Quantum critical behavior at the many-body-localization transition

Matthew Rispoli; Alexander Lukin; Robert Schittko; Sooshin Kim; M.; Eric Tai; Julian L\'eonard; and Markus Greiner

arXiv:1812.06959·cond-mat.quant-gas·November 1, 2019

Quantum critical behavior at the many-body-localization transition

Matthew Rispoli, Alexander Lukin, Robert Schittko, Sooshin Kim, M., Eric Tai, Julian L\'eonard, and Markus Greiner

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TL;DR

This paper reports the observation and characterization of quantum critical behavior at the many-body-localization transition in a disordered Bose-Hubbard system, revealing complex entanglement and transport phenomena.

Contribution

It provides the first experimental measurement of quantum criticality at the MBL transition, linking microscopic correlations with macroscopic critical behavior in a non-equilibrium quantum system.

Findings

01

Strong correlations emerge at the transition

02

Anomalous diffusive transport appears system-wide

03

Correlations extend to high orders and form a sparse network

Abstract

Phase transitions are driven by collective fluctuations of a system's constituents that emerge at a critical point. This mechanism has been extensively explored for classical and quantum systems in equilibrium, whose critical behavior is described by a general theory of phase transitions. Recently, however, fundamentally distinct phase transitions have been discovered for out-of-equilibrium quantum systems, which can exhibit critical behavior that defies this description and is not well understood. A paradigmatic example is the many-body-localization (MBL) transition, which marks the breakdown of quantum thermalization. Characterizing quantum critical behavior in an MBL system requires the measurement of its entanglement properties over space and time, which has proven experimentally challenging due to stringent requirements on quantum state preparation and system isolation. Here, we…

Equations21

\hat{H}

\hat{H}

+ \frac{U}{2} i \sum \overset{n}{^}_{i} (\overset{n}{^}_{i} - 1) + W i \sum h_{i} \overset{n}{^}_{i},

G_{c}^{(n)} (x) = G_{tot}^{(n)} (x) - G_{dis}^{(n)} (x),

G_{c}^{(n)} (x) = G_{tot}^{(n)} (x) - G_{dis}^{(n)} (x),

∣ ψ_{1} ⟩ =

∣ ψ_{1} ⟩ =

∣ ψ_{2} ⟩ =

G_{c}^{(3)} (i, j, k) =

G_{c}^{(3)} (i, j, k) =

-

-

G_{c}^{(n)} (i_{1}, i_{2}, ..., i_{n}) = ⟨ O_{i_{1}} O_{i_{2}} ... O_{i_{n}} ⟩ - g \in N \sum [\hat{P}_{({ k \in g } { i _{1} , i _{2} , ... , i _{n} })} (k_{i} \prod g G_{c}^{(k_{i})} [({ k _{i} \in k } { i _{1} , i _{2} , ... , i _{n} })])]

G_{c}^{(n)} (i_{1}, i_{2}, ..., i_{n}) = ⟨ O_{i_{1}} O_{i_{2}} ... O_{i_{n}} ⟩ - g \in N \sum [\hat{P}_{({ k \in g } { i _{1} , i _{2} , ... , i _{n} })} (k_{i} \prod g G_{c}^{(k_{i})} [({ k _{i} \in k } { i _{1} , i _{2} , ... , i _{n} })])]

G_{4} (i, j, k, l) = ⟨ O_{i} O_{j} O_{k} O_{l} ⟩

G_{4} (i, j, k, l) = ⟨ O_{i} O_{j} O_{k} O_{l} ⟩

-

-

-

-

-

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Full text

Quantum critical behavior at the many-body-localization transition

Matthew Rispoli

Alexander Lukin

Robert Schittko

Sooshin Kim

M. Eric Tai

Julian Léonard

Markus Greiner

[email protected]

Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

Phase transitions are driven by collective fluctuations of a system’s constituents that emerge at a critical point Täuber (2017). This mechanism has been extensively explored for classical and quantum systems in equilibrium, whose critical behavior is described by a general theory of phase transitions. Recently, however, fundamentally distinct phase transitions have been discovered for out-of-equilibrium quantum systems, which can exhibit critical behavior that defies this description and is not well understood Täuber (2017). A paradigmatic example is the many-body-localization (MBL) transition, which marks the breakdown of quantum thermalization Basko et al. (2006); Pal and Huse (2010); Serbyn et al. (2013); Huse et al. (2014); Alessio et al. (2016); Abanin et al. (2018); Schreiber et al. (2015); Smith et al. (2016); Choi et al. (2016); Lukin et al. (2018). Characterizing quantum critical behavior in an MBL system requires the measurement of its entanglement properties over space and time Serbyn et al. (2013); Huse et al. (2014); Abanin et al. (2018), which has proven experimentally challenging due to stringent requirements on quantum state preparation and system isolation. Here, we observe quantum critical behavior at the MBL transition in a disordered Bose-Hubbard system and characterize its entanglement properties via its quantum correlations. We observe strong correlations, whose emergence is accompanied by the onset of anomalous diffusive transport throughout the system, and verify their critical nature by measuring their system-size dependence. The correlations extend to high orders in the quantum critical regime and appear to form via a sparse network of many-body resonances that spans the entire system Potter et al. (2015); Khemani et al. (2017). Our results unify the system’s microscopic structure with its macroscopic quantum critical behavior, and they provide an essential step towards understanding criticality and universality in non-equilibrium systems Täuber (2017); Khemani et al. (2017); Abanin et al. (2018).

The many-body-localization (MBL) transition describes the breakdown of thermalization in an isolated quantum many-body system as disorder is increased beyond a critical value Schreiber et al. (2015); Smith et al. (2016); Choi et al. (2016); Lukin et al. (2018). It represents a novel type of quantum phase transition that fundamentally differs from both its classical and quantum ground-state counterparts Basko et al. (2006); Pal and Huse (2010); Abanin et al. (2018). Instead of being characterized by an instantaneous thermodynamic signature, it is identified by the system’s inherent dynamic behavior. In particular, the MBL transition manifests itself through a change in entanglement dynamics Lukin et al. (2018). Recent years have seen tremendous progress in our understanding of both the thermal and the MBL phases within the frameworks of quantum thermalization Alessio et al. (2016); Neill et al. (2016); Kaufman et al. (2016) and emergent integrability Serbyn et al. (2013); Huse et al. (2014); Schreiber et al. (2015); Smith et al. (2016); Choi et al. (2016); Lukin et al. (2018), respectively.

The quantum critical behavior at this transition, however, has remained largely unresolved Abanin et al. (2018). In particular, it is unclear whether the traditional association of collective fluctuations with static and dynamic critical behavior can be applied to this transition. The high amount of entanglement found at the MBL transition limits numerical studies due to the required computational power Agarwal et al. (2015); Setiawan et al. (2017). Several theoretical approaches, despite using disparate microscopic structures, suggest anomalous transport as the macroscopic behavior at the quantum critical point Vosk et al. (2015); Potter et al. (2015); Dumitrescu et al. (2017); Goremykina et al. (2018). Experimental studies indeed indicate a slowdown of the dynamics at intermediate disorder Lüschen et al. (2017a); Bordia et al. (2017). However, identifying anomalous transport as quantum critical dynamics is experimentally challenging, since similar behavior can also originate from stochastic effects: rare regions in the disorder potential De Roeck and Huveneers (2017); Nandkishore and Gopalakrishnan (2017); Agarwal et al. (2017), inhomogeneities in the initial state Luitz et al. (2016), or the coupling to a classical bath Nandkishore et al. (2014); Lüschen et al. (2017b). Our experimental protocol overcomes these challenges by using a quasi-periodic potential, which is rare-region free, as well as by evolving a pure, homogeneous initial state under unitary dynamics. Using this protocol, we observe quantum critical dynamics via anomalous transport, enhanced quantum fluctuations, and system-size dependent thermalization. In addition, we microscopically resolve and characterize the structure of the entanglement in the many-body states through their multi-particle quantum correlations.

