# Identifying Many-Body Localization in Realistic Dot Arrays

Alexander Nico-Katz  Department of Physics and Astronomy, University College
London, London WC1E 6BT, United Kingdom  Gulzat Jaliel  London Centre for
Nanotechnology, University College London, 17-19 Gordon Street, London WC1H
0AH, United Kingdom  Paola Atkinson  Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Sorbonne
Université, CNRS, Institut des NanoSciences de Paris, INSP, 4 place Jussieu,
F-75005 Paris, France  Thomas A. Mitchell  Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  David A.
Ritchie  Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue,
Cambridge CB3 0HE, United Kingdom  Charles G. Smith  Cavendish Laboratory,
University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United
Kingdom  Sougato Bose  Department of Physics and Astronomy, University College
London, London WC1E 6BT, United Kingdom

###### Abstract

We determine whether or not it is possible to identify many-body localization
in quantum dot arrays, given their current technological capacities. We
analyze the phase diagram of an extended Fermi-Hubbard model - a theoretical
system that quantum dot arrays are known to simulate - using several
quantities of varying experimental accessibility. By deriving the parameters
of our model from our experimental system, we find that many-body localization
can potentially be detected in current-generation quantum dot arrays. A
pitfall that we identify is that the freezing of a system due to strong
interactions yields signatures similar to conventional localization. We find
that the most widely-used experimental signature of localization - the
imbalance - is not sensitive to this fact, and may be unsuitable as the lone
identifier of the many-body localized regime.

##  I Introduction

Many-body localization (MBL), the breaking of ergodicity and corresponding
arrest of transport in disordered strongly-correlated quantum systems [1, 2,
3, 4], has become increasingly accessible in a wide array of experimental
systems. Devices in which MBL has been experimentally realized include
ultracold atoms and ions in optical lattices [5, 6, 7, 8], and superconducting
qubits [9], with recent evidence suggesting that extant transmon-based quantum
computers naturally tread a delicate line between localization and chaos [10].
A natural, yet hitherto unrealized, setting in which to explore MBL is that of
semiconducting quantum dot arrays (QDA). Such systems are promising simulators
of fermionic systems in both 1D and 2D [11, 12], they are highly tunable and
different lattice geometries can be readily fabricated: in short an ideal
testbed for MBL. Despite this, modern arrays are realistically limited to few
dots, readout can be noisy, and - as they are extremely sensitive to
environmental electrostatic discharge - they can be damaged during
fabrication, handling, or general use during the experimental process. Thus,
whilst the detection of MBL in realistic current-generation quantum dot arrays
is both a crucial proof-of-concept for such arrays as generic quantum
testbeds, it is also fraught with difficulties. A number of questions
naturally arise: can current-generation realistic arrays access MBL regimes?
How can we reliably identify MBL in such arrays? And - given how fragile these
systems are - what are the minimal measurements required to do so?

Figure 1:  (a) The six-dot design for the specific experimental device we
characterize. In section III we extract rough parameter ranges for the
theoretical model that the device simulates from the pair of dots defined by
the gates $G_{19}-B_{15}-B_{13}-B_{11}$ and addressed by plunger gates
$P_{14}$ and $P_{12}$ respectively. (b) An SEM image of an array we fabricated
which is similar to the design we characterize. Slight blurring of the SEM
image is due to a layer of protective PMMA with an approximate thickness of
200 nm.

In this article we address the above questions by first characterizing the
double-dot properties of a state-of-the-art device and extrapolating the rough
parameter ranges of an extended Fermi-Hubbard model that such a device can
simulate. We then analyze this model numerically, investigating a variety of
quantities in both bulk and local variants. These quantities require
measurements that range from density operator tomography of half the system to
simple charge sensing on two sites.

In section II we introduce and discuss the model that quantum dot arrays
simulate. In section III we discuss our experimental devices, namely one-
dimensional lateral arrays of electrostatically defined quantum dots, and then
characterize such a device to extract rough ranges for the theoretical model
parameters from experimental data. We then define several quantities in both
bulk and local variants in section IV which can be used to differentiate MBL
from the other phases of the model. Finally, in section V we analyze the model
across the extracted parameter ranges and use the aforementioned quantities to
develop a protocol for identifying MBL with minimal measurements on a
realistic device.

Figure 2: (a) A typical experimental honeycomb cell of the charge-stability
diagram obtained by differential conductance measurements on the two dots
defined by the gates $G_{19}-B_{15}-B_{13}-B_{11}$ in the middle of a device
similar to that shown in fig. 1. Bright (dark) regions indicate higher (lower)
measured values of thhe differential conductance as a function of the two
local plunger gate voltages $P_{14}$ and $P_{12}$. (b) A Gaussian smoothing of
the raw data shown in panel (a) with a schematic overlay of how the parameters
of the theoretical model that the device simulates are extracted from the
geometry of the honeycomb cell.

##  II Model and Theory

The theoretical model which our experimental quantum dot array simulates is
that of an extended Fermi-Hubbard model restricted to a single species of
electron, described by the Hamiltonian

$H=\tau\sum_{j}^{L-1}\left(c^{\dagger}_{j}c_{j+1}+\text{h.c.}\right)+V\sum_{j}^{L-1}n_{j}n_{j+1}+\sum_{j}^{L}h_{j}n_{j}$
(1)

where $n_{j}=c^{\dagger}_{j}c_{j}$ is the number operator at site $j$. The
parameters $\tau$ and $V$ are the tunnelling and nearest-neighbour coulomb
interaction energies respectively, and the $h_{j}$ are random energies drawn
uniformly from the interval $[-h,h]$; with $h$ tuning the overall disorder
strength. We have assumed (i) a single species of electron and (ii) a single
active energy level per site.

