However,
for BCS the size of the Cooper pairs is rather large, whereas it is omparatively small for
the TIP pairs.
Although the above mentioned "extrapolation" appears a priori unjustied, the results
presented in hapter 4 regarding the ground state transport properties in Hubbard models
at nite density suggest that the interplay of disorder and interations does indeed give
rise to an enhanement of transport for onsite interation eletrons. However, ontrary to
the TIP mehanism, the nite density ases | most of whih have only been studied in 1D
| show a lear dependene on the partiular type of interation studied [31, 108, 240℄ and
| as mentioned before | an be quite dierent for attrative and repulsive interation
[EPR97, 153, 240℄. Thus at present the explanation of the MIT observed in 2D annot be
explained by these simple onsiderations and remains an open and hallenging problem.
Chapter 4 is also onerned with the physis of integrable impurity systems. In setion 4.2,
I show that the absene of baksattering is the dominating feature of these systems suh
that no loalization an be expeted. A most promising numerial approah to disordered
systems in 1D is reviewed briey in setion 4.4: the DMRG. We have applied the method
to the problem of the interplay between the interation-indued Mott gap and the disorder
indued Anderson loalization. This problem is numerially quite hallenging and I have
spend some time explaining the diÆulties involved. Let me remark that the DMRG
approah to disordered systems in 1D represents a most promising route towards a deeper
understanding of the interplay of interations and disorder in low-dimensional quantum
systems.
In hapter 5, I explain the parallelization of TMM and CWI. At the heart of both algorithms
is a matrix-vetor multipliation whih an be straightforwardly adapted for a massively
parallel arhiteture. However, ommuniation for boundary terms or orthogonalization
subroutines is not negligible and leads to a redution in speedup for small system sizes.
Only for large system sizes is it really useful to use P-TMM and P-CWI. For the deimation
method of hapter 3, it turns out the a naive parallelization is suited best and we used this
approah for olleting the data of setions 3.5 and 3.6. Still, for ELS, WFA, and MFA,
the massively parallel approah is the only available method whih allows to go to huge
systems sizes as is important for the determination and haraterization of an MIT.
Aknowledgments
The sienti researh presented in the present thesis would not have been possible without
the onstant and ontinuing support of many olleagues, friends and teahers. Most of them
have already appeared prominently in the preeding pages as oauthors of a joint paper
or as authors of other sienti works ited herein. Nevertheless, I would like to thank
a few of them personally here. First, I should like to thank Alexander Punnoose and B.
Sriram Shastry for showing me that there is interesting physis to be done in disordered
systems and for taking part in the rst suh endeavor of mine. I am indebted to Dieter
Vollhardt for providing a room, desk and salary when it was really needed. I partiularly
thank Mihael Shreiber for giving me a job in disordered systems when all I had to show
was a single PRB in a 1D disordered model. Moreover, I thank him for trusting me with
managing dotoral and diploma students and for allowing me to pursue siene that I judge
to be interesting. Hans-Peter Ekle, Andrzej Eilmes, Uwe Grimm, Ramesh Pai, Gudrun
Shlieker and Jianxin Zhong I thank for their input and ideas that made our joint work
suh fun. Muh praise I also send to Thomas Vojta who never one omplained when I
bouned another of my ideas and problems of his shoulders. Ulrih Kleinekathofer, Mihael
Sternberg, and Frank Weih have kept the omputers, printers and networks working, the
neessary onditions for any type of omputer physis. Lastly, I would like to thank Philipp
Cain, Frank Milde, Ville Uski, and Cristine de los Reyes Villagonzalo for listening to me
when asked and for making me listen to them when it was needed. I have learned a lot
from all these individuals and only hope that I ould give them something, too.
Perfetion is ahieved, not
when there is nothing left to
add, but when there is
nothing left to take away.
