PHYSICAL REVIEW B
VOLUME 62, NUMBER 11
15 SEPTEMBER 2000-I
Dynamical response of a pinned two-dimensional Wigner crystal
Michael M. Fogler
School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
David A. Huse
Physics Department, Princeton University, Princeton, New Jersey 08544
共Received 20 April 2000兲
We reexamine a long-standing problem of the finite-frequency conductivity of a weakly pinned twodimensional classical Wigner crystal. In this system an inhomogeneously broadened absorption line 共pinning
mode兲 centered at disorder- and magnetic-field-dependent frequency ␻ p is known to appear. We show that the
relative linewidth ⌬ ␻ p / ␻ p of the pinning mode is of the order of 1 in weak magnetic fields, exhibits a
power-law decrease in intermediate fields, and eventually saturates at a small value in strong magnetic fields.
The linewidth narrowing is due to a peculiar mechanism of mixing between the stiffer longitudinal and the
softer transverse components of the collective excitations. The width of the high-field resonance proves to be
related to the density of states in the low-frequency tail of the zero-field phonon spectrum. We find a qualitative
agreement with recent experiments and point out differences from the previous theoretical work on the subject.
I. INTRODUCTION
The problem of pinning of an elastic solid by an external
random potential has been a long-standing problem of condensed matter physics, first addressed by Larkin1 in 1970.
Physical systems where this problem arises include chargedensity waves,2 vortex lattices in superconductors,3
interfaces,4 magnetic bubbles,5 liquid crystals in aerogels,6
and many others. An interesting example of the chargedensity wave is a Wigner crystal 共WC兲. Under terrestrial
conditions a stable WC has been realized in its twodimensional 共2D兲 form,7 on cryogenic surfaces and in semiconductor heterostructures with low carrier density and/or in
strong magnetic fields.8 In this work we examine the finitefrequency response of a collectively pinned 2D WC. The
collective pinning is the regime where individual defects of
the substrate are too weak to significantly deform the crystalline lattice, so that the WC is well ordered 共‘‘unpinned’’兲
at small length scales, but the cumulative effect of the disorder eventually dominates the elasticity at a pinning length
that is much larger than the lattice constant.
The linear-response conductivity ␴ ␣␤ (q, ␻ ) of the WC in
an external magnetic field is a tensor. We focus our study on
the real part of its diagonal component ␴ xx (q⫽0,␻ ), the
quantity that determines the power absorption in the presence of a uniform electric field. If the frequency ␻ is not too
small, the absorption is dominated by the collective excitations. In the absence of a magnetic field and random potential, a 2D WC possesses two branches of such excitations:
the longitudinal 共L兲 and transverse 共T兲 phonons. The longwavelength L phonon is also referred to as a plasmon. In a
finite magnetic field L and T phonons hybridize into magnetophonons 共gapless lower hybrid mode兲 and magnetoplasmons 共gapped higher hybrid mode兲. It is noteworthy that,
without the pinning potential, the frequency and oscillator
strength of the magnetophonons vanish at q⫽0 by virtue of
Kohn’s theorem. The absorption of a spatially uniform ac
field is possible only by exciting the magnetoplasmon mode,
0163-1829/2000/62共11兲/7553共18兲/$15.00
PRB 62
at the cyclotron frequency ␻ c ⫽eB/m e c set by the magnetic
field B. The pinning shifts the original q⫽0 magnetophonon
mode to a nonzero disorder-dependent frequency ␻ p , broadens it, and imparts to it a finite oscillator strength. The resulting absorption line, which can be called the pinning
mode, was first discussed by Fukuyama and Lee 共FL兲.9,10 In
those early works the linewidth ⌬ ␻ p of the pinning mode
was predicted to be of the order of ␻ p itself. Strictly speaking, FL considered not a WC but a more conventional
charge-density wave where the charge modulation has only
one harmonic, i.e., is cosinelike. Experiments on such materials, e.g., K0.3Mo03 共‘‘blue bronze’’兲, TaS3 , etc., performed
in zero magnetic field indeed revealed broad absorption lines
at disorder-dependent frequencies.11 A later work of Normand et al.12 devoted specifically to the WC in a strong B
revised some results of FL but left unchanged the prediction
that ⌬ ␻ p ⬃ ␻ p . At the time, experiments seemed to confirm
that.13 It came as a surprise when most recent measurements
in strong magnetic fields14–17 demonstrated that ⌬ ␻ p can be
more than an order of magnitude smaller than ␻ p . Such
unexpected findings revived interest in this long-standing
problem.
An important step toward resolution of the puzzle has
been made by Fertig,18 who showed that long-range Coulomb interaction plays an important role in reducing the inhomogeneous broadening of the pinning mode. However,
Fertig used an oversimplified model 共commensurate pinning兲
and did not calculate ⌬ ␻ p directly. His results for another
quantity may be interpreted as an indication that ⌬ ␻ p / ␻ p is
exponentially small for weak pinning. A more realistic
model was studied by Chitra et al.19 who considered pinning
by a short-range Gaussian random potential of a general
type. We comment on their results later in this section.
In the present paper we study essentially the same model
as FL and Chitra et al. except we treat the electrons classically. This limits the applicability of our results to the case
␰ ⬎l B , where ␰ is the correlation length of the pinning po7553
©2000 The American Physical Society
M. M. FOGLER AND DAVID A. HUSE
7554
PRB 62
We propose that the correct approach should be based on
considering certain low-probability disorder configurations
共‘‘optimal fluctuations’’兲, which leads to
3
s⫽ .
2
FIG. 1. Relative linewidth of the pinning mode as a function of
the cyclotron frequency.
tential and l B ⫽ 冑បc/eB is the magnetic length. The classical
approximation enables us to focus on the interaction of the
WC with disorder, which is really the essence of the pinning
mode phenomenon. We critically reexamine the previous
work on the subject, identify the physical mechanism responsible for the line narrowing, and spell out the conditions
under which it occurs: 共1兲 the lattice dynamics is inertial 共not
overdamped兲; 共2兲 the magnetic field is sufficiently strong,
such that ␻ c is larger than ␻ p ; 共3兲 the pinning is sufficiently
weak so that ␻ p is much smaller than the magnetophonon
bandwidth; 共4兲 the compression modulus ␭ of the WC evaluated at the pinning length is larger than its shear modulus ␮
共satisfied for long-range interaction兲. If any of these conditions is violated, then the line is conventionally broad, ⌬ ␻ p
⬃␻p .
Although we were unable to find an exact expression for
⌬ ␻ p , we show that there is an asymptotically exact relation
between ⌬ ␻ p at intermediate B and the low-frequency tail of
the phonon spectral function in zero B 共we remind the reader
here that we study classical electrons, which form the WC at
any B). Establishing such a connection and elucidating the
physical mechanism of the line narrowing are our main
achievements.
The phonon spectral function at low frequencies is expected to have a power-law form,
Im D ␣␣ 共 q⫽0,␻ 兲 ⬀ ␻ 2s⫹1 ,
共1兲
with a nontrivial exponent s. This additional input and further considerations enable us to predict that, as ␻ c increases
and becomes larger than ␻ p , ⌬ ␻ p / ␻ p first decreases in a
power-law fashion
⌬ ␻ p / ␻ p⬃共 ␻ p / ␻ c 兲
s
共2兲
but then eventually saturates at a value 共see Fig. 1兲
⌬ ␻ p / ␻ p ⬃ 共 ␮ /␭ 兲 s
共3兲
共here ␭ is meant to be evaluated at the pinning length兲. The
recent experiments14–17 have probed the high-field regime
where Eq. 共3兲 is supposed to apply.
In the aforementioned work of Chitra et al.,19 a set of
equations was derived that, when analyzed further, also yield
Eq. 共3兲, with s⫽ 21 . This specific value of s can be traced
down to the fact that Chitra et al. unknowingly rederive a
common form of the self-consistent Born approximation
共SCBA兲, where Eq. 共1兲 with s⫽ 21 is satisfied. Since the
SCBA is uncontrolled at small ␻ , these results are unreliable.
Moreover, appealing to the known properties of Lifshitz
tails20 in other disordered systems, we argue that the SCBA
strongly overestimates the low-frequency spectral function.
共4兲
The paper is organized as follows. In Sec. II we discuss
the ground state of the pinned WC, placing emphasis on the
parameters that provide the input for the dynamical problem
of interest. In Sec. III we formulate the model and the general framework for study of the finite-frequency response. In
Sec. IV we derive Eq. 共2兲. In Sec. V we study the soft phonon modes at B⫽0 and obtain the above formula for s. The
comparison with the experiments is given in Sec. VII after a
brief discussion of quantum and finite-temperature effects in
Sec. VI.
II. STATIC PROPERTIES
The distortions of the lattice in the ground state of a collectively pinned Wigner crystal accumulate gradually over
length scales much larger than the lattice constant a. Such
distortions can be described in terms of a smooth displacement field u(0) (r). The Hamiltonian of the system H⫽H el
⫹H p is the sum of the elastic term,
H el ⫽
1
2
冕
q
冉
0
u ␣(0) 共 ⫺q兲 H ␣␤
共 q兲 u ␤(0) 共 q兲 ,
0
⫽ ␦ ␣␤ ⫺
H ␣␤
q ␣q ␤
q
2
冊
H T0 共 q 兲 ⫹
q ␣q ␤
q2
共5兲
共6兲
H L0 共 q 兲 ,
H T0 共 q 兲 ⫽ ␮ q 2 , H L0 共 q 兲 ⫽␭ 共 q 兲 q 2 ,
共7兲
and the pinning term,21
H p ⫽n e
冕 再兺
r
U 共 r兲
K
冎
cos K关 r⫺u(0) 共 r兲兴 ⫺“u(0) ,
具 U 共 r兲 U 共 0 兲 典 ⫽C 共 r 兲 .
共8兲
共9兲
The notations here are as follows. The q integral is taken
over the Brillouin zone with measure d 2 q/(2 ␲ ) 2 ; the r integral is taken over the area of the system with measure d 2 r.
Parameters ␮ and ␭ represent the shear and bulk elastic
moduli of the WC, respectively. In strong magnetic fields,
i.e., at low filling factors, they approach their classical
values22 ␮ ⬇0.245e 2 n 3/2
e / ␬ and ␭(q)⫽⫺5 ␮ ⫹Y / 兩 q 兩 . Here
n e ⫽2/ 冑3a 2 is the average electron concentration, ␬ is the
dielectric constant of the medium, and Y ⫽2 ␲ e 2 n 2e / ␬ . The
infrared divergence of ␭(q) originates from the long-range
Coulomb interaction. The K’s are the reciprocal lattice vectors. Finally, the correlator C(r) is assumed to rapidly decay
at r⬎ ␰ where ␰ ⬍a is the correlation length of the shortrange Gaussian random potential U(r).
The static properties of the system defined by Eqs. 共5兲–共9兲
can be investigated using classical statistical mechanics
methods. Problems of this type have been studied extensively in the past 共see Ref. 3 for review and references兲, and
there are also some recent contributions.21,24,25,23,26 The 2D
DYNAMICAL RESPONSE OF A PINNED TWO- . . .
PRB 62
FIG. 2. Roughness W(r).
WC generally conforms to the d⫽2, m⫽2 class of elastic
media (d is the total number of spatial dimensions and m is
the number of components of u). However, some caution is
required in transferring the known results to the WC case
because the L and T components are not equivalent. The
large disparity ␭Ⰷ ␮ of the elastic moduli causes strong suppression of the longitudinal distortions compared to the
transverse ones. Therefore, in certain formulas one has to use
m⫽1. In this respect, the WC is similar to the Abrikosov
vortex lattice where the analogous inequality c 11Ⰷc 66
exists.3
Let us define the two-point correlation function 共roughness兲 W(r)⫽ 具 关 u(0) (r)⫺u(0) (0) 兴 2 典 1/2. Many static properties
of the system can be deduced from the behavior of W(r).
Three regimes can be distinguished: random-force, randommanifold, and the asympotic 共Fig. 2兲.
The random-force or Larkin regime appears for small r
where W(r)⬍ ␰ . The last inequality legitimizes expanding
the exponential in Eq. 共8兲 and keeping only the linear in u (0)
terms. The Hamiltonian becomes quadratic, and its ground
state is easily found to be
u ␣(0) 共 q兲 ⫽⫺in e
⫺1
共 q兲 K ␤ Ũ 共 q⫹K兲 .
兺K 关 H0 兴 ␣␤
共10兲
Here and below tildes denote Fourier transforms.
Averaging over the disorder, we arrive at
W 2 共 r兲 ⯝V ⬘ 共 0 兲
冕
q
共 1⫺e iqr兲关共 H T0 兲 ⫺2 ⫹ 共 H L0 兲 ⫺2 兴
共12兲
⬀r 2 ln共 R c /r 兲 ,
V 共 z 兲 ⬅⫺
n 2e
2
共11兲
兺K C̃ 共 K 兲 e ⫺zK /2 .