Prototypically, these assumptions are rarely made a priori, and an on-site
Coulomb interaction tuned by $U$ is additionally considered. This, however,
introduces an array of new tunnelling and interaction terms which dramatically
confuse the process of extracting Hamiltonian parameters from experimental
data; for this reason we assume that $U$ is sufficiently large that it
integrates out of our model entirely (we will later demonstrate that this
assumption is well-founded in the context of our physical devices). Together,
assumptions(i) and (ii) restrict us to charge number fluctuations of one
electron per site: the simplest experimental setting in which we can find MBL,
and the ideal setting in which to analyze the accessibility of MBL in QDA. In
experiment, these two assumptions can be imposed by (i) applying a magnetic
field during initialization to spin-polarize the electrons and (ii) ensuring
that all energy scales $\tau,V,h$ are kept much lower than the on-site
charging energy - which is related to $U$.

We note here several features of the Hamiltonian of eq. 1 which are of direct
relevance to our analysis. Firstly it is entirely a function of the number
operators for $\tau=0$, and as such is trivially diagonalizable at this point
in the number basis; this corresponds to the ’classical’ picture of quantum
dots in the constant interaction model, wherein the system’s ground states are
classical ground states of charges on a network [13]. Secondly, the system
maps directly onto an extended XXZ model (see appendix A) via the Jordan-
Wigner transformation, and so inherits an insulating phase at $V/\tau=2$. For
$V/\tau=0$ the system is non-interacting and, in the thermodynamic limit,
should localize for all $h/\tau>0$; this is Anderson localization - which we
do not consider here. For $0<V/\tau<2$ the system is conducting and
interacting and so should many-body localize for sufficient $h/\tau$, whilst
at $V/\tau>2$ the system is insulating. For these reasons, the ergodic-MBL
transition can only be meaningfully discussed in the regime $0<V/\tau<2$, and
the ability to differentiate the ergodic regime, interaction-induced
insulation due to high $V/\tau$, and disorder-induced MBL becomes critically
important when we seek to definitively identify the last in an exploratory
experimental context. Finally, the systems we consider are very small in order
to model the realistic scale of fully tunable experimental quantum dot arrays,
as such the system is suspect to a range of pathologies. Edge effects are non-
trivial, the phase transitions are expected to smear out - with e.g. Anderson
localization visible for small, non-zero, $V/\tau$ \- and the nature of the
MBL transition being generally suspect, potentially not reflecting behaviour
in the thermodynamic limit at all.

In particular, recent research suggests that attempting to isolate the
ergodic-MBL transition in such small systems is difficult - it is hard to make
declarative statements about the thermodynamic transition without accessing
both exponential time and length scales in microscopic analyses [14, 15], and
the small-system transition may belong to a different universality class than
the transition in the thermodynamic limit [16]. Rather it is better to
identify different regimes and investigate their properties away from the
ergodic-MBL critical line. For this reason, we do not attempt to
systematically investigate criticality in this article, rather determine the
conditions under which the different regimes can be differentiated in
realistic experimental quantum dot arrays.

(a) Bulk von Neumann Entropy

(b) Bulk Imbalance

(c) Bulk Number Entropy

(d) Local von Neumann Entropy

(e) Local Imbalance

(f) Local Number Entropy

Figure 3: Disorder-averaged late-time quantities across the full $V-h$ phase
diagram for the realistic system; initialized in a charge-density wave
configuration. The realistic system consists of $L=8$ sites, total evolution
times of $\tau t_{f}=150$, no time-averaging (the final value at $t_{f}$ is
simply read out), and disorder-averaging over only $50$ samples per $(V,h)$
coordinate.

##  III Experimental System and Accessible Parameter Ranges

The experimental devices we consider are lateral arrays of electrostatically
defined quantum dots formed by selectively depleting electrons using nano-
fabricated gate electrodes on the surface of a GaAs/AlGaAs heterostructure.
The specific device we use as a benchmark has a gate pattern which defines a
linear array of six quantum dots, the design of which is shown in fig. 1(a). A
scanning electron microscope (SEM) image of a similar (eight-dot) device we
fabricated is shown in fig. 1(b), where the positions of the quantum dots are
illustrated by red circles. These are both simply larger versions of the
designs often used in multi-dot experiments (see e.g. Refs. [17, 11]).

Tunnelling rates between adjacent dots are controlled through the voltages
applied to the barrier gates $B_{j}$ between neighbouring dots, where $j$ is
the gate number. On-site chemical potentials are controlled by voltages
applied to the plunger gates $P_{j}$. The leftmost and the rightmost dots are
also tunnel-coupled to the left and right reservoirs, respectively. The long
middle bar gate, labelled $G_{19}$ in fig. 1(a)), is the top barrier for all
wire gates below; and the three sensing gates, labelled $S_{1(2)(3)}$ in fig.
1(a), are quantum point contact charge detectors. The non-linear conductance
characteristics of these detector gates can be used as a sensitive probe of
the local electrostatic environment [18], which in turn can be used to measure
local charge fluctuations. In practice, individual gate voltages affect not
only the parameters they are designed to control but, through capacitive
cross-talk, also affect other electrochemical potentials and tunnel barriers.
However, this effect can be compensated for by using virtual gates: linear
combinations of multiple gate voltages chosen such that only a single
electrochemical potential or tunnel barrier is addressed [11, 19].

