Logarithmic growth of local entropy and total correlations in
many-body localized dynamics
Fabio Anza
1
, Francesca Pietracaprina
2
, and John Goold
3
1
Complexity Sciences Center, Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616
2
Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France
3
School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland.
The characterizing feature of a many-
body localized phase is the existence of
an extensive set of quasi-local conserved
quantities with an exponentially localized
support. This structure endows the sys-
tem with the signature logarithmic in
time entanglement growth between spa-
tial partitions. This feature differenti-
ates the phase from Anderson localiza-
tion, in a non-interacting model. Exper-
imentally measuring the entanglement be-
tween large partitions of an interacting
many-body system requires highly non-
local measurements which are currently
beyond the reach of experimental technol-
ogy. In this work we demonstrate that the
defining structure of many-body localiza-
tion can be detected by the dynamics of
a simple quantity from quantum informa-
tion known as the total correlations which
is connected to the local entropies. Cen-
tral to our finding is the necessity to prop-
agate specific initial states, drawn from the
Hamiltonian unbiased basis (HUB). The
dynamics of the local entropies and to-
tal correlations requires only local mea-
surements in space and therefore is poten-
tially experimentally accessible in a range
of platforms.
The study of transport properties of quantum
systems is a topic of paramount importance in
condensed matter physics. A crucial aspect is
the presence of disorder due to defects and irreg-
ularities in the material under study. In a cele-
brated work [8] Anderson showed how the pres-
ence of strong disorder can completely suppress
Fabio Anza: fanza@ucdavis.edu
Francesca Pietracaprina: pietracaprina@irsamc.ups-tlse.fr
John Go old: gooldj@tcd.ie
transport of non-interacting electrons in a tight-
binding model. Understanding the fate of this
localisation phenomenon in the presence of inter-
actions has seen an unprecedented revival in re-
cent years [2, 4]. In a seminal contribution Basko
et. al. [12] argued that such phenomenon is sta-
ble when interactions between particles are intro-
duced, showing the existence of a new dynam-
ical phase of matter, the Many-Body Localized
(MBL) phase [5, 7, 30] which, like its single parti-
cle counterpart exhibits a lack of both transport
and thermalization [5]. From the experimental
perspective, signatures of MBL physics have re-
cently been observed in a number of different lab-
oratories in cold atoms [15, 27, 34], ion traps [41]
and NMR [45].
As the system fails to thermalize, local observ-
ables retain memory of their initial conditions. In
the last ten years there has been a large amount
of effort devoted to the understanding of the MBL
phase [1, 4, 6, 21, 30, 33, 43]. The defining feature
of an MBL state (e.g. as opposed to Anderson
localization) has been identified in the fact that,
while the transport of energy and local quantities
is suppressed [3, 24, 25, 28, 35, 42, 48, 49], there is
transport of quantum information, occurring on a
logarithmic time scale manifested in the growth
of the half-chain entanglement entropy [11, 37].
This behaviour can be explained by the emer-
gence of an extensive set of quasi-local integrals
of motions (Q-LIOMs) [19, 21, 32, 37]. Such ob-
jects have a support which is exponentially local-
ized, the localization length being ξ. In the high-
disorder regime the tails become more suppressed
and the Q-LIOMs approach local quantities.
The canonical models to study MBL, which we
will also use in this letter, is the XXZ spin chain
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arXiv:1907.00291v3 [quant-ph] 28 Mar 2020
with random fields:
H =
L
X
i=1
s
x
i
s
x
i+1
+ s
y
i
s
y
i+1
+ s
z
i
s
z
i+1
+
L
X
i=1
h
i
s
z
i
,
(1)
where s
α
= 1/2 σ
α
, α = x, y, z are 1/2-spins, the
fields h
i
are random variables with uniform prob-
ability distribution in [W, W ] and W [0, )
is the disorder strength. For zero disorder this
model is the quantum Heisenberg model; it has
a many-body localization transition at W
c
3.72 [25] at ∆ = 1 and conserves the total magne-
tization s
z
=
P
i
s
z
i
. The emergence of integrabil-
ity corresponds to recasting (1) into the effective
Hamiltonian
H
MBL
=
X
j
1
λ
(1)
j
1
τ
z
j
1
+
X
i
1
<i
2
λ
(2)
i
1
i
2
τ
z
i
1
τ
z
i
2
+ . . . , (2)
where the τ
z
i
are the Q-LIOMs and the n-th or-
der interaction constants λ
(n)
i
1
,...,i
n
are expected to
fall off exponentially with distance. For exam-
ple, λ
(2)
i
1
,i
2
e
d(i
1
,i
2
)
, where ξ is the localiza-
tion length and d(i
1
, i
2
) is the distance between
site i
1
and site i
2
. The predicted logarithmic
growth of entanglement, the marker of a genuine
MBL phase, is a direct consequence of the exis-
tence of the exponentially small non-local tails of
the Q-LIOMs, and of their interaction [32, 37].
