We present zero-temperature simulations for the single-particle density of
states of the Coulomb glass. Our results in three dimensions are consistent
with the Efros and Shklovskii prediction for the density of states.
Finite-temperature Monte Carlo simulations show no sign of a thermodynamic
glass transition down to low temperatures, in disagreement with mean-field
theory. Furthermore, the random-displacement formulation of the model undergoes
a transition into a distorted Wigner crystal for a surprisingly broad range of
the disorder strength.Comment: 4 pages, 2 figures, 1 tabl

Allgood, Brandon A.

Blatter, Gianni

Katzgraber, Helmut G.

Surer, Brigitte

Zimanyi, Gergely T.

English

arXiv

Brigitte Surer

Helmut G. Katzgraber

Gergely T. Zimanyi

Brandon A. Allgood

Gianni Blatter

Crossref

Density of States and Critical Behavior of the Coulomb Glass

Texas A&M University

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Density of States and Critical Behavior of the Coulomb Glass
Brigitte Surer,1 Helmut G. Katzgraber,1, 2 Gergely T. Zimanyi,3 Brandon A. Allgood,4 and Gianni Blatter1
1Theoretische Physik, ETH Zurich, CH-8093 Zurich, Switzerland
2Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA
3Department of Physics, University of California, Davis, California 95616, USA
4Numerate Inc., 1150 Bayhill Drive, San Bruno, CA 94066, USA
We present zero-temperature simulations for the single-particle density of states of the Coulomb glass. Our
results in three dimensions are consistent with the Efros and Shklovskii prediction for the density of states.
Finite-temperature Monte Carlo simulations show no sign of a thermodynamic glass transition down to low
temperatures, in disagreement with mean-field theory. Furthermore, the random-displacement formulation of
the model undergoes a transition into a distorted Wigner crystal for a surprisingly broad range of the disorder
strength.
PACS numbers: 75.50.Lk, 75.40.Mg, 05.50.+q, 64.60.-i
The Coulomb glass (CG) is the earliest paradigm for under-
standing the effects of strong disorder in electronic systems
with long-ranged interactions. Among its applications are the
space-energy correlations in transistors, the magnetization-
switched metal-insulator transitions in tunnel devices, the
cotunneling magnetoresistance in ferromagnetic devices, the
ambipolar gate effect, the huge magnetoresistance in semi-
conductor stacks, and the transparent refractory oxides, to
name a few. In the CG, the Coulomb interaction remains
long ranged because the disorder localizes the electrons and
thus impedes screening. Therefore, the system forms its
low energy states by long-range configurational changes and
avalanches. After some early approaches [1, 2], Efros and
Shklovskii (ES) argued that the stability of the low-energy
states against the long-ranged single-particle dynamics re-
quires the formation of a soft “Coulomb gap” in the single-
particle density of states (DOS) of the form ρ(E) ∼ |E|δ
with δ = D − 1 as an upper bound and where the energy
E is measured from the Fermi level [3]; D being the space
dimension. This Coulomb gap leads to a typical variable
range hopping form of the low-temperature conductivity, i.e.,
σ(T ) = σ0 exp [−(T0/T )−1/2]; σ0 and T0 constant.
There is considerable experimental evidence in support
of these predictions from transport measurements of σ(T )
[4, 5, 6, 7], as well as from tunneling conductance measure-
ments of ρ(E) [8, 9, 10, 11]. However, subsequent theoreti-
cal considerations arrived at an exponential form of the DOS
by considering multi-electron “polaronic” processes: ρ(E) ∼
exp [−(E0/E)1/2] [12, 13], throwing the status of theoretical
predictions for the DOS in question.
A large number of nonequilibrium glassy phenomena have
been observed in disordered electronic systems. These in-
