PHYSICAL REVIEW B 97, 201105(R) (2018)
Rapid Communications
Many-body localization transition: Schmidt gap, entanglement length, and scaling
Johnnie Gray,
1,*
Sougato Bose,
1
and Abolfazl Bayat
2,1
1
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
2
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China,
Chengdu, People’s Republic of China
(Received 6 April 2017; revised manuscript received 30 January 2018; published 7 May 2018)
Many-body localization has become an important phenomenon for illuminating a potential rift between
nonequilibrium quantum systems and statistical mechanics. However, the nature of the transition between ergodic
and localized phases in models displaying many-body localization is not yet well understood. Assuming that this
is a continuous transition, analytic results show that the length scale should diverge with a critical exponent ν 2
in one-dimensional systems. Interestingly, this is in stark contrast with all exact numerical studies which find
ν ∼ 1. We introduce the Schmidt gap, new in this context, which scales near the transition with an exponent
ν>2 compatible with the analytical bound. We attribute this to an insensitivity to certain finite-size fluctuations,
which remain significant in other quantities at the sizes accessible to exact numerical methods. Additionally, we
find that a physical manifestation of the diverging length scale is apparent in the entanglement length computed
using the logarithmic negativity between disjoint blocks.
DOI: 10.1103/PhysRevB.97.201105
Introduction. It has become apparent that the Anderson
localization [1] of disordered models can survive in the
presence of interactions [2] with rigorous proof now found
for one-dimensional systems [3,4]. This phenomenon, known
as many-body localization (MBL), has attracted much interest
[5–10] in fundamental physics due to the fact that such systems
generically break ergodicity and fail to thermalize—thus lying
beyond the scope of statistical mechanics. Additionally, MBL
occurs throughout the energy spectrum, implying that its
fingerprint can be observed at all temperatures. These facts
combined have significant practical implications for quantum
transport [2] and information storage [11–13]. Experimental
advances have allowed the controlled observation of MBL
phenomena [14,15] further driving interest.
Considerable progress has been made in understanding
the strongly localized phase particularly in terms of local
integrables of motion [7,16–20], which permit a matrix-
product state description of all eigenstates [21–26]. However,
eigenstates in the ergodic phase generally have volume law en-
tanglement, restricting one to exact diagonalization techniques
and small system sizes (up to ∼ 20 spins)—this has constrained
the development of a clear picture of the nature of the transition
from ergodic to MBL (the MBLT). For example, questions
that still require attention include: (i) Which quantities can
best characterize the transition? (ii) Is it valid to treat the
MBLT using the same framework, based on the emergence
of a diverging length scale, developed for zero-temperature
quantum phase transitions? (iii) If so, what is the universal
critical exponent ν governing this length scale? And (iv) what
is the physical picture of the said length scale?
An extensive exact numerical analysis of the MBLT, using
a variety of quantities, can be found in Ref. [27] in which
*
john.gray.14@ucl.ac.uk
finite-size scaling analysis throughout the spectrum allows the
observation of a mobility edge. In fact, it is now commonplace
to diagnose the MBLT with the mean energy level statistics and
the block entanglement entropy [6,10,27–31]. These works are
largely based on the assumption that the MBLT is continuous,
and their exact numerical analyses have consistently found
ν ∼ 1. This is in striking contrast with analytic results,
found by Chayes-Chayes-Fisher-Spencer (CCFS) [32] and
Chandran-Laumann-Oganesyan (CLO) [33], which would
demand ν 2/d for system dimension d (the CCFS/CLO
bound). A recent explanation [30] posits that, at the finite
system sizes available for exact studies, the fluctuations in these
quantities are not yet dominated by the true disorder. Thus it
is highly desirable to use a new quantity better able to capture
the real disorder-induced transition properties.
In this Rapid Communication, we bring in new tools to
understand the nature of the MBLT. First, the Schmidt gap,
which has been successfully employed as an order parameter in
quantum phase transitions [34–36]. Second, an entanglement
length computed from the logarithmic negativity [37–41],
quantifying the bipartite entanglement between two disjoint
blocks [42–46], which has been previously used to probe the
extension of the Kondo screening cloud [47,48]. We find that,
unlike previously used quantities, the Schmidt gap reveals a
critical exponent ν 2, consistent with the CCFS/CLO bound,
although, curiously as opposed to previous studies, it does
not act as an order parameter. Moreover, we find that the
entanglement length witnesses the emergence of a diverging
length scale at the transition from the ergodic to the MBL
phase.