Measuring the slow growth of entanglement en-
tropy which is responsible for the unique struc-
ture of MBL phases is extremely challenging, al-
though recent progress has been made [26]. This
is mainly due to the highly non-local character of
the half-chain entanglement entropy, which is not
an easily measurable quantity beyond the small
systems [22, 26]. Such difficulty inspired alter-
native ways to witness such dynamical behav-
ior [13, 14, 16, 17, 20, 36, 38].
The purpose of this letter is to demonstrate
that the logarithmic spread is encoded in the
behavior of the single-site density matrices and
the local total correlations when an initial state
for propagation is carefully chosen. We believe
that this behaviour is experimentally accessible
with the available techniques of local tomogra-
phy; thus, we argue that local measurements are
able to distinguish Anderson Localisation (AL)
from its interacting counterpart i.e. MBL. This
provides a strategy for experimental detection of
unique MBL phenomenology. The letter is orga-
nized as follows. First, we provide a theoretical
argument supporting a logarithmic growth of the
local entropy and its relation to the notion of to-
tal correlations. Secondly, we discuss the optimal
initial states, drawn from the HUB, to use in the
time evolution. These special states allow us to
obtain the longest transient dynamics in the lo-
calized state (for systems of finite size). Finally,
we give numerical results for the local entropy
and total correlations in the time evolution of the
model (1).
Logarithmic growth of entanglement. Let
us consider the one-dimensional, disordered, iso-
lated quantum system of L spin-1/2 defined by
Eq. (1). Calling |ψ
t
i the state of the whole sys-
tem at time t, the reduced state of the nth site
is obtained by tracing out the complement L/n:
ρ
(n)
t
:
= Tr
L/n
|ψ
t
ihψ
t
|. The bipartite entangle-
ment between the nth site and the rest is quanti-
fied by the Von Neumann entropy of the reduced
state:
S
n
(t)
:
= Tr ρ
(n)
t
log ρ
(n)
t
(3)
This quantity is experimentally accessible by
means of local tomography. Moreover, one can
also consider the average entropy over all sites:
S(t) =
1
L
L
X
n=1
S
n
(t) . (4)
The latter quantity has the following opera-
tional meaning. Let P H be the set of all tensor
product states of an L-partite quantum system.
The total correlations T (ρ) of a (possibly mixed)
state ρ is defined as [29]
T (ρ) = min
π∈P
S(ρ||π) , (5)
where S(ρ||σ)
:
= Tr ρ log σ S(ρ) is the rela-
tive entropy. T (ρ) is an extensive quantity that
measures the distinguishability between ρ and the
closest product state π
ρ
P. It turns out that,
for each ρ, such state is unique. It is the prod-
uct state of the reduced density matrices obtained
from ρ: π
ρ
= ρ
1
. . . ρ
L
where ρ
i
:
= Tr
L/i
ρ.
In our case of a pure state |ψ
t
i it is easy to see
that the total correlations T
t
:
= T (|ψ
t
ihψ
t
|) are
simply a rescaling of S(t)
T
t
=
L
X
n=1
S
n
(t) S(|ψ
t
ihψ
t
|) = LS(t) (6)
It was recently shown that the study of the
total correlations in the diagonal ensemble can
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signal the transition from ergodic to the MBL
phase [17, 31] but since the diagonal ensemble
is a mixed state this requires knowledge of the
global state. Since the states here are pure the
total correlations can be probed dynamically by
means of only local operations.