clude slow dynamics [14, 15, 16, 17], aging, and memory
effects [18, 19, 20], as well as changes in the noise spectrum
[21]. However, the existence of a thermodynamic glass tran-
sition cannot be directly surmised from glassy dynamics. In
fact, no well-defined thermodynamic glass transition has been
found in association with these phenomena to date in three-
dimensional (3D) systems.
These different theoretical predictions merged with the in-
sight of Pastor and Dobrosavljevic, who described a disor-
dered electron system with long-range interactions using the
replica-based theoretical framework of glass physics [22, 23].
Their work offered a unified platform to analyze both the DOS
and the glassy characteristics of these systems. This pro-
gramme was subsequently expanded by the work of Mu¨ller,
Ioffe and Pankov who included replica symmetry breaking
technology into their calculations [24, 25, 26]. All these
studies concluded that—within a mean-field approach—a soft
Coulomb gap exists in the single-particle DOS at T = 0. Fur-
thermore, for T ≤ Tc ∼ W−1/2, where W is a measure of
the disorder, the system freezes into a “Coulomb glass” state.
Note that the Coulomb glass is analogous to a spin glass in a
(random) field [27] which is known to not order.
The Coulomb glass has attracted considerable attention nu-
merically as well. The initial work by Davies, Lee and
Rice reported the observation of a soft gap, but the data
were not conclusive with respect to the detailed functional
form of the DOS [28]. Subsequent numerical studies rep-
resented the disorder either by random site energies (CG)
[29, 30, 31, 32, 33, 34] or by random displacements (RD) be-
tween the sites [35, 36, 37, 38, 39, 40]. While the CG and RD
models have different symmetries (and thus possibly different
universality classes [41]), it has nevertheless been argued that
they both adequately capture the key aspects of the real elec-
tronic system [38]. Different studies of the DOS in 3D have
reported a DOS vanishing at the Fermi level with ρ(E) ∼ |E|δ
with δ = 2.1 – 2.6 [30, 31]. Even more surprising was the
claim of a strongly disorder-dependent exponent δ [38]. Fur-
thermore, studies attempting to locate a transition to a glassy
state were only successful in the RD model [36, 38, 39, 40].
The state of the field can be summarized as follows: A
soft gap in the DOS has been widely confirmed, but the pre-
dicted ES exponent δ = D − 1 is consistent only with exper-
imental data, not with numerical simulations. A true finite-
temperature transition to a glassy state has numerical support
in the RD model but lacks evidence in the CG model and in
experiments.
Our results show that in the 3D CG model δ is close to 2 and
weakly disorder dependent [26] and we find no signature of
2a finite-temperature glass transition. Furthermore, in the RD
model the low-temperature ordering is indicative of a distorted
Wigner crystal.
Model and numerical details.— The Coulomb glass (CG)
Hamiltonian is given by [3]:
HCG = 1
2
∑
i6=j
(ni − ν) e
2
κrij
(nj − ν) +
∑
i
niεi, (1)
where ni ∈ {0, 1} is the electron number at site i, ν the fill-
ing factor, and e2/κrij the Coulomb repulsion. The sites lie
on a three-dimensional lattice of size N = L3, and the elec-
tron number is coupled to Gaussian-distributed random site
energies εi with zero mean and standard deviation W , i.e.,
P(εi) = (2piW 2)−1/2 exp(−ε2i /2W 2). In the RD model,
instead of random site energies, the disorder is represented
by Gaussian-distributed random displacements of the lattice
sites with standard deviation
√
3W . The DOS is given by
the disorder average of ρ(E) = (1/N)
∑
i δ(E − Ei) with
Ei =
∑
j 6=i(nj − ν)(e2/κrij) + εi the local single-particle
energy [3].
For the simulations we use particle-conserving dynamics
