Chaotic Spin Chains Research Articles

Hayden-Preskill recovery in Hamiltonian systems

Information scrambling refers to the unitary dynamics that quickly spreads and encodes localized quantum information over an entire many-body system and makes the information accessible from any small subsystem. While information scrambling is the key to understanding complex quantum many-body dynamics and is well-understood in random unitary models, it has been hardly explored in Hamiltonian systems. In this Letter, we investigate the information recovery in various time-independent Hamiltonian systems, including chaotic spin chains and Sachdev-Ye-Kitaev models. We show that information recovery is possible in certain, but not all, chaotic models, which highlights the difference between information recovery and quantum chaos based on the energy spectrum or the out-of-time-ordered correlators. We also show that information recovery probes transitions caused by the change of information-theoretic features of the dynamics. Published by the American Physical Society 2024

Equilibration of isolated systems: Investigating the role of coarse-graining on the initial state magnetization

Many theoretical and experimental results show that even isolated quantum systems evolving unitarily may equilibrate, since the evolution of some observables may be around an equilibrium value with negligible fluctuations most of the time. There are rigorous theorems giving the conditions for such equilibration to happen. In particular, initial states prepared with a lack of resolution in the energy will equilibrate. We investigate how equilibration may be affected by a lack of resolution, or coarse-graining, in the magnetization of the initial state. In particular, for a chaotic spin chain and using exact diagonalization, we show that the level of equilibration of an initial state with a coarse, not well-defined magnetization is different from the level of an initial state with well-defined magnetization. This difference will depend on the degree of coarse-graining and the direction of magnetization. We also analyze the time for the system to reach equilibrium, showing good agreement with theoretical estimates and with some evidence that less resolution leads to faster equilibration. Our study highlights the crucial role of initial state preparation in the equilibration dynamics of quantum systems and provides new insights into the fundamental nature of equilibration in closed systems.

Overcoming the entanglement barrier in quantum many-body dynamics via space-time duality

Describing nonequilibrium properties of quantum many-body systems is challenging due to high entanglement in the wave function. We describe the evolution of local observables via the influence matrix (IM), which encodes the effects of a many-body system as an environment for local subsystems. Recent works found that in many dynamical regimes the IM of an infinite system has low temporal entanglement and can be efficiently represented as a matrix-product state (MPS). Yet, direct iterative constructions of the IM encounter highly entangled intermediate states---a temporal entanglement barrier (TEB). We argue that TEB is ubiquitous, and elucidate its physical origin via a semiclassical quasiparticle picture that exactly captures the behavior of integrable spin chains. Further, we show that a TEB also arises in chaotic spin chains, which lack well-defined quasiparticles. Based on these insights, we formulate an alternative light-cone growth algorithm, which provably avoids TEB, thus providing an efficient construction of the thermodynamic-limit IM as a MPS. This work uncovers the origin of the efficiency of the IM approach for studying thermalization and transport.

Universal entanglement of mid-spectrum eigenstates of chaotic local Hamiltonians

In systems governed by chaotic local Hamiltonians, my previous work [7] conjectured the universality of the average entanglement entropy of all eigenstates by proposing an exact formula for its dependence on the subsystem size. In this note, I extend this result to the average entanglement entropy of a constant fraction of eigenstates in the middle of the energy spectrum. The generalized formula is supported by numerical simulations of various chaotic spin chains.

The quasi-particle picture and its breakdown after local quenches: mutual information, negativity, and reflected entropy

We study the dynamics of (Rényi) mutual information, logarithmic negativity, and (Rényi) reflected entropy after exciting the ground state by a local operator. Together with recent results from ref. [1], we are able to conjecture a close-knit structure between the three quantities that emerges in states excited above the vacuum, including both local and global quantum quenches. This structure intimately depends on the chaoticity of the theory i.e. there exist distinct sets of equivalences for integrable and chaotic theories. For rational conformal field theories (RCFT), we find all quantities to compute the quantum dimension of the primary operator inserted. In contrast, we find the correlation measures to grow (logarithmically) without bound in all c > 1 conformal field theories with a finite twist gap. In comparing the calculations in the two classes of theories, we are able to identify the dynamical mechanism for the breakdown of the quasi-particle picture in 2D conformal field theories. Intriguingly, we also find preliminary evidence that our general lessons apply to quantum systems considerably distinct from conformal field theories, such as integrable and chaotic spin chains, suggesting a universality of entanglement dynamics in non-equilibrium systems.

