Possible experimental manifestations of the many-body localization
Possible experimental manifestations of the many-body localization
D. M. Basko,1, ∗ I. L. Aleiner,1 and B. L. Altshuler1, 2
Physics Department, Columbia University, New York, NY 10027, USA
NEC-Laboratories America, Inc., 4 Independence Way, Princeton, NJ 085540 USA
Recently, it was predicted that if all one-electron states in a non-interacting disordered system are
localized, the interaction between electrons in the absence of coupling to phonons leads to a finite-
temperature metal-insulator transition. Here, we show that even in the presence of a weak coupling
to phonons the transition manifests itself (i) in the nonlinear conduction, leading to a bistable I-V
curve, (ii) by a dramatic enhancement of the nonequilibrium current noise near the transition.
Introduction.— Low-temperature charge transport in
disordered conductors is governed by the interplay be-
tween elastic scattering of electrons off static disorder
(impurities) and inelastic scattering (electron-electron,
electron-phonon, etc.). At low dimensions an arbitrarily
weak disorder localizes [1] all single-electron states [2, 3],
and there would be no transport without inelastic pro-
cesses. For the electron-phonon scattering, the dc con-
ductivity σ(T ) at low temperatures T is known since long
ago [4]: in d dimensions
lnσ(T ) ∝ −1/T γ, γ = 1/(d+ 1) . (1)
What happens if the only possible inelastic process is
electron-electron scattering? The answer to this ques-
tion was found only recently [5]: σ(T ) = 0 identically for
T < Tc, the temperature of a metal-insulator transition.
Here we discuss experimental manifestations of this tran-
sition in real systems, where both electron-electron and
electron-phonon interactions are present. We show that
(i) the I-V characteristic exhibits a bistable region, and
(ii) non-equilibrium current noise is enhanced near Tc.
The notion of localization was originally introduced
for a single quantum particle in a random potential [1].
Subsequently, the concept of Anderson localization was
shown to manifest itself in a broad variety of phenomena
in quantum physics. This concept can also be extended to
many-particle systems. Statistical physics of many-body
systems is based on the microcanonical distribution, i. e.,
all states with a given energy are assumed to be realized
with equal probabilities. This assumption means delocal-
ization in the space of possible states of the system. It
does not hold for non-interacting particles; however, it is
commonly believed that an arbitrarily weak interaction
between the particles eventually equilibrates the system
and establishes the microcanonical distribution.
Many-body dynamics of interacting systems and its re-
lation to Anderson localization has been discussed in the
context of nuclear [6] and molecular [7] physics. For in-
teracting electrons in a chaotic quantum dot this issue
was raised in Ref. [8], where it was shown that electron-
electron interaction may not be able to equilibrate the
system. This corresponds to Anderson localization in the
many-body space. Recently it was demonstrated that
in an infinite low-dimensional system of (weakly) inter-
Tc δζ/λ δζ/λ
σ(T )
Insulator
Metal
FIG. 1: Schematic temperature dependence of the dc con-
ductivity σ(T ) for electrons subject to a disorder potential
localizing all single particle eigenstates, in the presence of
weak short-range electron-electron λδζ , λ ≪ 1, established in
Ref. [5]. Below the point of the many-body metal-insulator
transition, T < Tc ∼ δζ/|λ lnλ|, no inelastic relaxation occurs
and σ(T ) = 0. At T ≫ δζ/λ
2 the high-temperature metallic
perturbation theory[9] is valid, and corrections to the Drude
conductivity σ∞ are small. In the interval δζ/λ ≪ T ≪ δζ/λ
electron-electron interaction leads to electron transitions be-
tween localized states, and the conductivity depends on tem-
perature as a power-law.
acting electrons, subject to a static disorder, Anderson
transition in the many-body space manifests itself as a
finite-temperature metal-insulator transition [5].