Model. We consider a periodic spin-1/2 Heisenberg chain
with random magnetic fields in the z direction,
H =
L
i=1
J S
i
· S
i+1
− h
i
S
z
i
, (1)
2469-9950/2018/97(20)/201105(5) 201105-1 ©2018 American Physical Society
JOHNNIE GRAY, SOUGATO BOSE, AND ABOLFAZL BAYAT PHYSICAL REVIEW B 97, 201105(R) (2018)
(a) (b)
A B
A
E
FIG. 1. Schematic of the two main quantities studied here:
(a) the Schmidt gap across a bipartition of the system and (b)
the logarithmic negativity E between disjoint blocks separated by
length l
G
.
with J as the exchange coupling, S
i
=
1
2
(σ
x
i
,σ
y
i
,σ
z
i
)asa
vector of Pauli matrices acting on spin i, and dimensionless
parameter h
i
as the random magnetic field at site i drawn from
the flat distribution [−h,h]. We diagonalize the Hamiltonian
in either the spin-0 or spin-1/2 subspaces for even and odd
L’s, respectively. For each random instance we extract 50
eigenvectors {|E
k
} in the middle of the energy spectrum
[49,50]. Since there is evidence of a mobility edge in MBL [27],
at least for finite sizes, this targeting sharpens any transition
observed. The choice of 50 is a reasonable compromise on
numerical efficiency while being statistically representative.
Characterizing the MBLT. The main quantity we compute,
new in the context of MBL, is the Schmidt gap. For two
chain halves (or as close to for odd L), A and B as shown
in Fig. 1(a), an eigenvector’s reduced density matrix is ρ
A,k
=
Tr
B
(|E
k
E
k
|) for a particular sample of the random fields.
The disorder-averaged Schmidt gap is then defined as =
λ
k
1
− λ
k
2
k
, where λ
k
1
,λ
k
2
refer to the largest eigenvalues of
the reduced density-matrix ρ
A,k
, ·
k
denotes the average over
eigenstates and
· denotes the average over many samples.
The Schmidt gap has previously been shown to act as an
order parameter for quantum phase transitions [34,36]. We
explore the possibility of using it for characterizing the MBLT.
Unlike entanglement entropy, t he Schmidt gap ignores most
of the spectrum of ρ
A,k
, describing only the relationship
between the two dominant states across the A-B cut. This
is pertinent in light of the recent finding that, although the
Schmidt values decay polynomially in the MBL phase [22],
finite-size corrections are stronger for small Schmidt values.
In the ergodic phase we expect strong entanglement to produce
multiple, equally likely orthogonal states, thus ∼ 0. In the
MBL phase, however, a single dominant state should appear
on either side of the cut with rising towards 1 as h →∞,
implying a tensor product. This behavior is shown in Fig. 2(a)
and becomes sharper with increasing L. To see this more
vividly, we plot the derivative of with respect to h in Fig. 2(b).
The derivative has a peak at h =
˜
h
c
, which not only becomes
more pronounced, but also shifts to the right with L.We
infer this to be the finite-size precursor to the transition point,
which suggests that, in the thermodynamic limit L →∞,the
derivative of the Schmidt gap diverges at the MBLT and
˜
h
c
asymptotically approaches the t ransition point h
c
.
For reference, we consider the normalized half chain
entropy, widely employed to herald the MBLT [6,10,27–30].
The von Neumann entropy of subsystem A is defined as
S
vn
=−Tr(ρ
A,k
log
2
ρ
A,k
). This is normalized by the Page
entropy [51] S
P
= (1/ln 2)
mn
i=n+1
1
i
−
m−1
2n
with m, n as the
Hilbert space dimensions of subsystems A and B, yielding
dh
0.0
0.2
0.4
0.6
0.8
(a)
12 13 14 15 16 17 18 19 20
0 1 2 3 4 5 6 7 8
0.0
0.1
0.2
0.3
(b)
0 2 4 6 8
0.2
0.8
(c)
12
14
16
18
20
0 2 4 6 8
0.4
0.5
(d)
12
14
16
18
20
FIG. 2. (a) and (b) The Schmidt gap and its derivative as a
function of disorder h across the MBLT for varying chain length L.
