Stark Many-Body Localization

Stark Many-Body Localization in Interacting Infinite Dimensional Systems

We study bulk particle transport in a Fermi-Hubbard model on an infinite-dimensional Bethe lattice, driven by a constant electric field. Previous numerical studies showed that one dimensional analogs of this system exhibit a breakdown of diffusion due to Stark many-body localization at least up to time that scales exponentially with the system size. Here, we consider systems initially in a spin density wave state using a combination of numerically exact and approximate techniques. We show that for sufficiently weak electric fields, the wave's momentum component decays exponentially with time in a way consistent with normal diffusion. By studying different wavelengths, we extract the dynamical exponent and the generalized diffusion coefficient at each field strength. Interestingly, we find a nonmonotonic dependence of the dynamical exponent on the electric field. As the field increases toward a critical value proportional to the Hubbard interaction strength, transport slows down, becoming subdiffusive. At large interaction strengths, however, transport speeds up again with increasing field, exhibiting superdiffusive characteristics when the electric field is comparable to the interaction strength. Eventually, at the large field limit, localization occurs and the current through the system is suppressed.

Scaling and Localization in Multipole-Conserving Diffusion

We study diffusion in systems of classical particles whose dynamics conserves the total center of mass. This conservation law leads to several interesting consequences. In finite systems, it allows for equilibrium distributions that are exponentially localized near system boundaries. It also yields an unusual approach to equilibrium, which in d dimensions exhibits scaling with dynamical exponent z=4+d. Similar phenomena occur for dynamics that conserves higher moments of the density, which we systematically classify using a family of nonlinear diffusion equations. In the quantum setting, analogous fermionic systems are shown to form real-space Fermi surfaces, while bosonic versions display a real-space analog of Bose-Einstein condensation.

Spectral Response of Disorder-Free Localized Lattice Gauge Theories

We show that certain lattice gauge theories exhibiting disorder-free localization have a characteristic response in spatially averaged spectral functions: a few sharp peaks combined with vanishing response in the zero frequency limit. This reflects the discrete spectra of small clusters of kinetically active regions formed in such gauge theories when they fragment into spatially finite clusters in the localized phase due to the presence of static charges. We obtain the transverse component of the dynamic structure factor, which is probed by neutron scattering experiments, deep in this phase from a combination of analytical estimates and a numerical cluster expansion. We also show that local spectral functions of large finite clusters host discrete peaks whose positions agree with our analytical estimates. Further, information spreading, diagnosed by an unequal time commutator, halts due to real space fragmentation. Our results can be used to distinguish the disorder-free localized phase from conventional paramagnetic counterparts in those frustrated magnets which might realize such an emergent gauge theory.

Reviving product states in the disordered Heisenberg chain

When a generic quantum system is prepared in a simple initial condition, it typically equilibrates toward a state that can be described by a thermal ensemble. A known exception is localized systems that are non-ergodic and do not thermalize; however, local observables are still believed to become stationary. Here we demonstrate that this general picture is incomplete by constructing product states that feature periodic high-fidelity revivals of the full wavefunction and local observables that oscillate indefinitely. The system neither equilibrates nor thermalizes. This is analogous to the phenomenon of weak ergodicity breaking due to many-body scars and challenges aspects of the current phenomenology of many-body localization, such as the logarithmic growth of the entanglement entropy. To support our claim, we combine analytic arguments with large-scale tensor network numerics for the disordered Heisenberg chain. Our results hold for arbitrarily long times in chains of 160 sites up to machine precision.

Stark Localization as a Resource for Weak-Field Sensing with Super-Heisenberg Precision

Gradient fields can effectively suppress particle tunneling in a lattice and localize the wave function at all energy scales, a phenomenon known as Stark localization. Here, we show that Stark systems can be used as a probe for the precise measurement of gradient fields, particularly in the weak-field regime where most sensors do not operate optimally. In the extended phase, Stark probes achieve super-Heisenberg precision, which is well beyond most of the known quantum sensing schemes. In the localized phase, the precision drops in a universal way showing fast convergence to the thermodynamic limit. For single-particle probes, we show that quantum-enhanced sensitivity, with super-Heisenberg precision, can be achieved through a simple position measurement for all the eigenstates across the entire spectrum. For such probes, we have identified several critical exponents of the Stark localization transition and established their relationship. Thermal fluctuations, whose universal behavior is identified, reduce the precision from super-Heisenberg to Heisenberg, still outperforming classical sensors. Multiparticle interacting probes also achieve super-Heisenberg scaling in their extended phase, which shows even further enhancement near the transition point. Quantum-enhanced sensitivity is still achievable even when state preparation time is included in resource analysis.

