arXiv subjects

Tibor Rakovszky

Publications and source records attributed to Tibor Rakovszky.

32 records · Page 2Linked to original sources

arXiv2021

Operator backflow and the classical simulation of quantum transport

Tensor product states have proved extremely powerful for simulating the area-law entangled states of many-body systems, such as the ground states of gapped Hamiltonians in one dimension. The applicability of such methods to the \emph{dynamics} of many-body systems is less clear: the memory required grows exponentially in time in most cases, quickly becoming unmanageable. New methods reduce the memory required by selectively discarding/dissipating parts of the many-body wavefunction which are expected to have little effect on the hydrodynamic observables typically of interest: for example, some methods discard fine-grained correlations associated with $n$-point functions, with $n$ exceeding some cutoff $\ell_*$. In this work, we present a theory for the sizes of `backflow corrections', i.e., systematic errors due to discarding this fine-grained information. In particular, we focus on their effect on transport coefficients. Our results suggest that backflow corrections are exponentially suppressed in the size of the cutoff $\ell_*$. Moreover, the backflow errors themselves have a hydrodynamical expansion, which we elucidate. We test our predictions against numerical simulations run on random unitary circuits and ergodic spin-chains. These results lead to the conjecture that transport coefficients in ergodic diffusive systems can be captured to a given precision $ε$ with an amount of memory scaling as $\exp[\mathcal{O}(\log(ε)^2)]$, significantly better than the naive estimate of memory $\exp[\mathcal{O}(\mathrm{poly}(ε^{-1}))]$ required by more brute-force methods.

cond-mat.str-el↗

arXiv2021

Dynamics in Systems with Modulated Symmetries

We extend the notions of multipole and subsystem symmetries to more general {\it spatially modulated} symmetries. We uncover two instances with exponential and (quasi)-periodic modulations, and provide simple microscopic models in one, two and three dimensions. Seeking to understand their effect in the long-time dynamics, we numerically study a stochastic cellular automaton evolution that obeys such symmetries. We prove that in one dimension, the periodically modulated symmetries lead to a diffusive scaling of correlations modulated by a finite microscopic momentum. In higher dimensions, these symmetries take the form of lines and surfaces of conserved momenta. These give rise to exotic forms of sub-diffusive behavior with a rich spatial structure influenced by lattice-scale features. Exponential modulation, on the other hand, can lead to correlations that are infinitely long-lived at the boundary, while decaying exponentially in the bulk.

cond-mat.stat-mech↗

arXiv2020

Comment on "Entanglement growth in diffusive systems"

In a recent paper (Commun. Phys. 3, 100) Znidaric studies the growth of higher Renyi entropies in diffusive systems and claims that they generically grow ballistically in time, except for spin-1/2 models in d=1 dimension. Here, we point out that the necessary conditions for sub-ballistic growth of Renyi entropies are in fact much more general, and apply to a large class of systems, including experimentally relevant ones in arbitrary dimension and with larger local Hilbert spaces.

cond-mat.str-el↗

arXiv2020

Statistical localization: from strong fragmentation to strong edge modes

Certain disorder-free Hamiltonians can be non-ergodic due to a \emph{strong fragmentation} of the Hilbert space into disconnected sectors. Here, we characterize such systems by introducing the notion of `statistically localized integrals of motion' (SLIOM), whose eigenvalues label the connected components of the Hilbert space. SLIOMs are not spatially localized in the operator sense, but appear localized to sub-extensive regions when their expectation value is taken in typical states with a finite density of particles. We illustrate this general concept on several Hamiltonians, both with and without dipole conservation. Furthermore, we demonstrate that there exist perturbations which destroy these integrals of motion in the bulk of the system, while keeping them on the boundary. This results in statistically localized \emph{strong zero modes}, leading to infinitely long-lived edge magnetizations along with a thermalizing bulk, constituting the first example of such strong edge modes in a non-integrable model. We also show that in a particular example, these edge modes lead to the appearance of topological string order in a certain subset of highly excited eigenstates. Some of our suggested models can be realized in Rydberg quantum simulators.

cond-mat.str-el↗

arXiv2020

Dissipation-assisted operator evolution method for capturing hydrodynamic transport

We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators. We represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision.

