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arXiv:2107.04505 · Physical review. B./Physical review. B

Quasiconserved quantities in the perturbed spin- 1 2 XXX model

Denis V. Kurlov, Savvas Malikis, Vladimir Gritsev

Published Mar 3, 2022• 20 citations
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Abstract

We consider the isotropic spin-$\frac{1}{2}$ Heisenberg spin chain weakly perturbed by a local translationally and SU(2)-invariant perturbation. Starting from the local integrals of motion of the unperturbed model, we modify them in order to obtain quasiconserved integrals of motion (charges) for the perturbed model. Such quasiconserved quantities are believed to be responsible for the existence of the prethermalization phase at intermediate timescales. We find that for a sufficiently local perturbation the quasiconserved quantities indeed exist, and we construct an explicit form for the first few of them.

DOI
10.1103/physrevb.105.104302

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We consider the isotropic spin-12\frac{1}{2}21​ Heisenberg spin chain weakly perturbed by a local translationally and SU(2)-invariant perturbation. Starting from the local integrals of motion of the unperturbed model, we modify them in order to obtain quasiconserved integrals of motion (charges) for the perturbed model. Such quasiconserved quantities are believed to be responsible for the existence of the prethermalization phase at intermediate timescales. We find that for a sufficiently local perturbation the quasiconserved quantities indeed exist, and we construct an explicit form for the first few of them.
We consider the isotropic spin-1/2 Heisenberg spin chain weakly perturbed by a local translationally- and SU(2)-invariant perturbation. Starting from the local integrals of motion of the unperturbed model, we modify them in order to obtain quasi-conserved integrals of motion (charges) for the perturbed model. Such quasi-conserved quantities are believed to be responsible for the existence of the prethermalization phase at intermediate timescales. We find that for a sufficiently local perturbation only the first few integrals of motion can be promoted to the quasi-conserved charges, whereas higher-order integrals of motion do not survive.
We consider the spin-12\frac{1}{2}21​ Heisenberg chain (XXX model) weakly perturbed away from integrability by an isotropic next-to-nearest neighbor exchange interaction. Recently, it was conjectured that this model possesses an infinite tower of quasiconserved integrals of motion (charges) [D. Kurlov et al., Phys. Rev. B 105, 104302 (2022)]. In this work we first test this conjecture by investigating how the norm of the adiabatic gauge potential (AGP) scales with the system size, which is known to be a remarkably accurate measure of chaos. We find that for the perturbed XXX chain the behavior of the AGP norm corresponds to neither an integrable nor a chaotic regime, which supports the conjectured quasi-integrability of the model. We then prove the conjecture and explicitly construct the infinite set of quasiconserved charges. Our proof relies on the fact that the XXX chain perturbed by next-to-nearest exchange interaction can be viewed as a truncation of an integrable long-range deformation of the Heisenberg spin chain.
A quantum integrable system slightly perturbed away from integrability is typically expected to thermalize on timescales of order $\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}2}$, where $\ensuremath{\lambda}$ is the perturbation strength. We here study classes of perturbations that violate this scaling, and exhibit much longer thermalization times $\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}2\ensuremath{\ell}}$ where $\ensuremath{\ell}>1$ is an integer. Systems with these ``weak integrability breaking'' perturbations have an extensive number of quasiconserved quantities that commute with the perturbed Hamiltonian up to corrections of order ${\ensuremath{\lambda}}^{\ensuremath{\ell}}$. We demonstrate a systematic construction to obtain families of such weak perturbations of a generic integrable model for arbitrary $\ensuremath{\ell}$. We then apply the construction to various models, including the Heisenberg, XXZ, and XYZ chains, the Hubbard model, models of spinless free fermions, and the quantum Ising chain. Our analytical framework explains the previously observed evidence of weak integrability breaking in the Heisenberg and XXZ chains under certain perturbations.
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.
The isotropic Heisenberg chain represents a particular case of an integrable many-body system exhibiting superdiffusive spin transport at finite temperatures. Here, we show that this model has distinct properties also at finite magnetization $m\ensuremath{\ne}0$, even upon introducing the SU(2) invariant perturbations. Specifically, we observe nonmonotonic dependence of the diffusion constant ${\mathcal{D}}_{0}(\mathrm{\ensuremath{\Delta}})$ on the spin anisotropy $\mathrm{\ensuremath{\Delta}}$, with a pronounced maximum at $\mathrm{\ensuremath{\Delta}}=1$. The latter dependence remains true also in the zero magnetization sector, with superdiffusion at $\mathrm{\ensuremath{\Delta}}=1$ that is remarkably stable against isotropic perturbation (at least in finite-size systems), consistent with recent experiments with cold atoms.
We outline a procedure for counting and identifying a complete set of local and quasilocal conserved operators in integrable lattice systems. The method yields a systematic generation of all independent, conserved quasilocal operators related to the time average of local operators with a support on up to M consecutive sites. As an example, we study the anisotropic Heisenberg spin-1/2 chain and show that the number of independent conserved operators grows linearly with M. In addition to the known local operators, there exist novel quasilocal conserved quantities in all the parity sectors. The existence of quasilocal conserved operators is shown also for the isotropic Heisenberg model. Implications for the anomalous relaxation of quenched systems are discussed as well.