In Ref. [37], an argument has been proposed
to explain the logarithmic growth of the bipar-
tite entanglement entropy S
L/2
(t) that was pre-
viously numerically observed in Ref. [11]. Here we
summarize this argument and explore its conse-
quences for the growth of the local entropy. The
intuition is based on the presence of the expo-
nentially suppressed tails of the Q-LIOMs, which
decay on a length scale given by the localization
length ξ. Calling the coupling constant of the
s
z
i
s
z
i+1
interaction term, if there are no interac-
tions ( = 0) all energy eigenstates are single-
particle excitations, and there is Anderson Lo-
calization. In presence of interactions we have
many-body localization. In this case, if two par-
ticles are placed at a distance x
ij
they would have
an interaction energy which is exponentially sup-
pressed because of the exponentially small tails
V
ij
e
x
ij
. The dephasing time between
them is therefore t
ij
~/V
ij
= ~e
x
ij
/. This
implies that S
L/2
(t) should grow in time with a
logarithmic law, as outlined in Refs. [11, 37].
We now look at the implications of this argu-
ment for the bipartite entanglement between a
single site and the rest and hence for the total
correlations. The degrees of freedom on a lattice
site n will become entangled with the degrees of
freedom living on the n + k site on a time-scale
t
k
t
min
e
ka/ξ
where a is the lattice spacing. As
time evolves the n-th site will become entangled
with an increasing number of sites. The higher
the number of sites which have entanglement with
the n-th one, the higher S
n
(t), which will accu-
mulate on a logarithmic time scale. From this we
expect a logarithmic growth of S
n
(t). We would
like to stress here that this argument holds for
each individual site, thus requiring probing only
a single site (especially important e.g. in an ex-
perimental setup). Although in the following we
will show results for the average single site en-
tanglement entropy (4), we checked that the re-
sult for each individual sites is quantitatively the
same. S(t) quantifies the average growth of bi-
partite entanglement between one site and the
rest of the chain. Since S(t) is a rescaling of the
total correlations, it additionally provides an up-
per bound to the amount of multipartite entan-
glement present in the system once rescaled with
the system size.
Initial states. In the study of the dynamics
of isolated quantum systems, an crucial ingre-
dient is the choice of the initial state. A typ-
ical criterion driving this choice is experimen-
tal feasibility; a well-known example is the anti-
ferromagnetic (Néel) state, which can be pre-
pared as the ground state of a local Hamilto-
nian. From the information-theory point of view
the Néel state (polarized e.g. along the z direc-
tion) is part of the so-called computational basis
B
z
:
= {|s
z
1
i|s
z
2
i. . . |s
z
L
i}, the tensor-product ba-
sis of the z components of the local spins |s
z
i
i
{| ↑i
z
, | ↓i
z
}. Here, we additionally request that
the chosen initial state allows for sufficiently long
dynamics in the MBL phase before reaching sat-
uration. Since our goal is to probe dynamical fea-
tures of an isolated quantum system, our initial
state should not be too close to being a single en-
ergy eigenstate. Indeed, if this was the case, the
state dynamics would always occur in the proxim-
ity of the initial state, resulting in rapid dynamics
for observables before saturation that is unlikely
to be seen. The choice of initial state is there-
fore crucial and by starting with a state which
is a superposition of as many energy eigenstates
as possible, we can access a longer transient dy-
namics which explores a larger part of the Hilbert
space.
We focus on the strong disorder regime where
the system is characterized by an extensive num-
ber of Q-LIOMs and the energy eigenstates are
close to elements of the computational basis along
the z direction (basis in which the magnetic field
term in Eq. (1) is diagonal). Therefore, in the
MBL phase, the Néel state polarized along the
z direction is very close to an energy eigenstate
and has very short dynamics before saturation.