2
共13兲
To obtain Eq. 共12兲 we cut off the log-divergent integral in
Eq. 共11兲 at q⫽q c where q c ⫽ ␲ /R c and R c is the Larkin
length defined by the relation W(R c )⫽ ␰ . For a weak random
potential R c is much larger than a and is given by
R c ⬃ ␮ a ␰ 2 关 C 共 0 兲兴 ⫺1/2.
7555
The elastic-manifold regime is realized at intermediate
distances R c ⬍r⬍R a where ␰ ⬍W(r)⬍a. Here the roughness grows more slowly than in the Larkin regime W(r)
⬀r ␨ , ␨ ⬍1. For the 2D Wigner crystal the Flory estimate ␨
⫽ 13 共as appropriate for the d⫽2, m⫽2 case兲 should be accurate but perhaps not exact.3 The slower growth rate of
W(r) is due to a less efficient mechanism by which the lattice adjusts to the random potential on length scales exceeding R c . The steepest descent toward the nearest local minimum of the potential landscape in the Larkin regime gets
replaced by selection among a set of many possible yet
roughly equivalent nearby minima.
The third and last regime begins when W(r) reaches a at
the length scale R a ⫽R c (a/ ␰ ) 1/ ␨ . The accumulation of the
pinning energy density at such large length scales becomes
additionally suppressed by lattice periodicity effects. Indeed,
a uniform shift u→u⫹a reduces to relabeling the lattice
sites with no physical significance. As a result, the roughness
grows only logarithmically with distance, W(r)⬃a ln(r/Ra).
The asymptotic ln r roughness corresponds to a ‘‘quasi-shortrange’’ order: W(r) increases so slowly that it is barely
enough to remove the algebraic divergencies 共Bragg peaks兲
of the static structure factor
F 共 q兲 ⫽n 2e
冕 冉
1
exp i 共 q⫺K兲 r⫺ K 2 W 2 共 r 兲
2
r
冊
共15兲
at the reciprocal lattice vectors q⫽K. Remarkably, it also
creates enough elastic stress to generate unbound dislocations, on the length scale26 ␰ D ⬃R a exp关C0 ln1/2(R a /a) 兴 . At
this scale u becomes multivalued and Eq. 共5兲 must be modified.
Although very interesting, the random-manifold and
asymptotic regimes do not play much role in the further discussion. We mentioned them here for the sake of a more
comprehensive introduction and also to draw attention to the
difference between two characteristic length scales R c and
R a . R a is the length that seems to be important only for
rather subtle properties mentioned above: the destruction of
the Bragg peaks and the appearance of topological defects. In
contrast, R c determines most of the physically important response characteristics such as the threshold electric field in
dc transport and the frequency ␻ p of the pinning mode.19
This role of R c originates from the fact that it is the length
scale where the dominant contribution ⬃ ␮ ␰ 2 /R 2c to the pinning energy density is accumulated. In this sense R c plays
the role of the pinning length mentioned in Sec. I.
Let us now elaborate on the difference between the T and
L correlation functions mentioned above. To do so we decompose the elastic displacement into its T and L components,
共14兲
The same estimate for R c can be obtained in a simpler
way,3,10 by equating the elastic energy density ␮ ␰ 2 /R 2c to the
pinning energy density 关 C(0) 兴 1/2冑N/R 2c . In the last expression N⫽n e R 2c is the number of electrons within the area R c
⫻R c and the square root takes into account that individual
pinning energies have random signs.
u共 q兲 ⫽q̂u L 共 q兲 ⫹ 关 ẑ⫻q̂兴 u T 共 q兲 ,
共16兲
where q̂⫽q/ 兩 q兩 , and define the two components of roughness
W ␣ ⫽ 具 u ␣(0) (r)⫺u ␣(0) (0) 典 1/2, ␣ ⫽L,T. First of all, let us examine the short-distance behavior using Eq. 共10兲. For W L (r)
we obtain an expression similar to Eq. 共11兲:
7556
M. M. FOGLER AND DAVID A. HUSE
W L2 共 r 兲 ⯝V ⬘ 共 0 兲
冕
1⫺e iqr
q
关 H L0 共 q 兲兴
⯝
2
V ⬘共 0 兲
2␲Y
ln
2
冉冊
r
,
a
共17兲
and so W L (r) is much smaller than W T (r)⯝W(r) given by
Eq. 共12兲. Formula 共17兲 is certainly valid for r⬍R c . For
larger r we can no longer treat the elastic distortion
兩 u(0) (r)⫺u(0) (0) 兩 as small. Nonetheless, we expect that the
region of validity of Eq. 共17兲 extends somewhat beyond r
⫽R c . Indeed, instead of doing the perturbation theory
around the ideal lattice state, we can do it around the ground
state u⬁(0) of the incompressible crystal, ␭⫽⬁. In the second
approach we have to account for a finite pinning energy density, but for the longitudinal elastic stiffness Y q where q
⬃ ␲ /r exceeds the characteristic curvature ␮ q 2c of the pinning energy density landscape, i.e., for rⱗR L ⫽(Y / ␮ )R 2c
⬃R 2c /a, this should be unimportant. Substituting r⫽R L into
Eq. 共17兲 we find that W L (R L )Ⰶ ␰ , contrary to some previous
suggestions12 that W L (R L )⬃W T (R L ) 关recall that W T (R L )
⯝W(R L )ⰇW(R c )⫽ ␰ 兴.
The large-distance (r⬎R L ) behavior of W L (r) is inaccessible within the perturbation theory. However, we should get
qualitatively correct results by using the Gaussian variational
replica method 共GVRM兲.21 Assuming for simplicity that R L
ⰇR a and suitably modifying the GVRM for the present
problem, we find
W L2 共 r 兲 ⬀V ⬘ 共 0 兲
冕
1⫺cos共 qr兲
0
0
q H L 共 q 兲关 H L 共 q 兲 ⫹⌸ a 兴
,
共18兲
where ⌸ a ⫽ ␮ R ⫺2
a . It is easy to see that W L (r) tends to a
finite limiting value as r→⬁. So long-range positional order
is absent in the T but is preserved in the L part of the elastic
displacement field.27
To conclude the review of the statics, let us consider a
matrix of the second derivatives of H p ,
S共 r兲 ⫽
⳵ 2H p
⳵ u(0) ⳵ u(0)
⫽n e
兺K KKU 共 r兲 cos K关 r⫺u(0)共 r兲兴 .
共19兲
The matrix S plays an important role in the dynamics and we
need to understand its properties.
In the case of short-range disorder we study here, the
correlations in U(r) decay more rapidly with r than those of
u(0) (r). This enables us to find the two-point correlator of
S ␣␤ with little effort. It suffices to use the approximation
u(0) (r)⫽0 in Eq. 共19兲, which leads to
具 S̃ ␣␮ 共 k1 兲 S̃ ␯ ␤ 共 k2 兲 典 ⫺ 具 S̃ ␣␮ 典具 S̃ ␯ ␤ 典
⯝⫺
The parameter S 0 is positive, in accordance with the pinning
phenomenon: the crystal in its ground state is distorted in
such a way that the electrons are preferentially situated near
the minima of the random potential U(r) where its curvature
is positive. S 0 can be estimated to be
S 0 ⯝⫺
冉 冊
V ⬙共 0 兲
Rc
ln
2␲␮
a
共22兲
using the same procedure as for deriving Eq. 共10兲. Assuming
that 兩 u(0) (r)⫺u(0) (0) 兩 is ‘‘small,’’ we expand the exponential in Eq. 共19兲 to obtain
S̃ ␣␤ 共 q兲 ⯝⫺
in e
L
2
兺K K ␣ K ␤ K ␥ 冕q u ␥(0)共 q1 兲 Ũ 共 q⫺q1 ⫺K兲 .
1
共23兲
Combining Eqs. 共10兲 and 共23兲, averaging over the disorder,
and using the inequality H L0 (q)ⰇH T0 (q), we get
S 0 ⯝⫺V ⬙ 共 0 兲
冕
q
关 H T0 共 q 兲兴 ⫺1 ,
共24兲
which leads to Eq. 共22兲. Here we again had to cut off the
infrared logarithmic divergence by hand, at q⫽q c ⫽ ␲ /R c .
More sophisticated approximation schemes such as the
GVRM would implement this cutoff more gracefully but, as
long as we are not interested in the numerical factor in the
argument of the logarithm, the result is the same.
At this point we conclude the discussion of the ground
state properties of the pinned WC, as we are now ready to
address its dynamical response.
III. FINITE-FREQUENCY RESPONSE: PERTURBATION
THEORY AND QUALITATIVE CONSIDERATIONS
To study the dynamics of the WC we introduce a timedependent displacement field u, which is the deviation of the
total displacement from its ground state value u(0) (r). We
will restrict ourselves to the harmonic approximation where
the action is quadratic in u. The response of a harmonic
oscillator is the same in quantum and classical mechanics, so
we can use the convenient imaginary-time formalism without
compromising our original intention to treat the system classically. The resultant action contains two terms: one that describes a uniformly pinned WC,
A 0⫽
1 1
2 ប␤
冕
兺
␻
q
n
u† 共 q,i ␻ n 兲关 D0 兴 ⫺1 u共 q,i ␻ n 兲 ,
共25兲
and the other that takes into account fluctuations in local
pinning strength,
V ⬙共 0 兲
共 ␦ ␣␮ ␦ ␯ ␤ ⫹ ␦ ␣␤ ␦ ␮ ␯ ⫹ ␦ ␣ ␯ ␦ ␮␤ 兲 ␦ k1 k2 L 2
2
共20兲
for the WC with a hexagonal lattice and in the longwavelength limit k 1 ,k 2 Ⰶ1/a. Here L 2 is the area of the system. Another important property of S ␣␤ is a nonzero mean,
具 S ␣␤ 共 r兲 典 ⫽ ␦ ␣␤ S 0 .
PRB 62
共21兲
A 1⫽
1 1
2 ប␤
兺
␻
n
冕冕
q1
q2
u† 共 q1 ,i ␻ n 兲 ␦ S̃共 q1 ⫺q2 兲 u共 q2 ,i ␻ n 兲 .
共26兲
Here ␤ ⫽1/(k B T) is the inverse temperature, ␻ n
⫽2 ␲ n/(ប ␤ ) are the bosonic Matsubara frequencies, ␦ S̃⬅S̃
⫺S 0 I, I is the identity matrix, D0 ⫽D0 (q,i ␻ n ) is the phonon
propagator of a uniformly pinned WC 共cf. Refs. 10 and 22兲
PRB 62
DYNAMICAL RESPONSE OF A PINNED TWO- . . .
0 ⫺1
关D 兴
⫽Rq†
冋
H T0 ⫹S 0 ⫹ ␳␻ 2n
⫺ ␳␻ n ␻ c
␳␻ n ␻ c
H L0 ⫹S 0 ⫹ ␳␻ 2n
册
Rq , 共27兲
␳ ⫽m e n e is the average mass density, and Rq is the O(2)
rotation by angle arg(q x ⫹iq y ). We also define the disorderaveraged propagator D(q,i ␻ n ),
D共 q,i ␻ n 兲 ⫽
冓
冕
DuDu* u共 q,i ␻ n 兲 u* 共 q,i ␻ n 兲 e ⫺A
共ប␤L2兲
冕
DuDu* e ⫺A
冔
,
共28兲
where A⬅A 0 ⫹A 1 and L 2 is again the area of the system.
The quantity we set out to calculate is the ac conductivity
␴(q, ␻ ), which in this model is given by10
␴共 q, ␻ 兲 ⫽⫺ie 2 n 2e ␻ D共 q,i ␻ n 兲 兩 i ␻ n → ␻ ⫹i0 .
共29兲
The conductivity can also be expressed in terms of the phonon self-energy,
⌸共 q, ␻ 兲 ⬅S 0 I⫹ 共 D⫺1 ⫺ 关 D0 兴 ⫺1 兲 兩 i ␻ n → ␻ ⫹i0 .
共30兲
We are interested primarily in the case q⫽0 where the most
general form of ⌸(q, ␻ ) consistent with rotational symmetry
is
⌸ ␣␤ 共 0,␻ 兲 ⫽ ␦ ␣␤ ⌸ 共 ␻ 兲 ⫺i␧ ␣␤ ␳␻␻ c f xy 共 ␻ 兲 ,
共31兲
f xy ( ␻ ) being the relative correction ⌬ ␳ xy / ␳ 0xy to the bare
Hall resistivity ␳ 0xy ( ␻ )⫽B/(n e ec). Approximate calculations presented in this paper give f xy ⫽0, which leads us to
believe that f xy ( ␻ ) must be small for weak pinning. We
choose to neglect such fine details and to assume that f xy
vanishes.28 In this case ⌸(0,␻ ) is a scalar and ␴ xx ( ␻ ) is
given by
⌸ 共 0,⑀ 兲 ⫺ ⑀
,
Re ␴ xx 共 ␻ 兲 ⫽e 2 n 2e ␻ Im
关 ⌸ 共 0,⑀ 兲 ⫺ ⑀ 兴 2 ⫺ ⑀⑀ c
共32兲
where we introduced convenient ‘‘energy’’ variables
⑀ ⬅ ␳␻ 2 , ⑀ c ⬅ ␳␻ 2c .