and periodic boundary conditions. To cope with the long-
range Coulomb interactions we perform a resummation tech-
nique in which we sum all interactions over periodic im-
ages and renormalize the energy scales such that the nearest-
neighbor distance is a = 1. To compute the ground-state
DOS (T = 0) we use extremal optimization [42]. For the
CG model we perform 219N updates and study systems of up
to N = 143 sites in 3D for W = 0.2 and 0.4 and average over
3000 disorder samples forL ≤ 12 and 1800 (800) samples for
L = 14 for W = 0.2 (W = 0.4). For the RD model we study
N = 143 sites and average over 100 disorder samples (fluc-
tuations are small). For the study at finite temperatures we
use exchange Monte Carlo [43, 44]. Equilibration is tested
by a logarithmic data binning. Once the last three bins agree
within errors, the system is in thermal equilibrium. Simula-
tion parameters can be found in Table I.
Results for the density of states.— Figure 1 (top and cen-
ter panels) show the DOS at T = 0 for the 3D CG model
for two disorder strengths close to the Fermi level (E = 0)
at half filling (ν = 1/2); the insets show the whole func-
tional shape. The data can be fit very well with a form∼ |E|δ
with δ = 2.01(2) (L = 14) for W = 0.2 and δ = 1.83(3)
(L = 14) for W = 0.4 (restricted to |E| ≤ 0.3), which is
close to the ES value of δ ≈ D − 1.
Fig. 1 (bottom) shows the DOS of the RD model for L =
14. The DOS shows a pronounced double-peak, the width of
the peaks dependent on W . There is no sign of the character-
istic Coulomb gap shape, moreover the peaks at |E| ∼ 1 are
typical of a Wigner crystal (WC). Thus the DOS of the RD
model is indicative of the formation of a moderately-distorted
WC at T = 0.
Results at finite temperature.— At half filling (ν = 1/2)
the ground state of the clean system (W = 0) is a WC with
a bipartite charge pattern. For a WC the DOS is expected to
FIG. 1: (Color online) Top: DOS for the 3D CG model for W =
0.20. The data are well fit by ρ(E) ∼ Eδ , δ ≈ 2 (dashed lines are
a guide to the eye) around the Fermi level. Center: Same as in the
top panel for W = 0.40. The insets show the full DOS. Both panels
have the same horizontal range. Bottom: DOS for the 3D RD model.
For all W studied the data show a bimodal structure with peaks at
|E| ∼ 1 and a hard gap of size ∼ 2, in stark contrast to the CG
model.
3TABLE I: Top: Simulation parameters for the simulations of the
3D Coulomb glass model with Gaussian disorder of strength W at
finite temperature. L is the system size, Nsa is the number of disor-
der samples, Nsw is the number of equilibration sweeps, Tmin is the
lowest temperature, Tmax = 0.455 the highest temperature and Nr
the number temperatures used in the exchange Monte Carlo method.
Temperatures are measured in units of e2/κa, a = 1 being the lattice
constant. Bottom: Parameters for the 3D RD model simulations.
W L Nsa Nsw Tmin Nr
0.20 6 4 290 218 0.030 27
0.20 10 388 218 0.030 27
0.20 14 251 218 0.083 17
0.40 6 4 955 218 0.030 27
0.40 10 148 218 0.030 27
0.40 14 98 218 0.083 17
W L Nsa Nsw Tmin Nr
0.10 8 173 220 0.030 27
0.20 8 133 220 0.030 27
0.40 8 93 220 0.030 27
0.80 8 143 2
20
0.030 27
be two delta functions, separated by a charge gap EWC. The
energy required to move a particle from a site on the occupied
sublattice to a site on the unoccupied sublattice isEWC ≈ 2 in
units of e2/κa. Since the peaks of the DOS of the CG are ap-
proximately centered around |E| ∼ 1 (Fig. 1, inset), it needs
to be verified that the observed DOS is indeed representative
of a glassy phase and not only that of a distorted WC. There-
fore we study the nature of the phase at finite T by computing
both an order parameter for a glassy state,
qGL =
4
N
N∑
i=1
(nαi − 1/2)(nβi − 1/2), (2)
and an order parameter for the competing Wigner crystal
mWC =
2
N
N∑
i=1