A set of initial states which can be used to avoid
this issue is given by the elements of a Hamil-
tonian Unbiased Basis [9, 10] (HUB). A basis
B
:
= {|v
µ
i}
D
µ=1
is called a HUB when
|hv
µ
|E
ν
i|
2
=
1
D
µ, ν , (7)
where |E
ν
i are the Hamiltonian eigenstates and
D is the dimension of the Hilbert space. If our
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initial state is part of a HUB, its decomposition in
the Hamiltonian basis will include all eigenstates
and will be as far away as possible from being an
Hamiltonian eigenstate. In the case of the XXZ
model with disordered magnetic field along the z
direction, deep in the MBL phase the Q-LIOMs
will be almost diagonal in B
z
. Hence, deep in the
MBL phase, states that are a tensor product of
the local spins polarized along the x or y direction
are close to HUBs:
B
x
:
= {|s
x
1
i. . . |s
x
L
i} B
y
:
= {|s
y
1
i. . . |s
y
L
i} , (8)
where |s
x
i
i {| ↑i
x
, | ↓i
x
} and |s
y
i
i {| ↑i
y
, | ↓i
y
}
for all i = 1, . . . , L.
Here we will consider initial states that are el-
ements of the bases B
x
and B
y
, and specifically
the Néel state along the x direction | ↑↓↑↓ . . .i
x
.
These states have contributions from all sub-
spaces that would conserve total S
z
magnetiza-
tion and have already been prepared and used for
experiments in Refs.[18, 23, 40, 44, 46, 47]. The
results do not depend on this choice, and in the
Supplementary materials we show that the Néel
state along the y direction | ↑↓↑↓ . . .i
y
, the ferro-
magnetic states | ↑↑↑↑ . . .i
x
and | ↑↑↑↑ . . .i
y
and
the states with two polarized domains | ↑↑ . . . ↓↓
. . .i
x
and | ↑↑ . . . ↓↓ . . .i
y
along the x and y di-
rections all give the same logarithmic growth and
phenomenology outlined in the next paragraph.
Results. We consider the time evolution of the
spin chain (1) with = 1 and four values of
the disorder strength in the delocalized (very low
W = 0.1 and low W = 1 disorder) and local-
ized (W = 5 and W = 10) phases. We con-
sider systems up to L = 20 spins, averaging
over a sufficient number of disorder realizations
(1000, 100 and 50 realizations for sizes L 12,
13 L 19, L = 20 respectively). To per-
form the time evolution, we used an iterative
Krylov subspace method with the Lanczos algo-
rithm that avoids full diagonalization [39, 42].
With the initial states outlined above, we
clearly obtain a logarithmic growth for the single-
site entanglement entropy: in Fig. 1 we plot the
behaviour of S(t) at L = 20 for the different val-
ues of W , showing that a logarithmic envelope is
present only in the MBL phase (W = 5, 10). In
the thermal phase (W = 0.1, 1) we simply observe
a quick thermalization towards the maximum en-
tropy configuration. The inset shows that the
L = 5
L = 10
L = 15
L = 20
40 80
t
0.60
0.64
0.68
S(t)
W = 0.1
W = 1
W = 5
W = 10
0 20 40 60 80 100
12.0
12.5
13.0
13.5
t
L S(t)
Figure 1: Time-dependent behaviour of S(t) for L =
20 and W = 0.1, 1, 5, 10. In the MBL phase S(t) is
modulated on a logarithmic time-scale (the dashed line
is a logarithmic fit of the minima). The inset shows the
behaviour of S(t) at W = 10 for L = 20, compared to
the one for system sizes L = 5, 10, 15, showing very weak
system size dependence up to very small sizes. Further
checks have been performed with systems of size L = 5
through L = 20, all showing quantitatively the same
time-dependent profile.
same behavior emerges already for sizes as small
as L = 5. More details can be found in Section C
of the Supplemental materials. At each L we can
isolate the local minima of S
i
(t) and extract the
slope c(W ) of the logarithmic growth as function
of W . The data are shown in Fig.2, showing that
c/L is constant within the error. Understanding
the relation of c with quantities of phenomeno-
logical relevance, as the localization length, goes
beyond the purpose of this letter and it is left for
future investigation.
Due to its quasi-periodic behaviour, we per-
formed a discrete Fourier transform F(ω) on S(t)
and studied its power spectrum P (ω) := |F(ω)|
to understand its oscillations. In Fig. 3, we show
the behavior of the power spectrum for W = 10
and different sizes L = 5, . . . , 20. As for S(t), its
structure is independent on the size of the system.