共33兲
From Eq. 共32兲 one can see that in strong magnetic fields,
␻ c Ⰷ ␻ p , the power absorption in the uniform electric field
takes place mainly within a frequency interval of width
Im ⌸ 共 0,⑀ p 兲
⌬ ␻ p ⫽⫺
␳␻ c
共34兲
centered at ␻ ⫽ ␻ p , where
␻ p⫽
冑⑀
p
␳
⫽
Re ⌸ 共 0,⑀ p 兲
.
␳␻ c
共35兲
The last equation is the implicit definition of ⑀ p . In the following we will work predominantly with the ‘‘energy’’ variable ⑀ rather than the frequency ␻ .
At small ⑀ the calculation of ⌸ is a difficult problem but
at large energies, ⑀ Ⰷ ⑀ p , the first Born approximation suffices. The only parameters needed to implement it are the
7557
mean and the variance of the matrix elements of S given by
Eqs. 共20兲 and 共21兲. For qⰆ1/a we obtain that ⌸ ␣␤ (q, ⑀ )
⫽ ␦ ␣␤ ⌸( ⑀ ), where
⌸ 共 ⑀ 兲 ⫽S 0 ⫹V ⬙ 共 0 兲
冕
k
tr D0 共 k, ⑀ 兲 .
共36兲
⌸ has both imaginary and real parts, corresponding to the
broadening and frequency shift of the pinning mode, Eqs.
共34兲 and 共35兲, respectively. The imaginary part comes solely
from the pole共s兲 of D0 (k, ⑀ ), i.e., the solutions of 关 H T0 (k)
⫹S 0 ⫺ ⑀ 兴关 H L0 (k)⫹S 0 ⫺ ⑀ 兴 ⫺ ⑀⑀ c ⫽0. A more accurate expression for ⌸ can be obtained using the self-consistent Born
approximation,
⌸ 共 ⑀ 兲 ⫽S 0 ⫹V ⬙ 共 0 兲
冕
k
tr D共 k, ⑀ 兲 ,
共37兲
as long as the full propagator D(k, ⑀ ) has a pole at 兩 k 兩 Ⰷq c
⫽ ␲ /R c . Under this condition the diagrams with intersecting
lines not included in the SCBA are suppressed, much as they
are suppressed in a dirty metal with k F Ⰷl ⫺1 , k F being the
Fermi momentum and l the mean free path. The indicated
condition is satisfied when
再
⑀ Ⰷmin ⌸ 0 ,
冎
␭ ⌸ 20
.
␮ ⑀c
共38兲
Here and below ␭ is meant to be evaluated at q⫽q c unless it
is indicated otherwise. The parameter ⌸ 0 , which can be estimated to be
⌸ 0 ⬃ ␮ q 2c ,
共39兲
represents the real part of the self-energy at the lowest ⑀
allowed by inequality 共38兲. ⌸ 0 is smaller than S 0 by a logarithmic factor due to the partial cancellation between the first
and the second terms in Eq. 共37兲. This unfortunate cancellation leaves us with only the order of magnitude estimate
共39兲.
When inequality 共38兲 is violated, the SCBA becomes an
uncontrolled approximation. The only SCBA result that can
presumably be trusted at such ⑀ is a slow dependence of
Re ⌸( ⑀ ) on energy. This is because the real part of the integral in Eq. 共37兲 is dominated by large k’s where the lowestorder perturbation theory is valid. To estimate ⑀ p we can take
Re ⌸( ⑀ )⬇⌸ 0 , which yields10,12
再
⑀ p ⬃min ⌸ 0 ,
⌸ 20
⑀c
冎
,
共40兲
represented graphically in Fig. 3 together with the SCBA
domain.
As one can see from this graph, for ⑀ c ⬍⌸ 0 共weak magnetic fields兲 there is no parametric separation between the
SCBA domain and the curve ⑀ ⫽ ⑀ p ; therefore, the SCBA is
qualitatively correct even along this curve where it predicts
Im ⌸( ⑀ p )⬃⫺⌸ 0 . Hence, ⌬ ␻ p ⬃ ␻ p . The resultant broad absorption maximum is sketched in Fig. 4.
If ␭ and ␮ are comparable, the same argument works also
in strong magnetic fields ( ⑀ c Ⰷ⌸ 0 ) where ␴ xx ( ␻ ) looks
qualitatively similar, except the position of the maximum
M. M. FOGLER AND DAVID A. HUSE
7558
PRB 62
S T qq⬘ ⫽ 共 ␦ ␣␤ trS̃⫺S̃ ␣␤ 兲 q̂ ␣ q̂ ␤⬘ ,
共42b兲
S X qq⬘ ⫽S̃ ␣␤ 共 q⫺q⬘ 兲 ⑀ ␤␥ q̂ ␣ q̂ ␥⬘ .
共42c兲
Finally, SX⫹ ⬅SX ⫹ 冑⑀⑀ c I. Integrating out uL is an exact algebraic transformation, which preserves the spectrum of the
collective modes: the resolvent DT ( ⑀ ) of the operator HT
⫺ ⑀ I has poles at the same ⑀ as the full propagator. In fact,
DT , which can be written in the form
DT ⫽ 共关 DT* 兴 ⫺1 ⫺S* 兲 ⫺1 ,
共43兲
DT* ⫽ 共 HT0 ⫹ST ⫺ ⑀ I兲 ⫺1
共44兲
†
S* ⫽SX⫹
共 HL0 ⫹SL ⫺ ⑀ I兲 ⫺1 SX⫹ ,
共45兲
where
FIG. 3. Top: Pinning mode location as a function of the magnetic field. Bottom: SCBA domain of applicability.
depends on the magnetic field: ␻ p ⫽ ␻ 2p0 / ␻ c , where ␻ p0 is
the pinning frequency at B⫽0. On the other hand, if ␭Ⰷ ␮ ,
as for the WC, the pinning mode is situated in the interior of
a parametrically wide range of ⑀ where the SCBA fails.
Thus, the calculation of ⌬ ␻ p requires other methods. Later
we will show that ⌬ ␻ p in strong magnetic fields is related to
the low-frequency tail of ␴ xx ( ␻ ) in the absence of the magnetic field. Although calculating the functional form of such
a tail is another nontrivial problem 共after all, it is beyond the
SCBA兲, this intriguing relation is sufficient to establish that
the absorption line narrows down, in agreement with the
experiments.14–17
We unfold our argument gradually over the remaining
sections. In this section we start implementing this task by
clarifying why ␴ xx ( ␻ ) is virtually independent of the longitudinal stiffness ␭ provided ␭Ⰷ ␮ and B⫽0.
Since action A is quadratic in uL , the L phonon degrees of
freedom can easily be integrated out, leading to the effective
Hamiltonian for the T phonons,
†
HT ⫽HT0 ⫹ST ⫺SX⫹
共 HL0 ⫹SL ⫺ ⑀ I兲 ⫺1 SX⫹ ,
共41兲
where HT0 , HL0 , and Si should be understood as operators.
The first two are diagonal in the basis of plane waves, with
matrix elements H T0 (q) and H L0 (q), respectively; the Si ’s
have both diagonal and off-diagonal matrix elements:
S L qq⬘ ⫽S̃ ␣␤ 共 q⫺q⬘ 兲 q̂ ␣ q̂ ␤⬘ ,
共42a兲
FIG. 4. Frequency dependence of the conductivity in weak magnetic fields.
is nothing other than the T-T component of the full phonon
propagator before the disorder averaging. The self-energy of
the averaged propagator at q⫽0 is a scalar 共see above兲;
therefore, the T-T component determines the entire propagator for this particular q.
Let us now show that in zero magnetic field the mixing
between the T and L phonons represented by S* has little
effect on DT . Indeed, in the diagrammatic expansion of
具 DT ( ⑀ ) 典 in powers of ␦ ST and S* , the typical momentum
transfer k for ⑀ ⱗ⌸ 0 is of the order of q c . A single occurrence of S* contributes ⬃k 2 兩 V ⬙ (0) 兩 /␭k 2 ⬃( ␮ /␭)⌸ 0 to the
self-energy, compared to ⬃⌸ 0 from ␦ ST . Diagrams with
multiple occurrences of S* are suppressed by even higher
powers of the small parameter ␮ /␭. Thus, in the parameter
range relevant for the observation of the pinning mode,
具 DT 典 共 q, ⑀ 兲 ⯝ 具 DT* 典 共 q, ⑀ 兲 ⬅
1
H T0 共 q 兲 ⫹⌸ * 共 q, ⑀ 兲 ⫺ ⑀
. 共46兲
Note that DT* is determined only by the shear modulus ␮ and
the disorder in the T-T channel ␦ ST , while bulk modulus ␭
drops out. It follows from this discussion that for ␭Ⰷ ␮ we
can calculate the q⫽0 response by pretending that the L
degree of freedom does not exist.
Let us indeed imagine that the L phonons are forbidden
and try to investigate the nature of the T phonon eigenstates
that would compose the pinning mode, i.e., the states that
would respond to the uniform electric field 共still at B⫽0).
To do so we need to analyze the solutions of the eigenvalue
problem for the single-particle Hamiltonian H* ⫽HT0 ⫹ST .
In the uniformly pinned WC, ST ⫽S 0 I and the eigenstates are
just the plane waves labeled by momenta k of the Brillouin
zone. In the actual random system the eigenstates are wave
packets of the plane waves with characteristic spread of momenta of the order of L T⫺1 , where L T ⬃ v T / 兩 Im ⌸ 兩 is the T
phonon mean free path, v T ⫽ 冑⑀␮ playing the role of T phonon ‘‘velocity.’’ Since 兩 Im ⌸( ⑀ p ) 兩 ⬃ ⑀ p ⬃ ␮ q 2c , at ⑀ ⬃ ⑀ p we
have L T ⬃R c . Hence, the eigenstates that respond appreciably to a q⫽0 external drive are wave packets built from 0
⬍k⬍q c plane waves. For such wave packets the average k is
of the order of the inverse mean free path and the Ioffe-Regel
criterion29 maintains that these states are localized. In other
DYNAMICAL RESPONSE OF A PINNED TWO- . . .
PRB 62
7559
Concluding this section, we would like to reiterate one of
its main results, that the presence 共or absence兲 of the stiff L
phonon degree of freedom affects the B⫽0, q⫽0 response
only weakly. Naturally, it makes the fate of the L phonons
seem quite mysterious. The finite-B dynamical response has
also been barely touched upon. These gaps in understanding
will be filled in the next section.
FIG. 5. Sketch of spatial distribution of T phonon states contributing to the pinning mode at B⫽0.
words, R c is not only the mean free path but also the localization length of the states within the pinning mode.
Let us define the B⫽0 T phonon density of states,
␯共 ⑀ 兲⫽
1
L
2
冓兺
i
冔 冕
␦共 ⑀⫺⑀i兲 ⫽
1
␲
k
具 DT* 典 共 k, ⑀ 兲 .
共47兲
At ⑀ Ⰷ ⑀ p it tends to the bare value ␯ 0 ⫽(4 ␲ ␮ ) ⫺1 , and at ⑀
⬃ ⑀ p it is only slightly smaller. It is easy to see then that,
within an area R c ⫻R c and energy interval 0⬍ ⑀ ⬍ ⑀ p , there
is typically only one 共localized兲 state. The picture that
emerges is illustrated in Fig. 5: we have a collection of localized states of roughly the same size 共i.e., the inverse participation ratio兲 and roughly the same distance from each
other in real space, both of the order of R c . The broad distribution of shapes and sizes characteristic of localized states
in disordered systems yields a large inhomogeneous broadening ⌬ ␻ p ⬃ ␻ p of the absorption line, as we found earlier
based on a different line of reasoning.
In early works on pinning a heuristic imagery of ‘‘domains’’ was sometimes invoked12 to describe the ground
state structure. However, it was never clear where to draw
the boundaries between different domains. This difficulty is
partially resolved in our picture of phonon localization,
where domains can be defined as areas where individual localized phonon wave functions are appreaciable. However,
modern understanding of the subject reviewed in the previous section no longer appeals to any ‘‘domains,’’ and so this
interpretation may be regarded as a historic sentiment.