Our experiments start with a pure state of up to twelve unentangled lattice sites at unity filling (see Methods) and study its out-of-equilibrium evolution under the bosonic, interacting Aubry-André Hamiltonian:

[TABLE]

where $\hat{a}_{i}^{\dagger}$ ( $\hat{a}_{i}$ ) is the creation (annihilation) operator for a boson on site $i$ , and $\hat{n}_{i}$ is the corresponding particle number operator. The tunneling time $\tau=\hbar/J=4.3(1)\,\text{ms}$ (with the reduced Planck constant $\hbar$ ) between neighboring sites and the pair-wise interaction energy $U=2.87(3)J$ remain constant for all experiments. The chemical potential $h_{i}=\cos\left(2\pi\beta i+\phi\right)$ on site $i$ follows a quasi-periodic distribution of amplitude $W$ , period $1/\beta\approx 1.618$ lattice sites, and phase $\phi$ . After a variable evolution time, we obtain full counting statistics of the quantum state through a fluorescence imaging technique. The applied unitary evolution preserves the initial purity of $99.1(2)\%$ per site, such that all correlations are expected to stem from entanglement in the system Kaufman et al. (2016); Lukin et al. (2018).

We first characterize the system’s dynamical behavior by studying its transport properties for different disorder strengths. Since the initial state has exactly one atom per site, the system starts with zero density correlations at all length scales. However, during the Hamiltonian evolution, tunneling dynamics build up anti-correlated density fluctuations between coupled sites of increasing distance (Fig. 2a). Motivated by this picture, we quantify the particle dynamics by defining the transport distance, $\Delta x=2\sum_{d}d\times\langle G_{\text{c}}^{(2)}(i,i+d)\rangle_{i}$ , as the first moment of the disorder-averaged two-point density correlations, $G_{\text{c}}^{(2)}(i,i+d)=\langle\hat{n}_{i}\hat{n}_{i+d}\rangle-\langle\hat{n}_{i}\rangle\langle\hat{n}_{i+d}\rangle$ (Fig. 2a). With increasing disorder, we observe a slowdown of particle transport that is consistent with a power-law growth $\Delta x\sim t^{\alpha}$ (Fig. 2b, see Methods) Lucioni et al. (2011). We extract the anomalous diffusion exponent $\alpha$ from a subset of the data points which excludes the initial transient dynamics (Fig. 2b inset). The exponent $\alpha$ approaches zero for successively higher disorder, demonstrating the absence of transport in the MBL regime.

In order to identify the anomalous diffusion as a signature of quantum critical dynamics, we measure the system-size dependence of two observables in the long-time limit $(t=100\tau)$ : the on-site number fluctuations $\mathcal{F}\equiv G_{\text{c}}^{(2)}(d=0)$ as a probe of local thermalization, and the transport distance $\Delta x$ as a localization measure (Fig. 2c, see Methods). At low disorder, the fluctuations agree with those predicted by a thermal ensemble and particles are completely delocalized for both system sizes. This demonstrates that local quantum thermalization in our system is not subject to finite-size effects. At strong disorder, we find sub-thermal fluctuations and a transport distance $\Delta x\ll L$ . This indicates that the physics is governed by a system-size independent, intrinsic length scale, namely the localization length $\Delta x$ Choi et al. (2016); Lukin et al. (2018). However, at intermediate disorder, we find a system-size dependence for both observables, demonstrating the absence of an intrinsic length scale and the presence of finite-size-limited local fluctuations. Our measurements can be visualized as two horizontal cuts in a finite-size phase diagram, whose finite-size dependence agrees with the physics associated with a shrinking quantum critical cone (Fig. 2c inset) Täuber (2017).

We then investigate the multi-particle correlations in the system to probe the presence of enhanced quantum fluctuations in the quantum critical regime (Fig. 2d). For this study, we employ the $n$ -point connected density-correlation functions Liu (2016); Schweigler et al. (2017); Hodgman et al. (2017),

[TABLE]

which act on lattice sites with positions $\textbf{x}=(x_{1},...,x_{n})$ . The disconnected part of this function, $G_{\text{dis}}^{(n)}$ , is fully determined by all lower-order correlation functions, and therefore does not contain new information at order $n$ . By removing it from the total measured correlation function, $G_{\text{tot}}^{(n)}$ , we isolate all $n$ -order correlations that are independent of lower-order processes (see Methods). This approach gives a direct handle on the level of complexity of the underlying many-body wave function and characterizes its non-seperabilty into sub-systems of size $<n$ . We quantify the relevance of order $n$ processes by computing the mean absolute value of all correlations arising from both contiguous and non-contiguous $n$ sites in the system (Fig. 2e). We find that in the thermal and the many-body-localized regimes, the system becomes successively less correlated at higher order. The behavior in the quantum critical regime is strikingly different: we observe that the system is strongly correlated at all measured orders.