The positions of the first (ω
1
) and the second (ω
2
)
peaks are related to each other by a simple rela-
tion ω
2
= 2ω
1
. This is due to the fact that, in
the MBL phase, the eigenvalues λ
(n)
±
(t) of ρ
(n)
t
have a periodic structure which is modulated on
a logarithmic time-scale. Because of that, when
we perform the Fourier transform of S
n
(t), the
terms in the Taylor expansion of S
n
(t) as a func-
tion of λ
(n)
±
(t) are responsible for the presence of
peaks at frequencies that are multiple integers of
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W = 6
W = 7
W = 8
W = 9
W = 11
W = 15.
10 20 50 100
10.8
11.0
11.2
11.4
11.6
11.8
12.0
t
L S(t)
L = 10 L = 12 L = 14
L = 16 L = 18
5 6 7 8 9 10 11
0.016
0.018
0.020
0.022
W
c/L
Figure 2: Logarithmic growth of S(t) for different disor-
der strengths. Top panel: Local minima (obtained as in
Fig. 1) as a function of time for size L = 18. Bottom
panel: Coefficient c of the fit LS(t) = a + c log(t) as a
function of disorder strength, rescaled by system size L,
for values L = 10, 12, 14, 16, 18.
the lowest frequency: ω
n
=
1
. In Figure 3 only
the first two are visible.
Finally, we remark once again that all the S
n
(t)
have the same time-dependent profile and power
spectrum as their average S(t).
As a final check of the consistency of the local
entropy results with the known log(t) behavior of
the half-chain entropy S
L/2
(t), in Fig. 4 we com-
pare their behaviour for strong disorder W = 10,
showing that they both exhibit the logarithmic
growth in time.
Discussion. We stress that the logarithmic
spread of entanglement is one of the character-
istic features of genuine MBL, as opposed to
AL: indeed, in Fig. 5 we show the two quali-
tatively different behaviors in the time evolution
of our model (1). Experimentally, the detection
of genuine MBL through the measurement of en-
tanglement is a daunting proposition. Here we
have demonstrated that this definitive signature
of MBL can be obtained from local measurements
alone. We obtained this result through a care-
ful choice of the initial state, building on the no-
tion of Hamiltonian Unbiased Basis (HUB), and
through both a theoretical argument and numer-
ical evidence that the logarithmic spread of en-
tanglement is encoded in the behavior of each in-
L = 5
L = 10
L = 15
L = 20
1 2 3
ω
0.01
0.1
1
10
P(ω)
W = 0.1
W = 1
W = 5
W = 10
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.01
0.10
1
10
ω
P(ω)
ω
1
ω
2
=2ω
1
Figure 3: Power spectrum of S(t) for L = 20 and W =
0.1, 1, 5, 10. There are two visible peaks, whose frequen-
cies are related by ω
2
= 2ω
1
. The inset shows the power
spectrum of S(t), at W = 10, for L = 5, 10, 15, 20,
which is quantitatively independent on the size of the
system.
L = 5
L = 10
L = 15
L = 20
1 10 50 100
t
0.1
0.2
S(t)
S
L/2
(t)
S(t)
S
n
(t)
0.5 1 5 10 50 100
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t
Figure 4: Time-dependent behaviour of the half chain
entropy S
L/2
(t) compared to S(t) and the local S
n
(t),
for L = 20 and W = 10. The inset shows how S
L/2
(t)
behaves for L = 20, 12, 6 and W = 10. We found no
system size dependence: sizes L = 5 through 20 exhibit
the same time-dependent profile.
dividual single-site entropies.
There are several unique features of the anal-
ysis that deserved to be highlighted. Firstly, the
logarithmic modulation in time of the single-site
entanglement entropy is quite evident for systems
of sizes as small as L = 5, very far away from the
thermodynamic limit. Secondly, the study of the
power spectrum of S
n
(t) (and S(t)) shows that
the presence of the peaks in Fig.2 is a distinctive
feature of the MBL phase. Again, this is clear
already for systems of size as small as L = 5.
This points towards the fact that the power spec-
trum of the entanglement entropy can be a useful
tool to investigate quantum systems, beyond its
relevance for the physics of MBL systems.