The design of the specific six-dot device use to extract our model parameters
is shown in fig. 1(a). The fabrication of this device uses a Si-doped
$\text{GaAs}/\text{Al}_{x}\text{Ga}_{1-x}\text{As}$ heterostructure, with a
two-dimensional electron gas $90\text{nm}$ below the surface, a mobility of
$9\times 10^{5}$ $\text{cm}^{2}/\text{Vs}$, and an electron concentration of
$1.62\times 10^{11}\text{cm}^{-2}$. All gates are fabricated in a layer of
Ti/Au of thickness $5/20\text{nm}$, evaporated on the bare substrate. The
device was cooled in a dilution refrigerator with a base temperature of $T\sim
70\text{mK}$. The electron temperature, however, is estimated to remain at
$\sim 100\text{mK}$. Extracting from their individual Coulomb diamonds, the
on-site charging energy $E_{C}$ is estimated to be $\sim 1.3\text{meV}$, and
the zero-dimensional level spacing of an individual dot to be $\sim
400\mu\text{eV}$.

We do not characterize charge-qubit coherence times, but take them to be
approximately $1-10$ns, which is in conservatively line with other
characterizations in similar systems [20, 21]. Though we note that silicon-
based devices and spin-qubits have coherence times orders of magnitude greater
[22, 23, 24, 25], and may be a lucrative setting in which to investigate MBL
as well.

We characterize the two dots in the middle of our array, defined by the gates
$G_{19}-B_{15}-B_{13}-B_{11}$, by scanning the applied plunger gate voltages
$P_{12}$ and $P_{14}$ and measuring the differential conductance across the
double-dot system. This yields a charge-stability diagram comprised of
‘honeycomb’ cells wherein the boundaries between stable electronic
configurations admit the flow of current and appear as bright regions (high
differential conductance). A typical honeycomb cell we obtained from this
analysis is shown in fig. 2 (a). From this cell we can determine the
properties of the extended Fermi-Hubbard model that these two dots simulate
(for a detailed discussion, see Ref. [13] and the thesis of T. Hensgens in
Ref. [26]).

We now turn our attention to the extraction of the parameter ranges (for
$U/\tau$ and $V/\tau$) of our theoretical model eq. 1 from the experimental
data shown in fig. 2(a). This informs the model regimes that the experimental
device can access. To facilitate this process we first process the data using
a Gaussian kernel-density estimate, producing the smoothed honeycomb cell
shown in fig. 2(b).

The energy $U$ is the simplest parameter to extract, it is simply the cost of
adding a new electron to the dot, including both charging and zero-dimensional
energies [13]. This is simply the vertical distance between two classical
ground state electron configurations, i.e. the potential we need to overcome
to add a single electron to a single site [11]. In fig. 2, we have selected a
honeycomb cell without a contribution from the zero-dimensional level spacing;
thus this article works with a ‘worst case’ scenario where $U/\tau$ is not
strengthened by zero-dimensional effects. Even in this regime, we later find
$U/\tau$ sufficiently large to justify our assumption (ii) in section II.

The nearest-neighbour Coulomb repulsion $V$ is related to the shortest
distance between two phases which differ by a single additional electron on
_both_ sites, i.e. the energy required to add two electrons to neighbouring
sites after overcoming the necessary on-site energy requirements. Due to
hybridization caused by $\tau$, this distance is proportional to $V+2\tau$
[26].

The tunelling energy $\tau$ affects the classical charge stability diagram
mainly through the hybridization of neighbouring classical ground states of
identical total electron number; this broadens the distance between triple
points and phase boundaries, and causes rounding of the phase boundaries near
the classical triple points. Thus $\tau$ can be extracted in several ways.
Firstly, by analyzing the extent to which phase boundaries are curved [27].
Secondly, by extracting the tunnelling rate from Larmor oscillations observed
over time as in [28, 29]. Thirdly, by numerically fitting a line cut of the
charge-stability diagram along a detuning axis $V_{1}-V_{2}$ (the red double-
headed arrow in fig. 3(f)) to the analytic form of the steady-state
conductance through an open double-dot model, or the response of a sensing
dot; both of which should broaden with increasing $\tau$ [13, 11, 12]. As
triple points are separated by a distance proportional to $V+2\tau$, we
estimate rough ranges on this broadening (and thus $\tau$) by estimating the
maximum and minimum radii of the smeared-out triple points (shown as black
rings in fig. 3(f)). Whilst a detailed characterization of $\tau$ is critical
in experiment, these rough estimates are sufficient to give us the range of
possible $\tau$ values across which we must understand MBL in order to
determine its accessibility in QDA 111Moreover, $\tau$ can be directly
controlled by tuning the voltages applied to the barrier gates which separate
dots, as evidenced by the ability to ‘pinch off’ a dot (isolate it) before
e.g. a measurement. This makes $\tau$ the parameter which is easiest to
control in situ \- without having to fabricate a new device..

Altogether, approximate ranges for $U/\tau$, and $V/\tau$ can be determined by
the features of the charge-stability diagram. The relationships between these
parameters and the geometry of a typical honeycomb cell are shown as an
overlay onto the smoothed data in fig. 2(b). We manually identify suspected
triple points - denoted by black crosses - and take minimum (inner) and
maximum (outer) broadening radii - denoted by the inner and outer black
circles respectively. The phase boundaries are derived from lines connecting
these triple points, and the range of the radii yields ranges of values for
$\tau$, $V+2\tau$, and $U$ as annotated, and as discussed above. Our resulting
estimated ranges are summarized in table 1. In each case, the on-site Coulomb
interaction $U$ is found to be much larger than both $\tau$ and $V$,
justifying our assumption that the corresponding model has one active energy
level per site. These estimated ranges encompass extant characterizations of
multi-dot arrays in e.g. Ref. [11]. Finally, we note that the $h/\tau$ is
freely tunable by altering the on-site chemical potentials by changing plunger
gate voltages, and is thus only limited by the restriction that $U\gg h$.