cond-mat.str-el↗

arXiv2020

Ergodicity-breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians

We show that the combination of charge and dipole conservation---characteristic of fracton systems---leads to an extensive fragmentation of the Hilbert space, which in turn can lead to a breakdown of thermalization. As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total $S^z$) and its associated dipole moment. First, we consider a minimal model including only three-site terms and find that the infinite temperature auto-correlation saturates to a finite value---showcasing non-thermal behavior. The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local $S^z$ basis, arising from the interplay of dipole conservation and local interactions. Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: the system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states. More generally, for any finite range of interactions, the system still exhibits non-thermal eigenstates appearing throughout the entire spectrum. We compare our results to charge and dipole moment conserving random unitary circuit models for which we reach identical conclusions.

cond-mat.str-el↗

arXiv2019

Sub-ballistic growth of Rényi entropies due to diffusion

We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Rényi entropies. We argue that the latter generically grow \emph{sub-ballistically}, as $\propto\sqrt{t}$, in systems with diffusive transport. We provide strong evidence for this in both a U$(1)$ symmetric random circuit model and in a paradigmatic non-integrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second Rényi entropy and show that it exhibits hydrodynamic tails with \emph{three distinct power laws} occurring for different classes of initial states.

cond-mat.str-el↗

arXiv2019

Entanglement growth after inhomogenous quenches

We study the growth of entanglement in quantum systems with a conserved quantity exhibiting diffusive transport, focusing on how initial inhomogeneities are imprinted on the entropy. We propose a simple effective model, which generalizes the minimal cut picture of \textit{Jonay et al.} in such a way that the `line tension' of the cut depends on the local entropy density. In the case of noisy dynamics, this is described by a Kardar-Parisi-Zhang (KPZ) equation coupled to a diffusing field. We investigate the resulting dynamics and find that initial inhomogeneities of the conserved charge give rise to features in the entanglement profile, whose width and height both grow in time as $\propto\sqrt{t}$. In particular, for a domain wall quench, diffusion restricts entanglement growth to be $S_\text{vN} \lesssim \sqrt{t}$. We find that for charge density wave initial states, these features in the entanglement profile are present even after the charge density has equilibrated. Our conclusions are supported by numerical results on random circuits and deterministic spin chains.

cond-mat.str-el↗

arXiv2019

Hamiltonian truncation approach to quenches in the Ising field theory

In contrast to lattice systems where powerful numerical techniques such as matrix product state based methods are available to study the non-equilibrium dynamics, the non-equilibrium behaviour of continuum systems is much harder to simulate. We demonstrate here that Hamiltonian truncation methods can be efficiently applied to this problem, by studying the quantum quench dynamics of the 1+1 dimensional Ising field theory using a truncated free fermionic space approach. After benchmarking the method with integrable quenches corresponding to changing the mass in a free Majorana fermion field theory, we study the effect of an integrability breaking perturbation by the longitudinal magnetic field. In both the ferromagnetic and paramagnetic phases of the model we find persistent oscillations with frequencies set by the low-lying particle excitations not only for small, but even for moderate size quenches. In the ferromagnetic phase these particles are the various non-perturbative confined bound states of the domain wall excitations, while in the paramagnetic phase the single magnon excitation governs the dynamics, allowing us to capture the time evolution of the magnetisation using a combination of known results from perturbation theory and form factor based methods. We point out that the dominance of low lying excitations allows for the numerical or experimental determination of the mass spectra through the study of the quench dynamics.

cond-mat.stat-mech↗

arXiv2019

Signatures of information scrambling in the dynamics of the entanglement spectrum

We examine the time evolution of the entanglement spectrum of a small subsystem of a non-integrable spin chain following a quench from a product state. We identify signatures in this entanglement spectrum of the distinct dynamical velocities (related to entanglement and operator spreading) that control thermalization. We show that the onset of level repulsion in the entanglement spectrum occurs on different timescales depending on the "entanglement energy", and that this dependence reflects the shape of the operator front. Level repulsion spreads across the entire entanglement spectrum on timescales close to when the mutual information between individual spins at the ends of the subsystem reaches its maximum; this timescale is much shorter than that for full thermalization of the subsystem. We provide an analytical understanding of this phenomenon and show supporting numerical data for both random unitary circuits and a microscopic Hamiltonian.