Finally, we highlight once more that a num-
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J
z
= 0
J
z
= 1
0 20 40 60 80 100
10.5
11.0
11.5
12.0
12.5
t
L S(t)
Figure 5: Time-dependent behaviour of S(t) for system
size L = 18 and disorder W = 10 for the interacting
(J
z
= 1) and the non-interacting (J
z
= 0) systems,
showing the different behavior of the MBL phase (loga-
rithmic growth, Q-LIOMs oscillations) and the AL phase
(saturation, damped oscillations).
ber of experimental setups are especially suited
to measure the local quantities that we propose
here; these are trapped ions [41] and nuclear mag-
netic resonance setups [45].
Acknowledgements. F.A. would like to ac-
knowledge many discussions on the phyiscs of
MBL systems with X. Lei and J. Crutchfield.
F.A. acknowledges that this project was made
possible through the support of a grant from
Templeton World Charity Foundation, Inc.. The
opinion expressed in this publication are those
of the authors and do not necessarily reflect
the views of Templeton World Charity Founda-
tion, Inc. F.P. acknowledges the support of the
project THERMOLOC ANR-16-CE30-0023-02 of
the French National Research Agency (ANR) and
thanks F. Alet for useful suggestions. This work
was supported by an SFI-Royal Society Univer-
sity Research Fellowship (J.G.). This project
received funding from the European Research
Council (ERC) under the European Union’s Hori-
zon 2020 research and innovation program (grant
agreement No. 758403.
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A Initial states, Hamiltonian Unbiased Basis and Power Spectrum
As stated in the main text, we prepare the system in an initial state which is close to an element of an
Hamiltonian Unbiased Basis (HUB). This offers a substantial advantage in our case: the initial state
will be a linear superposition of as many energy eigenstates as possible, allowing both a longer and a
wider exploration of the Hilbert space.
In the left panel of Fig. 6 we show that the Néel state along the x direction closely approximates
a HUB at high disorder, in contrast with the Néel state along the z direction which has overlap only
with very few eigenstates.
In the main text we only showed the results obtained with the initial state | ↑↓↑↓ . . .i
x
. In the right
panel of Fig. 6 we show that the results are quantitatively the same if we consider any one of the
following additional initial states, all of which closely approximate a HUB: | ↑↓↑↓ . . .i
y
, | ↑↑↑↑ . . .i
x
,
| ↑↑↑↑ . . .i
y
, | ↑↑ . . . ↓↓ . . .i
x
and | ↑↑ . . . ↓↓ . . .i
y
.
|↑↓↑↓⋯>
x
|↑↓↑↓⋯>
y
|↑↑⋯>
x
|↑↑⋯>
y
|↑↑⋯↓↓>
x
|↑↑⋯↓↓>
y
0 20 40 60 80 100
0.58
0.60
0.62
0.64
0.66
0.68
t
S(t)
Figure 6: Average local entropy S(t) as a function of time for various initial states that closely approximate a member
of a HUB at strong disorder, for a system of size L = 18 and disorder W = 10.
With all the initial states outlined above, we clearly obtain a logarithmic growth for the single-site
entanglement entropy. Moreover, in Fig. 1 of the main paper we plotted the behaviour of S(t) at
L = 20 for different values of W , showing that a logarithmic envelope (red line) is present only in the
MBL phase (W = 5, 10).
B Lattice-variance of the single-site entanglement entropy
In the main text we claim that the local entropies S
n
(t) and their average S(t) are quantitatively the
same behavior. To show this, in Figure 7 we plot the square root of the average difference-squared
between the local entropies S
n
(t) and S(t)
δ(t) =
v
u
u
t
1
L
L
X
n=1
(S
n
(t) S(t))
2
, (9)
This quantity is the lattice-variance of the single-site entanglement entropy. It estimates, at each time,
how much each of the S
n
(t) differ from their average S(t). As we can see in Figure 7, this appears
to be consistently small for all the parameters of the model and at any time. Indeed, we find that
δ(t) . 10
3
, for all system sizes, values of the disorder and at any time.
C Position of the peaks in the Fourier transform of the single-site entropy
In Figure 3 of the main text we plot the power spectrum of the entanglement entropy, which has a
peculiar behavior in the MBL phase.
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0 10 20 30 40 50 60 70 80 90 100
t/J
10
-5
10
-4
10
-3
10
-2
10
-1
(t)
L=20 L=11 L=3
Figure 7: Variance of the local entropy across the lattice (Eq.9), for three system sizes L = 20, 11, 3 at W = 10.