Let single-particle eigenstates be localized on a spatial
scale ζloc (localization length). The characteristic energy
scale of the problem is the level spacing within the lo-
calization volume: δζ = 1/(νζ
loc), ν being the density of
states per unit volume. Below we neglect energy depen-
dences of ν, ζloc, and δζ . According to Ref. [5], as long
as electrons are not coupled to any external bath (such
as phonons), a weak short-range electron-electron inter-
action (of typical magnitude λδζ , with the dimensionless
coupling constant λ <∼ 1) does not cause inelastic relax-
ation unless the temperature T exceeds a critical value:
|λ lnλ|
. (2)
http://arxiv.org/abs/0704.1479v2
The small denominator |λ ln λ| represents the character-
istic matrix element of the creation of an electron-hole
pair. The ratio Tc/δζ is the number of states available
for such a pair (in other words, the phase volume) at
T = Tc. Only provided that this number is large enough
to compensate the smallness of the matrix element, the
interaction delocalizes the many-body states and thus
leads to an irreversible dynamics. As a result, the finite-
temperature dc conductivity σ(T ) vanishes identically if
T < Tc, while σ(T > Tc) is finite, i. e. at T = Tc a metal-
insulator transition occurs. The overall dependence σ(T )
is summarized in Fig. 1.
In any real system the electron-phonon interaction is
always finite. This makes σ(T ) finite even at T < Tc:
σ(T ) is either exponentially small [Eq. (1)] at T ≪ δζ ,
or follows a power-law [10] at δζ ≪ T ≪ Tc. At the
transition point the phonon-induced conductivity is not
exponentially small, i. e. phonons smear the transition
into a crossover. Are there any experimental signatures
of the many-body localization? In what follows we show
that if the electron-phonon coupling is weak enough, a
qualitative signature of the metal-insulator transition can
be identified in the nonlinear conduction. Namely, in a
certain interval of applied electric fields E and phonon
temperatures Tph both metallic and insulating states of
the system turn out to be stable! As a result, the I-
V curve exhibits an S-shaped bistable region (Fig. 3).
Moreover, we show that the many-body character of
the electron conduction dramatically modifies the non-
equilibrium noise near the transition [Eq. (12)].
Bistable I-V curve.— Our arguments are based on two
observations. First, in the absence of phonons a weak but
finite electric field cannot destroy the insulating state – it
rather shifts the transition temperature. Let us neglect
the effect of the field on the single-particle wave functions,
representing it as a tilt of the local chemical potential of
electrons. Then at T = 0 the role of the field in the insu-
lating regime is increase the energy of the electron-hole
(e-h) excitation of a size L by eEL. This provides in addi-
tional phase volume of the order of eEL/δζ. However, for
L > ζloc the matrix element for creation of such an ex-
citation quickly vanishes. In the diagrammatic language
for the effective model of Ref. [5] this means that each
electron-electron interaction vertex must be accompanied
by tunneling vertices which describe coupling between lo-
calization volumes and whose number is (i) at least one
in order to gain phase volume (in contrast to the finite-T
case when tunneling had to be included only to overcome
the finiteness of the phase space in a single grain [5]), and
(ii) not much greater than one, otherwise the diagram is
exponentially small. As a result, at T = 0 the insulator
state is stable provided that E < Ec ∼ Tc/(eζloc).
In the same way one can analyze the finite-temperature
correction to the critical field, and the finite-field correc-
tion to the critical temperature can be found by taking
into account the extra phase volume in the calculation of
TelTph Tc(E)
(Tel − Tph)/eE ,
Lph(Tel)
FIG. 2: Sketch of the dependences (Tel − Tph)/(eE) (dashed
line) and Lph(Tel) (solid lines, electron-phonon coupling
strength being weaker for higher curves). Actual value of Tel
is determined by their crossing.
Ref. [5]. One obtains σ(T ) = 0 for T < Tc(E), where
Tc(E) = Tc − c1eEζloc . (3)
with a model-dependent factor c1 ∼ 1, weakly dependent
on E [here and below Tc without the argument E is the
zero-field value given by Eq. (2)]. As a consequence, at
T < Tc(E) the nonlinear transport, as well as the linear
one, has to be phonon-assisted.