Information scrambling at finite temperature in local quantum systems

This paper investigates the temperature dependence of quantum information scrambling in local systems with an energy gap, $m$, above the ground state. We study the speed and shape of growing Heisenberg operators as quantified by out-of-time-order correlators, with particular attention paid to so-called contour dependence, i.e. dependence on the way operators are distributed around the thermal circle. We report large scale tensor network numerics on a gapped chaotic spin chain down to temperatures comparable to the gap which show that the speed of operator growth is strongly contour dependent. The numerics also show a characteristic broadening of the operator wavefront at finite temperature $T$. To study the behavior at temperatures much below the gap, we perform a perturbative calculation in the paramagnetic phase of a 2+1D O($N$) non-linear sigma model, which is analytically tractable at large $N$. Using the ladder diagram technique, we find that operators spread at a speed $\sqrt{T/m}$ at low temperatures, $T\ll m$. In contrast to the numerical findings of spin chain, the large $N$ computation is insensitive to the contour dependence and does not show broadening of operator front. We discuss these results in the context of a recently proposed state-dependent bound on scrambling.

Speed of quantum information spreading in chaotic systems

We present a general theory of quantum information propagation in chaotic quantum many-body systems. The generic expectation in such systems is that quantum information does not propagate in localized form; instead, it tends to spread out and scramble into a form that is inaccessible to local measurements. To characterize this spreading, we define an information speed via a quench-type experiment and derive a general formula for it as a function of the entanglement density of the initial state. As the entanglement density varies from zero to one, the information speed varies from the entanglement speed to the butterfly speed. We verify that the formula holds both for a quantum chaotic spin chain and in field theories with an AdS/CFT gravity dual. For the second case, we study in detail the dynamics of entanglement in two-sided Vaidya-AdS-Reissner-Nordstrom black branes. We also show that, with an appropriate decoding process, quantum information can be construed as moving at the information speed, and, in the case of AdS/CFT, we show that a locally detectable signal propagates at the information speed in a spatially local variant of the traversable wormhole setup.

Quantum vs. classical information: operator negativity as a probe of scrambling

We consider the logarithmic negativity and related quantities of time evolution operators. We study free fermion, compact boson, and holographic conformal field theories (CFTs) as well as numerical simulations of random unitary circuits and integrable and chaotic spin chains. The holographic behavior strongly deviates from known non- holographic CFT results and displays clear signatures of maximal scrambling. Intriguingly, the random unitary circuits display nearly identical behavior to the holographic channels. Generically, we find the “line-tension picture” to effectively capture the entanglement dynamics for chaotic systems and the “quasi-particle picture” for integrable systems. With this motivation, we propose an effective “line-tension” that captures the dynamics of the logarithmic negativity in chaotic systems in the spacetime scaling limit. We compare the negativity and mutual information leading us to find distinct dynamics of quantum and classical information. The “spurious entanglement” we observe may have implications on the “simulatability” of quantum systems on classical computers. Finally, we elucidate the connection between the operation of partially transposing a density matrix in conformal field theory and the entanglement wedge cross section in Anti-de Sitter space using geodesic Witten diagrams.

Accessing scrambling using matrix product operators

Scrambling, a process in which quantum information spreads over a complex quantum system becoming inaccessible to simple probes, happens in generic chaotic quantum many-body systems, ranging from spin chains, to metals, even to black holes. Scrambling can be measured using out-of-time-ordered correlators (OTOCs), which are closely tied to the growth of Heisenberg operators. In this work, we present a general method to calculate OTOCs of local operators in local one-dimensional systems based on approximating Heisenberg operators as matrix-product operators (MPOs). Contrary to the common belief that such tensor network methods work only at early times, we show that the entire early growth region of the OTOC can be captured using an MPO approximation with modest bond dimension. We analytically establish the goodness of the approximation by showing that if an appropriate OTOC is close to its initial value, then the associated Heisenberg operator has low entanglement across a given cut. We use the method to study scrambling in a chaotic spin chain with $201$ sites. Based on this data and OTOC results for black holes, local random circuit models, and non-interacting systems, we conjecture a universal form for the dynamics of the OTOC near the wavefront. We show that this form collapses the chaotic spin chain data over more than fifteen orders of magnitude.

Complexity of thermal states in quantum spin chains

We study the quantum correlations and complexity of simulation, characterized by quantum mutual information and entanglement entropy in operator space, respectively, for thermal states in critical, noncritical, and quantum chaotic spin chains. A simple general relation between the two quantities is proposed. We show that in all cases mutual information and entanglement entropy saturate with system size, whereas as a function of the inverse temperature, we find logarithmic divergences for critical cases and uniform bounds in noncritical cases. A simple efficient quasiexact method for computation of arbitrary entropy-related quantities in thermalized $XY$ spin chains is proposed.

Broken Ergodicity in Classically Chaotic Spin Systems

A one dimensional classically chaotic spin chain with asymmetric coupling and two different inter-spin interactions, nearest neighbors and all-to-all, has been considered. Depending on the interaction range, dynamical properties, as ergodicity and chaoticity are strongly different. Indeed, even in presence of chaoticity, the model displays a lack of ergodicity only in presence of all to all interaction and below an energy threshold, that persists in the thermodynamical limit. Energy threshold can be found analytically and results can be generalized for a generic XY model with asymmetric coupling.