- Antoine de St. Exupery (1900-1944)
60
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84
Curriulum vitae
Rudolf A. Romer
Degrees:
12. Jun. 94
16. De. 91
Employment:
Jan. 1996 { today
Ot. 1995 { De. 1995
Aug. 1994 { Sep. 1995
Dotor of Philosophy (Ph.D.) in physis
Diplom-Physiker
Wissenshaftliher Mitarbeiter with Prof. Dr. Mihael Shreiber
at the Institut fur Physik, TU Chemnitz
Wissenshaftliher Mitarbeiter with Prof. Dr. Dieter Vollhardt
at the Institut fur Theoretishe Physik, RWTH Aahen
Researh Assoiate with Prof. Dr. B. Sriram Shastry at the ondensed matter theory unit of the Jawaharlal Nehru Centre for
Advaned Sienti Researh, Indian Institute of Siene ampus, Bangalore
Teahing Assistant (University of Utah)
Studentishe Hilfskraft (FU Berlin)
Teahing Assistant (Duke University)
Jan. 1992 { Jun. 1994
May 1990 { De. 1991
Jan. 1990 { May 1990
Eduation:
Jan. 1992 { Jun. 1994 Ph.D. thesis work with Prof. Dr. Bill Sutherland at the University of Utah, Salt Lake City, on an exatly soluble quantum
many-body problem with ompeting long-range interation (SCmodel), grade: A (sehr gut)
De. 1990 { De. 1991 Diploma thesis with Prof. Dr. Robert Shrader at the Freie Universitat Berlin (Germany) on ground state properties of spinlattie models, grade: sehr gut (A)
May 1990 { De. 1990 Physis student at the Freie Universitat Berlin
Aug. 1989 { Apr. 1990 Exhange student of the Freie Universitat Berlin at Duke University in Durham, North Carolina
Ot. 1988 { Jul. 1989 Physis student at the Freie Universitat Berlin
22. Ot. 1988
Vordiplom examinations, grade: sehr gut (A)
Ot. 1986 { Ot. 1988 Physis student at the Freie Universitat Berlin
12. Jun. 1985
Abitur (high shool diploma) at the Wolfgang-Ernst Gymnasium
Budingen, Hessen
Prizes / Honors:
May 1994
Thomas J. Parmley researh prize of the Department of Physis,
University of Utah
Finanial Support / Sholarships:
Jul. 1999 { today
DAAD/ARC projet-based personnel exhange program with
the Imperial College London (Prof. Dr. A. MaKinnon)
Jan. 1999 { today
DAAD/NSF projet-based personnel exhange program with the
University of Utah (Prof. Dr. M. Raikh)
85
Aug. 1999 { Sep. 1999
Jul. 1997 { Aug. 1997
Jan. 1996 { today
Jul. 1994 { Aug. 1995
Jul. 1993 { Jun. 1994
2 DAAD short term letureships at the University of Ruhuna in
Matara, Sri Lanka
Deutshe Forshungsgemeinshaft (SFB 393)
Feodor Lynen fellow of the Alexander von Humboldt foundation
Quadrille Ball Sholarship of the Germanisti Soiety of Ameria
further support through the Saxonian Ministry of Siene and
Art (1998,1999), the WE Heraeus foundation (1993, 1998) and
MINERVA (1991)
Conferene Organization:
29. Jul. { 3. Aug. 1999 Loalization 99, Universitat Hamburg, loal oorganizer with B.
Kramer (hairman)
5. Feb. { 7. Mar. 1999 Cooperative Phenomena in Statistial Physis: Theory and Appliations, MPI PKS Dresden, with M. Baake and U. Grimm
6. Ot. { 9. Ot. 1998 218. WE Heraeus seminar on Perolation, Interation, and Loalization: Simulations of transport in disordered materials,
Magnus-Haus Berlin, with M. Shreiber and T. Vojta
Teahing Experiene:
Aug. 1999 { Sep. 1999 Leture Computational physis in disordered materials inluding training sessions for students at the Department of Physis,
University of Ruhuna in Matara, Sri Lanka
Apr. 1998 { Jul. 1998 Seminar Kooperative Phanomene in Statistisher Physik with
Dr. T. Vojta and Prof. Dr. M. Shreiber at the TU Chemnitz
Ot. 1998 { Feb. 1999 2 seminars Neues aus der Physik | einfah dargestellt with Dr.
Ot. 1997 { Feb. 1998 U. Grimm and Prof. Dr. M. Shreiber at the TU Chemnitz
Jul. 1997 { Aug. 1997 Leture Quantum many-body physis in one dimension inluding training sessions for students at the Department of Physis,
University of Ruhuna in Matara, Sri Lanka
Ot. 1996 { Jan. 1997 Leture Vielteilhenphysik in einer Dimension inluding training
sessions for students together with Prof. Dr. M. Shreiber at the
TU Chemnitz
Jan. 1990 { Jun. 1994 Teahing Assistant for physis and non-physis majors at the
University of Utah (92{94), the FU Berlin (90{91) and at Duke
University (90), topis: mehanis, eletrodynamis, statistial
physis, laboratory experiments for the basi physis ourses,
topis: basi siene ourses and omputer aided laboratory
experiments
Personal:
1. Sep. 1993
Marriage to Mrs. Yoginie A. Wijesekara of Dondra (Sri Lanka)
Ot. 1985 { Sep. 1986 Military servie
22. Feb. 1966
Born as son of Rudolf Romer, teaher and Helga Romer, nee
Koh, in Gedern, Hessen (Germany)
Nationality:
German
86
Erklarung gema Habilitationsordnung x5 (1) 5/6
Ih versihere, dass die vorliegende Arbeit selbststandig und ohne andere als die darin
angegebenen Hilfsmittel angefertigt wurde. Die Stellen in der Arbeit, die in Wortlaut
oder Sinn anderen Werken entnommen wurden, sind entsprehend gekennzeihnet. Die
Habilitationsshrift wurde an keiner anderen Fakultat zur Beurteilung vorgelegt und ist
noh niht anderweitig veroentliht worden. Die derzeit gultige Habilitationsordnung der
Fakultat fur Naturwissenshaften der Tehnishen Universitat Chemnitz vom 15. Januar
1996 ist mir bekannt.
Chemnitz, Marh 26, 2000
Rudolf A. Romer