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arXiv:1811.00029 · Physical Review Letters

Evolution of Entanglement Spectra under Generic Quantum Dynamics

Po-Yao Chang, Xiao Chen, Sarang Gopalakrishnan, J. H. Pixley

Published Nov 6, 2019• 30 citations
arXivHTMLPDF

Abstract

We characterize the early stages of the approach to equilibrium in isolated quantum systems through the evolution of the entanglement spectrum. We find that the entanglement spectrum of a subsystem evolves with three distinct timescales. First, on an o(1) timescale, independent of system or subsystem size and the details of the dynamics, the entanglement spectrum develops nearest-neighbor level repulsion. The second timescale sets in when the light cone has traversed the subsystem. Between these two times, the density of states of the reduced density matrix takes a universal, scale-free 1/f form; thus, random-matrix theory captures the local statistics of the entanglement spectrum but not its global structure. The third time scale is that on which the entanglement saturates; this occurs well after the light cone traverses the subsystem. Between the second and third times, the entanglement spectrum compresses to its thermal Marchenko-Pastur form. These features hold for chaotic Hamiltonian and Floquet dynamics as well as a range of quantum circuit models.

DOI
10.1103/physrevlett.123.190602

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We characterize the early stages of the approach to equilibrium in isolated quantum systems through the evolution of the entanglement spectrum. We find that the entanglement spectrum of a subsystem evolves with three distinct timescales. First, on an o(1) timescale, independent of system or subsystem size and the details of the dynamics, the entanglement spectrum develops nearest-neighbor level repulsion. The second timescale sets in when the light cone has traversed the subsystem. Between these two times, the density of states of the reduced density matrix takes a universal, scale-free 1/f form; thus, random-matrix theory captures the local statistics of the entanglement spectrum but not its global structure. The third time scale is that on which the entanglement saturates; this occurs well after the light cone traverses the subsystem. Between the second and third times, the entanglement spectrum compresses to its thermal Marchenko-Pastur form. These features hold for chaotic Hamiltonian and Floquet dynamics as well as a range of quantum circuit models.
We examine the time evolution of the entanglement spectrum of a small subsystem of a nonintegrable 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 a timescale that is parametrically shorter than that for full thermalization of the subsystem. This timescale is also close to when the mutual information between individual spins at the ends of the subsystem reaches its maximum. We provide an analytical understanding of this phenomenon and show supporting numerical data for both random unitary circuits and a microscopic Hamiltonian.
Abstract Entanglement entropy has long served as a central diagnostic of complexity and information scrambling in quantum many-body systems, and its behavior in random quantum circuits is by now well characterized. However, as a single scalar, entanglement entropy captures only the total uncertainty of the reduced density matrix spectrum while discarding the detailed structure of that spectrum — including the shape of the eigenvalue distribution, level repulsion signatures, and the transition from integrable to chaotic behavior. In this work, we propose spectral entropy — the Shannon entropy of the eigenvalue distribution of the reduced density matrix — as a structured diagnostic that goes beyond standard entanglement entropy. Using numerical simulations, we compute the spectral entropy of subsystem reduced states in Haar-random and Clifford circuits as a function of circuit depth, and compare its behavior against von Neumann entanglement entropy, Rényi-2 entropy, and entanglement spectrum level-spacing statistics. We expect to find three key results: (i) spectral entropy exhibits a growth timescale distinct from that of entanglement entropy, potentially reflecting spectral restructuring more sensitively; (ii) in the depth regime where entanglement entropy has already saturated, spectral entropy may continue to evolve, revealing late-time complexification that standard entanglement measures cannot resolve; (iii) spectral entropy may display a distinctive response to measurement-induced or noise-induced disentanglement that differs qualitatively from entanglement entropy. This work positions spectral entropy as a complementary — not replacement — tool to entanglement entropy, offering a finer spectral lens on information scrambling, thermalization, and complexity growth in random quantum circuits.