One more historic comment is in order. Previous work30
on phonon localization found that the phonon eigenstates
remain extended even in the ␻ →0 limit, seemingly in contradiction to our results. In fact, there is no contradiction
because the model we study here and the conventional formulations, where disorder originates from the defects of the
crystalline lattice, e.g., substitutions by impurity atoms, are
quite different. While in those conventional models the defects move with the lattice and their influence rapidly diminishes at small k’s, the pinning potential for the WC is static,
so it is not affected by the lattice motion. It couples to the
phonons via the matrix S whose fluctuations have approximately the same rms magnitude at all k’s within the Brillouin zone; see Eq. 共20兲. This is the reason why in our case
the disorder effects are much stronger, causing the localization of the phonon eigenstates. Actually, our model has more
in common with those of noninteracting electrons in dirty
metals and semiconductors. 共At the one-particle level the distinction between the bosonic statistics of phonons and the
fermionic statistics of electrons is irrelevant.兲
IV. SCATTERING OF L PHONONS AND PINNING MODE
LINEWIDTH IN A FINITE MAGNETIC FIELD
In the previous section we discussed the scattering and
localization of the T component of WC lattice vibrations. Let
us now turn to the L component. This time we integrate out
uT to obtain the effective Hamiltonian for the L phonons, the
corresponding Green’s function, and the self-energy:
†
,
HL ⫽HL0 ⫹SL ⫺SX⫹ 共 HT0 ⫹ST ⫺ ⑀ I兲 ⫺1 SX⫹
共48兲
DL ⫽ 共 HL ⫺ ⑀ I兲 ⫺1 ,
共49兲
⌸L 共 ⑀ 兲 ⫽ 具 DL 典 ⫺1 ⫺ 共 HL0 ⫺ ⑀ I兲 .
共50兲
Since the self-energy ⌸(q, ⑀ ) of the original system 共before
integrating out uT ) is a scalar at q⫽0, ⌸ L must obey the
relation
⌸ L 共 0,⑀ 兲 ⫽⌸ 共 0,⑀ 兲 ⫺
⑀⑀ c
.
⌸ 共 0,⑀ 兲 ⫺ ⑀
共51兲
Our first task is to reproduce the results of Sec. III by demonstrating that in sufficiently weak magnetic fields ⌸ L indeed has the above form with ⌸⯝⌸ * .
There are two types of process that contribute to ⌸ L : the
intraband scattering 共L-L channel兲 and the scattering through
an intermediate T state 共L-T-L channel兲. They originate from
the second and third terms of HL , respectively. The intraband scattering will be treated in more detail in Appendix A,
where using a combination of the renormalization group and
instanton methods we show that it is exponentially small. For
our current purposes a weaker result is sufficient, that the
intraband contribution ⌸ L→L to ⌸ L is much smaller than ⌸ 0 .
This can be established without detailed calculations. Indeed,
the effective random potential SL in the L-L channel has
basically the same mean and rms fluctuation as ST ; therefore, SL and ST can be regarded as disorder of approximately
the same strength. Then the inequality 兩 ⌸ L→L 兩 Ⰶ 兩 ⌸ 兩 ⬃⌸ 0
mentioned above follows simply from the fact that L
phonons have a much steeper bare dispersion relation ⑀ (q)
⫽␭(q)q 2 than T phonons and are scattered much more
weakly by disorder of the same strength. It is legitimate to
neglect the L-L scattering altogether by replacing SL with its
average value S 0 I, and to concentrate exclusively on the
L-T-L processes. In this case, much as for dirty semiconductors with two bands of carriers,31 the diagrammatic expansion of ⌸ L can be formulated as the sum of all one-particle
irreducible graphs involving SX⫹ and DT 关see Eqs. 共43兲–共45兲
and Figs. 6共a兲 and 6共b兲兴.
The crucial point substantiated below is that S* can be
neglected, i.e., that DT can be replaced by DT* , provided ␭
Ⰷ ␮ and the magnetic field is not too strong,
M. M. FOGLER AND DAVID A. HUSE
7560
PRB 62
we found previously in Sec. III. However, if ⑀ c ⫽0, these
diagrams also generate terms of the order of
冑⑀⑀ c
1
H L0 共 k 兲
冑⑀⑀ c ⬃
⑀⑀ c
␭q 2c
,
共57兲
which act as an additional self-energy correction to DT* in
Fig. 6共c兲. They can be approximately accounted for if we
replace ⑀ in the argument of ⌸ * in Eq. 共56兲 by a renormalized value ˜⑀ ,
FIG. 6. 共a兲 Diagrams for ⌸ * . Filled circles and solid lines represent ␦ ST and DT0 ⬅(HT0 ⫹S 0 I⫺ ⑀ I) ⫺1 , respectively. Dashed lines
symbolize contractions. 共b兲 Diagrammatic definitions of DT and
⌸ L . 共c兲 Diagrams without S* . Diamonds represent factors i 冑⑀⑀ c .
共d兲 Some representive diagrams for ⌸ L not included in 共c兲.
⑀ c Ⰶ 共 ␭/ ␮ 兲 ⌸ 0 .
共52兲
For such ⑀ c only two graphs contribute to ⌸ L 关Fig. 6共c兲兴,
which is a significant simplification. However, we still have
to specify how to average over the disorder. Naive averaging
of SX and DT* separately from each other yields
⌸ L 共 q, ⑀ 兲 ⫽S 0 ⫺ ⑀⑀ c 具 DT* 典 共 q, ⑀ 兲 ⫹V ⬙ 共 0 兲
冕具
k
DT* 典 共 k, ⑀ 兲
共53兲
but it is certainly not accurate because SX and ␦ ST are not
independent. Indeed, their two-point correlator
具 S Xq1 qS Tq2 q3 典 ⬀ 共 q̂1 q̂2 兲共 q̂3 q̂兲 ⫹ 共 q̂1 q̂3 兲共 q̂2 q̂兲 ⫹ 共 q̂1 q̂兲共 q̂3 q̂2 兲
共54兲
is in general nonvanishing.
In the special case of q⫽0 further progress is possible.
Comparing the above equation with
具 S Tq1 qS Tq2 q3 典 ⬀ 共 q̂1 q̂2 兲关 q̂⫻q̂3 兴 ẑ⫹ 共 q̂1 q̂3 兲关 q̂⫻q̂2 兴 ẑ⫹ 共 q̂1 q̂兲
⫻关 q̂3 ⫻q̂2 兴 ẑ,
共55兲
we observe that the former is transformed into the latter by
the replacement q→ 关 ẑ⫻q兴 . If the incoming momentum q is
zero, rotating it by ␲ /2 has no effect. Thus, for q⫽0 we can
replace SX in the second diagram of Fig. 6共c兲 by ␦ ST , after
which it becomes identical to the totality of diagrams in Fig.
6共a兲 for ⌸ * ; therefore,
⌸ L 共 0,⑀ 兲 ⫽⌸ * 共 0,⑀ 兲 ⫺
⑀⑀ c
⌸ * 共 0,⑀ 兲 ⫺ ⑀
.
共56兲
As explained in Sec. III, ⌸ * ⯝⌸; hence, we will succeed in
reproducing Eq. 共51兲 as soon as we show that inequality 共52兲
is indeed the relevant condition for dropping S* in the diagrammatic series for ⌸ L . To do so we develop one step
further the argument of Sec. III. One-particle irreducible diagrams containing S* have at least one L phonon line with
typical momentum transfer k⬃q c 关see Fig. 6共d兲兴. If ⑀ c ⫽0
such diagrams are suppressed by at least a factor of ␮ /␭, as
⌸ L 共 0,⑀ 兲 ⫽⌸ * 共 0,˜⑀ 兲 ⫺
˜⑀ ⫽ ⑀ ⫹C 1
⑀⑀ c
␭q 2c
⑀⑀ c
,
⌸ * 共 0,˜⑀ 兲 ⫺ ⑀
, C 1 ⬃1.
共58兲
共59兲
The difference between ⑀ and ˜⑀ can be safely ignored as long
as the inequality 共52兲 is satisfied, in which case Eq. 共56兲 is
asymptotically exact. In stronger fields Eq. 共58兲 applies,
which gives only the order of magnitude of ⌸ L because of
the uncertainty in the parameter C 1 .
The meaning of Eqs. 共56兲 and 共58兲 can be elucidated by
returning to the original formulation where both T and L
degrees of freedom are present:
⌸ 共 0,⑀ 兲 ⫽
再
⌸ * 共 0,⑀ 兲 , ⑀ c Ⰶ 共 ␭/ ␮ 兲 ⌸ 0
⌸ * 共 0,˜⑀ 兲 , ⑀ c Ⰷ 共 ␭/ ␮ 兲 ⌸ 0 .
共60a兲
共60b兲
We see that ⌸(0,⑀ ) remains magnetic field independent up
to rather high B. As explained above, this occurs because the
enhancement of the L-T mixing due to the Lorentz force is
strongly impeded by the disparity of the two elastic moduli.
Let us now evaluate the consequences of the obtained
equations 共60a兲 and 共60b兲. In the intermediate field range,
⌸ 0 Ⰶ ⑀ c Ⰶ(␭/ ␮ )⌸ 0 , where Eq. 共60a兲 applies, the consequences are twofold: the suppression of ⑀ p , which is well
known, and the suppression of the pinning mode linewidth,
which is nontrivial. Indeed, since ⑀ p becomes much less than
⌸ 0 关Eq. 共40兲兴, it slips into the low-energy tail of the zerofield phonon spectrum where Im ⌸Ⰶ⌸ 0 ; hence, ⌬ ␻ p Ⰶ ␻ p .
The connection between the linewidth of the absorption line
in strong magnetic fields and the low-frequency phonon
modes in zero magnetic field, which we just established, is
the keystone of the present paper.
The properties of the zero-field soft modes will be discussed in more detail in Sec. V where we argue that they
give rise to the power-law dependence of Im ⌸ on ⑀ ,
Im ⌸ 共 ⑀ 兲 ⯝⫺C 2 ⌸ 0 共 ⑀ /⌸ 0 兲 s , ⑀ Ⰶ⌸ 0 ,
共61兲
with exponent s⫽3/2 and numerical prefactor C 2 ⬃1. Combining Eqs. 共34兲,共35兲 and 共60兲,共61兲, we find for the
intermediate-B regime
␴ xx 共 ␻ 兲 ⫽⫺i
e 2n e␻
1⫺i f 1 共 ␻ 兲
m ␻ 2p0
关 1⫺i f 1 共 ␻ 兲兴 2 ⫺ 共 ␻␻ c / ␻ 2p0 兲 2
, 共62兲
PRB 62
DYNAMICAL RESPONSE OF A PINNED TWO- . . .
f 1共 ␻ 兲 ⯝
再
C 2 共 ␻ / ␻ p0 兲 2s ,
␻ Ⰶ ␻ p0
const,
␻ p0 Ⰶ ␻ Ⰶ ␻ c ,
␻ p0 Ⰶ ␻ c Ⰶ ␻ p0 冑␭/ ␮ .
共63兲
共64兲
In this regime the absorption line narrows down according to
Eq. 共2兲, which we reproduce here for convenience:
⌬ ␻ p / ␻ p⬃共 ␻ p / ␻ c 兲s.
共65兲
The high-field regime is described by Eq. 共60b兲. In this case
the relative linewidth is field independent and is of the order
of
⌬ ␻ p / ␻ p ⬃ 共 ␮ /␭ 兲 s
共66兲
as given by Eq. 共3兲 and illustrated in Fig. 1. Although Eq.
共60b兲 was derived with much less rigor than Eq. 共60a兲, the
saturation of ⌬ ␻ p / ␻ p in strong fields is certainly to be expected. Indeed, in strong fields the dynamics is dominated by
the Lorentz force; therefore, ⑀ must enter in the combination
⑀⑀ c , not by itself. But then a phonon eigenstate, which for a
2
given 共large兲 ⑀ c ⫽ ⑀ c* ⫽ ␳ ( ␻ *
c ) has an energy ⑀ i , is also an
eigenstate of the system for larger ⑀ c , with the eigenvalue
⑀ i ( ⑀ c* / ⑀ c ). Hence, as the magnetic field increases, all relevant eigenfrequencies scale inversely proportional to ␻ c ,
while the conductivity varies in the self-similar way
␴ xx 共 ␻ 兲 ⫽
冉 冊
␻*
␻␻ c
c
␴*
␻ c xx ␻ c*
共67兲
and possesses constant ⌬ ␻ p / ␻ p . The explicit expression for
␴ xx ( ␻ ) is similar to Eq. 共63兲,
e n e␻
1⫺i f 2 共 ␻ 兲
2
␴ xx 共 ␻ 兲 ⫽⫺i
m ␻ 2p0
f 2共 ␻ 兲 ⯝
再
关 1⫺i f 2 共 ␻ 兲兴
2
⫺ 共 ␻␻ c / ␻ 2p0 兲 2
共 ␻ /⍀ 兲 2s ,
␻ Ⰶ⍀
const,
⍀Ⰶ ␻ Ⰶ ␻ c ,
␻ 2p0
冑␭/ ␮ , ␻ c Ⰷ ␻ p0 冑␭/ ␮ ,
⍀⫽C 3
␻c
into the form of the effective Hamiltonian HLef f after the
coarse graining is given by the spectral decomposition of the
matrix element,
冓冏
r
1
HT0 ⫹ST ⫺ ⑀ I
共70兲
where C 3 ⬃1.
