Critical parameters for the
Anderson transition in BCC
and FCC lattices
Andrea M. Fischer
Department of Physics/ Centre for Scientific
Computing, University of Warwick, Coventry, UK
arXiv: 0801.3338v1 [cond-mat.dis-nn]
Where is Warwick?
-founded 1965
-17,000 students
-5000 staff
-57th worldwide
THES 2007
-5th nationally in
RAE
Perfect Crystalline Solids
• Non-interacting electrons in a periodic potential
- Bloch electrons

 nk r   unk r e
ik.r
infinite conductivity of metals!?
Felix Bloch
electron
Fermi energy
periodic potential
Real Crystalline Solids
• Real crystalline materials contain vacancies and
impurity atoms
disorder
• Electron-electron interactions can also be important,
although these are shielded and this depends on the
electron density
• Ions are not fixed, but vibrate about their mean
positions – avoid by going to low temperatures
(< 10K)
electron
Fermi energy
random potential
The Anderson Transition
• At a certain critical disorder
there is a transition between
metallic and insulating states
- the Anderson transition
[P.W. Anderson, Phys. Rev. 109, 1492 (1958)]
• Can think of it as a second
order phase transition
• An entirely quantum
mechanical effect that controls
macroscopic, measurable
properties of the system!!
i.e. resistance
Philip Anderson
Mechanism for Localisation
• Localisation due to quantum interference of
scattered electron wavefunction
• Also occurs for classical waves e.g. water waves
regular array of scatterers
disordered array of
scatterers
[P.E. Lindelof, J. Noregaard and I. Hanberg, Phys. Scr. T14 17]
Distinguishing between metallic
and insulating states
Experimentalist - resistivity  T 
• metal:  T  increases with increasing
T
• Insulator:  T  decreases with
increasing T
 T 
Theoretician - localisation length 
• metal:  increases as a function
of system size
• insulator:  decreases as a
function of system size
 r   f r  e
insulator
metal
T
[M.P. Sarachik and S.V. Kravchenko, Proc.
Natl. Acad. Sci. U.S.A. 96, 5900 (1999)]

 r
Anderson Model
• Anderson Hamiltonian
H

i
i i   tij i j
i j
i
potential energy
• Electronic wavefunction
   i i
i
hopping term
Anderson model
• No disorder in hopping term,
nearest neighbour hopping
only i.e.
• Consider case of uniform
disorder in the onsite potential
energy
P 
i
1 j  i  1, j  i  1
tij  
0 otherwise
1
W

i
W
2
W
2
• Under these conditions get lattice Schrödinger equation for SC lattice
H  l  l 1  l 1  E l
( 2)
l
Outline
• Anderson transition has been studied extensively for
SC lattice
• study the more complex BCC and FCC lattices,
which are more common in nature – involves
introduction of connectivity matrices
• transfer-matrix method to obtain localisation lengths
as a function of system size, energy and disorder
• finite-size scaling to analyse data and find the
critical disorders, energies and exponents
• show that BCC and FCC lattices are in the same
universality class as the SC lattice
Transfer-matrix method (TMM)
• Model a 3D system as a quasi-1D bar of cross-sectional area M x M and
length N, where N >> M.
M
• Re-write lattice Schrödinger equation:
M
 l 1   E1  H l  1   l 

  


Tl
0  l 1 
 l   1
N
• Lyapunov exponents obtained from eigenvalues of
transfer matrix
1
1
 l  Tl Tl 1...T1
[V.I. Oseledec, Trans. Moscow. Math. Soc. 19, 197 (1968)]
• Localisation length obtained as inverse of
smallest Lyapunov exponent
 M , E ,W  
1
 min
, reduced localisation length
 M E , W  
 M , E ,W 
M
What about other lattice structures?
• Hopping terms in the 2D Hamiltonian representing in-layer
connections are different and depend on boundary conditions
• connectivity matrices Cl describe the connections between
layers l-1 and l:
(Cl)ij=1 if sites i and j are connected and 0 otherwise
  Cl11 E1  H l   Cl11Cl 

Tl  

1
0


• Challenge: choose boundary conditions, system sizes and
lattice planes so connectivity matrices are non-singular
Example of connections resulting
in a singular connectivity matrix
1
12
1
2D
23
12
34
23
4
34
4
1 0 0 1