cond-mat.stat-mech↗

arXiv2018

Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation

The scrambling of quantum information in closed many-body systems, as measured by out-of-time-ordered correlation functions (OTOCs), has lately received considerable attention. Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits, aspects of which are conjectured to be universal to ergodic many-body systems, even without randomness. Here we extend this approach to systems with locally conserved quantities (e.g., energy). We do this by considering local random unitary circuits with a conserved U$(1)$ charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time ordered {\textit{and}} out-of-time-ordered correlation functions, both can have a diffusively relaxing component or "hydrodynamic tail" at late times. We verify the presence of such tails also in a deterministic, peridocially driven system. We show that for OTOCs, the combination of diffusive and ballistic components leads to a wave front with a specific, asymmetric shape, decaying as a power law behind the front. These results also explain existing numerical investigations in non-noisy ergodic systems with energy conservation. Moreover, we consider OTOCs in Gibbs states, parametrized by a chemical potential $μ$, and apply perturbative arguments to show that for $μ\gg 1$ the ballistic front of information-spreading can only develop at times exponentially large in $μ$ -- with the information traveling diffusively at earlier times. We also develop a new formalism for describing OTOCs and operator spreading, which allows us to interpret the saturation of OTOCs as a form of thermalization on the Hilbert space of operators.

cond-mat.stat-mech↗

arXiv2017

Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws

Thermalization and scrambling are the subject of much recent study from the perspective of many-body quantum systems with locally bounded Hilbert spaces (`spin chains'), quantum field theory and holography. We tackle this problem in 1D spin-chains evolving under random local unitary circuits and prove a number of exact results on the behavior of out-of-time-ordered commutators (OTOCs), and entanglement growth in this setting. These results follow from the observation that the spreading of operators in random circuits is described by a `hydrodynamical' equation of motion, despite the fact that random unitary circuits do not have locally conserved quantities (e.g., no conserved energy). In this hydrodynamic picture quantum information travels in a front with a `butterfly velocity' $v_{\text{B}}$ that is smaller than the light cone velocity of the system, while the front itself broadens diffusively in time. The OTOC increases sharply after the arrival of the light cone, but we do \emph{not} observe a prolonged exponential regime of the form $\sim e^{λ_\text{L}(t-x/v)}$ for a fixed Lyapunov exponent $λ_\text{L}$. We find that the diffusive broadening of the front has important consequences for entanglement growth, leading to an entanglement velocity that can be significantly smaller than the butterfly velocity. We conjecture that the hydrodynamical description applies to more generic ergodic systems and support this by verifying numerically that the diffusive broadening of the operator wavefront also holds in a more traditional non-random Floquet spin-chain. We also compare our results to Clifford circuits, which have less rich hydrodynamics and consequently trivial OTOC behavior, but which can nevertheless exhibit linear entanglement growth and thermalization.

cond-mat.str-el↗

arXiv2016

Detecting topological invariants in chiral symmetric insulators via losses

We show that the bulk winding number characterizing one-dimensional topological insulators with chiral symmetry can be detected from the displacement of a single particle, observed via losses. Losses represent the effect of repeated weak measurements on one sublattice only, which interrupt the dynamics periodically. When these do not detect the particle, they realize negative measurements. Our repeated measurement scheme covers both time-independent and periodically driven (Floquet) topological insulators, with or without spatial disorder. In the limit of rapidly repeated, vanishingly weak measurements, our scheme describes non-Hermitian Hamiltonians, as the lossy Su-Schrieffer-Heeger model of Phys. Rev. Lett. 102, 065703 (2009). We find, contrary to intuition, that the time needed to detect the winding number can be made shorter by decreasing the efficiency of the measurement. We illustrate our results on a discrete-time quantum walk, and propose ways of testing them experimentally.

cond-mat.mes-hall↗

arXiv2015

Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

Quantum walks are promising for information processing tasks because on regular graphs they spread quadratically faster than random walks. Static disorder, however, can turn the tables: unlike random walks, quantum walks can suffer Anderson localization, whereby the spread of the walker stays within a finite region even in the infinite time limit. It is therefore important to understand when we can expect a quantum walk to be Anderson localized and when we can expect it to spread to infinity even in the presence of disorder. In this work we analyze the response of a generic one-dimensional quantum walk -- the split-step walk -- to different forms of static disorder. We find that introducing static, symmetry-preserving disorder in the parameters of the walk leads to Anderson localization. In the completely disordered limit, however, a delocalization sets in, and the walk spreads subdiffusively. Using an efficient numerical algorithm, we calculate the bulk topological invariants of the disordered walk, and interpret the disorder-induced Anderson localization and delocalization transitions using these invariants.

cond-mat.mes-hall↗