In order to reveal the microscopic origin for the anomalous transport, we now investigate the site-resolved structure of the many-body state (Fig. 3a). We first study how much each lattice site contributes to the transport by considering the site-resolved two-point correlations in the long-time limit. In the thermal regime, we find similar correlations between all lattice sites, which correspond to uniformly delocalized atoms. In contrast, density correlations are restricted to nearby sites in the MBL regime due to localization. Intriguingly, we observe a sparse structure of correlations at intermediate disorder, which involves only specific distances between lattice sites, yet spans the entire system size.

The sparse structure is expected to be linked to the applied quasi-periodic potential. The average energy offsets of sites $d$ apart in the system are correlated by this potential. This correlation is then inherited by the system’s fluctuations when the interaction energy $U$ compensates for these correlated offsets. To investigate this structure, we compare the two-point density correlations with the autocorrelation function, $A(d)=\langle h_{i}h_{i+d}\rangle_{i}$ , of the quasi-periodic potential. Indeed, we find that the site-averaged density correlations $G_{\text{c}}^{(2)}(d)=\langle G^{(2)}(i,i+d)\rangle_{i}$ inherit their spatial structure from $A(d)$ (Fig. 3b). We find that this contribution is maximal in the critical regime but is strongly reduced in the thermal and MBL regimes (Fig. 3c). These observations contrast with the behavior of a non-interacting system, where the sign of the structure is opposite since resonant tunneling is favored for zero potential energy difference (Fig. 3c). These results illustrate microscopically how the interplay of strong interactions and disorder can lead to anomalous diffusion. However, this picture of effective single-particle hopping that couples distant sites neglects the many-body nature of these systems.

In order to investigate the system’s many-body structure, we examine the site-resolved contributions of the three-point correlations. Since all non-zero contributions to the three-point correlations involve correlated hopping of at least two particles, they are a signature for multi-particle entanglement and therefore demonstrate a breakdown of mean-field approximations Schweigler et al. (2017). In the quantum critical regime, we find that these correlations span the entire system and are highly structured, taking on both positive and negative values (Fig. 4a). In contrast to the pattern in the second-order correlation function, this third-order structure is not directly recognizable as the quasi-periodic-potential correlations. In order to gain further insight into the structure, we analyze the contributions of all possible particle configurations in Fig. 4b. In particular, for $G^{(3)}_{c}(d_{1}=3,d_{2}=1)$ , which is positive, we see that the dominant contribution comes from a particular process that favors multiple atoms hopping to the same site. In contrast, $G^{(3)}_{c}(d_{1}=3,d_{2}=2)$ , which is negative, has a dominant process that favors all atoms leaving the three sites considered. While this provides some intuition for the emergent many-body resonances, the three-point correlations are, in fact, the result of a superposition of many correlated processes. These observations further demonstrate how the interactions between multiple atoms can compensate for the disorder via correlated tunneling of several atoms. In this way, we can see the additional role interactions play in the disordered system: they supply higher-order many-body resonances that preserve transport where lower-order processes are energetically suppressed.

Our results demonstrate how a many-body, sparse resonant structure drives the quantum critical behavior at the MBL transition. This observed microscopic description is consistent with the theoretically suggested mechanisms of a sparse backbone of resonances that can act as a functional bath for the system Potter et al. (2015); Khemani et al. (2017). However, our results provide a new perspective on this description by mapping out the prevalence of high-order processes in the system that facilitate this critical thermalization.