Absence of localization in interacting spin chains with a discrete symmetry

Novel paradigms of strong ergodicity breaking have recently attracted significant attention in condensed matter physics. Understanding the exact conditions required for their emergence or breakdown not only sheds more light on thermalization and its absence in closed quantum many-body systems, but it also has potential benefits for applications in quantum information technology. A case of particular interest is many-body localization whose conditions are not yet fully settled. Here, we prove that spin chains symmetric under a combination of mirror and spin-flip symmetries and with a non-degenerate spectrum show finite spin transport at zero total magnetization and infinite temperature. We demonstrate this numerically using two prominent examples: the Stark many-body localization system (Stark-MBL) and the symmetrized many-body localization system (symmetrized-MBL). We provide evidence of delocalization at all energy densities and show that delocalization persists when the symmetry is broken. We use our results to construct two localized systems which, when coupled, delocalize each other. Our work demonstrates the dramatic effect symmetries can have on disordered systems, proves that the existence of exact resonances is not a sufficient condition for delocalization, and opens the door to generalization to higher spatial dimensions and different conservation laws.

Discrete Time Crystal Enabled by Stark Many-Body Localization

Discrete time crystals (DTCs) have recently attracted increasing attention, but most DTC models and their properties are only revealed after disorder average. In this Letter, we propose a simple disorder-free periodically driven model that exhibits nontrivial DTC order stabilized by Stark many-body localization (MBL). We demonstrate the existence of the DTC phase by analytical analysis from perturbation theory and convincing numerical evidence from observable dynamics. The new DTC model paves a new promising way for further experiments and deepens our understanding of DTCs. Since the DTC order does not require special quantum state preparation and the strong disorder average, it can be naturally realized on the noisy intermediate-scale quantum hardware with much fewer resources and repetitions. Moreover, in addition to the robust subharmonic response, there are other novel robust beating oscillations in the Stark-MBL DTC phase that are absent in random or quasiperiodic MBL DTCs.

Exploring the Regime of Fragmentation in Strongly Tilted Fermi-Hubbard Chains

Intriguingly, quantum many-body systems may defy thermalization even without disorder. One example is so-called fragmented models, where the many-body Hilbert space fragments into dynamically disconnected subspaces that are not determined by the global symmetries of the model. In this Letter we demonstrate that the tilted one-dimensional Fermi-Hubbard model naturally realizes distinct effective Hamiltonians that are expected to support nonergodic behavior due to fragmentation, even at resonances between the tilt energy and the Hubbard on site interaction. We find that the effective description captures the observed dynamics in experimentally accessible parameter ranges of moderate tilt values. Specifically, we observe a pronounced dependence of the relaxation dynamics on the initial doublon fraction, which directly reveals the microscopic processes of the fragmented model. Our results pave the way for future studies of nonergodic behavior in higher dimensions.

Restoring Ergodicity in a Strongly Disordered Interacting Chain

We consider a chain of interacting fermions with random disorder that was intensively studied in the context of many-body localization. We show that only a small fraction of the two-body interaction represents a true local perturbation to the Anderson insulator. While this true perturbation is nonzero at any finite disorder strength W, it decreases with increasing W. This establishes a view that the strongly disordered system should be viewed as a weakly perturbed integrable model, i.e., a weakly perturbed Anderson insulator. As a consequence, the latter can hardly be distinguished from a strictly integrable system in finite-size calculations at large W. We then introduce a rescaled model in which the true perturbation is of the same order of magnitude as the other terms of the Hamiltonian, and show that the system remains ergodic at arbitrary large disorder.

Localization and Melting of Interfaces in the Two-Dimensional Quantum Ising Model

We study the nonequilibrium evolution of coexisting ferromagnetic domains in the two-dimensional quantum Ising model-a setup relevant in several contexts, from quantum nucleation dynamics and false-vacuum decay scenarios to recent experiments with Rydberg-atom arrays. We demonstrate that the quantum-fluctuating interface delimiting a large bubble can be studied as an effective one-dimensional system through a "holographic" mapping. For the considered model, the emergent interface excitations map to an integrable chain of fermionic particles. We discuss how this integrability is broken by geometric features of the bubbles and by corrections in inverse powers of the ferromagnetic coupling, and provide a lower bound to the timescale after which the bubble is ultimately expected to melt. Remarkably, we demonstrate that a symmetry-breaking longitudinal field gives rise to a robust ergodicity breaking in two dimensions, a phenomenon underpinned by Stark many-body localization of the emergent fermionic excitations of the interface.