Parameters | Minimum $\tau$ | Maximum $\tau$
---|---|---
$V/\tau$ | 3.727 | 0.603
$U/\tau$ | 12.941 | 4.792
Table 1: Table of upper and lower bounds on the considered parameter ranges
for eq. 1 extrapolated from the features of the experimental charge stability
diagram of fig. 2.

##  IV Probing Quantities

We consider three quantities: the von Neumann entropy $S$, the widely-used
imbalance $\mathcal{I}$, and the number entropy $S_{N}$, which vary first in
their experimental accessibility and secondly - as we show in section V \-
their ability in differentiating regimes in the phase diagram of eq. 1.

The first quantity of interest is the von Neumann entropy which - in a
bipartite pure state - unambiguously quantifies the entanglement across the
bi-partition. It has been used extensively in the context of MBL theory and
experiment, and serves here as a benchmark for the other quantities [31, 32,
33, 34, 35]. For a given subsystem $\rho(t)$ of size $L_{\rho}$ it is defined
as

$S\left(\rho(t)\right)=-\frac{1}{L_{\rho}}\text{Tr}\left[\rho(t)\log_{2}\rho(t)\right]$
(2)

where we have chosen a logarithm base 2 such that the resulting quantity is
measured in bits, and that it saturates (for a single species of fermion with
local dimension 2, and given that $L_{\rho}/L\leq 1/2$) to unity. In
experiment, the calculation of eq. 2 would require full tomography of the
density operator of the region of interest - a prohibitively expensive and
difficult task - but it serves as crucial benchmarks nonetheless.

The second quantity we consider is the imbalance, widely used in MBL [5, 36,
37, 38, 39]. The imbalance is used to determine how much a system has deviated
from an initial charge configuration, it is directly related to standard auto-
correlation functions. We define it as

$\mathcal{I}(\rho(t))=\frac{2}{L_{\rho}}\sum_{j}\text{Tr}\left[\rho(0)n_{j}\right]\text{Tr}\left[\rho(t)n_{j}\right]$
(3)

where $j$ runs over the physical sites of the subsystem $\rho(t)$ and where -
in the case of an initial charge density wave state such that $n_{j}$ has
$\rho(0)$ as an eigenstate - it reduces to the conventional statement of the
imbalance as the difference between occupancy numbers on odd and even sites.
If the charge configuration of $|\psi(t)\rangle$ becomes uncorrelated to the
initial configuration of $|\psi(0)\rangle$ then the imbalance saturates to
zero $\mathcal{I}(t)=0$, whereas if they remain (anti-)correlated it persists
as a finite non-zero value $\mathcal{I}(t)>0$ ($\mathcal{I}(t)<0$).In
experiment $\mathcal{I}(\rho(t))$ requires only charge sensing on the relevant
sites, which is significantly easier than the state tomography required by the
von Neumann entropy.

Finally we consider the number entropy, a quantity which has seen some use in
MBL [34, 40, 41], and which is simply the entropy of the discrete probability
distribution $p(\rho(t),n)$ of finding $n$ particles in the subsystem
$\rho(t)$, defined as

$S_{N}(t)=-\frac{1}{L_{\rho}}\sum_{N}p(\rho(t),n=N)\log_{2}p(\rho(t),n=N).$
(4)

This quantity is directly related to the von Neumann entropy by
$S=S_{N}+S_{C}$ where $S_{C}$ is the configurational entropy: the contribution
to the entanglement due to configurational correlations. We can compute the
distributions from the density operator by constructing projectors
$P_{N}=\sum_{r}|N_{r}\rangle\langle N_{r}|$ where $|N_{r}\rangle$ are the
$N$-particle states in the number basis that span the reduced Hilbert space of
$\rho(t)$. The probability is then given by
$p(\rho(t),n=N)=\text{Tr}[\rho(t)P_{N}]$. We calculate an ergodic limit of the
number entropy in appendix D, which we use to benchmark our numerical results.

We consider both bulk and local variants of the quantities by considering
three different subsystems, the full system
$\rho_{\text{f}}(t)=|\psi(t)\rangle\langle(t)\psi|$, half of the system
$\rho_{\text{hc}}(t)$, and two sites in the middle of the system
$\rho_{\text{2}}(t)$. The quantities are computed using the equations in
section IV using the bulk or local reduced density operators according to
Table table 2.

Quantity | Bulk | Local
---|---|---
VN Entropy | $S^{(b)}(t)=S(\rho_{\text{hc}}(t))$ | $S^{(l)}=S(\rho_{\text{2}}(t))$
Imbalance | $\mathcal{I}^{(b)}(t)=\mathcal{I}(\rho_{\text{f}}(t))$ | $\mathcal{I}^{(l)}(t)=\mathcal{I}(\rho_{\text{2}}(t))$
Number Entropy | $S_{N}^{(b)}(t)=S_{N}(\rho_{\text{hc}}(t))$ | $S_{N}^{(l)}(t)=S_{N}(\rho_{\text{2}}(t))$
Table 2:  Summary of bulk and local variants of the quantities discussed in
section IV. Bulk quantities are calculated over the state of the full system
$\rho_{\text{f}}(t)$ or half the chain $\rho_{\text{hc}}(t)$, local quantities
are calculated over two sites in the middle of the device $\rho_{2}(t)$.
Experimentally, the imbalance and number entropy require identical local
charge-sensing measurements only, and highly experimentally tractable.