The second observation is that when both σ and E
are finite, there is Joule heating. The thermal balance is
qualitatively different in the insulating and the metallic
phases. Deep in the insulating phase (T ≪ Tc) each
electron transition is accompanied by a phonon emis-
sion/absorption, i. e. electrons are always in equilibrium
with phonons whose temperature Tph ≪ Tc we assume
to be fixed. On the contrary, in the metallic phase elec-
trons gain energy when drifting in the electric field, i. e.
they are heated. Due to this Joule heating the effective
electron temperature Tel deviates from the bath temper-
ature. The role of phonons is then to stabilize Tel. For
weak electron-phonon coupling Tel and Tph can differ sig-
nificantly. A self-consistent estimate for Tel follows from
Tel − Tph ∼ eELph(Tel) , (4)
Lph(Tel) =
D(Tel) τph(Tel) . (5)
Here τph(Tel) is the time it takes an electron to emit
or absorb a phonon, Lph(Tel) is the typical electron dis-
placement during this time, and D(Tel) = σ(Tel)/(e
is the electron diffusion coefficient.
We sketch in Fig. 2 (Tel − Tph)/(eE) and Lph(Tel)
for different electron-phonon coupling strengths as func-
tions of Tel. It is taken into account that (i) Lph coin-
cides with variable range hopping length at Tel ≪ δζ ,
(ii) Lph ∼ ζloc at δζ ≪ Tel ≪ Tc(E), (iii) D(Tel)
quickly rises to its large metallic value D∞ ∼ δζζ2loc near
eEζloc
∼ Tc∼ (ζloc/L
ph)Tc
eEζloc
FIG. 3: (a) sketch of the bistable I-V curve for a fixed value
of Tph; (b) (E , Tph)-plane with the bistable region schemat-
ically shown by shading. The dashed line represents the
crossover between the metallic state at high electric field E
or high phonon temperature Tph, and the insulating state at
low E and low Tph.
Tc(E), (iv) τph(Tel) decreases as a power law with in-
creasing Tel. The peak of the curve rises with decreas-
ing electron-phonon coupling strength, and eventually
the curve crosses the straight line. After that, in ad-
dition to Tel = Tph, Eq. (4) acquires two more solutions,
both with Tel > Tc(E), of which only the rightmost so-
lution is stable. The maximum of Lph can be estimated
as L∗ph ∼
D∞τph(Tc), so three solutions appear when
Tc(E) − Tph ≪ eEL∗ph. At the same time, as seen from
Eq. (3), electric field is unable to break down the insu-
lator as long as eEζloc ≪ Tc − Tph. Thus, the interval
of electric fields where both regimes are stable, is deter-
mined by
Tc(E)− Tph
eL∗ph
≪ E ≪ Tc − Tph
eζloc
. (6)
The two conditions are compatible provided that
L∗ph ≫ ζloc , (7)
which is realistic when electron-phonon coupling is weak.
In the bistable region (6), for a given value of E one
finds two stable solutions for Tel, giving two possible
values of the conductivity and the current, which corre-
sponds to an S-shape current-voltage characteristic [11],
the third (unstable) solution corresponding to the neg-
ative differential conductivity branch. The macroscopic
consequences of such behavior depend on the dimension-
ality. In a 2d sample the two phases of different electronic
temperature and current density can coexist, separated
by a boundary of the width ∼ L∗ph, parallel to the di-
rection of the electric field. The particular state of the
system determined by the boundary conditions (proper-
ties of the external circuit), as well as by the history.
Noise enhancement.— In the vicinity of the criti-
cal point conduction is dominated by correlated many-
electron transitions (electronic cascades). Each cascade
is triggered by a single phonon. As Tel → Tc(E), the typ-
ical value n̄ of the number n of electrons in the cascade
diverges together with the time duration of a cascade.
The results of Ref. [5], adapted for a finite electric field,
give the following probability for an n-electron transition
to go with the rate Γ:
Pn(Γ) =
e−Γ̄n/(4Γ)
, Γ̄n ∝
Tel + c1eEζloc
which gives
1/n̄ ∼ ln[Tc/(Tel + c1eEζloc)]. (9)
The divergence in n̄ is cut off when electron-phonon cou-
pling is finite. The largest n̄ is such that the phonon
broadening of the single-electron levels, 1/τph(Tc), is
comparable to n̄-particle level spacing (in other words,
time duration of a cascade cannot exceed τph):
τph(Tc)
δζ/n̄
⇒ n̄max ∼
ln(δζτph)
ln(Tc/δζ)
with logarithmic precision; n̄α represents the divergent
spatial extent of the cascade (correlation length) [12].