(−1)i(ni − 1/2). (3)
In Eq. (2) α and β refer to two copies of the system with
the same disorder [45]. If the system forms a Wigner crys-
tal we expect [〈mWC〉]av → 1 for T ≤ Tc, whereas if the
system freezes into a glass we expect [〈qGL〉]av → 1 and
[〈mWC〉]av → 0 for T → 0. Here [· · · ]av denotes the average
over disorder and 〈· · · 〉 is a thermal average.
To locate the putative glass transition we compute the two-
point finite-size correlation length [46] given by
ξGL(L, T ) =
1
2 sin(|kmin|/2)
[
χ(0)
χ(kmin)
− 1
]1/2
, (4)
where kmin = (2pi/L, 0, 0) is the smallest nonzero wave vec-
tor and χ(k) is the Fourier transform of the susceptibility
χ = [〈q2
GL
〉 − 〈qGL〉2]av. We use four replicas to compute
χ to avoid biases. Because ξ/L ∼ X [L1/ν(T − Tc)], a phase
transition at Tc is signaled by the correlation lengths for dif-
ferent L’s crossing at the same T = Tc.
Figure 2 (top panel) shows the q2
GL
(T ) and m2
WC
(T ) or-
der parameters as a function of temperature for different dis-
order strengths in the CG model. The glass order parameter
FIG. 2: (Color online) Top: Wigner crystal order parameter
[〈m2WC〉]av and glass order parameter [〈q2GL〉]av as a function of tem-
perature T for different disorder strengths W for the CG model. In
all cases [〈m2WC〉]av ≪ [〈q2GL〉]av . Center: Finite-size correlation
length as a function of T for different disorder strengths and sys-
tem sizes for the CG model. The data show no crossing, i.e., the
absence of a thermodynamic transition for the studied temperature
range. Bottom: Wigner crystal order parameter for the RD model
(L = 8) as a function of temperature for different W . For T . 0.1,
which quantitatively agrees with the critical temperatures estimated
in Ref. [38], crystalline order emerges.
4increases as the T → 0, whereas the Wigner crystal order pa-
rameter does not exhibit any ordering tendency, mWC(T ) re-
maining ∼ 40 (140) times smaller than qGL(T ) for W = 0.2
(W = 0.4) at T = 0.08. Figure 2 (center panel) shows
the correlation length for the glass order parameter as a func-
tion of T for the CG model. The data do not cross for the
studied temperatures and thus there is no sign of a transi-
tion for T ≥ 0.03, disagreeing with mean-field predictions
[25, 26]. The lack of a transition is mirrored by the small cor-
relation length and the proximity to the ground-state energy
(not shown).
In Fig. 2 (bottom panel) we show m2
WC
for the RD model
for disorder strengths up to W = 0.8 covering the disorder
range studied in Ref. [38]. For all W studied, m2
WC
rises
noticeably (in contrast to the CG model). This further under-
lines that—for the studied disorder range—the phase transi-
tion in the RD model occurs into a surprisingly robust dis-
torted Wigner crystal phase.
Conclusions.— We have analyzed the Coulomb glass at
low and zero temperature and find that the gap exponent of the
density of states is close to δ ≈ D−1 in 3D systems. Further-
more, we find no evidence of a finite-temperature transition
into a CG phase in 3D forW = 0.2 and 0.4. This suggests that
the CG in 3D is at or below its lower critical dimension, which
would explain the discrepancy with the mean-field results pre-
dicting a finite transition temperature. Finally, we have shown
that in a broad disorder range the random-displacement ver-
sion of the CG model orders into a distorted Wigner crystal
and not into a glassy state.
We thank S. Boettcher, V. Dobrosavljevic, E. Grannan,
N. Jensen, M. Mu¨ller, M. Palassini, S. Pankov, D. Popovic,
V. Oganesyan, M. Schechter, D. Sherrington, B. Shklovskii,
T. Vojta, and A. P. Young for discussions. The simulations
have been done on the ETH Zu¨rich clusters. H.G.K. acknowl-
edges support from the SNF under Grant No. PP002-114713.
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Physical Review Letters