(c) and (d) The normalized half chain entropy S and the mean energy
level spacing ratio r as a function of disorder for varying L. Error bars
shown where visible.
the disorder-averaged S = S
vn
k
/S
P
.S
P
is the expected
entropy for a subsystem of a random pure state; since these
overwhelmingly have entropy that scales as their enclosed
volume, S gives a measure of how far |E
k
has departed
towards area law behavior. In Fig. 2(c) the behavior of S
across the MBLT is shown. In the ergodic phase its value
approaches 1 (showing the volume law), whereas in the MBL
phase it falls to 0 (representing the area law) as expected.
For reference we also compute the mean energy level spac-
ing ratio r. For energy eigenvalues E
k
with gaps δ
k
= E
n
−
E
n−1
, this is defined as r = min(δ
k
,δ
n+1
)/ max(δ
k
,δ
n+1
)
k
.In
the ergodic phase, energy-level repulsion yields statistics for
r that match those of Gaussian orthonormal ensemble random
matrices [52] with r = 0 .5307(1). In the MBL phase, however,
the eigenenergies are no longer correlated, and the energy
levels are simply spaced according to Poisson statistics, giving
r ≈ 0.386 29. In Fig. 2(d) the behavior of r is shown across the
MBLT, clearly varying between these two statistical regimes.
For all of these quantities, we average over between 10 000
for L = 10 and 1000 for L = 20 samples of random fields
and compute errors using statistical bootstrapping across these
samples.
Scaling. The behavior in Figs. 2(a) and 2(b) suggests that
the MBLT is a continuous transition in which a diverging
length scale ξ ∝|h − h
c
|
−ν
emerges near the transition point,
consistent with Ref. [10]. In order to estimate the exponent ν,
finite-size scaling analysis [53] has previously been employed
for various quantities, including the entanglement entropy
S. These analyses, based on exact numerical methods, find
ν ∼ 1[27,30,54], contradicting the CCFS/CLO bound. A
recently proposed explanation [30] suggests that there are
two universality classes at play here with that of intersample
randomness not yet dominant for the system sizes studied.
In order to estimate ν for both models we consider the
following finite-size scaling ansatz,
= f (L
1/ν
x), (2)
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−10 0 10
0.00
0.25
0.50
0.75
(a)
12
13
14
15
16
17
18
19
20
3 4 5 6
1
2
3
4
(b)
10
0
10
1
2 3 4 5
0.00
0.05
0.10
(c)
−10 0 10 20
0.00
0.25
0.50
0.75
(d)
12
13
14
15
16
17
18
19
20
FIG. 3. (a) Schmidt gap data collapse with fitting parameters as
shown. (b) Quality of data collapse Q (lower is better) across the
whole parameter space. The dashed lines denote the minimum point
which yields the parameters shown in (a). (c) Pseudocritical points
˜
h
c
as a function of inverse length 1/L. (d) Schmidt gap data collapse
using
˜
h
c
directly and optimizing for ν only—the value of which is
shown. Error bars shown where visible.
where f (·) is an unspecified function and x is ideally the
scaled coordinate h-h
c
. Given the ansatz of Eq. (2), one can
then find the best fit of h
c
and ν, using an objective function
quantifying quality. We use such a quality measure Q as refined
in Ref. [55], which is discussed in the Supplemental Material
[56]. In Fig. 3(a) we show the optimal data collapse of for
various L’s, which is found to occur for h
c
= 5.06 ± 0.09 and
ν = 2.35 ± 0.21. Remarkably, this value for ν is consistent
with the CCFS/CLO bound, in contrast to finite-size scaling
analyses for S and r, which previous studies [27,30,54]have
generally shown to yield values of ν ∼ 1—a finding also
reproduced in our analyses (data not shown). We show the
quality of collapse Q for all possible combinations of h
c
and
ν in Fig. 3(b), the minimum point of which defines the best-fit
values of ν and h
c
. To define errors on ν and h
c
, we perform
the scaling with various subsets of data ( see the Supplemental
Material [56]) and compute the variance among all those which
achieve a good quality.