In experiment the number entropy requires the same charge sensing measurements
as the imbalance, but yields signatures similar to the von Neumann entropy.
This becomes useful in section V where we find that the number entropy can
identify phases that the imbalance cannot without requiring the prohibitively
difficult state tomography of the full von Neumann entropy. Moreover, the
local variants of the quantities shown in the last column of table 2, require
measurements only on two sites; making their calculation even simpler in
experiment.

(a) Bulk Imbalance and Number Entropy against $h/\tau$

(b) Bulk Imbalance and Number Entropy against $V/\tau$

(c) Local Imbalance and Number Entropy against $h/\tau$

(d) Local Imbalance and Number Entropy against $V/\tau$

Figure 4:  Slices of the phase diagrams of fig. 3(f) (the realistic $L=8$
model) for (a) and (c) fixed values of $V/\tau$ on both sides of the
transition point between thermal and insulating phases at $V/\tau=2$, and (b)
and (d) fixed values of $h/\tau$ on both sides of the crossover from thermal
to MBL regimes at $h/\tau=3.5$; error bars shown where visible.

##  V Identifying Many-Body Localization in Realistic Dot Arrays

Here we investigate the phase diagram of eq. 1 at time $t_{\text{f}}$ after
initialization in the charge density wave state wherein every other site is
occupied at time $t=0$. We vary $V/\tau$ and $h/\tau$, and analyze the
quantities (both bulk and local variants) discussed in section IV. Given the
upper and lower bounds extracted in section III, we investigate the phase
diagram across the intervals $V/\tau\in[0,4]$ and $h/\tau\in[0,6]$. In the
main text, we restrict ourselves to realistic simulations of the system,
namely: system sizes of $L=8$ sites, intermediate time scales of $\tau
t_{\text{f}}/h=150$ which (for reasonable tunnelling energies $\tau\sim
50-100\mu eV$) are within typical charge-qubit coherence times for these
systems ($t_{\text{f}}\sim 1-10\text{ns}$) [20, 21], no time-averaging - we
simply read out the value at the final time $t_{\text{f}}$ \- and the number
of disorder samples limited to $50$ realizations. Additionally, we introduce
large absolute $\pm 0.1$ and relative $\pm 5\%$ uncertainties (box-
distributed) in the parameters $h/\tau$ and $V/\tau$ such that the couplings
are non-isotropic and random for each disorder realization. This accounts for
small changes in these couplings due to cross-talk, and errors in the
fabrication and characterization of the device. In the appendix (see appendix
B), we additionally consider an ideal system of $L=10$ sites in which all the
above restrictions are lifted; this serves to benchmark the realistic analysis
presented here.

The results of this analysis are shown in fig. 3(f), wherein panels (a), (b),
and (c) show the bulk von Neumann entropy, Imbalance, and Number entropy at
$t_{\text{f}}$ respectively, and panels (d), (e), and (f) show the local
variants. Red dashes lines indicate the expected transition at $V/\tau=2$ and
an ergodic-MBL crossing at $h/\tau\sim 3$. The phase diagrams of the von
Neumann entropy and number entropy are qualitatively similar, with a build-up
of both entropies in the thermal regime that falls off as $h/\tau$ or $V/\tau$
increase. Surprisingly, the imbalance doesn’t seem to register the metallic-
insulator transition at $V/\tau=2$, instead staying close to its thermal value
of zero for all values of $V/\tau$ for sufficiently low disorder strength
$h/\tau$. This suggests that the imbalance alone cannot unambiguously detect
the presence of the MBL regime: if a transition is seen in $h/\tau$, it is
entirely possible that the system is already in an insulating regime that the
imbalance is simply agnostic to. As such, differentiation between the thermal,
MBL, and insulating regimes requires a different quantity. The number entropy
requires the same measurements as the imbalance (i.e. charge sensing as
opposed to full state tomography) but is also sensitive to the transition in
$V/\tau$. Thus by post-processing the results of a large number of charge
measurements in two different ways, the imbalance and number entropy can both
be calculated and compared, with the former sensitive only to the MBL
transition, and the latter sensitive to both the MBL and insulating
transition.

Perhaps the most striking feature of fig. 3(f) is that the qualitative
features of the bulk and local phase diagrams for individual quantities are
consistent. For the purpose of differentiating the thermal, MBL, and
insulating regimes, local measurements on a few sites seem to suffice. In
conjunction with the above discussion, this leads to a simple protocol for
detecting MBL in realistic QDA: perform charge-sensing measurements on a few
dots in the middle of the array, then post-process these measurements
differently to construct the local imbalance and local number entropy.
Reapeating this protocol as we scan $h$ from a low to a high value will show a
transition in both the number entropy and imbalance if the system is MBL, and
only the imbalance if the system is insulating.