Formulas 共63兲 and 共69兲 are our final results. They describe
the entire line shape of the pinning mode in both intermediate and strong magnetic fields. However, their derivation was
presented in diagrammatic rather than physical terms. Next,
we will give an alternative derivation, which elucidates the
physics of the line narrowing and also helps to clarify the
structure of the phonon eigenstates that compose the pinning
mode.
We will start with the qualitative picture of localized T
phonons developed in Sec. III and add a new ingredient, the
stiff L degree of freedom. It is clear that an admixture of the
L component produced by a joint action of the disorder and
the Lorentz force makes phonon eigenstates much more extended in real space. Unlike the softer T phonons, the stiffer
L ones cannot be confined in small areas of size R c . In order
to understand the large-scale structure of the eigenstates, we
can coarse-grain the system by integrating out the degrees of
freedom on the spatial scales between a and R c . An insight
冏冔
r⬘ ⫽
兺i
* 共 r⬘ 兲
u Ti 共 r兲 u Ti
⑀ i⫺ ⑀
.
共71兲
Here ⑀ i and u Ti (r) are the eigenvalues and eigenfunctions of
localized T phonons. Since each of u Ti is localized within an
area of size R c , we expect that after the coarse graining the
* (r⬘ ) transform into local operators
numerators u Ti (r)u Ti
␦
(r⫺r
)
␦
(r⫺r
R ⫺2
⬘
i ), where ri is the center of gravity of the
c
ith mode. As for the denominators, the discussion in Sec. III
shows that in weak and intermediate magnetic fields 关defined
by the inequality 共52兲兴 ⑀ i ’s should remain unaffected. Finally, naive averaging of SL in Eq. 共7兲 yields S 0 I but the
interaction with high-k T modes, which can be calculated
within SCBA-type perturbation theory, renormalizes S 0 to
⌸ 0 . The resultant coarse-grained Hamiltonian is
HLef f ⫽HL0 ⫹⌸ 0 I⫺
兺i
c i ⌸ 20 ⫹ ⑀⑀ c
⑀ i⫺ ⑀
R 2c ␦ 共 r⫺ri 兲 ,
共72兲
where c i ⬃1 are random and ⑀ i 苸(0,2⌸ 0 ). Without losing
essential physics, we can assume that the ri ’s form a regular
square lattice with lattice constant R c .
What are the properties of the obtained lattice model?
One of them is the Lorentzian tails of the distribution function P(U) of the on-site disorder terms U i ⬅(c i ⌸ 20
⫹ ⑀⑀ c )/( ⑀ ⫺ ⑀ i ). For example, if ⑀ ⱗ ⑀ p , then
P共 U 兲⬃
, 共68兲
共69兲
7561
⌸ 20 R 2c ␯ 共 ⑀ 兲
U2
, 兩 U 兩 →⬁,
共73兲
where ␯ ( ⑀ ) is the T phonon density of states introduced in
Sec. III. Clearly, the variance 具 U 2 典 of the on-site disorder is
unbounded. One can therefore expect much stronger disorder
effects than in the case of a Gaussian distribution with the
same typical values of U i , i.e., the same 具 ln兩U兩典. Let L L be
the mean free path of the L component of the low-energy
phonons. As explained in Sec. III, the Ioffe-Regel criterion29
suggests that L L is simultaneously their localization length.
For weak Gaussian disorder and unscreened Coulomb interactions L L is exponentially large 共see Appendix A兲. Now we
will show that, in the presence of the long tails 共73兲, L L is
only a power-law function of the disorder strength. The crucial point is that the scattering is dominated by the few largest U i ’s. As soon as we realize this, we can replace P(U) by
the Cauchy distribution
P共 U 兲⫽
⌫
1
, ⌫ 共 ⑀ 兲 ⬃⌸ 20 R 2c ␯ 共 ⑀ 兲 ,
␲ 共 U⫺U 0 兲 2 ⫹⌫ 2
共74兲
on the grounds that it has the same tails. At this point we
make the reasonable assumption that correlations among
U i ’s at different sites can be neglected, and arrive at the
famous Lloyd model. The self-energy can now be calculated
exactly,32
⌸ L 共 ⑀ 兲 ⫽⌸ 0 ⫺U 0 共 ⑀ 兲 ⫺i⌫ 共 ⑀ 兲 ,
共75兲
M. M. FOGLER AND DAVID A. HUSE
7562
FIG. 7. Schematic structure of the localized phonon eigenstate
in the crystal with both T and L degrees of freedom present. Circles
represent the T modes of Fig. 5; the resonant modes are shaded.
while the mean free path is the solution of the equation
H L0 (L L⫺1 )⫽ 兩 Im ⌸ L 兩 , which gives L L ⬃Y /⌫.
To compare with our earlier results we need to know ␯
and U 0 . The integral in Eq. 共47兲 is determined by 兩 k兩 ⱗq c ;
therefore,
␯ 共 ⑀ 兲 ⬃q 2c
Im ⌸ * 共 ⑀ 兲
⌸ 20
⬃ ␮ ⫺1
冉 冊
⑀ s
, ⑀ ⱗ⌸ 0 .
⌸0
共76兲
The estimate of U 0 is U 0 ⬃ ⑀⑀ c /⌸ 0 . Combining these together, we find that Eq. 共75兲 is in agreement with Eq. 共56兲,
which validates our mapping onto the Lloyd model.
To describe the dominant scattering mechanism in the
Lloyd model notice that Im ⌸ L comes from the inhomogeneous broadening of the phonon energies, i.e., the energy
shifts ␦ ⑀ caused by on-site disorder U i . For each localized L
phonon the largest energy shift comes from the site with
largest U i within the localization area L L ⫻L L , which is
typically U i ⬃(L L /R c ) 2 ⌫ for the Cauchy distribution 共74兲.
Since the amplitude of the L phonon wave function at the
position of the oscillator is u L ⬃1/L L , the resultant energy
shift is ␦ ⑀ ⬃U i u L2 R 2c ⬃⌫. The energy shifts from other sites
are subdominant; thus, Im ⌸ L ⬃⌫, in agreement with the exact result, Eq. 共75兲. In the original formulation large U i come
from the localized T oscillators whose energy ⑀ i is almost in
resonance with ⑀ , i.e., 兩 ⑀ i ⫺ ⑀ 兩 ⬃1/ ␯ ( ⑀ )L L2 . A few of such
resonant scatterers will localize the L component on the scale
of the mean free path L L . The proposed picture resembles
the situation in a crystal pinned by strong dilute impurities,
the role of impurities being played by the resonant sites.
The structure of a typical phonon is illustrated in Fig. 7. It
involves of the order of (L L /R c ) 2 T modes of Fig. 5 oscillating with the same frequency imposed by the mutual coupling mediated by the L phonon. The oscillation amplitude is
roughly the same for all the T modes except for a few resonant ones, where it is much larger. These resonant sites
‘‘drain’’ the energy of the L degree of freedom, thereby localizing it.
PRB 62
Let us now describe the evolution of the phonon eigenstates as the magnetic field increases. Since the pinning
mode frequency goes down, lower- and lower-frequency
resonant sites are required to effectively scatter and localize
the mixed L-T phonon vibrations. Such sites are therefore
the soft modes, the oscillators of unusually low frequency,
which shape up the tails of the zero-field phonon spectrum
共Lifshitz tails兲. The soft modes appear due to very rare, peculiar disorder configurations. The lower the frequency, the
more dilute they are in real space. Correspondingly, the
mean free path 共localization length兲 of the magnetophonons
becomes larger and larger. The narrowing of the absorption
line is then similar to the motional narrowing phenomenon.
Eventually, in very strong magnetic fields, the enhancement
of the L-T mixing by the Lorentz force start to diminish the
frequencies of the localized T phonons. At this point slipping
of the pinning frequency ␻ p deeper into the soft-mode tail
stops. The wave functions of the phonons cease to change;
only their eigenfrequencies continue decreasing in inverse
proportion to the magnetic field. As a result, ⌬ ␻ p / ␻ p remains constant.
The physical structure of the pinned phonon modes being
clarified, the only important questions that remain to be considered are the nature of the soft modes and the derivation of
Eq. 共61兲. This will be the subject of the next section.
V. SOFT MODES
It is clear that the soft phonon modes must come from
rare places where pinning is unusually weak. The actual calculation of the density of states ␯ ( ⑀ ) in the low-⑀ tail is,
however, nontrivial. The method most suitable for the task
seems to be the method of optimal fluctuation. It has been
successfully employed for calculation of the Lifshitz tails in
other disordered systems,20 where it proves to be asymptotically exact. The method is based on the following idea. The
energy ⑀ of a given eigenstate is determined in a complicated
way by the distribution of the random potential within the
entire area supporting the eigenstate and so there are many
different random potential configurations, which give the
same eigenenergy. If ⑀ is small, any such configuration is
untypical, i.e., a fluctuation of some sort. The method is
based on the assumption that certain fluctuations have much
higher probability than the rest and dominate the quantity of
interest, ␯ ( ⑀ ) in this case. The objective is then to design
such an optimal fluctuation and evaluate its probability.
Let us mention one type of fluctuation, which is not optimal. This is a configuration where the random potential is
strongly suppressed in a large area L⫻L, where L⬃ 冑␮ / ⑀ .
共It may be useful to recall at this point that we are investigating the soft modes of a system where L phonons are integrated out.兲 The probability of such a fluctuation can be
estimated by mentally dividing the area into uncorrelated
blocks of size R c ⫻R c and multiplying together the probabilities ⬃ ⑀ /⌸ 0 that the random potential is suppressed within
each block. The total probability is exponentially small
⬃exp关⫺(⌸0 /⑀)ln(⌸0 /⑀)兴. This is a general situation for fluctuations spread over a large area in real space. Thus, the
optimal fluctuation must be of the smallest possible size. It is
easy to see that this size is R c . Indeed, to get a small phonon
eigenenergy ⑀ in a small volume, the positive ‘‘kinetic en-
DYNAMICAL RESPONSE OF A PINNED TWO- . . .
PRB 62
ergy’’ ␮ L ⫺2 must be accurately compensated by the negative ‘‘potential energy’’ 具 ␦ ST 典 L . The fluctuations of the latter averaged over the area L⫻L are of the order of
⌸ 0 (R c /L). Thus, for L⬃R c the cancellation of ␮ L ⫺2 can be
done by a typical fluctuation of ␦ ST and the probability of
such an event will not have any exponential suppression.
Instead, we expect the power-law dependence of ␯ ( ⑀ ) on ⑀ ,
Eq. 共76兲.
In order to possess a soft mode, the system must have a
soft direction, i.e., a certain collective coordinate X such that
the total energy of the system E as a function of X has a very
shallow minimum. Let X⫽0 be the ground state. In its vicinity E should be Taylor expandable,
␣
E 共 X 兲 ⫽ X 2 ⫹ ␤ X 3 ⫹ ␥ X 4 ⫹••• .
2
FIG. 8. Energy of the system as a function of the collective
coordinate X.
P 共 ␣ 兲 ⫽ 具 ␦ 共 E ⬙ ⫺ ␣ 兲 典 兩 E ⬘ ⫽0
共77兲
The coefficient ␣ has the meaning of an effective spring
constant of the local oscillator, while the phonon energy ⑀ is
determined by the ratio of the spring constant and the effective mass. The latter is proportional to the area involved in
the oscillations. As we argued above, this area is of the order
of R 2c for all ⑀ ⬍ ⑀ p . Therefore, ␣ should scale linearly with
⑀ in the limit ⑀ →0,
␣ ⫽C 4 ⑀ .
7563
共78兲
If we find a way to estimate the probability of such untypically small ␣ , we will succeed in calculating the exponent s
of the power-law function ␯ ( ⑀ ).