The critical h we find with is slightly higher than that
generally reported. One possible explanation is that a lower
effective ν fits best with a lower effective h
c
, a relation that
can be seen in Fig. 3(b). Thus it is possible that in other studies
using S and r, where ν ∼ 1,h
c
is artificially lower due to the
finite-size effects. We note that a standard method of extracting
h
c
independently—plotting the pseudocritical points against
inverse length, shown in Fig. 3(c)—does not give a decisive
value for the real critical point. In fact h
c
∼ 3.7 would seem
to be a lower bound on the transition point with a value
between 4.5 and 5.5 more consistent. Additionally, if one were
to identify an intersection point for all lengths in Fig. 2—which
should occur at h = h
c
as implied by Eq. (2)—this would also
be at h ∼ 5. In contrast, the point of intersection for S and r
shifts significantly as L increases—implying a deviation from
the finite-size ansatz. As a final cross validation, to estimate
ν independently from h
c
, we take the pseudocritical points
˜
h
c
directly to define the scaled coordinate x and find the best
quality of fit Q solely as a function of ν. This approach yields
2 3 4
0.0
0.1
0.2
(a)
12
14
16
18
20
2 3 4
(b)
12
14
16
18
20
2 3 4
0.05
0.06
(c)
12
14
16
18
20
FIG. 4. Standard deviation between samples for (a) the Schmidt
gap σ
, (b) normalized block entropy σ
S
, and (c) mean energy-level
spacing ratio σ
r
. Shown as a function disorder h and length L. Error
bars shown where visible.
ν = 2.13 ± 0.15—in accordance with the first estimate—for
which data collapse is shown in Fig. 3. In contrast to previous
ground-state quantum phase transitions [34,36], here we find
that the Schmidt gap is a scaling function rather than an order
parameter. Namely, it corresponds to β = 0ifEq.(2) had
prefactor L
β/ν
.
Sample fluctuations. In order to understand why the
Schmidt gap is more successful than typical quantities, we
study the fluctuation of , S, and r between samples. Mo-
tivated by Ref. [30], we consider how the size of these
fluctuations scales with L. We define the standard devi-
ations as σ
2
= Var [λ
k
1
− λ
k
2
k
],σ
2
S
= Var [S
vn
k
/S
P
], and
σ
2
r
= Var [min(δ
k
,δ
n+1
)/ max(δ
k
,δ
n+1
)
k
] with the variance
Var [ ·] taken across samples. These are shown across the MBLT
for various system sizes in Figs. 4(a)–4(c). All three quantities
must lie between 0 and 1, thus their standard deviation is capped
at 0.5. As the figures show however, the peaks of σ
S
and σ
r
are both still rising significantly with L and not yet saturated,
whereas the peak of σ
is almost constant. The implication
is that, for S and r, the effect of the small system sizes is to
suppress the amount of fluctuations driven by the true disorder.
On the other hand, changing the length L seems to have little
effect on σ
—suggesting that it already experiences the full
disorder-driven thermodynamic-limit fluctuations. A possible
explanation is that finite-size effects are dominantly confined
to the smaller Schmidt coefficients, which still contribute
significantly to σ
S
. This could also be phrased in terms of
the presence of various length scales, not yet very small
compared to the correlation length, that the Schmidt gap is
largely insensitive to.
Entanglement length. The nature of the diverging length
scale ξ in the context of MBLT is mysterious and a physical
picture is lacking. To shed light on this, we introduce an
entanglement length as previously used for detecting the
Kondo screening cloud [47,48]. Specifically, we consider the
entanglement between a small subsystem A, here a single
spin, and an environment E, separated by a gap of length
l
G
, a geometry shown in Fig. 1(b). The reduced state of the
two blocks is ρ
AE,k
= Tr
G
(|E
k
E
k
|), where Tr
G
removes the
2l
G
spins not in A or E. We use the logarithmic negativity
[37–41] to quantify the entanglement between systems A and
E, defining E (l
G
) = log
2
||ρ
AE,k
||
1
k
with as the partial
transpose and ·
1
as the trace norm. Since we are only
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JOHNNIE GRAY, SOUGATO BOSE, AND ABOLFAZL BAYAT PHYSICAL REVIEW B 97, 201105(R) (2018)
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
(a)
0
1
2
3
4
5
6
0 2 4 6
1
2
3
4
5
(b)
10
12
14
16
18
20
FIG. 5. (a) Average logarithmic negativity as a function of the gap
between two disjoint blocks as depicted in Fig. 1, the first block being a
single spin, and the second being the rest of the system. Here, L = 20 ,
which has a pseudocritical point at h ∼ 3. (b) Bipartite entanglement
length as computed with Eq. (3) across the MBLT for varying chain
length L.
concerned with the relative decay of entanglement we also
define the normalized entanglement as
˜
E(l
G
) = E (l
G
)/E(0).