We suggest that both the sensitivity of the number entropy and insensitivity
of the imbalance to the transition in $V/\tau$ can be explained by a type of
‘rolling’ behaviour, in which the charge-density wave moves coherently through
the system. In essence, $V/\tau$ is so strong that excitations cannot exist in
neighbouring sites (similar to Rydberg-blockaded systems [42, 43, 44]) and due
to the fact that total particle number is conserved: the dynamical state of
the system oscillates between the states $|\circ,\bullet,\circ\cdots\rangle$
and $|\bullet,\circ,\bullet\cdots\rangle$ (where $\circ$ corresponds to an
empty dot, and $\bullet$ to an occupied dot). This would register an imbalance
which looks thermal, but a probability distribution $p(\rho(t),n=N)$ sharply
peaked around $N=L_{\rho}/2$ \- and thus a near-zero number entropy. We
support this suggestion with a brief numerical analysis in appendix C in which
we investigate the overlap of a dynamical state in a single, typical, disorder
profile, with both charge-density wave states. Interestingly, we see similar
behaviour in systems with an odd number of sites, suggesting a more
complicated explanation which we defer to future study.

We also examine individual slices of the phase diagrams of fig. 3(f) in fig.
4(d), to emulate the kind of results we would expect to see from an experiment
which scans $h/\tau$ or $V/\tau$ whilst holding the other constant. Such an
experiment is far simpler (especially in the case of scanning $h/\tau$ which
can be freely tuned by modifying plunger gate voltages) than determining the
full phase diagram. Panels (a) and (b) of fig. 4(d) show the bulk imbalance
and number entropy as we hold either $V/\tau$ and $h/\tau$ fixed on both sides
of the MBL and metal-insulator transition and scan the other parameter
respectively. Panels (c) and (d) of fig. 4(d) show the local variants. The
black dashed line in each panel shows the thermal value of the number entropy,
a benchmark of ergodicity in the system, which we derive in appendix D.

We find that, as expected, the form of the imbalance is agnostic to changes in
the interaction strength $V/\tau$; with each pair of imbalance curves in each
panel having similar functional forms. The only difference is in panels (b)
and (d) in which the imbalance is significantly higher for high $h/\tau$ and
easily differentiated from the low $h/\tau$ case; though they are roughly
invariant as we scan $V/\tau$.

The number entropy is clearly sensitive to both parameters, staying fixed for
sufficiently high $V/\tau$ or $h/\tau$ as the other parameter is scanned; but
showing a clear decrease with increasing $V/\tau$ or $h/\tau$ as the other is
held at a low constant value. Panel (b) shows this difference in the behaviour
of the number entropy for different values of $V/\tau$ most clearly. The
high-$h/\tau$ number entropy curve is roughly constant, whilst the
low-$h/\tau$ curve shows a decrease with increasing $V/\tau$. Moreover, the
number entropies are most readily differentiated in the thermal regime where
they diverge, meeting again for sufficiently $h/\tau$. Importantly, this
difference in behaviour becomes much harder to distinguish in (c); in which
both number entropy curves are much closer together and their functional forms
are harder to differentiate. This means that fixing $V/\tau$ and extracting
local imbalance and number entropy values for different $h/\tau$ may be - by
itself - insufficient to differentiate between MBL and insulating behaviour.
The experimentalist may have to supplement this analysis by either considering
bulk quantities instead, or by finding a way to vary $V/\tau$ such that the
imbalance and number entropy curves in panels (b) or (d) can be directly
differentiated instead.

We note that our decision to restrict our analysis to a small number of
disorder realizations, and our use of large absolute and relative
uncertainties in all parameters, represents a worst-case scenario. Thus,
experimental results are likely to be clearer than those presented in this
article. Results for an ideal system are shown in appendix B, in which
differentiating between the MBL and insulating regimes becomes a much easier
task. Despite this, differentiation between these regimes is possible even in
our conservative model. Provided that the charging energies are high enough
that $h/\tau$ can be freely tuned, local charge sensing is enough to identify
MBL, and unambiguously differentiate it from insulating behaviour due to
strong electron-electron interactions, in current-generation QDA.

##  VI Conclusions

Our findings show that MBL is accessible in state-of-the-art QDA, but that its
identification is not a simple task. The key limitation to detecting MBL in
modern quantum dot arrays is ensuring high enough on-site charging energies
such that they cannot be surmounted by the large applied random chemical
potentials required to localize the system. This may be improved by e.g.
decreasing $\tau$, but that requires a corresponding increase in charge-qubit
coherence times such that total evolution times are large enough to see
localization. This could perhaps be achieved by designing free-standing dots
that isolate the system from phonons. Working within these limitations we
first characterized an experimental device and extrapolated that
characterization into a worst-case model. We then numerically determined the
phase diagram of the device as a function of disorder and interaction
strength; identifying an insulating regime which may be mistaken for MBL. We
find that the widely-used imbalance is agnostic to this phase, and propose an
alternative protocol based on the number entropy - which requires the same
measurements as the imbalance - and which successfully differentiates MBL from
the thermal and insulating regimes. This protocol relies only on local charge-
sensing measurements, which are readily accessible in modern quantum dot
experiments. In addition. we find that performing these measurements on two
sites in the middle of the system yields qualitatively similar results as bulk
analyses, drastically reducing the number and complexity of measurements
required.

##  VII Acknowledgements

The authors acknowledge the EPSRC grant Nonergodic quantum manipulation
EP/R029075/1. The authors also thank Dr. G. Stefanou for his advice and
technical experimental support.

##  VIII Author Contributions

A.N.-K., G.J., C.G.S., and S.B. devised the project and the main conceptual
ideas. A.N.-K. developed the theoretical model and carried out the numerical
simulations. G.J. designed, planned, and carried out the experiments. P.A.,
T.A.M., and D.A.R. contributed to the fabrication of the devices. A.N.-K. and
G.J. wrote the manuscript with support and contributions from C.G.S. and S.B.