Quantum quench is a non-equilibrium process where the Hamiltonian is suddenly changed during the quantum evolution. The change can be made by spatially local perturbations (local quench) or globally switching to a completely different Hamiltonian (global quench). This thesis investigates the post-quench non-equilibrium dynamics with an emphasis on the time dependence of the quantum entanglement. We inspect the scaling of entanglement entropy (EE) to learn how correlation and entanglement built up in a quench. We begin with two local quench examples. In Chap. 2, we apply a local operator to the groundstate of the quantum Lifshitz model and monitor the change of the EE. We find that the entanglement grows according to the dynamical exponent z = 2 and then saturates to the scaling dimension of the perturbing operator -- a value representing its strength. In Chap. 3, we study the evolution after connecting two different one-dimensional critical chains at their ends. The Loschmidt echo which measures the similarity between the evolved state and the initial one decays with a power law, whose exponent is the scaling dimension of the defect (junction). Among other conclusions, we see that the local quench dynamics contain universal information of the (critical) theory. In the global quench scenario, the change of the Hamiltonian affects all parts of the system. In this thesis, we focus on the global chaotic quench driven by generic non-integrable Hamiltonians. In Chap. 4, we propose to use the operator entanglement entropy of the unitary operator as a probe. Its fast linear entanglement production is sharply contrasted to the slow logarithmic spreading of the many-body localized system. The entanglement saturation suggests that the evolution operator in the long time can be modeled by a random unitary matrix. In Chap. 5, we construct a random tensor network which consists of random unitary matrices connected locally to model chaotic evolution with local interactions. We find that the entanglement dynamics is mapped to the statistical mechanics of interacting random walks. This appealing emergent picture allows us to understand the universal linear growth as well as the fluctuations of entanglement in a chaotic quench.
We investigate the evolution of entanglement spectra under a global quantum quench from a short-range correlated state to the quantum critical point. Motivated by the conformal mapping, we find that the dynamical entanglement spectra demonstrate distinct finite-size scaling behaviors from the static case. As a prototypical example, we compute real-time dynamics of the entanglement spectra of a one-dimensional transverse-field Ising chain. Numerical simulation confirms that the entanglement spectra scale with the subsystem size l as ∼l −1 for the dynamical equilibrium state, much faster than ∝ ln−1 l for the critical ground state. In particular, as a byproduct, the entanglement spectra at the long time limit faithfully gives universal tower structure of underlying Ising criticality, which shows the emergence of operator-state correspondence in the quantum dynamics.
We investigate the distribution of the eigenvalues of the reduced density matrix (entanglement spectrum) after a global quantum quench. We show that in an appropriate scaling limit the lower part of the entanglement spectrum exhibits ``universality''. In the scaling limit and at asymptotically long times the distribution of the entanglement spectrum depends on two parameters that can be determined from the Rényi entropies. We show that two typical scenarios occur. In the first one, the distribution of the entanglement spectrum levels is similar to the one describing the ground-state entanglement spectrum in Conformal Field Theories. In the second scenario, the lower levels of the entanglement spectrum are highly degenerate and their distribution is given by a series of Dirac deltas. We benchmark our analytical results in free-fermion chains, such as the transverse field Ising chain and the XX chain, in the rule 54 chain, and in Bethe ansatz solvable spin models.
Exactly solvable models provide an opportunity to study different aspects of reduced quantum dynamics in detail. We consider the reduced dynamics of a single spin in finite XX and XY spin 1/2 chains. First we introduce a general expression describing the evolution of the reduced density matrix. This expression proves to be tractable when the combined closed system (i.e. open system plus environment) is integrable. Then we focus on comparing decoherence and thermalization timescales in the XX chain. We find that for a single spin these timescales are comparable, in contrast to what should be expected for a macroscopic body. This indicates that the process of quantum relaxation of a system with few accessible states can not be separated in two distinct stages - decoherence and thermalization. Finally, we turn to finite-size effects in the time evolution of a single spin in the XY chain. We observe three consecutive stages of the evolution: regular evolution, partial revivals, irregular (apparently chaotic) evolution. The duration of the regular stage is proportional to the number of spins in the chain. We observe a "quiet and cold period" in the end of the regular stage, which breaks up abruptly at some threshold time.