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arXiv:0805.4640v2  [cond-mat.dis-nn]  12 Feb 2009Density of States and Critical Behavior of the Coulomb GlassBrigitte Surer,1 Helmut G. Katzgraber,1, 2 Gergely T. Zimanyi,3 Brandon A. Allgood,4 and Gianni Blatter11Theoretische Physik, ETH Zurich, CH-8093 Zurich, Switzerland2Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA3Department of Physics, University of California, Davis, California 95616, USA4Numerate Inc., 1150 Bayhill Drive, San Bruno, CA 94066, USAWe present zero-temperature simulations for the single-particle density of states of the Coulomb glass. Ourresults in three dimensions are consistent with the Efros and Shklovskii prediction for the density of states.Finite-temperature Monte Carlo simulations show no sign of a thermodynamic glass transition down to lowtemperatures, in disagreement with mean-field theory. Furthermore, the random-displacement formulation ofthe model undergoes a transition into a distorted Wigner crystal for a surprisingly broad range of the disorderstrength.PACS numbers: 75.50.Lk, 75.40.Mg, 05.50.+q, 64.60.-iThe Coulomb glass (CG) is the earliest paradigm for under-standing the effects of strong disorder in electronic systemswith long-ranged interactions. Among its applications are thespace-energy correlations in transistors, the magnetization-switched metal-insulator transitions in tunnel devices, thecotunneling magnetoresistance in ferromagnetic devices, theambipolar gate effect, the huge magnetoresistance in semi-conductor stacks, and the transparent refractory oxides, toname a few. In the CG, the Coulomb interaction remainslong ranged because the disorder localizes the electrons andthus impedes screening. Therefore, the system forms itslow energy states by long-range configurational changes andavalanches. After some early approaches [1, 2], Efros andShklovskii (ES) argued that the stability of the low-energystates against the long-ranged single-particle dynamics re-quires the formation of a soft “Coulomb gap” in the single-particle density of states (DOS) of the form ρ(E) ∼ |E|δwith δ = D − 1 as an upper bound and where the energyE is measured from the Fermi level [3]; D being the spacedimension. This Coulomb gap leads to a typical variablerange hopping form of the low-temperature conductivity, i.e.,σ(T ) = σ0 exp [−(T0/T )−1/2]; σ0 and T0 constant.There is considerable experimental evidence in supportof these predictions from transport measurements of σ(T )[4, 5, 6, 7], as well as from tunneling conductance measure-ments of ρ(E) [8, 9, 10, 11]. However, subsequent theoreti-cal considerations arrived at an exponential form of the DOSby considering multi-electron “polaronic” processes: ρ(E) ∼exp [−(E0/E)1/2] [12, 13], throwing the status of theoreticalpredictions for the DOS in question.A large number of nonequilibrium glassy phenomena havebeen observed in disordered electronic systems. These in-clude slow dynamics [14, 15, 16, 17], aging, and memoryeffects [18, 19, 20], as well as changes in the noise spectrum[21]. However, the existence of a thermodynamic glass tran-sition cannot be directly surmised from glassy dynamics. Infact, no well-defined thermodynamic glass transition has beenfound in association with these phenomena to date in three-dimensional (3D) systems.These different theoretical predictions merged with the in-sight of Pastor and Dobrosavljevic, who described a disor-dered electron system with long-range interactions using thereplica-based theoretical framework of glass physics [22, 23].Their work offered a unified platform to analyze both the DOSand the glassy characteristics of these systems. This pro-gramme was subsequently expanded by the work of Mu¨ller,Ioffe and Pankov who included replica symmetry breakingtechnology into their calculations [24, 25, 26]. All thesestudies concluded that—within a mean-field approach—a softCoulomb gap exists in the single-particle DOS at T = 0. Fur-thermore, for T ≤ Tc ∼ W−1/2, where W is a measure ofthe disorder, the system freezes into a “Coulomb glass” state.Note that the Coulomb glass is analogous to a spin glass in a(random) field [27] which is known to not order.The Coulomb glass has attracted considerable attention nu-merically as well. The initial work by Davies, Lee andRice reported the observation of a soft gap, but the datawere not conclusive with respect to the detailed functionalform of the DOS [28]. Subsequent numerical studies