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##  Appendix A Mapping the Fermi-Hubbard Model to the XXZ Spin Chain

The spinless Fermi-Hubbard model of eq. 1 can be mapped onto an spin-1/2 XXZ
model by means of a standard Jordan-Wigner transformation. We start by
identifying the two states of each site - occupied or unoccupied - with the
two spin-1/2 states - spin-up and spin-down. We then relate the fermionic
creation and annihilation operators with the spin-1/2 raising and lowering
operators $c^{\dagger}_{j}\to S^{+}_{j}$ and $c_{j}\to S^{-}_{j}$, and impose
the standard fermionic anti-commutators by applying highly non-local Jordan-
Wigner string operators such that the final transformation takes the form:

$\displaystyle
c_{j}^{\dagger}=\left(\prod_{k<j}\sigma^{z}_{j}\right)S^{+}_{j}$ (5)
$\displaystyle c_{j}=\left(\prod_{k<j}\sigma^{z}_{j}\right)S^{-}_{j}$ (6)

where the $\sigma^{z}_{j}$ is the standard Pauli Z operator. In practice, due
to the fact that $(\sigma^{z}_{j})^{2}=1$, many of these Jordan-Wigner strings
cancel and one can often manipulate the final Hamiltonian into a local form.
Substituting eq. 5 and eq. 6 into the Hamiltonian eq. 1, and simplifying the
Jordan-Wigner strings, yields the model

$\displaystyle H=\tau$
$\displaystyle\sum_{j}^{L-1}\left(S_{j}^{+}\sigma_{j}^{z}S_{j+1}^{-}+\text{h.c.}\right)$
$\displaystyle+V\sum_{j}^{L-1}S_{j}^{+}S_{j}^{-}S_{j+1}^{+}S_{j+1}^{-}+\sum_{j}^{L}h_{j}S_{j}^{+}S_{j}^{-}$

which we can simplify further by noting that
$S_{j}^{+}S_{j}^{-}=S^{z}_{j}+1/2$, that $S_{j}^{+}\sigma_{j}^{z}=-S_{j}^{+}$,
and by invoking the definition of $S^{\pm}_{j}=S_{j}^{x}\pm iS^{y}_{j}$; where
$S_{j}^{\alpha}$ are the standard spin-1/2 operators. Under these
simplifications, the model now takes the form

$\displaystyle H=-2\tau$
$\displaystyle\sum_{j}^{L-1}\left(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}\right)$
$\displaystyle+V\sum_{j}^{L-1}\left(S_{j}^{z}S_{j+1}^{z}+S_{j}^{z}+S_{j+1}^{z}+1\right)$
$\displaystyle+\sum_{j}^{L}h_{j}\left(S_{j}^{z}+1/2\right)$

wherein the negative overall phase of the tunnelling term proportional to $t$
is irrelevant to the physics of the system: we can e.g. simply relabel every
other site such that, for odd $j$, $S^{x/y}_{j}\to-S_{j}^{x/y}$, and
$S_{j}^{z}$ is left unchanged. We eliminate all constant terms which do not
contribute to the physics of the system, and note that - since the Hamiltonian
conserved total particle number, its Jordan-Wignerization equivalently
conserves total spin. Thus we can neglect all terms proportional to
$\sum_{j}^{L}S_{j}^{z}$, which appear in the sum over single-site operators
proportional to $V$ such that
$\sum_{j}^{L-1}S_{j}^{z}+S_{j+1}^{z}=-S_{1}^{z}-S_{L}^{z}+\sum_{j}^{L}S_{j}^{z}$.
Collecting terms, and performing these substitutions gives us an extended XXZ
Hamiltonian

$\displaystyle H=2\tau$
$\displaystyle\sum_{j}^{L-1}\left(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}+\frac{V}{2t}S_{j}^{z}S_{j+1}^{z}\right)$
$\displaystyle-V\left(S_{1}^{z}+S_{L}^{z}\right)+\sum_{j}^{L}h_{j}S_{j}^{z}$

which exhibits an XXZ transition at $V=2\tau$, makes clear the importance of
edge effects in small systems by explicit inclusion of the dangling
$S_{1}^{z}$ and $S_{L}^{z}$ operators.

(a) Bulk von Neumann Entropy

(b) Bulk Imbalance

(c) Bulk Number Entropy

(d) Local von Neumann Entropy

(e) Local Imbalance

(f) Local Number Entropy

Figure 5: Time and disorder-averaged steady-state quantities across the full
$V-h$ phase diagram for the ideal system; initialized in a charge-density wave
configuration. The ideal system consists of $L=10$ sites, exponential total
evolution times of $\tau t_{f}=10^{10}$, time-averaging over the final third
of the total evolution time, and disorder-averaging over 512 samples per
$(V,h)$ coordinate.

##  Appendix B Ideal phase diagrams for a system of size L=10

In section V of the main text, we analyzed the von Neumann entropy, the
imbalance, and the number entropy in both bulk and local variants (see section
IV of the main text for definitions and discussion) respectively. There we
considered a highly conservative realistic scenario: with constraints placed
on evolution times, number of disorder realizations, and parameter
characterization errors. Here we present similar results but for an ’ideal’
system of $L=10$ sites, exponential time scales of $\tau
t_{\text{f}}=10^{10}$, late-time time-averaging - wherein we average over the
final third of the evolution time to eliminate remaining fluctuations - and
$512$ disorder realizations. Additionally, we eliminate the absolute and
relative uncertainties in the parameters $h/\tau$ and $V/\tau$ in the main
text.

