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arXiv:1410.4186 · Physical Review E

Slowest local operators in quantum spin chains

HyungWon Kim, Mari Carmen Bañuls, J. I. Cirac, Matthew B. Hastings, David A. Huse

Published Jul 22, 2015• 58 citations
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Abstract

We numerically construct slowly relaxing local operators in a nonintegrable spin-1/2 chain. Restricting the support of the operator to M consecutive spins along the chain, we exhaustively search for the operator that minimizes the Frobenius norm of the commutator with the Hamiltonian. We first show that the Frobenius norm bounds the time scale of relaxation of the operator at high temperatures. We find operators with significantly slower relaxation than the slowest simple "hydrodynamic" mode due to energy diffusion. Then we examine some properties of the nontrivial slow operators. Using both exhaustive search and tensor network techniques, we find similar slowly relaxing operators for a Floquet spin chain; this system is hydrodynamically "trivial," with no conservation laws restricting their dynamics. We argue that such slow relaxation may be a generic feature following from locality and unitarity.

DOI
10.1103/physreve.92.012128

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We numerically construct slowly relaxing local operators in a nonintegrable spin-1/2 chain. Restricting the support of the operator to M consecutive spins along the chain, we exhaustively search for the operator that minimizes the Frobenius norm of the commutator with the Hamiltonian. We first show that the Frobenius norm bounds the time scale of relaxation of the operator at high temperatures. We find operators with significantly slower relaxation than the slowest simple "hydrodynamic" mode due to energy diffusion. Then we examine some properties of the nontrivial slow operators. Using both exhaustive search and tensor network techniques, we find similar slowly relaxing operators for a Floquet spin chain; this system is hydrodynamically "trivial," with no conservation laws restricting their dynamics. We argue that such slow relaxation may be a generic feature following from locality and unitarity.
In this paper we study one-dimensional quantum Ising spin chains in external magnetic field close to an integrable point. We concentrate on the dynamics of the slowest operator, that plays a key role at the final period of thermalization. We introduce two independent definitions of the slowest operator: local and translationally-invariant ones. We construct both operators numerically using tensor networks and extensively compare their physical properties. We find that the local operator has a significant overlap with energy flux, it does not correspond to an integral of motion, and, as one goes away from the integrable point, its revivals get suppressed and the rate of delocalization changes from extremely slow to slower than diffusion. The translationally-invariant operator corresponds to an integral of motion; as the system becomes less integrable, at some point this operator changes its nature: from no overlap with any magnetization and fast rate of delocalization, to non-zero overlap with magnetizations σxσ_{x}σx​ and σzσ_{z}σz​ and slow rate of delocalization.
In this paper we study one-dimensional quantum Ising spin chains in an external magnetic field close to an integrable point. We concentrate on the dynamics of the slowest operator that plays a key role at the final period of thermalization. We introduce two independent definitions of the slowest operator: local and translationally invariant ones. We construct both operators numerically using tensor networks and extensively compare their physical properties. We find that the local operator has a significant overlap with energy flux, it does not correspond to an integral of motion, and, as one goes away from the integrable point, its revivals get suppressed and the rate of delocalization changes from extremely slow to slower than diffusion. The translationally invariant operator corresponds to an integral of motion; as the system becomes less integrable, at some point this operator changes its nature: from no overlap with any magnetization and fast rate of delocalization, to nonzero overlap with magnetizations σ_{x} and σ_{z} and slow rate of delocalization.
We construct almost conserved local operators, that possess a minimal commutator with the Hamiltonian of the system, near the many-body localization transition of a one-dimensional disordered spin chain. We collect statistics of these slow operators for different support sizes and disorder strengths, both using exact diagonalization and tensor networks. Our results show that the scaling of the average of the smallest commutators with the support size is sensitive to Griffiths effects in the thermal phase and the onset of many-body localization. Furthermore, we demonstrate that the probability distributions of the commutators can be analyzed using extreme value theory and that their tails reveal the difference between diffusive and subdiffusive dynamics in the thermal phase.
We consider Lindbladian operator dynamics in many-body quantum systems with one or more integrals of motion (IOMs), subject to weak local dissipation. We demonstrate that IOMs with small support become slow modes of these dynamics, in the sense that their Frobenius norms decay more slowly compared to generic operators. As a result, the eigenoperators of such Lindbladians with slowest decay rates have a large overlap with the IOMs of the underlying Hamiltonian. We demonstrate this correspondence between slow modes and IOMs numerically for a number of many-body models and further corroborate it with perturbative arguments. These results open up another method for the identification of IOMs and provide insights into the dissipative many-body dynamics.
Out-of-time-ordered correlators (OTOCs) help characterize the scrambling of quantum information and are usually studied in the context of nonintegrable systems. In this work, we compare the relaxation dynamics of OTOCs in interacting integrable and nonintegrable spin-12\frac{1}{2}21​ XYZ chains in regimes without a classical counterpart. In both kinds of chains, using the presence of symmetries such as U(1)U(1)U(1) and supersymmetry, we consider regimes in which the OTOC operators overlap or not with the Hamiltonian. We show that the relaxation of the OTOCs is slow (fast) when there is (there is not) an overlap, independently of whether the chain is integrable or nonintegrable. When slow, we show that the OTOC dynamics follows closely that of the two-point correlators. We study the dynamics of OTOCs using numerical calculations, and gain analytical insights from the properties of the diagonal and of the off-diagonal matrix elements of the corresponding local operators in the energy eigenbasis.
We numerically construct translationally invariant quasiconserved operators with maximum range MMM, which best commute with a nonintegrable quantum spin chain Hamiltonian, up to M=12M=12M=12. In the large coupling limit, we find that the residual norm of the commutator of the quasiconserved operator decays exponentially with its maximum range MMM at small MMM, and turns into a slower decay at larger MMM. This quasiconserved operator can be understood as a dressed total ``spin-zzz'' operator, by comparing with the perturbative Schrieffer-Wolff construction developed to high order reaching essentially the same maximum range. We also examine the operator inverse participation ratio of the operator, which suggests its localization in the operator Hilbert space. The operator also shows an almost exponentially decaying profile at short distance, while the long-distance behavior is not clear due to limitations of our numerical calculation. Further dynamical simulation confirms that the prethermalization-equilibrated values are described by a generalized Gibbs ensemble that includes such quasiconserved operator.