1 1 0 0
SINGULAR!
0 1 1 0


0 0 1 1


• M=4, periodic boundary conditions SINGULAR
• M=4, hard wall boundary conditions NON-SINGULAR
• M=3, periodic boundary conditions NON-SINGULAR
BCC and FCC lattice structures
Body centred cubic
• 8 nearest neighbours,
0 in-plane and 4 in each
nearest neighbour plane
• TMM direction: <100>
Face centred cubic
• 12 nearest neighbours,
6 in-plane and 3 in each nearest
neighbour plane
• TMM direction: <111>
i.e. close packed planes
Phase diagrams
BCC
FCC
dashed lines = theoretical band edges  Z  W / 2
dotted lines = analytical estimation for critical disorder based on self-consistent
theory of localisation [E. Kotov and M. Sadovskii, Z. Phys. B 51, 17 (1983)]
Creation of phase diagrams
BCC
FCC
W
W
-E
-E
• Grid of E-W points with separation 0.5
• Extended (red)  7   9 , localised (green)  7   9
• Average over 3 outermost points separately for extended and localised regions to
get two boundaries
• Do a spline fit for these points
Symmetry of phase diagrams
• BCC phase diagram symmetric about E=0, whereas FCC lattice is not
• Reason: BCC lattice bipartite, FCC lattice not
• Bipartite: A lattice is bipartite if it can be split into 2 sublattices s.t. for a site
on one sublattice all its nearest neighbours are on the other sublattice.
• Why does bipartiteness result in symmetry?
H  E
 THT 1T  ET , where
 h11  h12

h22
 h
THT 1   21
h
 h32
 31
 



HT    ET
h13
 h23
h33

1



 1

T

1







bipartite transformation
... 

...   H bipartite lattice, zero disorder

...   H bipartite lattice, disorder



TMM results for E=0
BCC
FCC
• Colours represent different system sizes M=3,…,15
• Error bars within symbol size
• Lines are fits obtained using finite size scaling
Finite-Size Scaling (FSS)
• The Anderson transition is characterised by a divergent correlation
length  in the thermodynamic limit
 W   W  Wc

  E   E  Ec

for fixed energy E and
for fixed disorder W, where  is the
critical exponent
• Main idea is one-parameter scaling hypothesis
 M  f E , W , M   f
  
M
 E ,W
 M  f  r M 1/ ,  i M y  ,
~
• A better fit can be obtained by using
where  r and  i are the relevant and irrelevant scaling variables
respectively, y < 0.
Finite-Size Scaling (FSS)
• Proceed via a Taylor expansion
ni
M    M
n 0
n
i
ny
~

fn r M
1

,
~
f n   ani  M
• Include non-linear dependence on W,
mr
 r (w)   bn wn ,
n 1
nr
i 0
i
r
1

mi
 i ( w)   cn wn
n 0
where w=(Wc-W)/W
• Aim: to obtain the best fit whilst keeping no. of parameters
reasonably small
no. params = (ni+1)(nr+1)+mr+mi+1(or 2)
• Non-linear fit performed using Levenberg-Marquardt
method
Scaled localisation lengths for FCC lattice, W=18
M=9,11,13,15
metallic
insulating
FSS
Scaling parameter for BCC lattice, W=17.5
   r E 

   r E 
   r E 
  1.45
BCC results
(b)
• All errors are standard errors
• (a) 91 data points, 41 best fit
models
• (b) 108 data points, 8 best fit
models
• Compare to   1.57  0.02 found
for SC lattice using similar
numerical techniques.
[K. Slevin and T. Ohtsuki, Phys. Rev. Lett. 82, 382 (1999)]
FCC results at barycentre (E=0)
• 91 data points, 31 best fit models
• No irrelevant scaling necessary
FCC results for W=18
q
• 83 data points, 8 best fit models
• Good q values
•  = no. of degrees of freedom
= no. of data points – no. of parameters
2
•    indicates a good fit
Conclusions
•   1.5 in agreement with
previous results confirming that
the BCC and FCC lattices are
in the same universality class
as the SC lattice
• The critical disorder values are
non-universal and increase with
increasing no. of nearest
neighbours
• Agreement with results of
classical site and bond
percolation models for SC,
BCC and FCC lattices
For more details please see
[S. Galam and A. Mauger, Phys. Rev. E
arXiv: 0801.3338v1 [cond53, 2177 (1996)]
mat.dis-nn]
Thanks
Rudolf A. Römer
Andrzej Eilmes
• The work was carried out
in collaboration with
Andrzej Eilmes
(Jagiellonian University,
Krakow, Poland) and my
PhD supervisor, Rudolf A.
Römer (University of
Warwick, Coventry, UK).
• We are grateful to Tom
Wright for producing the
M=7 and M=9 data used in
the phase diagrams.
• We acknowledge funding
from the EPSRC.
Aharonov-Bohm effect for an
interacting system
•AB effect:
Energy of a charged particle in a ring
oscillates as a function of
perpendicular magnetic flux
•Look at AB effect for neutral exciton
in a nanoring with perpendicular
magnetic field and in-plane electric
field
Electric field
•Electric field destroys translational
symmetry of Schrödinger equation so
can’t solve exactly
What happens to amplitude of oscillation?
Oscillation amplitude versus
electric field for different
interaction strengths
Excitonic Wavefunctions
Magnetic flux
Electric field