Excitations of Quantum Many-Body Systems via Purified Ensembles: A Unitary-Coupled-Cluster-Based Approach

State-average calculations based on a mixture of states are increasingly being exploited across chemistry and physics as versatile procedures for addressing excitations of quantum many-body systems. If not too many states should need to be addressed, calculations performed on individual states are also a common option. Here we show how the two approaches can be merged into one method, dealing with a generalized yet single pure state. Implications in electronic structure calculations are discussed and for quantum computations are pointed out.

Quantum many-body scars and Hilbert space fragmentation: a review of exact results

The discovery of quantum many-body scars (QMBS) both in Rydberg atom simulators and in the Affleck-Kennedy-Lieb-Tasaki spin-1 chain model, have shown that a weak violation of ergodicity can still lead to rich experimental and theoretical physics. In this review, we provide a pedagogical introduction to and an overview of the exact results on weak ergodicity breaking via QMBS in isolated quantum systems with the help of simple examples such as the fermionic Hubbard model. We also discuss various mechanisms and unifying formalisms that have been proposed to encompass the plethora of systems exhibiting QMBS. We cover examples of equally-spaced towers that lead to exact revivals for particular initial states, as well as isolated examples of QMBS. Finally, we review Hilbert space fragmentation, a related phenomenon where systems exhibit a richer variety of ergodic and non-ergodic behaviors, and discuss its connections to QMBS.

Hilbert Space Shattering and Disorder-Free Localization in Polar Lattice Gases

Emerging dynamical constraints resulting from intersite interactions severely limit particle mobility in polar lattice gases. Whereas in absence of disorder hard-core Hubbard models with only strong nearest-neighbor interactions present Hilbert space fragmentation but no many-body localization for typical states, the 1/r^{3} tail of the dipolar interaction results in Hilbert space shattering, as well as in a dramatically slowed down dynamics and eventual disorder-free localization. Our results show that the study of the intriguing interplay between disorder- and interaction-induced many-body localization is within reach of future experiments with magnetic atoms and polar molecules.

Stark Many-Body Localization on a Superconducting Quantum Processor

Quantum emulators, owing to their large degree of tunability and control, allow the observation of fine aspects of closed quantum many-body systems, as either the regime where thermalization takes place or when it is halted by the presence of disorder. The latter, dubbed many-body localization (MBL) phenomenon, describes the nonergodic behavior that is dynamically identified by the preservation of local information and slow entanglement growth. Here, we provide a precise observation of this same phenomenology in the case where the quenched on-site energy landscape is not disordered, but rather linearly varied, emulating the Stark MBL. To this end, we construct a quantum device composed of 29 functional superconducting qubits, faithfully reproducing the relaxation dynamics of a nonintegrable spin model. At large Stark potentials, local observables display periodic Bloch oscillations, a manifesting characteristic of the fragmentation of the Hilbert space in sectors that conserve dipole moments. The flexible programmability of our quantum emulator highlights its potential in helping the understanding of nontrivial quantum many-body problems, in direct complement to simulations in classical computers.

Observation of Stark many-body localization without disorder

Thermalization is a ubiquitous process of statistical physics, in which a physical system reaches an equilibrium state that is defined by a few global properties such as temperature. Even in isolated quantum many-body systems, limited to reversible dynamics, thermalization typically prevails¹. However, in these systems, there is another possibility: many-body localization (MBL) can result in preservation of a non-thermal state^2,3. While disorder has long been considered an essential ingredient for this phenomenon, recent theoretical work has suggested that a quantum many-body system with a spatially increasing field-but no disorder-can also exhibit MBL⁴, resulting in 'Stark MBL'⁵. Here we realize Stark MBL in a trapped-ion quantum simulator and demonstrate its key properties: halting of thermalization and slow propagation of correlations. Tailoring the interactions between ionic spins in an effective field gradient, we directly observe their microscopic equilibration for a variety of initial states, and we apply single-site control to measure correlations between separate regions of the spin chain. Furthermore, by engineering a varying gradient, we create a disorder-free system with coexisting long-lived thermalized and non-thermal regions. The results demonstrate the unexpected generality of MBL, with implications about the fundamental requirements for thermalization and with potential uses in engineering long-lived non-equilibrium quantum matter.

Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains

The thermalization of isolated quantum many-body systems is deeply related to fundamental questions of quantum information theory. While integrable or many-body localized systems display non-ergodic behavior due to extensively many conserved quantities, recent theoretical studies have identified a rich variety of more exotic phenomena in between these two extreme limits. The tilted one-dimensional Fermi-Hubbard model, which is readily accessible in experiments with ultracold atoms, emerged as an intriguing playground to study non-ergodic behavior in a clean disorder-free system. While non-ergodic behavior was established theoretically in certain limiting cases, there is no complete understanding of the complex thermalization properties of this model. In this work, we experimentally study the relaxation of an initial charge-density wave and find a remarkably long-lived initial-state memory over a wide range of parameters. Our observations are well reproduced by numerical simulations of a clean system. Using analytical calculations we further provide a detailed microscopic understanding of this behavior, which can be attributed to emergent kinetic constraints.

Proposal for Realizing Quantum Scars in the Tilted 1D Fermi-Hubbard Model

Motivated by recent observations of ergodicity breaking due to Hilbert space fragmentation in 1D Fermi-Hubbard chains with a tilted potential [Scherg et al., arXiv:2010.12965], we show that the same system also hosts quantum many-body scars in a regime U≈Δ≫J at electronic filling factor ν=1. We numerically demonstrate that the scarring phenomenology in this model is similar to other known realizations such as Rydberg atom chains, including persistent dynamical revivals and ergodicity-breaking many-body eigenstates. At the same time, we show that the mechanism of scarring in the Fermi-Hubbard model is different from other examples in the literature: the scars originate from a subgraph, representing a free spin-1 paramagnet, which is weakly connected to the rest of the Hamiltonian's adjacency graph. Our work demonstrates that correlated fermions in tilted optical lattices provide a platform for understanding the interplay of many-body scarring and other forms of ergodicity breaking, such as localization and Hilbert space fragmentation.

Emergent Bloch Oscillations in a Kinetically Constrained Rydberg Spin Lattice

We explore the relaxation dynamics of elementary spin clusters in a kinetically constrained spin system. Inspired by experiments with Rydberg lattice gases, we focus on the situation in which an excited spin leads to a "facilitated" excitation of a neighboring spin. We show that even weak interactions that extend beyond nearest neighbors can have a dramatic impact on the relaxation behavior: they generate a linear potential, which under certain conditions leads to the onset of Bloch oscillations of spin clusters. These hinder the expansion of a cluster and, more generally, the relaxation of many-body states toward equilibrium. This shows that nonergodic behavior in kinetically constrained systems may occur as a consequence of the interplay between reduced connectivity of many-body states and weak interparticle interactions. We furthermore show that the emergent Bloch oscillations identified here can be detected in experiment through measurements of the Rydberg atom density and discuss how spin-orbit coupling between internal and external degrees of freedom of spin clusters can be used to control their relaxation behavior.

Polynomially Filtered Exact Diagonalization Approach to Many-Body Localization

Polynomially filtered exact diagonalization method (POLFED) for large sparse matrices is introduced. The algorithm finds an optimal basis of a subspace spanned by eigenvectors with eigenvalues close to a specified energy target by a spectral transformation using a high order polynomial of the matrix. The memory requirements scale better with system size than in the state-of-the-art shift-invert approach. The potential of POLFED is demonstrated examining many-body localization transition in 1D interacting quantum spin-1/2 chains. We investigate the disorder strength and system size scaling of Thouless time. System size dependence of bipartite entanglement entropy and of the gap ratio highlights the importance of finite-size effects. We discuss possible scenarios regarding the many-body localization transition obtaining estimates for the critical disorder strength.

Nonequilibrium Properties of Berezinskii-Kosterlitz-Thouless Phase Transitions

We employ a novel, unbiased renormalization-group approach to investigate nonequilibrium phase transitions in infinite lattice models. This allows us to address the delicate interplay of fluctuations and ordering tendencies in low dimensions out of equilibrium. We study a prototypical model for the metal to insulator transition of spinless interacting fermions coupled to electronic baths and driven out of equilibrium by a longitudinal static electric field. The closed system features a Berezinskii-Kosterlitz-Thouless transition between a metallic and a charge-ordered phase in the equilibrium limit. We compute the nonequilibrium phase diagram and illustrate a highly nonmonotonic dependence of the phase boundary on the strength of the electric field: for small fields, the induced currents destroy the charge order, while at higher electric fields it reemerges due to many-body Wannier-Stark localization physics. Finally, we show that the current in such an interacting nonequilibrium system can counter-intuitively flow opposite to the direction of the electric field. This nonequilibrium steady state is reminiscent of an equilibrium distribution function with an effective negative temperature.