We observe the emergence of two major peaks at frequencies, ω
1
, ω
2
which are related via ω
2
= 2ω
1
.
Here we provide an analytic argument to understand why there is such relation.
The first step is to notice that the behavior emerges already for very small system sizes. This,
together with the fact that the period of the oscillation is relatively small, suggests that the existence
of the oscillations should be due to short-range interactions between nearest neighbour spins. To verify
this idea we choose L = 10 and W = 10 and studied what happens to the power spectrum of the
average single-site entropy S(t) when changes. In particular we focus on ∆ = 0.2, 0.4, 0.6, 0.8, 1.
Figure 8: Here we show how the power spectrum of the average single-site entropy, S(t), changes as a function of
the interaction strength , for a system with L = 10 spins and W = 10. In the main figure we show the various
power spectra at different values ∆ = 0.2, 0.4, 0.6, 0.8, 1. In the inset we plot the location of the largest peak of the
power spectrum as a function of , which exhibits a linear dependence.
As we can see from Figure 8 the frequency of the oscillations is determined by the interaction strength
, which in the main text is fixed to 1. Intuitively, the existence of the localized phase suppresses
the hopping term favouring the magnetic field term and the σ
z
σ
z
nearest neighbour interaction,
whose intensity is regulated by . For this reason, we believe the behaviour of the oscillations can be
understood by looking at the following two-spins model with phenomenological Hamiltonian
H
phen
= σ
z
1
+ σ
z
2
+ V
int
σ
z
1
σ
z
2
(10)
The dynamics of such simple theoretical model can be solved exactly. In particular, the Hamiltonian
has three distinct eigenvalues: E
min
, E
max
and E
C
. The central one, E
C
, has degeneracy 2. So we
have
E
max
= 2 + V
int
E
C
= V
int
E
min
= 2 + V
int
(11)
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Respectively, their eigenstates are
|E
max
i = | ↑↑i |E
C
(α, θ)i = α| ↑↓i +
q
1 |α|
2
e
| ↓↑i |E
min
i = | ↓↓i (12)
We can easily write down the propagator:
U(t) = e
i
~
H
eff
t
= e
i
~
(2+V
int
)t
| ↑↑ih↑↑ | + e
i
~
V
int
t
(| ↑↓ih↑↓ | + | ↓↑ih↓↑ |) + e
i
~
(2+V
int
)t
| ↓↓ih↓↓ | (13)
As in our simulation, the initial state that we choose is the Neel state, polarized along the X direction:
|ψ
0
i = |Neel
X
i = |
x
,
x
i =
1
4
(| ↑↑i + | ↑↓i | ↓↑i | ↓↓i) (14)
Which gives the following exact expression for the time-dependent pure state:
|ψ
t
i =
1
4
h
e
i
~
(2+V
int
)t
| ↑↑i + e
i
~
V
int
t
(| ↑↓i | ↓↑i) e
i
~
(2+V
int
)t
| ↓↓i
i
(15)
Eventually, the time-dependent density matrix is obtained via the outer product of |ψ
t
i with hψ
t
|:
ρ
t
=
1
4
1 e
i
~
(2+2V
int
)t
e
i
~
(2+2V
int
)t
e
i
~
4t
c.c. 1 1 e
i
~
(22V
int
)t
c.c c.c. 1 e
i
~
(22V
int
)t
c.c. c.c. c.c. 1
(16)
From this, it is easy to compute the partial trace over the firs spin and obtain the reduced density
matrix of the second one. For the sake of simplicity, here we do not make explicit the value of the
energy eigenvalues:
ρ
(2)
t
=
1
4
1 + 1 e
i
~
(E
max
E
C
)t
+ e
i
~
(E
C
E
min
)t
e
i
~
(E
max
E
C
)t
+ e
i
~
(E
C
E
min
)t
1 + 1
!