rep-resented the disorder either by random site energies (CG)[29, 30, 31, 32, 33, 34] or by random displacements (RD) be-tween the sites [35, 36, 37, 38, 39, 40]. While the CG and RDmodels have different symmetries (and thus possibly differentuniversality classes [41]), it has nevertheless been argued thatthey both adequately capture the key aspects of the real elec-tronic system [38]. Different studies of the DOS in 3D havereported a DOS vanishing at the Fermi level with ρ(E) ∼ |E|δwith δ = 2.1 – 2.6 [30, 31]. Even more surprising was theclaim of a strongly disorder-dependent exponent δ [38]. Fur-thermore, studies attempting to locate a transition to a glassystate were only successful in the RD model [36, 38, 39, 40].The state of the field can be summarized as follows: Asoft gap in the DOS has been widely confirmed, but the pre-dicted ES exponent δ = D − 1 is consistent only with exper-imental data, not with numerical simulations. A true finite-temperature transition to a glassy state has numerical supportin the RD model but lacks evidence in the CG model and inexperiments.Our results show that in the 3D CG model δ is close to 2 andweakly disorder dependent [26] and we find no signature of2a finite-temperature glass transition. Furthermore, in the RDmodel the low-temperature ordering is indicative of a distortedWigner crystal.Model and numerical details.— The Coulomb glass (CG)Hamiltonian is given by [3]:HCG = 12∑i6=j(ni − ν) e2κrij(nj − ν) +∑iniεi, (1)where ni ∈ {0, 1} is the electron number at site i, ν the fill-ing factor, and e2/κrij the Coulomb repulsion. The sites lieon a three-dimensional lattice of size N = L3, and the elec-tron number is coupled to Gaussian-distributed random siteenergies εi with zero mean and standard deviation W , i.e.,P(εi) = (2piW 2)−1/2 exp(−ε2i /2W 2). In the RD model,instead of random site energies, the disorder is representedby Gaussian-distributed random displacements of the latticesites with standard deviation√3W . The DOS is given bythe disorder average of ρ(E) = (1/N)∑i δ(E − Ei) withEi =∑j 6=i(nj − ν)(e2/κrij) + εi the local single-particleenergy [3].For the simulations we use particle-conserving dynamicsand periodic boundary conditions. To cope with the long-range Coulomb interactions we perform a resummation tech-nique in which we sum all interactions over periodic im-ages and renormalize the energy scales such that the nearest-neighbor distance is a = 1. To compute the ground-stateDOS (T = 0) we use extremal optimization [42]. For theCG model we perform 219N updates and study systems of upto N = 143 sites in 3D for W = 0.2 and 0.4 and average over3000 disorder samples forL ≤ 12 and 1800 (800) samples forL = 14 for W = 0.2 (W = 0.4). For the RD model we studyN = 143 sites and average over 100 disorder samples (fluc-tuations are small). For the study at finite temperatures weuse exchange Monte Carlo [43, 44]. Equilibration is testedby a logarithmic data binning. Once the last three bins agreewithin errors, the system is in thermal equilibrium. Simula-tion parameters can be found in Table I.Results for the density of states.— Figure 1 (top and cen-ter panels) show the DOS at T = 0 for the 3D CG modelfor two disorder strengths close to the Fermi level (E = 0)at half filling (ν = 1/2); the insets show the whole func-tional shape. The data can be fit very well with a form∼ |E|δwith δ = 2.01(2) (L = 14) for W = 0.2 and δ = 1.83(3)(L = 14) for W = 0.4 (restricted to |E| ≤ 0.3), which isclose to the ES value of δ ≈ D − 1.Fig. 1 (bottom) shows the DOS of the RD model for L =14. The DOS shows a pronounced double-peak, the width ofthe peaks dependent on W . There is no sign of the character-istic Coulomb gap shape, moreover the peaks at |E| ∼ 1 aretypical of a Wigner crystal (WC). Thus the DOS of the RDmodel is indicative of the formation of a moderately-distortedWC at T = 0.Results at finite temperature.— At half filling (ν = 1/2)the ground state of the clean system (W = 0) is a WC witha bipartite charge pattern. For a WC the DOS is expected toFIG. 