This kind of calculation is hardly possible without specifying X. Actually, ascribing a precise meaning to X amounts
more or less to designing the optimal fluctuation. At the moment we do not have a good understanding of how to do it in
the 2D case. However, in one dimension the general structure of the calculation is clear. We are guided by the earlier
work on the subject by Feigelman et al.33 and Aleiner and
Ruzin.34
In the case of a weakly pinned 1D elastic chain, the role
of the desired collective coordinate X can be played by the
elastic displacement field u at an arbitrary point in the middle
of the chain. This point divides the system into two parts,
which communicate only through the single variable u. For
any fixed u serving as a boundary condition, we can find
separately the ground state energies E ⬍ (u) and E ⬎ (u) of the
left and right halves of the chain. The ground state of the
whole system is determined by minimizing the sum E(u)
⫽E ⬍ (u)⫹E ⬎ (u) with respect to u. Both E ⬍ (u) and E ⬎ (u)
are periodic functions of u with period equal to the lattice
constant a. On their period they have several 共typically, of
the order of a/ ␰ ) minima and maxima. The beautiful idea of
Aleiner and Ruzin34 共totally overlooked in Ref. 33兲 was that
the most efficient way to generate a soft mode is via a frustration, when a maximum of E ⬍ (u) occurs near a minimun
of E ⬎ (u). In this case the second derivative of E, i.e., ␣
⫽E ⬙ , can be very small even though each of the second
derivatives of E ⬍ (u) and E ⬎ (u) are typical. The next step is
to show that the probability density distribution P( ␣ ) of ␣ at
local minima of E vanishes linearly with ␣ in the limit ␣
→0. Indeed,
⫽
冓冕
1
a
a
0
du ␦ 共 E ⬙ ⫺ ␣ 兲 ␦ 共 E ⬘ 兲
冏 冏冔
⳵E⬘
⳵u
→兩␣兩具␦共 E ⬙ 兲␦共 E ⬘ 兲典
⬙ ⫹E ⬎
⬙ 兲␦共 E⬍
⬘ ⫹E ⬎
⬘ 兲典.
⫽兩␣兩具␦共 E⬍
共79兲
The last equation explicitly demonstrates the aforementioned
cancellations between the derivatives of E ⬍ and E ⬎ . The
value of the ␦ -function product average is determined by
properties of typical configurations. Hence, the low probability of having unusually shallow local minima is entirely due
to the first factor on the left-hand side of Eq. 共79兲 and so
P( ␣ )⬀ 兩 ␣ 兩 . In particular, P⬀ ⑀ in the case of interest 共78兲.
But this is not yet the end of the story. Following Aleiner and
Ruzin34 we argue that the probability of having a shallow
global minimum is additionally suppressed. Indeed, if the
coefficient ␤ in front of the the cubic term in Eq. 共77兲 is too
large, E(X) will have a second minimum that is deeper than
that at the ground state X⫽0. To avoid that 兩 ␤ 兩 must not
exceed 冑3 ␣␥ ⬀ 冑⑀ . The typical dependence of the resultant
E(X) is illustrated in Fig. 8. The total probability density of
the optimal fluctuation is therefore proportional to ⑀ 冑⑀ ⫽ ⑀ s
with s⫽3/2, as we claimed. It is very likely that s⫽3/2 is in
fact the general result independent of the number of dimensions because the optimal fluctuation will always have only
one soft direction, i.e., the system is essentially one
dimensional.34
Finally, let us compare our results with those in the literature. One group of work, by Feigelman et al.,33 has already
been mentioned. We borrowed from them the idea of splitting the system into two statistically independent parts. The
major mistake of these authors is overlooking the possibility
of frustrations in the system. In other words, they did not
realize that it is not necessary to require that both E ⬍ and E ⬎
have a small second derivative with respect to u when only
their sum E ⬍ ⫹E ⬎ needs to be so. Another point of disagreement between us and them is the size of the localized
phonons. Feigelman et al. take for granted that it is of the
order of 冑␮ / ⑀ , while we gave an argument that it should be
R c , i.e., much smaller.
In another large group of papers, Refs. 9, 10, 19, and 35,
␯ ␻ ( ␻ )⬀ ␻ 2 for the density of states, and a similar dependence Re ␴ xx ( ␻ )⬀ ␻ 2 for the conductivity were calculated.
To facilitate the comparison we should point out that the
conventionally defined phonon density of states9 ␯ ␻ ( ␻ ) is
related to our ␯ ( ⑀ ) by
7564
M. M. FOGLER AND DAVID A. HUSE
␯ ␻ 共 ␻ 兲 ⫽2 ␯ 共 ␳␻ 2 兲 ␳␻ .
共80兲
Hence, our result is Re ␴ xx ( ␻ )⬀ ␯ ␻ ( ␻ )⬀ ␻ 4 , the same as in
Ref. 34 共see also Ref. 36兲. All the papers in the second group
are explicitly or implicitly based on the SCBA 共see Appendix B for more details兲. The second power of ␻ can be traced
to the unphysical square-root singularity ␯ ( ⑀ )⬀ 兩 ⑀ ⫺ ⑀ th兩 1/2
near a band edge ⑀ th ( ⑀ th⫽0 in our case兲, which is a wellknown basic flaw of the SCBA.20 Note, however, that even
within the SCBA ␯ ( ⑀ ) conforms to the generic form 共76兲,
except that it corresponds to s⫽1/2 instead of what we believe is the correct result, s⫽3/2.
VI. QUANTUM AND THERMAL EFFECTS
So far we have neglected any effects of a quantum nature
or due to a finite temperature. Some of them will be addressed in this section but their systematic, in-depth treatment is deferred for future work.
Let us begin by noting that the electrons of the pinned
WC constantly fluctuate around their equilibrium positions.
The typical size of such fluctuations is easy to find. For example, in strong magnetic fields and at low temperatures
(k B TⰆប ␻ c ) where all the electrons are confined to the lowest Landau level, we have
具 u2 典 ⫽l B2 ⫹
冉
ប
␳␻ c
⫻ ⌸ 0⫹
冕
ប⍀ 共 q兲
1
coth
2k
T
␳␻
q
B
c ⍀ 共 q兲
冊
␮ 共 q兲 ⫹␭ 共 q兲 2
q ,
2
共81兲
⍀ 共 q兲 ⫽ 共 ␳␻ c 兲 ⫺1 冑关 ⌸ 0 ⫹ ␮ 共 q兲 q 2 兲 ] 关 ⌸ 0 ⫹␭ 共 q兲 q 2 兴 . 共82兲
This formula can be used away from the thermal or quantum
melting transitions, where the phenomenological criterion
具 u2 典 1/2⬍␧a is satisfied. Here ␧⬃0.2 is the Lindemann
parameter.8 Incidentally, at T⫽⌸ 0 ⫽0, Eq. 共81兲 gives the
variance of the electron fluctuations in the correlated WC of
Lam and Girvin.37
Let us focus on the range of temperatures k B TⰆប ␻ B
where ␻ B ⬃4 ␲ ␮ /m e ␻ c is the magnetophonon bandwidth
共usually, ប ␻ B /k B ⬃2 K). At such temperatures 具 u2 典 1/2 is totally dominated by quantum fluctuations and is of the order
of l B . We expect that as long as 具 u2 典 1/2 is smaller than the
correlation length of the pinning potential ␰ , which is the
smallest length scale in the classical theory, all the results
obtained in the previous sections acquire at most minor corrections.
Recently Fertig18 and Chitra et al.19 studied the opposite
limit, ␰ Ⰷl B , and found a novel dependence of the pinning
frequency ␻ p on the magnetic field, although they disagree
with each other on the functional form of such a dependence.
To sort things out we offer a different perspective on this
question. Recall that the classical theory predicts the 1/B
behavior10
␻ p⬃
C共 0 兲c
e ␮ ␰ 4B
,
共83兲
which follows from Eqs. 共14兲, 共35兲, and 共39兲. We propose
that a reasonably accurate estimate for ␻ p in the regime ␰
PRB 62
ⰆlB can be obtained within a quasiclassical approximation.
The electrons are visualized as compact wave packets whose
centers of gravity perform classical motion, while the quantum effects are presumed to be contained in the form factor
of the wave packets. This type of quantum effect amounts to
replacing the bare pinning potential U(r) by its convolution
with the form factor F e (r) of the wave packets. For small
fluctuations l B Ⰶa the appropriate form factor is Gaussian,
F e (r)⫽( ␲ 具 u2 典 ) ⫺1 exp(⫺r2/具u2 典 ). Upon the convolution, we
obtain an effective random potential, with the correlation
length l B and variance C(0) ␰ 2 /l B2 . In view of such modifications, Eq. 共83兲 transforms into
␻ p⬃
e2
␮ ប 3c 2
C 共 0 兲 ␰ 2 B 2 , ␰ ⬍l B .
共84兲
The quasiclassical approximation is certainly not exact;
nonetheless, when used away from the quantum Hall fractions, at sufficiently low T, and for robust quantities such as
␻ p , it should be qualitatively correct.
The linear in B dependence of ␻ p derived by Fertig 关see
Eq. 共44兲 in Ref. 18兴 is at odds with the B 2 dependence found
above 关Eq. 共84兲兴 and in Ref. 19. Although Fertig considers a
Poissonian disorder 共potential wells or ‘‘pits’’ of size s
⬍l B , energy ⌬V, and areal density n i ⬍1/ ␲ l B2 ) rather than a
Gaussian one, the real source of the discrepancy is his assertion that the elastic distortion at the Larkin length scale is of
the order of n ⫺1/2
. Instead, l B should be used. Upon this
i
correction, the formula for ␻ p acquires the above form 共84兲
with C(0) ␰ 2 replaced by n i s 4 ⌬V 2 .
The finite amplitude of the quantum fluctuations has another important repercussion: nonvanishing high order in u
terms in the pinning energy. As a result, the absorption line
acquires extra broadening even at T⫽0. It originates from
the finite widths ⌫ n of the energy levels n⫽1,2, . . . of the
localized magnetophonon oscillators 共the ground state n⫽0
remains unbroadened兲. At zero T only the transitions between n⫽0 and n⫽1 levels contribute to the absorption.
The level width ⌫ 1 of the n⫽1 state is determined by the
decay of the given magnetophonon into two other magnetophonon excitations of lower frequencies, ប ␻ 1 →ប ␻ 2
⫹ប ␻ 3 共see Fig. 9兲.
Such a process is always possible in a truly random system, the necessary three-phonon matrix elements
M ␣␤␥ 共 r兲 ⫽⫺n e
兺K K ␣ K ␤ K ␥ U 共 r兲 sin K关 r⫺u(0)共 r兲兴
being provided by the cubic anharmonicities. The corresponding contribution Im ⌸ 3 to the imaginary part of the
self-energy can be obtained by evaluating the diagram in Fig.
9,
Im ⌸ 3 共 q, ␻ 兲 ⫽⫺
⫻
2ប
L2
冕
冕冕 具
k
k⬘
M̃ ␣␮ ␴ M̃ ␤ ␯ ␶ 典
d␻⬘
关 n 共 ␻ ⬘ 兲 ⫹n B 共 ␻ ⫺ ␻ ⬘ 兲 ⫹1 兴
2␲ B
⫻Im D ␮ ␯ 共 k, ␻ ⬘ 兲 Im D ␴ ␶ 共 k⬘ , ␻ ⫺ ␻ ⬘ 兲 ,
共85兲
DYNAMICAL RESPONSE OF A PINNED TWO- . . .
PRB 62
FIG. 9. Top: low-temperature absorption in a system with energy levels broadened by anharmonicities. Bottom: the diagram
used for estimating ⌫ 1 .
n B共 ␻ 兲 ⫽
1
.
exp共 ប ␻ /k B T 兲 ⫺1
共86兲
At k B TⰆប ␻ p only the frequency interval 0⬍ ␻ ⬘ ⬍ ␻ contributes to the integral, which leads us to the estimate
Im ⌸ 3 共 ␻ p 兲 ⬃Im ⌸ 共 ␻ p /2兲
l B2 a 2 兩 Im ⌸ 共 ␻ p /2兲 兩
.
⌸0
␰ 2 R 2c
共87兲
This formula is derived assuming that ␰ Ⰷl B ; otherwise, in
the spirit of the quasiclassical approximation, we have to
replace ␰ by l B . Even in that case Im ⌸ 3 ( ␻ p ) is smaller than
the previously found Im ⌸( ␻ p ) by a large factor
⬃(R c /a) 2 (␭/ ␮ ) s . At higher temperatures, the anharmonic
effects are enhanced by a factor of k B T/ប ␻ p originating
from the Bose distribution functions n B in Eq. 共85兲. They
should start to affect the linewidth when k B T
⬃(R c /a) 7/2ប ␻ p 共cf. Ref. 38兲.
A word of caution is in order. The diagram in Fig. 9 does
not contain vertex corrections, which contain information
about, e.g., the soft modes. The soft modes are crucial for the
dynamical response in strong magnetic fields and at the same
time are potential sources of much larger anharmonicities.
Thus, the response of a weakly pinned quantum WC may
prove to be more nontrivial.
Concluding this section, we would like to reemphasize
that for the problem studied in the main part of the paper
共zero-temperature linear-response of a classical WC兲, the anharmonicities are not important.
VII. COMPARISON WITH EXPERIMENT
In this section we will attempt to analyze the most recent
experimental data16,17 in the light of our understanding of the
pinning mode. Let us first discuss the position of the pinning
resonance.