(a) Bulk Imbalance and Number Entropy against $h/\tau$

(b) Bulk Imbalance and Number Entropy against $V/\tau$

(c) Local Imbalance and Number Entropy against $h/\tau$

(d) Local Imbalance and Number Entropy against $V/\tau$

Figure 6: Slices of the phase diagrams of fig. 5(f) (the ideal $L=10$ model)
for (a) and (c) fixed values of $V/\tau$ on both sides of the transition point
between thermal and insulating phases at $V/\tau=2$, and (b) and (d) fixed
values of $h/\tau$ on both sides of the crossover from thermal to MBL regimes
at $h/\tau=3.5$; error bars shown where visible.

The results of this analysis of an ideal system are shown in fig. 5(f), and
are qualitatively similar to the corresponding results in the main text: the
imbalance is agnostic to changes in $V/\tau$ and cannot detect the insulating
regime at $V/\tau>2$, this is in contrast to the von Neumann and number
entropies which identify both this transition and the ergodic-MBL crossover at
$h/\tau\sim 3.5$. Strikingly, the local variants which use measurements on
only two sites in the middle of the array are qualitatively similar to the
bulk variants, indicating that local measurements are enough to identify the
different regimes.

Figure 7:  Numerical evidence of coherent oscillation of a charge-density wave
between two configurations as $V/\tau$. For a system initialized in the state
$|\bullet,\circ,\bullet,\cdots\rangle$ (a) shows overlap with the initial
state and (b) shows the overlap with the inverted state
$|\circ,\bullet,\circ,\cdots\rangle$. System hamiltonian is a typical
disorder-realization of strength $h/\tau=1$.

Individual slices of the phase diagrams of the ideal system shown in fig. 5(f)
are shown in fig. 6(d), wherein we fix $V/\tau$ or $h/\tau$ and scan the other
respectively. The behaviours noted in section V of the main text are much
clearer in the larger, ideal system. The imbalance is clearly agnostic to
changes in $V/\tau$, as evidenced by its constancy in panels (b) and (d) of
fig. 6(d). Moreover, the imbalance curves in panels (a) and (c) respectively
are almost identical. This supports the protocol we propose which requires
different post-processing on charge measurements to construct both the
imbalance and number entropy in tandem. The number entropy has clearly
different behaviour as both $V/\tau$ and $h/\tau$ vary, staying fixed for
sufficiently high $V/\tau$ or $h/\tau$. Panel (b) shows this behaviour most
clearly, with both the imbalance and number entropy constant with increasing
$V/\tau$ for high disorder, and with imbalance constant but number entropy
decreasing with increasing $V/\tau$ for low disorder.

##  Appendix C Oscillating Insulating State

In the main text we note that a potential explanation for observing thermal
values for the imbalance, but low number entropies, is by a ‘rolling’
behaviour in which the state of the system moves coherently from left to
right. Here we provide some preliminary evidence for this explanation.

We initialize a system in the charge-density wave
$|\psi(0)\rangle=|\bullet,\circ,\bullet,\cdots\rangle$ (where $\circ$
corresponds to an empty dot, and $\bullet$ to an occupied dot) and select a
typical disorder profile of strength $h/\tau=1$. In fig. 7 we then investigate
the overlap of the state $|\psi(t)\rangle$ with $|\psi(0)\rangle$ in panel (a)
and its overlap with the complementary charge-density wave state
$|\circ,\bullet,\circ,\cdots\rangle$ in panel (b). We compute the overlap as
the expectation value squared:
$|\langle\psi(t)|\psi_{\text{target}}\rangle|^{2}$. For low $V/\tau$ (thermal
regime), we see both curves drop rapidly to low values and remain there. But
as we increase $V/\tau$ we see sharp revivals in the overlap of
$|\psi(t)\rangle$ with both charge-density wave states. The first peak in
panel (a) (after $t=0$) occurs after the first peak in panel (b) and at
approximately the same time difference as the first peak in panel (b) does
from $t=0$. This suggests that our ‘rolling’ behaviour explanation may be a
good approximation to the dynamics of the state; though a more detailed
investigation is warranted. Such an investigation is beyond the scope of this
article.

##  Appendix D Derivation of the Infinite-Temperature Thermal Number Entropy

We start by assuming that, given $L$ sites populated by $N_{0}$ electrons,
computational microstates which conserve $N_{0}$ are equally probable i.e.
that the microcanonical ensemble gives the correct physical description of the
equilibrated system at late time [45]. The problem of deriving the probability
distribution $p_{k}(n)$ of observing $n$ electrons within $k$ selected sites
becomes straightforward. The probability $p_{k}(n=N)$ is simply the
probability of detecting $N$ occupied sites and $k-N$ empty sites, multiplied
by the multiplicity $k\choose N$ of such microstates:

$p_{k}(n=N)={k\choose
N}\underbrace{\left[\prod_{j=0}^{N-1}\frac{n_{0}-j}{L-j}\right]}_{\text{occupied}}\underbrace{\left[\prod_{j=0}^{k-N-1}1-\frac{n_{0}-N-j}{L-N-j}\right]}_{\text{empty}}$
(7)

The number entropy $S_{N}$ of such an infinite-temperature subsystem is then
readily calculated according to eq. 4 of the main text with
$\rho(t)\to\rho_{\text{th}}$. This quantity serves - in a similar capacity as
the page entropy - as a thermal limiting case for the number entropy [46].