1: (Color online) Top: DOS for the 3D CG model for W =0.20. The data are well fit by ρ(E) ∼ Eδ , δ ≈ 2 (dashed lines area guide to the eye) around the Fermi level. Center: Same as in thetop panel for W = 0.40. The insets show the full DOS. Both panelshave the same horizontal range. Bottom: DOS for the 3D RD model.For all W studied the data show a bimodal structure with peaks at|E| ∼ 1 and a hard gap of size ∼ 2, in stark contrast to the CGmodel.3TABLE I: Top: Simulation parameters for the simulations of the3D Coulomb glass model with Gaussian disorder of strength W atfinite temperature. L is the system size, Nsa is the number of disor-der samples, Nsw is the number of equilibration sweeps, Tmin is thelowest temperature, Tmax = 0.455 the highest temperature and Nrthe number temperatures used in the exchange Monte Carlo method.Temperatures are measured in units of e2/κa, a = 1 being the latticeconstant. Bottom: Parameters for the 3D RD model simulations.W L Nsa Nsw Tmin Nr0.20 6 4 290 218 0.030 270.20 10 388 218 0.030 270.20 14 251 218 0.083 170.40 6 4 955 218 0.030 270.40 10 148 218 0.030 270.40 14 98 218 0.083 17W L Nsa Nsw Tmin Nr0.10 8 173 220 0.030 270.20 8 133 220 0.030 270.40 8 93 220 0.030 270.80 8 143 2200.030 27be two delta functions, separated by a charge gap EWC. Theenergy required to move a particle from a site on the occupiedsublattice to a site on the unoccupied sublattice isEWC ≈ 2 inunits of e2/κa. Since the peaks of the DOS of the CG are ap-proximately centered around |E| ∼ 1 (Fig. 1, inset), it needsto be verified that the observed DOS is indeed representativeof a glassy phase and not only that of a distorted WC. There-fore we study the nature of the phase at finite T by computingboth an order parameter for a glassy state,qGL =4NN∑i=1(nαi − 1/2)(nβi − 1/2), (2)and an order parameter for the competing Wigner crystalmWC =2NN∑i=1(−1)i(ni − 1/2). (3)In Eq. (2) α and β refer to two copies of the system withthe same disorder [45]. If the system forms a Wigner crys-tal we expect [〈mWC〉]av → 1 for T ≤ Tc, whereas if thesystem freezes into a glass we expect [〈qGL〉]av → 1 and[〈mWC〉]av → 0 for T → 0. Here [· · · ]av denotes the averageover disorder and 〈· · · 〉 is a thermal average.To locate the putative glass transition we compute the two-point finite-size correlation length [46] given byξGL(L, T ) =12 sin(|kmin|/2)[χ(0)χ(kmin)− 1]1/2, (4)where kmin = (2pi/L, 0, 0) is the smallest nonzero wave vec-tor and χ(k) is the Fourier transform of the susceptibilityχ = [〈q2GL〉 − 〈qGL〉2]av. We use four replicas to computeχ to avoid biases. Because ξ/L ∼ X [L1/ν(T − Tc)], a phasetransition at Tc is signaled by the correlation lengths for dif-ferent L’s crossing at the same T = Tc.Figure 2 (top panel) shows the q2GL(T ) and m2WC(T ) or-der parameters as a function of temperature for different dis-order strengths in the CG model. The glass order parameterFIG. 2: (Color online) Top: Wigner crystal order parameter[〈m2WC〉]av and glass order parameter [〈q2GL〉]av as a function of tem-perature T for different disorder strengths W for the CG model. Inall cases [〈m2WC〉]av ≪ [〈q2GL〉]av . Center: Finite-size correlationlength as a function of T for different disorder strengths and sys-tem sizes for the CG model. The data show no crossing, i.e., theabsence of a thermodynamic transition for the studied temperaturerange. Bottom: Wigner crystal order parameter for the RD model(L = 8) as a function of temperature for different W . For T . 0.1,which quantitatively agrees with the critical temperatures estimatedin Ref. [38], crystalline order emerges.4increases as the T → 0, whereas the Wigner crystal order pa-rameter does not exhibit any ordering tendency, mWC(T ) re-maining ∼ 40 (140) times smaller than qGL(T ) for W = 0.2(W = 0.4) at T = 0.08. Figure 2 (center panel) showsthe correlation length for the glass order parameter as a func-tion of T for the CG model. The data do not cross for thestudied temperatures and thus there is no sign of a transi-tion for T ≥ 0.03, disagreeing with mean-field predictions[25, 26]. The lack of a transition is mirrored by the small cor-relation length and the proximity to the ground-state energy(not shown).In Fig. 2 (bottom panel) we show m2WCfor the RD modelfor disorder strengths up to W = 0.8 covering the disorderrange studied in Ref. [38]. For all W studied, m2WCrisesnoticeably (in contrast to the CG model). This further under-lines that—for the studied disorder range—the phase transi-tion in the RD model occurs into a surprisingly robust dis-torted Wigner crystal phase.Conclusions.— We have analyzed the Coulomb glass atlow and zero temperature and find that the gap exponent of thedensity of states is close to δ ≈ D−1 in 3D systems. 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Density of States and Critical Behavior of the Coulomb Glass

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