Slowest local operators in quantum spin chains

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

doi:10.1103/physreve.92.012128

Cited by 61 publications

(75 citation statements)

References 35 publications

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“…Our scheme constitutes the first scalable representation of the full set of eigenstates of FMBL systems by tensor networks: Local observables can be approximated with an error that decreases like an inverse polynomial of the computational cost. We use a figure of merit that is directly related to the qLIOMs and motivated by a procedure introduced by Kim et al to identify slow operators in disorder-free nonintegrable models [53]. This strongly reduces the computational cost of the tensor network (TN) contractions needed to optimize our unitaries compared to using the variance as a figure of merit (as in Ref.…”

Section: Introductionmentioning

confidence: 99%

Efficient Representation of Fully Many-Body Localized Systems Using Tensor Networks

Wahl¹,

Pal²,

Simon³

2017

Phys. Rev. X

125

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We propose a tensor network encoding the set of all eigenstates of a fully many-body localized system in one dimension. Our construction, conceptually based on the ansatz introduced in Phys. Rev. B 94, 041116(R) (2016), is built from two layers of unitary matrices which act on blocks of l contiguous sites. We argue that this yields an exponential reduction in computational time and memory requirement as compared to all previous approaches for finding a representation of the complete eigenspectrum of large many-body localized systems with a given accuracy. Concretely, we optimize the unitaries by minimizing the magnitude of the commutator of the approximate integrals of motion and the Hamiltonian, which can be done in a local fashion. This further reduces the computational complexity of the tensor networks arising in the minimization process compared to previous work. We test the accuracy of our method by comparing the approximate energy spectrum to exact diagonalization results for the random-field Heisenberg model on 16 sites. We find that the technique is highly accurate deep in the localized regime and maintains a surprising degree of accuracy in predicting certain local quantities even in the vicinity of the predicted dynamical phase transition. To demonstrate the power of our technique, we study a system of 72 sites, and we are able to see clear signatures of the phase transition. Our work opens a new avenue to study properties of the many-body localization transition in large systems.

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Section: Introductionmentioning

confidence: 99%

Efficient Representation of Fully Many-Body Localized Systems Using Tensor Networks

Wahl¹,

Pal²,

Simon³

2017

Phys. Rev. X

125

View full text Add to dashboard Cite

show abstract

“…We focus on translationally invariant Hermitian operators obtained as sums of local terms and adopt the formalism of Ref. [34]. We restate this approach as a problem in the operator Hilbert space as follows.…”

Section: Methods Of the Slowest Operatormentioning

confidence: 99%

“…In the system in the good ergodic regime studied in Ref. [34], in the nontranslationally invariant setting, the slowest operator can be understood as dressed energy density modulation operator. On the other hand, in the translationally invariant setting, the slowest operator does not have simple connection to the energy density modulation and its physical meaning remains an open question.…”

Section: B Relation To Operator Norm and Thermalization Time Scalementioning

confidence: 99%

“…Indeed, following Ref. [34], let us consider a quench setting where we start from some initial state |ψ ini . Using the Heisenberg representation of observables, Q H (t) ≡ e iH t Qe −iH t , and denoting the expectation value of the operator Q H (t) ≡ ψ ini |Q H (t)|ψ ini , the deviation of the expectation value from its initial value can be estimated as .…”

Section: B Relation To Operator Norm and Thermalization Time Scalementioning

confidence: 99%

“…Following the "slowest operator formalism" introduced in Ref. [34], we numerically construct the quasiconserved local operator for the nonintegrable spin model,…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Explicit construction of quasiconserved local operator of translationally invariant nonintegrable quantum spin chain in prethermalization

Lin

Motrunich

2017

Phys. Rev. B

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We numerically construct translationally invariant quasiconserved operators with maximum range M, which best commute with a nonintegrable quantum spin chain Hamiltonian, up to M = 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 M at small M, and turns into a slower decay at larger M. This quasiconserved operator can be understood as a dressed total "spin-z" 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.

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Equilibration Times in Closed Quantum Many-Body Systems

Wilming

Oliveira

Short

et al. 2018

Fundamental Theories of Physics

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One of the main questions of research on quantum many-body systems following unitary out of equilibrium dynamics is to find out how local expectation values equilibrate in time. For non-interacting models, this question is rather well understood. However, the best known bounds for general quantum systems are vastly crude, scaling unfavorable with the system size. Nevertheless, empirical and numerical evidence suggests that for generic interacting many-body systems, generic local observables, and sufficiently well-behaved states, the equilibration time does not depend strongly on the system size, but only the precision with which this occurs does. In this discussion paper, we aim at giving very simple and plausible arguments for why this happens. While our discussion does not yield rigorous results about equilibration time scales, we believe that it helps to clarify the essential underlying mechanism, the intuition and important figures of merit behind equilibration. We then connect our arguments to common assumptions and numerical results in the field of equilibration and thermalization of closed quantum systems, such as the eigenstate thermalization hypothesis as well as rigorous results on interacting quantum many-body systems. Finally, we complement our discussion with numerical results -both in the case of examples and counter-examples of equilibrating systems.

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Slowest local operators in quantum spin chains

Cited by 61 publications

References 35 publications

Efficient Representation of Fully Many-Body Localized Systems Using Tensor Networks

Efficient Representation of Fully Many-Body Localized Systems Using Tensor Networks

Explicit construction of quasiconserved local operator of translationally invariant nonintegrable quantum spin chain in prethermalization

Equilibration Times in Closed Quantum Many-Body Systems

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