In future experiments, the tunability of our system will allow us to address further open questions on the MBL transition, such as possible discontinuities of the entanglement entropy Khemani et al. (2017), the potential emergence of new dynamic phases near the critical point, and the influence of rare-regions in the disorder potential De Roeck and Huveneers (2017); Nandkishore and Gopalakrishnan (2017). Furthermore, the demonstrated techniques pave the way to explore the role of universality in non-equilibrium systems. From a quantum computing perspective, our system’s Hilbert space dimension exceeds the dimension of 22 spins with zero total magnetization, bringing numerically intractable sizes within experimental access.

Acknowledgements.

We acknowledge fruitful discussions with D. Abanin, E. Altman, H. Bernien, C. Chiu, S. Choi, E. Demler, A. Hébert, M. Heyl, W. W. Ho, V. Kasper, V. Khemani, J. Kwan, D. Luitz and L. Santos. We are supported by grants from the National Science Foundation, the Gordon and Betty Moore Foundations EPiQS Initiative, an Air Force Office of Scientific Research MURI program, an Army Research Office MURI program and the NSF Graduate Research Fellowship Program. J. L. acknowledges support from the Swiss National Science Foundation.

I Supplementary information

I.1 Experimental protocol

All of our experiments start with a unity-filling, two-dimensional Mott insulator of bosonic 87Rb atoms in a deep, blue-detuned optical lattice with lattice constant $a=680\text{nm}$ and $45E_{\text{r}}$ lattice depth, where $E_{\text{r}}=h\times 1.1\,\text{kHz}$ is the recoil energy for an atom with mass $m$ , and $h$ is the Planck constant. We then employ two digital micromirror devices (DMDs) in the Fourier plane of our microscope to optically confine a single chain of $N$ ( $N=8,12$ ) atoms chosen from the Mott insulator’s unity-filling shell and subsequently ramp down the power of the optical lattice. Upon waiting for a few tens of milliseconds to allow for a departure of the non-confined atoms surrounding the chain, we then turn on the lattice again and ramp down the confining DMD potentials, thereby finalizing the initialization of our unity-filling, 1x $N$ system. Taking into account the additional post-selection described below, this procedure results in a 99.1(2)% probability of finding exactly one atom on any given site. Notably, this value establishes lower bounds of 93% and 90% for the global quantum state purities of the 8-site and the 12-site system, respectively.

Upon initializing the system of our choice, we subsequently pursue three separate paths of action to initiate the dynamics we are interested in studying. In a first step, we use one of our DMDs to project an optical potential onto our atoms. This “wall-potential” provides a box-like confinement which is registered to the position of the optical lattice, and later defines the size of the one-dimensional system once the bare lattice depth has been lowered. Secondly, we simultaneously use the other DMD to project a custom, quasi-periodic disorder potential onto our atoms. Finally, after both of these potentials have been turned on, we rapidly lower the bare lattice depth along the atomic chain from $V_{x}=45E_{\text{r}}$ to $V_{x}=8E_{\text{r}}$ , thereby quenching the system and giving rise to many-body dynamics. After a variable evolution time in this lowered potential, we freeze said dynamics by quickly ramping the longitudinal lattice back up to $V_{x}=45E_{\text{r}}$ .

We then let the atom populations located on individual lattice sites expand into independent tubes and use fluorescence imaging to perform a site-resolved atom number measurement. The expansion step before the imaging procedure is employed to avoid parity projection during the imaging process. We subsequently post-select our data by excluding any images which do not contain the correct total number of atoms.

The steps briefly described above are conceptually identical to those employed in Kaufman et al. (2016); Lukin et al. (2018), where they are described in more detail in the methods sections.

I.2 High-Order Correlation Functions

Generically, a $n^{\mathrm{th}}$ order correlation function can be measured from a set of operators $\mathcal{O}_{i}$ by their joint expectation value $\langle\prod_{i=1}^{n}\mathcal{O}_{i}\rangle=\langle\mathcal{O}_{1}\mathcal{O}_{2}...\mathcal{O}_{n}\rangle$ . However, this joint expectation value captures two kinds of information: “disconnected” correlations that exist at $n^{\mathrm{th}}$ order due to existing lower order correlations, and “connected” correlations that only exist at order $n$ and can’t be described by factorization into correlations of lower order Liu (2016); Schweigler et al. (2017); Hodgman et al. (2017); Preiss et al. (2018).