Each many-electron transition can be characterized,
besides its rate Γ, by the total dipole moment ~d it
produces. The corresponding backward transition pro-
duces the dipole moment −~d and goes with the rate
~E·~d)/Tel [13]. The average current 〈I(t)〉 is deter-
mined by the difference between forward and backward
rates; obviously, it vanishes for E = 0. At the same time,
the noise power (second cumulant) S2 ≡
[〈I(t)I(t′)〉 −
〈I(t)〉〈I(t′)〉] dt′ is determined by the sum of the forward
and backward rates; at E = 0 it is given by the equilib-
rium Nyquist-Johnson expression.
Equilibrium noise carries no information about the na-
ture of conduction. To see a signature of many-electron
transitions it would be natural to analyze the shot noise,
whose power is proportional to the charge transferred in
a single event. Many-electron cascades would then cor-
respond to “bunching” of electrons, thus increasing the
shot noise. However, shot noise is observed in the limit
when transitions transferring charge only in one direc-
tion (namely, ~d · ~E > 0) are allowed, i. e. Tel ≪ eEζloc,
which is impossible to satisfy in the insulating state,
as Tel ∼ max{Tph, eEζloc} [14]. Thus, S2 inevitably
has both equilibrium and non-equilibrium contributions,
which are difficult to separate.
To see the “bunching” effect unmasked by a large ther-
mal noise at low fields one should study the third Fano
factor S3 of the current fluctuations [15]. Indeed, being
proportional to an odd power of the current, it vanishes
in equilibrium, so it is not subject to the problems de-
scribed above for S2. In a wire of length L the ratio
S3/〈I〉 is given by
L−1 〈〈Γd〉〉
. (11)
The double angular brackets on the right-hand side mean
the sum over all allowed transitions. For nearest-neighbor
single-electron transitions with d = ±eζloc Eq. (11) gives
S3/〈I〉 ∼ e2(ζloc/L)2, which is analogous to the Schot-
tky expression reduced by the effective number of tunnel
junctions in series, L/ζloc, for S2 [16].
Since d3 diverges stronger than d as n̄ → ∞, we ex-
pect a divergence in Eq. (11). The critical index of d
depends on the order of limits: d ∼
neζloc if the lin-
ear response limit E → 0 is taken prior to n̄ → ∞, while
d ∼ neζloc(eEζloc/Tc) for a small but finite E . As a result,
eζloc
n̄eEζloc
, n̄ <∼ n̄max ,
where n̄ is given by Eq. (9), and the saturation of the
divergence is determined by Eq. (10). Upon further in-
crease of the temperature, the system crosses over to the
metallic state, and n̄ starts to decrease. This decrease is
governed by the same Eq. (10) with the phonon inelas-
tic rate substituted by the typical value of the electron-
electron inelastic rate, which grows with temperature.
As the critical behavior of Γ on the metallic side of the
transition is unknown, we cannot give any quantitative
estimate of S3 above Tc.
Conclusions.— In conclusion, we have shown that the
finite-temperature metal-insulator transition, predicted
theoretically in Ref. [5], can manifest itself on the macro-
scopic level as an S-shape current-voltage characteristic
with a bistable region. In fact, the hysteretic behaviour
of the current in YxSi1−x [17] is a possible candidate for
the effect discussed in the present paper.
Besides, we have shown that the many-body nature of
the conduction near the transition manifests itself in the
dramatic increase of the non-equilibrium current noise:
the noise depends on the total charge transferred in each
random event, while the number of electrons, involved in
such an event, increases as one approaches the transition.
We acknowledge discussions with M. E. Gershenson,
C. M. Marcus, A. K. Savchenko, M. Sanquer, and thank
H. Bouchiat for drawing our attention to Ref. [17].
∗ Electronic address: basko@phys.columbia.edu
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mailto:basko@phys.columbia.edu
http://arxiv.org/abs/cond-mat/0602510
http://arxiv.org/abs/cond-mat/0111057