(17)
The Von Neumann entropy of this reduced state is the entanglement entropy. Hence, we need its
eigenvalues λ
±
2
(t). After some algebraic manipulation and using the fact that Tr H
eff
= 0 we have
λ
±
2
(t) =
1 ± cos θ(t)
2
θ(t) :=
2|E
C
|
~
t (18)
Eventually, here is the analytical expression for the Entanglement entropy of a L = 2 effective Hamil-
tonian model:
S
2
(t) =
1 + cos θ(t)
2
log
1 + cos θ(t)
2
1 cos θ(t)
2
log
1 cos θ(t)
2
(19)
This is a function with smallest period T =
~π
2|E
C
|
=
~π
2|V
int
|
. However, the entropy is a more complicated
function, which involves the natural logarithm of 1 + cos θ(t). The natural logarithm is responsible
for the presence of peaks which go beyond the first one. Indeed, if we plot the Power spectrum of
Eq.(19) (see Figure 9 right panel) with the ones obtained from the data, we see that the position of
the peaks are in perfect agreement, for the choice V
int
= 1/4. Incidentally, we notice that with such
choice, T =
~π
2V
int
2π~. Now, the reason why the secondary peaks are located at frequencies which
are multiple integers of the frequency of the first peak seems to be purely technical. The logarithm is
not a periodic function. However, the entropy in Eq.(19) involves the logarithm of a periodic function.
Thus, if we Taylor-expand the logarithm we obtain
log(1 + cos θ(t)) =
X
n=1
(1)
n+1
[cos θ(t)]
n
n
= cos θ(t)
[cos θ(t)]
2
2
+
[cos θ(t)]
3
3
+ . . . (20)
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0 0.1 0.2 0.3 0.4 0.5 0.6
0
2
4
6
P( )
Effective Model
Data - W=5 - L=2
Data - W=10 - L=2
Out[33]=
0.3
0.4
0.5
0.6
w
0
5
10
PHwL
H1+Cos 2tLLogH1+Cos 2tL
Figure 9: Left panel: Comparison between the power spectrum of the entanglement entropy that we obtain from
the effective model and the one that we obtain from the data on the Heisenberg model at L = 2. All the peaks are
located at the same position for V
int
= 1/4. The precise value of V
int
is empirical. Right panel: Power spectrum of
the function f(t) = (1 + cos 2t) log(1 + cos 2t). The secondary peaks are located at frequencies which are multiple
integers of the frequency of the major peak.
Each one of these terms will give a peak which is centered in frequencies that are multiple integers of
the original frequency of the cos θ(t), due to the fact that we have integer powers of cos θ(t). Indeed,
if we look at the power spectrum of (1 + cos 2t) log(1 + cos 2t) we obtain the right panel of Figure 9.
More in general, if we forget for a moment about the coefficients, in the Taylor expansion of Eq.20,
using Euler formulas for the cos θ we have
(cos θ)
n
=
e
+ e
2
!
n
=
1
2
n
n
X
k=0
n
k
!
e
(nk)
e
iθk
=
n
X
k=0
1
2
n
n
k
!
e
(n2k)
(21)
Hence, its Fourier transform is simply
F[(cos θ)
n
](ω) =
n
X
k=0
1
2
n
n
k
!
2πδ(ω (n 2k)) (22)
If we now consider that, in our case, θ is not the independent variable but it is linearly propor-
tional to it, θ(t) =
2|E
C
|
~
t, we conclude that the peaks will all be located at frequencies ω
n
=
2|E
C
|
~
, 2
2|E
C
|
~
, 3
2|E
C
|
~
, . . . , n
2|E
C
|
~
To help visualize the effect, in Figure 10 we plot the power spectrum
of the first four powers of cos t.
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Out[119]=
0.1
0.2
0.3
0.4
0.5
0.6
w
0
5
10
15
20
PHwL
Cos t
0.1
0.2
0.3
0.4
0.5
0.6
w
0
5
10
15
20
PHwL
HCos tL
2
0.1
0.2
0.3
0.4
0.5
0.6
w
0
5
10
15
20
PHwL
HCos tL
3
0.1
0.2
0.3
0.4
0.5
0.6
w
0
5
10
15
20
PHwL
HCos tL
4
Figure 10: Power spectrum of the functions f
n
(t) = (cos t)
n
for n = 1, . . . , 4. The power spectrum of the entangle-
ment entropy can be understood as a superposition of these objects, with certain coefficients.
Accepted in Quantum 2020-03-24, click title to verify. Published under CC-BY 4.0. 14