Entanglement and thermodynamic entropy in a clean many-body-localized system

Whether or not the thermodynamic entropy is equal to the entanglement entropy of an eigenstate is of fundamental interest, and is closely related to the eigenstate thermalization hypothesis (ETH). However, this has never been exploited as a diagnostic tool in many-body- localized (MBL) systems. In this work, we perform this diagnostic test on a clean interacting system (subjected to a static electric field) that exhibits three distinct phases: integrable, non-integrable ergodic and non-integrable MBL. We find that in the non-integrable phase, the equivalence between the thermodynamic entropy and the entanglement entropy of individual eigenstates holds. In sharp contrast, in the integrable and non-integrable MBL phases, the entanglement entropy shows large eigenstate-to-eigenstate fluctuations, and differs from the thermodynamic entropy. Thus the non-integrable MBL phase violates ETH similar to an integrable system; however, a key difference is that the magnitude of the entanglement entropy in the MBL phase is significantly smaller than in the integrable phase, where the entanglement entropy is of the same order of magnitude as in the non-integrable phase, but with a lot of eigenstate-to-eigenstate fluctuations. Quench dynamics from an initial CDW state independently supports the validity of the ETH in the ergodic phase and its violation in the MBL phase.

Thouless Time Analysis of Anderson and Many-Body Localization Transitions

Spectral statistics of disordered systems encode Thouless and Heisenberg timescales, whose ratio determines whether the system is chaotic or localized. We show that the scaling of the Thouless time with the system size and disorder strength is very similar in one-body Anderson models and in disordered quantum many-body systems. We argue that the two parameter scaling breaks down in the vicinity of the transition to the localized phase, signaling a slowing-down of dynamics.

Many-Body Quantum Dynamics of Initially Trapped Systems due to a Stark Potential: Thermalization versus Bloch Oscillations

We analyze the dynamics of an initially trapped cloud of interacting quantum particles on a lattice under a linear (Stark) potential. We reveal a dichotomy: initially trapped interacting systems possess features typical of both many-body-localized and thermalizing systems. We consider both fermions (t-V model) and bosons (Bose-Hubbard model). For the zero and infinite interaction limits, both systems are integrable: we provide analytic solutions in terms of the moments of the initial cloud shape and clarify how the recurrent dynamics (many-body Bloch oscillations) depends on the initial state. Away from the integrable points, we identify and explain the timescale at which Bloch oscillations decohere.

Intramolecular vibrational energy redistribution and the quantum ergodicity transition: a phase space perspective

The onset of facile intramolecular vibrational energy flow can be related to features in the connected network of anharmonic resonances in the classical phase space.

Bath-Induced Decay of Stark Many-Body Localization

We investigate the relaxation dynamics of an interacting Stark-localized system coupled to a dephasing bath, and compare its behavior to the conventional disorder-induced many body localized system. Specifically, we study the dynamics of population imbalance between even and odd sites, and the growth of the von Neumann entropy. For a large potential gradient, the imbalance is found to decay on a timescale τ that grows quadratically with the Wannier-Stark tilt. For the noninteracting system, it shows an exponential decay, which becomes a stretched exponential decay in the presence of finite interactions. This is different from a system with disorder-induced localization, where the imbalance exhibits a stretched exponential decay also for vanishing interactions. As another clear qualitative difference, we do not find a logarithmically slow growth of the von Neumann entropy as it is found for the disordered system. Our findings can immediately be tested experimentally with ultracold atoms in optical lattices.

From Bloch oscillations to many-body localization in clean interacting systems

In this work we demonstrate that nonrandom mechanisms that lead to single-particle localization may also lead to many-body localization, even in the absence of disorder. In particular, we consider interacting spins and fermions in the presence of a linear potential. In the noninteracting limit, these models show the well-known Wannier-Stark localization. We analyze the fate of this localization in the presence of interactions. Remarkably, we find that beyond a critical value of the potential gradient these models exhibit nonergodic behavior as indicated by their spectral and dynamical properties. These models, therefore, constitute a class of generic nonrandom models that fail to thermalize. As such, they suggest new directions for experimentally exploring and understanding the phenomena of many-body localization. We supplement our work by showing that by using machine-learning techniques the level statistics of a system may be calculated without generating and diagonalizing the Hamiltonian, which allows a generation of large statistics.