From Eqs. 共83兲 and 共84兲 we see that, depending on the
type of disorder, ␻ p can either decrease as 1/B or increase as
B 2 when the magnetic field increases.19 Weak magnetic field
dependence of ␻ p found in the experiments16,17 leads us to
7565
conclude that ␰ remains of the order of l B in the limited
range 10–15 T of the magnetic fields where the pinning
mode was detected, i.e., ␰ ⬇l B ⬇75 Å . The source of such a
short-range random potential is likely to be the roughness of
the heterostructure interface as suggested by Fertig.18 The
experimental value of ␻ p ⬇8⫻109 s⫺1 for n e ⫽5.4
⫻1011 cm⫺2 can then be reproduced with reasonable values
of fitting parameters 共see Sec. VI兲, s⫽30 Å , ⌬V⫽4.3 K,
and n i ⫽4⫻1011 cm⫺2 . Unfortunately, the disorder parameters are poorly known, and so we have to use ␻ p to estimate
the disorder characteristics, not the other way around. For
example, we can solve Eq. 共84兲 for the rms amplitude 冑C(0)
of the pinning potential, which gives a value of the order
0.2 K.
Let us now discuss the density dependence of ␻ p . Explicitly, it enters Eq. 共84兲 only through the shear modulus ␮ (n e ).
共see Sec. II兲 away from
The latter should behave as ␮ ⬀n 3/2
e
the quantum Hall fractions, e.g., the 31 filling. If C(0) is n e
independent, we therefore expect ␻ p ⬀n ⫺3/2
. Such a depene
dence was indeed observed by Li et al.16 for the concentration of holes (p-type samples were used兲 in the range
(3 –5)⫻1010 cm⫺2 , which corresponds to the filling factor
range 0.1–0.16. At lower concentrations the pinning frequency continued growing as n e decreased but less rapidly.
This comes presumably from the fact that the pinning is no
longer weak at such low n e : one can verify that R c is approximately 6a at the highest densities, but approaches a
共the lattice constant兲 at the lowest densities used in Ref. 16.
共In Ref. 17 the density dependence of ␻ p was not investigated.兲
The relationships among the empirical values of ␻ p and
⌬ ␻ p and the experimental parameters (n e and B) provide
another means to test the agreement between the theory and
the experiment. To this end we rewrite Eq. 共3兲 共with s
⫽3/2) in the following way:
Q⬅
␻p
⬃
⌬␻p
冋冉
ec n 3/2
e
26
␬ B f pk
冊 册
1/2
⫺5
3/2
,
共88兲
where f pk ⫽ ␻ p /(2 ␲ ) is the pinning frequency in cycles per
second. The largest quality factor Q⬇8 reported by Li et al.
was achieved for n e ⫽5.4⫻1010 cm⫺2 , B⫽13 T, where f pk
was measured to be 1.4 GHz. In Ref. 17 Q⬇45 was found
for comparable n e , B, and f pk . Our theoretical estimate
from Eq. 共88兲 is Q⬇260 and far exceeds both. We should
point out, however, that specific details of the experimental
setup become important for observability of such a high Q.
Henceforth we focus on the work of Li et al.,16 since most of
their data is available in the published record.
Because of the finite distance (30 ␮ m) between the plates
of the coplanar microwave waveguide 共see Fig. 10兲, these
were not the ideal q⫽0 measurements. We estimate that the
spread of q’s imposes an upper bound of about 30 for the
largest observable Q. The remaining discrepancy seems to be
rooted in finite-temperature and to a lesser degree finiteincident-power effects ignored in the derivation of Eq. 共88兲.
In the experiments, the linewidth was decreasing roughly
linearly as the temperature varied from T⫽200 mK to T
⫽50 mK with other parameters held fixed. Below 50 mK the
linewidth appeared to reach saturation, but only at the nomi-
7566
M. M. FOGLER AND DAVID A. HUSE
PRB 62
Our theory should literally apply to the pinning mode of a
WC formed by electrons on solid hydrogen 共see the second
book cited under Ref. 8兲. We hope that this paper will encourage further experiments on these or other numerous systems where the pinning manifests itself.
ACKNOWLEDGMENTS
FIG. 10. Top: schematics of the microwave setup used by Li
et al. The transmission line is shaded, the 2D electron 共or more
precisely, hole兲 gas is represented by the thick horizontal line underneath 共adapted from Ref. 16兲. Bottom: typical form of the electric field distribution in the plane of the WC 共Ref. 39兲. Solid line is
the signal in phase with the source, dash-dotted line is the signal in
quadrature.
nal level of incident microwave power (80 pW). When
lower power levels were used, the line continued sharpening
up. At the lowest experimental temperature of 25 mK, the
linewidth did not show any signs of saturation 共this time as a
function of power兲 even upon tenfold input power reduction,
which was near the sensitivity limit.
Several sources of the thermal broadening can be envisioned: 共a兲 thermally excited single-particle excitations of the
WC,12 共b兲 phonons of the host semiconductor, 共c兲 the anharmonisms of the collective modes of the pinned WC, and
perhaps some others.18 The classical activation energy of vacancies and interstitials is of the order of 0.5 K at the densities studied, and mechanism 共a兲 should be frozen out at
25 mK. Furthermore, for a WC of low density the bandwidth
of such excitations is very narrow and so they are easily
localized by the random potential. The weak electronphonon coupling in GaAs and small phonon phase space at
the frequencies involved ( ␻ p ) are likely to render the mechanism 共b兲 inefficient. The preliminary estimate of the broadening due to mechanism 共c兲 was given in the previous section. It is too small to explain the experimental observations.
We speculate that stronger anharmonic effects and a better
agreement with the experiments may be obtained if one
properly accounts for untypically large anharmonicities at
the locations of the soft modes, which, as we showed in this
paper, play a very important role in the response.
It is also quite possible that the actual pinning potential is
more complicated than the one studied here. In reality, it can
be due to a combination of the interface roughness18 and
dilute residual ions in the vicinity of the WC plane.40 In this
case ␻ p may be determined by the former, while Q could be
limited by the latter. Such more complicated models as well
as the dynamic response of a pinned quantum WC are interesting subjects awaiting further investigation.
It would also be interesting to investigate if the phenomena discussed appear in conventional charge-density waves
共see Sec. I兲. These materials are three dimensional, and some
important modifications of the present theory may be needed.
There are other complications such as an extra dissipation
due to uncondensed quasiparticles but those can be suppressed by lowering the temperature. In any case, it would be
remarkable if narrow pinning modes could be produced in
these materials simply by applying a sufficiently strong magnetic field.
This research is supported by U.S. Department of Energy
Grant No. DE-FG02-90ER40542 and NSF Grant No. DMR9802468. We thank R. Chitra, Mark Dykman, Lloyd Engel,
Herb Fertig, Gabi Kotliar, Chi-Chun Li, Leonid Pryadko,
Misha Raikh, Dan Tsui, and Valerii Vinokur for useful discussions. M.M.F. is grateful to Tito Williams for providng a
copy of Ref. 17.
APPENDIX A: MODEL CALCULATION
FOR L-L SCATTERING
In order to evaluate the importance of the L-L scattering
channel, let us consider the model action
A L共 ⑀ 兲 ⫽
1
2
冕
⫹
q
u L* 共 q兲关 H L0 共 q兲 ⫺ ⑀ 兴 u L 共 q兲
1
2
冕
r
共A1兲
u L* 共 r兲 S L 共 r兲 u L 共 r兲 ,
where S L is a local operator of a Gaussian white-noise type
with parameters
具 S L 典 ⫽⌸ 0 , 具 S L 共 r1 兲 S L 共 r2 兲 典 ⫽⌸ 20 ⫺V ⬙ 共 0 兲 ␦ 共 r1 ⫺r2 兲 .
Our objective is to calculate the disorder-averaged Green’s
function
D L 共 q, ⑀ 兲 ⬅
i
L2
冓
冕
Du L Du L* u L 共 q兲 u L* 共 q兲 e ⫺iA L ( ⑀ )
冕
Du L Du L* e
⫺iA L ( ⑀ )
冔
,
where an infinitesimal imaginary correction is assumed to be
included via ⑀ → ⑀ ⫹i0 to make the integrals convergent.
Using the ‘‘supersymmetry’’ technique,41 D L can be represented in the form
D L⫽
i
L2
冓冕
冔
D␾D␾* u L u L* e ⫺iA s [ ␾ ] ,
where ␾†i ⫽ 关 u L* v L* 兴 is a supervector, v L (q, ␻ ) is an auxiliary
fermionic field, and A s 关 ␾兴 is the sypersymmetric Euclidian
action
A s⫽
1
2
冕␾
q
†
共 q兲共 Y q⫹⌸ 0 ⫺ ⑀ 兲 ␾共 q兲
i
⫹ V ⬙共 0 兲
8
冕␾
r
关
†
共 r兲 ␾共 r兲兴 2 .
共A2兲
Note that the coefficient in front of the quartic term is imaginary, which makes the action non-Hermitian. The sign of the
imaginary part is such that excitations above the vacuum
DYNAMICAL RESPONSE OF A PINNED TWO- . . .
PRB 62
Im D L 共 q, ⑀ 兲 ⫽
FIG. 11. Diagrams for the 共a兲 coupling constant and 共b兲 mass
term.
state ␾⬅0 decay after traveling a certain distance. This distance is simply the phonon mean free path.
It is easy to establish some of the properties of the propagator D L right away. Since the field ␾ is massive for ⑀
ⱗ⌸ 0 at the tree level, the imaginary part of the propagator
D L is small at such ⑀ . However, Im D L should grow rapidly
as soon as ⑀ crosses the ⑀ ⫽⌸ 0 threshold. Below we are
going to verify this by explicit calculations. Although our
methods do not work in the immediate vicinity of ⑀ ⫽⌸ 0 , the
expressions obtained for ⑀ ⬍⌸ 0 and ⑀ ⬎⌸ 0 can be smoothly
matched onto each other.
Rescaling the fields in Eq. 共A2兲 by 冑2/Y , we obtain the
action with the Lagrangian
L⫽ ␾† K̂ ␾⫹
M †
g
␾ ␾⫺i 共 ␾† ␾兲 2 ,
2
4
共A3兲
where K̂ is the operator, which corresponds to multiplication
by 兩 q兩 in the momentum representation, M ⫽2(⌸ 0 ⫺ ⑀ )/Y
plays the role of mass, and g⬅⫺2V ⬙ (0)/Y 2 is the bare coupling constant. The coupling constant is dimensionless, and
it is possible to show that the field theory is renormalizable.
Let us derive the Wilson-style renormalization group 共RG兲
equations for the renormalized parameters g R and M R . As
usual, it is done by successive integrations over narrow momentum shells ⌳(l⫹ ␦ l)⬍q⬍⌳(l), where ⌳(l)⫽⌳ c e ⫺l
and ⌳ c ⬃1/a are the running and the bare high-momentum
cutoffs, respectively.
Consider the effect of such an integration on g R first.
Initially, g R remains small and we need to take into account
only the three diagrams of lowest order in g R 关see Fig. 11共a兲兴
to find
2
g R 共 l⫹ ␦ l 兲 ⫽g R 共 l 兲 ⫹ g R2
3
⫻
冉
冕
e ⫺ ␦ l ⌳⬍q⬍⌳
1
共 2 ␲ 兲 q⫹M R
冊
共A4兲
冕
e ⫺ ␦ l ⌳⬍q⬍⌳
d 2q
1
.
共 2 ␲ 兲 兩 q兩 ⫹M R
共A5兲
2
In addition to g R and M R , the kinetic term K̂ also gets
renormalized. We will neglect such an effect because it is of
higher 共second兲 order in g R . Consequently, the spectral
function Im D L is determined just by the mass,
共A6兲
In this equation M R ⫽M R (q, ⑀ ) stands for the renormalized
mass at such large negative l that the RG flow eventually
stops either because q⬃⌳(l) or because 兩 M R 兩 ⬃⌳(l). Below
we consider exclusively the q⫽0 case.
From Eq. 共A4兲 we find that in the limit 兩 qi 兩 , 兩 M R 兩 Ⰶ⌳ the
function ␤ ⬅ ⳵ g R / ⳵ l depends only on g R ,
␤⫽
1 2
g ,
␲ R
共A7兲
and so the RG flow equation is easy to solve:
g R⫽
␲
␲
, l L⬅ .
l L ⫺l
g
共A8兲
At l⬇l L ⫺ ␲ , g R becomes of the order of 1, and the theory
enters the strong-coupling regime. The corresponding spatial
scale is L L-L ⬃⌳ ⫺1
c exp(␲/g). Since the strength of coupling
maps to the strength of the effective disorder in the original
formulation, L L-L has the meaning of the L phonon localization length 共due to L-L scattering兲. Note that g R seems to
exhibit a divergence 共Landau pole兲 at l⫽l L . This is an artifact of the one-loop RG. We expect that instead g R flows
toward its fixed point value of the order of 1.