In the two-point case, this would mean comparing the measured value of $\langle\mathcal{O}_{i}\mathcal{O}_{j}\rangle$ to the product of their individual expectation values $\langle\mathcal{O}_{i}\rangle\langle\mathcal{O}_{j}\rangle$ . The “connected” part of the correlation between $i$ and $j$ is defined as the correlations that remain after removing the contributions from factorization into smaller groups. This motivates the definition of $G_{\text{c}}^{(2)}(i,j)=\langle\mathcal{O}_{i}\mathcal{O}_{j}\rangle-\langle\mathcal{O}_{i}\rangle\langle\mathcal{O}_{j}\rangle$ .

To provide some intuition, we describe two concrete examples in terms of two-point joint expectation values constructed from atom-number operators $\hat{n}_{i}$ as $\langle n_{1}n_{2}\rangle$ . The two example states are:

[TABLE]

For both $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ we see that they have the same local fluctuations $\langle n_{i}^{2}\rangle-\langle n_{i}\rangle^{2}=1$ and local on-site number $\langle n_{i}\rangle=1$ . However, we see, by construction, that $|\psi_{1}\rangle$ is created from an outter product of states with these local fluctuations and therefore should have uncorrelated joint fluctuations. This means that sites would give the joint expectation value of two incoherently fluctuating random variables, i.e. $G_{\text{c}}^{(2)}(|\psi_{1}\rangle)=1-1=0$ or site-1 and site-2 in $|\psi_{1}\rangle$ have no genuine “connected” two-point correlations. This differs from the second case of $|\psi_{2}\rangle$ where $\langle n_{i}\rangle=1$ and $\langle n_{i}^{2}\rangle-\langle n_{i}\rangle^{2}=1$ but the correlated fluctuations of the value bring $\langle n_{1}n_{2}\rangle=2$ which then gives a “connected” correlation value of $G_{\text{c}}^{(2)}(|\psi_{2}\rangle)=2-1=1$ . This shows that the state $|\psi_{2}\rangle$ has genuine two-point correlations that cannot be described by factorization into smaller groups.

For a three-point “connected” correlation function, we must subtract out contributions that come from “connected” two-point correlations that can look like three-point correlations when randomly combined with a residual 1-point correlation. This is how the “connected” three-point correlation function is defined in the main text as $G_{\text{c}}^{(3)}$ for the on-site number operator $\hat{n}_{i}$ .

[TABLE]

This leads to a recursive definition of $n$ -order “connected” correlation functions that then depend on the integer partitioning and permutations of all lower-order “connected” correlations. For convenience and compact notation, we will make a set of useful definitions:

•

$G_{\text{c}}^{(1)}(i)=\langle\mathcal{O}_{i}\rangle$ & $G_{\text{c}}^{(0)}(\emptyset)=1$

•

a function $\hat{\mathcal{P}}_{{\{i_{1},i_{2},...,i_{n}\}}\choose{\{k\in g\}}}\big{(}\hskip 1.42262pt...\hskip 1.42262pt\big{)}$ that finds all unique permutations of choosing all indices in the group $\{i_{1},i_{2},...,i_{n}\}$ separated into integer partitions $\{k\}=\{k_{1},k_{2},...,k_{g}\}$ that sum to $n$ : $\sum_{i}^{g}k_{i}=n$

•

$\sum_{g\in N}$ that defines a sum over all unique integer partitions $\{g\}$ of the order $n$

[TABLE]

In general, we must subtract all permutations of factorable groups at lower-order. This is the same as combining the integer partition problem and then finding all permutations of choosing those integer partitions from the number of indices equal to the sum. To exemplify how this looks for the next order, it is applied to finding all the correct factorizations of the four-point correlation function. The unique integer partitions of 4 with at least two non-zero integers are $\left\{(3,1),(2,2),(2,1,1),(1,1,1,1)\right\}$ .