This naturally gives information about bipartite entanglement
over a range of scales, unlike the two-site concurrence, for
example (which quickly goes to zero for large separation), and
unlike the widely used entanglement entropy (which cannot
quantify the entanglement of mixed states—which inevitably
arise when looking at two subsystems of a larger state). In the
ergodic phase, due to volume law entanglement, the eigenstates
are highly multipartite entangled between their spins. This
implies that any reduced state of two small blocks is close to the
identity and thus very weakly entangled. From this two features
can be inferred: (i)
˜
E(l
G
) is initially expected to decay slowly
with increasing l
G
, and (ii)
˜
E (l
G
) must go to zero as l
G
→ L/2.
Since this precludes a linear-type decay, it is expected that there
is a distance at which
˜
E(l
G
) rapidly decays—indeed we find
this to be the case with a sharp drop-off when half the system
is traced out, i.e., l
G
∼ L/4. In the MBL phase, however, A
will be weakly entangled with only spins close to it, and thus
˜
E (l
G
) should decay quickly even for small l
G
.InFig.5(a)
we plot
˜
E as a function of l
G
for various disorder strengths
h in a chain of length L = 20. As is clear from the figure
the location of the main drop in
˜
E varies significantly with h.
Whereas in the ergodic phase
˜
E this decay is concentrated at
l
G
∼ L/4, and in the MBL phase it is concentrated at l
G
∼ 1.
Interestingly, at the pseudocritical point [
˜
h
c
∼ 3forL = 20,
see Fig. 2(b)], entanglement decays close to linearly—each
spin lost contributes equally to the entanglement, implying that
the bipartite entanglement is equally spread over many sites.
This fits with a picture of a self-similar structure of entangled
clusters [29,57]. The detailed behavior of
˜
E as a function of
system size can be found in the Supplemental Material [56].
To extract a l ength scale from
˜
E (l
G
) we define a length η
from the maximum inverse gradient as such,
η = max
l
G
|d
˜
E/dl
G
|
−1
. (3)
Assuming the fastest decay is exponential-like, this quantity
naturally arises from expressions of the form
˜
E ∝ e
−l
G
/η
.This
is a more robust way of finding an exponential fit in the region
of the most rapid decay of
˜
E or a more general fit for the full
behavior. At the transition point, where
˜
E decays linearly, η
takes its maximum value since the gradient is always small, or
equivalently, a very slow exponential fit is needed.
The behavior of η as a function of h for varying L is
shown in Fig. 5(b) in which it can be seen to sharply peak
at h ∼
˜
h
c
for each L across the critical region—evidence that
the diverging length scale ξ is closely captured by the length
η. In the Supplemental Material [56] we show that taking the
initial block as two spins yields almost identical results. A
plausible explanation for the increase in η as one approaches
the MBLT from the ergodic side is that proximal spins become
off-resonant so that bonding (bipartite entanglement) takes
place at increasingly longer scales—a process that is not
possible if the spins are part of a large multipartite entangled
block. We note several interesting approaches that made use
of the two-site concurrence [58,59] or mutual information
[60], which despite revealing other interesting features, such
as scaling, do not show a divergence in the localization length
from both sides of the transition. An alternative approach
for identifying the diverging length scale on the ergodic side
based on the entanglement spectrum has been recently devel-
oped in Ref. [61]. It is an interesting open question whether
that length is related to the entanglement length proposed
here.
Conclusions. In this Rapid Communication we have ex-
plored the MBLT using the Schmidt gap and the entanglement
length. We show that the Schmidt gap not only exhibits
scaling at the MBLT, but also does so with a critical exponent
ν>2, compatible with analytic predictions. This compatibil-
ity is absent in all quantities studied with exact numerical
methods thus far, a fact that we attribute to the presence
of significant finite-size effects which the Schmidt gap is
less sensitive to. We have also considered an entanglement
length computed using the logarithmic negativity across two
disjoint blocks, which yields a diverging length scale at the
MBLT.
Acknowledgments. J.G. acknowledges funding from the
EPSRC Center for Doctoral Training in Delivering Quantum
Technologies at UCL (Grant No. EP/L015242/1). A.B. and
S.B. acknowledge the EPSRC Grant No. EP/K004077/1. S.B.
acknowledges financial support by the ERC under Starting
Grant No. 308253 PACOMANEDIA.
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