The other scaling equation, for M R , can be deduced from
Eq. 共A5兲,
␥M⬅
⳵MR
⌳ 2共 l 兲
g R共 l 兲
.
⫽⫺
⳵l
4 ␲ ⌳ 共 l 兲 ⫹M R ⫺i ␦
共A9兲
The solution cannot be expressed in elementary functions but
we can describe its asymptotics.
共1兲 (g/4␲ )⌳ c ⫽M Ⰶ 兩 M 兩 Ⰶ⌳ c . Here the renormalization
*
of the coupling constant is not important because the RG
flow is effectively cut off at l⬃ln(⌳c /兩M R兩) before g R manages to get large. Equation 共A9兲 reduces to the SCBA equation
M R ⫽M ⫺
2
where S⫽s2 ⫽(q1 ⫺q2 ) 2 , T⫽t2 ⫽(q1 ⫺q3 ) 2 , and U⫽u2
⫽(q1 ⫺q4 ) 2 are the Mandelstam variables with qi , i
⫽1, . . . ,4, being the incoming momenta. Similarly, the tadpole diagram shown in Fig. 11共b兲 determines the variation of
the renormalized mass M R ,
gR
2
Im M R
2
.
Y 共 q⫹Re M R 兲 2 ⫹ 共 Im M R 兲 2
d 2q
1
⫹ 共 s→t兲 ⫹ 共 s→u兲 ,
兩 q⫹s兩 ⫹M R
M R 共 l⫹ ␦ l 兲 ⯝M R 共 l 兲 ⫺
7567
g
4␲
冕
⌳c
⌳
dqq
,
q⫹M R ⫺i ␦
共A10兲
with the approximate solution
Re M R ⯝M ⫺M ,
共A11a兲
Im M R ⯝ 共 g/4 兲 ⌰ 共 ⫺Re M R 兲 Re M R ,
共A11b兲
*
where ⌰(x) is the Heaviside step function. A word of caution is in order here. The imaginary part of M R vanishes at
positive Re M R only at the level of the one-loop RG. Nonperturbative methods indicate that Im M R is nonzero albeit
exponentially small 共see below兲.
⫺1
Ⰶ 兩 M ⫺M 兩 ⰆM . Now instead of Eq. 共A11兲
共2兲 L L⫺L
*
*
we have
Im M R ⯝
1 ␲
⌰ 共 ⫺Re M R 兲 Re M R ,
4 l L ⫺l
*
共A12a兲
7568
M. M. FOGLER AND DAVID A. HUSE
Re M R ⯝
冉 冊
冉 冊
lL
l L ⫺l
1/4
共 M ⫺M 兲 ,
l ⫽ln ⫺
*
*
*
⌳c
.
Re M R
共A12b兲
共A12c兲
The above comment about small but nonzero Im M R at
Re M R ⬎0 applies here as well.
⫺1
. Since the one-loop RG cannot be
共3兲 兩 M ⫺M 兩 ⱗL L⫺L
*
trusted beyond the point l⫽l L ⫺ ␲ , we choose to terminate
the RG procedure at this stage. Unfortunately, this leaves us
with a theory that 共a兲 does not have any small parameters and
共b兲 includes not only the original quartic but also higherorder interaction terms generated by the RG. However, on
physical grounds we can expect that in this, essentially massless, limit, the mass scale must be set by the running cutoff
⌳(l L ), i.e.,
Im M R ⬃⌳ c g ⫺1/4 e ⫺ ␲ /g .
共A13兲
This can be compared with the solution of the SCBA equation 共A10兲 in the massless limit (Re M R ⫽0),
Im M RSCBA⬃⌳ c e ⫺4 ␲ /g .
共A14兲
Clearly, the SCBA misses the factor of 4 in the exponential.
As mentioned above, RG-enhanced perturbation theory
formulas 共A11b兲 and 共A12a兲 do not work at low energies.
Nonvanishing Im M R can be detected using the nonperturbative method of constrained instantons.39 The details of the
calculation will be presented elsewhere. Here we only quote
the result,
Im M R ⯝
4 ␲ e 冑6 ␲
9g R2
ln
冉
冊
⌳ cg R
⌳ e ⫺ ␲ /g ,
Re M R c
共A15兲
which is valid for Re M R Ⰷ1/L L-L , g R Ⰶ1. Equations 共A13兲
and 共A15兲 match for 兩 Re M R 兩 ⬃1/L L-L . Therefore, we expect
that the spectral function Im D L exhibits a quasi-Lorentzian
peak centered at ⑀ p ⫽Y M ⯝⌸ 0 of exponentially small
*
width
Y
⌬ ⑀ p ⫽ Im M R 共 0,⑀ p 兲 ⬃⌳ c Y g ⫺ ␻ e ⫺ ␲ /g .
2
共A16兲
APPENDIX B: DYNAMICAL RESPONSE WITHIN THE
SCBA
Within the SCBA the self-energy ⌸ ␣␤ ( ␻ ,q) is diagonal,
⌸ ␣␤ ( ␻ ,q)⫽ ␦ ␣␤ ⌸ and weakly q dependent for qaⰆ1. ⌸ is
to be found from the self-consistency equation 共37兲, which
we reproduce here for convenience,
⌸ 共 ␻ 兲 ⫽S 0 ⫹V ⬙ 共 0 兲
冕
k
tr D 共 k, ␻ 兲 .
共B1兲
As discussed in Secs. III and IV, the SCBA applies only at
relatively high frequencies. Moreover, in strong magnetic
fields and near the pinning frequency ␻ ⬃ ␻ p , it is not even
qualitatively correct. One may ask why we bother elaborating on this faulty approximation scheme here. The reason is
as follows. We discovered that as far as the dynamical re-
PRB 62
sponse is concerned, all alternative theories suggested so
far10,12,18,19 are merely different forms of the SCBA. Thus,
we deemed that it would be helpful to expose their interrelationship and correct a few calculational errors.
Let us briefly summarize the results obtained by previous
authors. Fukuyama and Lee10 concluded that ⌬ ␻ p ⬃ ␻ p but
did not present the details of the calculation. The perturbative analysis of Fertig42 yields an exponentially small
⌬ ␻ p / ␻ p 共for weak disorder兲. The Gaussian variational replica method of Chitra et al.19 reduces to the SCBA at nonzero
frequencies, and, when analyzed further, predicts a powerlaw dependence of ⌬ ␻ p on the disorder strength. Finally, the
SCBA analysis in the Appendix of Ref. 12 leads to yet another dependence. One reason for such a disarray of conflicting results is the extreme sensitivity of the solution ⌸( ␻ ) of
Eq. 共B1兲 to the precise value of the dimensionless parameter
C⬅⫺V ⬙ (0)/(4 ␲ ␮ ⌸ 0 ), where ⌸ 0 ⬅⌸(0). In principle, C is
fully determined by S 0 , but S 0 is known only approximately.
A strong partial cancellation of S 0 by the second term in Eq.
共B1兲 leaves us only the order of magnitude estimate C⬃1.
One could argue, as Fukuyama and Lee did in an analogous
situation,9 that C⫽1 is the ‘‘best’’ choice because others
lead to various physical absurdities. For example, if C⬎1,
then ⌸( ␻ ) acquires a finite imaginary part at complex ␻ ,
which means that the ground state is unstable; if C⬍1, then
⌸( ␻ ) has an imaginary part only above some threshold frequency ␻ th⬎0, i.e., the phonon spectrum has a gap, which
also appears to be unphysical. The truth is, of course, that the
SCBA is simply unable to correctly describe the properties
of the ␻ →0 tail; therefore, it cannot be relied upon for selecting a ‘‘good’’ value of C. A more sensible approach is to
investigate a certain range of C around unity, hoping that
certain quantities depend on C only weakly. Alas, the linewidth ⌬ ␻ p is not one of such quantities.
We found that the results of Chitra et al. are recovered for
the ‘‘Fukuyama-Lee choice,’’ C⫽1. 共Therefore, these two
groups of authors should have obtained the same results.兲 A
slight reduction of C from unity is sufficient to cross over to
the very different predictions of Fertig. Let us now demonstrate this in more detail.
Using the definition of C and Eq. 共B1兲, we obtain
⌸ 共 ␻ 兲 ⫺⌸ 0 ⫽2 ␮ C⌸ 0
冕
⬁
0
dkk 关 D T 共 k,0 兲 ⫺D T 共 k, ␻ 兲兴 .
共B2兲
Let us now describe the solution ⌸( ␻ ) of this equation for
different C in the weak-pinning regime R c Ⰷa, which corresponds to the inequality ␣ ⬅ 冑⌸ 0 ␮ /Y ⫽ ␮ /␭Ⰶ1.
共1兲 0⬍1⫺Cⱗ ␣ . In this case
˜ p 兲 2 ⫺ 共 1⫺C 兲 2 ,
⌸ 共 ␻ 兲 ⯝C⌸ 0 ⫹i⌸ 0 冑共 ␻ / ␻
共B3兲
˜ p ⫽ ␻ p /( ␲ ␣ ) 1/2 and ␻ p ⫽⌸ 0 / ␳␻ c . As one can see,
where ␻
˜ p . For C⫽1 Eq.
the threshold frequency is ␻ th⫽(1⫺C) ␻
共B3兲 coincides with that of Ref. 19. Strictly speaking, Eq.
共B3兲 describes the solution of Eq. 共B2兲 only for ␻ Ⰶ ␻ p .
Near the resonance ␻ ⬃ ␻ p , it is off by a numerical factor of
the order of unity. In principle, this factor can also be calculated. We limit ourselves to the order of magnitude estimate
Im ⌸( ␻ p )⬃ ␣ 1/2⌸ 0 , which leads to ⌬ ␻ p / ␻ p ⬃ ␣ 1/2
PRB 62
DYNAMICAL RESPONSE OF A PINNED TWO- . . .
⫽(␮/␭)1/2. Although this result is of the same form as Eq.
共66兲, with s⫽1/2, we believe that s⫽3/2 is correct 共see
Secs. IV and V兲.
共2兲 1⫺CⰇ ␣ . Equation 共B3兲 still applies as long as ␻ p0
⫺ ␻ is positive and not too small. At larger frequencies,
⌸( ␻ ) receives an additional contribution from the pole k
*
of D T in the complex plane of k, which moves close to the
real axis. The resonance linewidth ⌬ ␻ p is exponentially
small,
Im ⌸ 共 ␻ p 兲
2
⬃ ␻ p e ⫺F/4 ␣ ,
⌬ ␻ p⫽
␳␻ c
1⫽2 ␮ C⌸ 0
冕
⬁
0
7569
dkk 关 ⑀⑀ c ⫹ 共 ⌸⫹Y k 兲 2 兴
关共 ⌸⫹ ␮ k 2 兲共 ⌸⫹Y k 兲 ⫺ ⑀⑀ c 兴 2
.
共B5兲
At frequencies somewhat higher than ␻ p , Im ⌸( ␻ ) is dominated by the aforementioned pole at k ⫽( ⑀⑀ c ⫺⌸ 2 )/⌸Y
*
near the real axis, 兩 Re k 兩 Ⰶ 兩 Im k 兩 . Im ⌸ is given simply by
*12
*
the residue of this pole,
Im ⌸ 共 ␻ 兲 ⯝2 ␲ ␣ 2 C⌸ 0
共B4兲
冉 冊
␻2
␻ 2p
⫺1 .
共B6兲
where F⯝⌸( ␻ th)/(C⌸ 0 )⫺1. The threshold frequency ␻ th
is only slightly below ␻ p , by an amount of the order of
⌬ ␻ p . To be exact ␻ th can be found from the condition that
the solution of Eq. 共B2兲 is also the solution of the ‘‘derivative’’ of this equation with respect to ⌸,
In conclusion, we would like to reiterate that none of Eqs.
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system correctly. We derived them here just to facilitate the
comparison with the previous work on the subject. The correct expressions for Im ⌸ and ␴ xx are given in Sec. IV.
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W L,T (r) should not be confused with quantities B L,T (r) introduced in Ref. 21. For W L ⰆW T , B L ⯝B T ⯝W T2 /2.
28
Incidentally, this means that in the ␻ →0 limit the collectivemode contribution to the magnetotransport has the same form as
in the ‘‘Hall insulator,’’ i.e., vanishing ␴ xx but classical ␳ xy .
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pinned WC behaves as a Hall insulator just on the basis of the
phonon response studied here.
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Recently, H. Yi and H. A. Fertig 关Phys. Rev. B 61, 5311 共2000兲兴
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a quartic anharmonicity U(r)⫽ ␣ r 2 ⫺ ␤ r 4 . The coefficients ␣
and ␤ were taken to be identical for each lattice site. In such a
PRB 62
system the magnetophonon spectrum has a gap and the anharmonic broadening decays exponentially as T→0. In contrast,
here we showed that in a truly random system, where the phonon spectrum is gapless, the anharmonic effects are much stronger, persist even at zero temperature, and vary linearly with T.
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