[TABLE]

I.3 Distribution of n-point correlation functions

The probability distribution of the n-point correlation for each set of lattice sites is intrinsically asymmetric towards larger absolute values. While this bias is incorporated in the s.e.m. for a single correlation value, this results in a finite-sampling bias when calculating the mean-absolute-value. Calculating this quantity for a finite sample number therefore overestimates the expectation value in the limit of infinite sample number. We can include this effect in theory by Monte-Carlo sampling from the exact diagonalization calculations with the same number of measured shots ( $\sim 110/\mathrm{disorder}$ ). This finite-sample theory is plotted in Fig. 2e. In order to ensure that the qualitative feature of enhanced correlations remains unchanged for larger sample numbers, we compare the finite-sample theory with the exact calculations, see Fig. 5. We find that indeed the two theory curves are in qualitative agreement.

We further investigate the connected n-point correlations functions by calculating histograms of the distribution of all sets of lattice sites, see Fig. 6.

I.4 Thermal Ensemble Calculation

At $t=0$ , the system is quenched to the Hamiltonian $H_{q}(W)$ , which depends on disorder strength $W$ . The thermal prediction shown in Fig. 2c is calculated using a canonical ensemble, in which the eigenstates of the system are exponentially populated according to $P_{i}\sim e^{E_{i}/T}$ , where $E_{i}$ is the eigenenergy of state $i$ . The effective temperature of the system is determined by finding the canonical temperature $T(J)$ that yields the correct average energy $\langle E\rangle$ after quenching into the Hamiltonian $H_{q}(W)$ . We have additionally compared this canonical ensemble to a microcanonical one, finding excellent agreement between the two. The microcanonical ensemble we used is composed of an equal-probability statistical mixture of those 11 eigenstates of $H_{q}(W)$ which are closest to the average energy of the initial state, which is given by $E_{0}=\bra{\psi_{0}}H_{q}(W)\ket{\psi_{0}}$ . We additionally verified that these results do not depend on the exact number of included eigenstates.

I.5 Numerics

In order to get theoretical predictions for 8-site systems, we perform numerical exact-diagonalization (ED) calculations. For 12-site systems, matrix diagonalization is computationally challenging due to the large Hilbert space dimension. Instead, we implement an exact numerical integration of Schrödinger’s equation $\ket{\psi(t)}=e^{-iHt/\hbar}\ket{\psi_{0}}$ based on the Krylov-subspace method. Since the Hamiltonian is sparse, the Krylov-subspace method provides an efficient (both in terms of memory and CPU-run-time usage) way to numerically compute the time evolution while achieving high, controlled precision. All data points are averaged over 200 different disorder realizations. The computations are performed on the Harvard Odyssey computing cluster. For specifications see: https://www.rc.fas.harvard.edu/odyssey/

I.6 Determination of Time Scaling Behavior (Power Law 12-Sites)

In order to quantify the transport distance of the particles, we defined the first moment of the two-point-density-correlation distribution, $\Delta x=\sum_{d}G_{2}(d)\times d$ where $G_{2}(d)=\langle n_{i}n_{i+d}\rangle_{i,\phi}-\langle n_{i}\rangle_{i,\phi}\langle n_{i+d}\rangle_{i,\phi}$ , $i$ is the site index, and $\phi$ is the disorder realization. Numerical simulations were performed to determine time-scaling behavior of the first moment (Fig. 7), and it appears that in the critical disorder regime ( $5<W/J<8$ ), the transport distance follows a power-law time scaling rather than a logarithmic growth. Hence, in this paper, we assumed a power-law time scaling of the particle spread to analyze the data.

I.7 Data Analysis

For all experimental data we use 197 unique disorder patterns, each defined by a different phase $\phi$ of the quasi-periodic potential, and perform a running average over them by randomly sampling a given number of realizations and treating them as independent measurements of the same system.

We extract the anomalous diffusion exponent $\alpha$ in Fig. 2b in a fit-free manner as follows: we first exclude the data at times $t<L/2\tau$ , where the initial transient dynamics are still ongoing. We then calculate the slopes between all successive pairs of data points. The exponent $\alpha$ then corresponds to the average of those slopes.

The single-site atom number fluctuations $\mathcal{F}=G_{\text{c}}^{(2)}(d=0)$ are extracted from the edge sites for both system sizes. The edge sites are most insensitive to the introduction of additional (bulk) sites into the system and therefore allow for the fairest comparison between different total system sizes Khemani et al. (2017).

The number of samples for each experiment is summarized in the following table:

[TABLE]

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