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arXiv:2210.06494 · Physical review. B./Physical review. B

Open-system spin transport and operator weight dissipation in spin chains

Yongchan Yoo, Christopher David White, Brian G. Swingle

Published Mar 9, 2023• 7 citations
arXivHTMLPDF

Abstract

We use nonequilibrium steady states to study the effect of dissipation-assisted operator evolution (DAOE) on the scaling behavior of transport in one-dimensional spin chains. We consider three models in the XXZ family: the XXZ model with staggered anisotropy, which is chaotic; XXZ model with no external field and tunable interaction, which is Bethe-ansatz integrable and (in the zero interaction limit) free-fermion integrable; and the disordered XY model, which is free-fermion integrable and Anderson localized. We find evidence that DAOE's effect on transport is controlled by its effect on the system's conserved quantities. To the extent that DAOE preserves those symmetries, it preserves the scaling of the system's transport properties; to the extent it breaks those conserved quantities, it pushes the system towards diffusive scaling of transport.

DOI
10.1103/physrevb.107.115118

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We use nonequilibrium steady states to study the effect of dissipation-assisted operator evolution (DAOE) on the scaling behavior of transport in one-dimensional spin chains. We consider three models in the XXZ family: the XXZ model with staggered anisotropy, which is chaotic; XXZ model with no external field and tunable interaction, which is Bethe-ansatz integrable and (in the zero interaction limit) free-fermion integrable; and the disordered XY model, which is free-fermion integrable and Anderson localized. We find evidence that DAOE's effect on transport is controlled by its effect on the system's conserved quantities. To the extent that DAOE preserves those symmetries, it preserves the scaling of the system's transport properties; to the extent it breaks those conserved quantities, it pushes the system towards diffusive scaling of transport.
Interacting lattice Hamiltonians at high temperature generically give rise to energy transport governed by the classical diffusion equation; however, predicting the rate of diffusion requires numerical simulation of the microscopic quantum dynamics. For the purpose of predicting such transport properties, computational time evolution methods must be paired with schemes to control the growth of entanglement to tractably simulate for sufficiently long times. One such truncation scheme---dissipation-assisted operator evolution (DAOE)---controls entanglement by damping out components of operators with large Pauli weight. In this paper, we generalize DAOE to treat fermionic systems. Our method instead damps out components of operators with large fermionic weight. We investigate the performance of DAOE, the new fermionic DAOE (FDAOE), and another simulation method, density matrix truncation (DMT), in simulating energy transport in an interacting one-dimensional Majorana chain. The chain is found to have a diffusion coefficient scaling like interaction strength to the fourth power, contrary to naive expectations based on Fermi's Golden rule---but consistent with recent predictions based on the theory of weak integrability breaking. In the weak interaction regime where the fermionic nature of the system is most relevant, FDAOE is found to simulate the system more efficiently than DAOE.
In this work we analyze the simultaneous emergence of diffusive energy transport and local thermalization in a nonequilibrium one-dimensional quantum system, as a result of integrability breaking. Specifically, we discuss the local properties of the steady state induced by thermal boundary driving in a XXZ spin chain with staggered magnetic field. By means of efficient large-scale matrix product simulations of the equation of motion of the system, we calculate its steady state in the long-time limit. We start by discussing the energy transport supported by the system, finding it to be ballistic in the integrable limit and diffusive when the staggered field is finite. Subsequently, we examine the reduced density operators of neighboring sites and find that for large systems they are well approximated by local thermal states of the underlying Hamiltonian in the nonintegrable regime, even for weak staggered fields. In the integrable limit, on the other hand, this behavior is lost, and the identification of local temperatures is no longer possible. Our results agree with the intuitive connection between energy diffusion and thermalization.
In integrable one-dimensional quantum systems an infinite set of local conserved quantities exists which can prevent a current from decaying completely. For cases like the spin current in the XXZ\mathit{XXZ}XXZ model at zero magnetic field or the charge current in the attractive Hubbard model at half filling, however, the current operator does not have overlap with any of the local conserved quantities. We show that in these situations transport at finite temperatures is dominated by a diffusive contribution with the Drude weight being either small or even zero. For the XXZ\mathit{XXZ}XXZ model we discuss in detail the relation between our results, the phenomenological theory of spin diffusion, and measurements of the spin-lattice relaxation rate in spin chain compounds. Furthermore, we study the Haldane-Shastry model where a conserved spin current exists.
The nonequilibrium dynamics of isolated many-body quantum systems leads generically to the emergence of standard thermal ensembles. However, some models, denoted as integrable, can display nontrivial steady states with a large memory of the initial state. This is due to the presence of conserved quantities constraining the unitary dynamics. Here, the authors show that in interacting integrable spin chains there exist also conserved operators that are responsible for ballistic spin transport. Such conserved operators highly influence both the equilibrium and nonequilibrium physics: at equilibrium they are responsible for a nonvanishing spin Drude weight even at half filling, while in the nonequilibrium case they are responsible for out-of-equilibrium steady states characterized by persistent spin currents or dynamically expanding magnetic domains.
In the thesis we present an analytic approach towards exact description for steady state density operators of nonequilibrium quantum dynamics in the framework of open systems. We employ the so-called quantum Markovian semi-group evolution, i.e. a general form of time-autonomous positivity and trace-preserving dynamical equation for reduced density operators, by only allowing unitarity-breaking dissipative terms acting at the boundaries of a system. Such setup enables to simulate macroscopic reservoirs for different values of effective thermodynamic potentials, causing incoherent transitions between quantum states which are modeled with aid of the Lindblad operators. This serves as a simple minimalistic model for studying quantum transport properties, either in the linear response domain or in more general regimes far from canonical equilibrium. We are mainly exploring possibilities of identifying nonequilibrium situations which are amenable to exact description within matrix product state representation, by exclusively focusing on steady states, i.e. fixed points of the Lindblad equation, of certain prototypic interacting integrable spin chains driven by incoherent polarizing processes. Finally, we define a concept of pseudo-local extensive almost-conserved quantities by allowing a violation of time-invariance up to boundary-localized terms. We elucidate the role of such quantities on non-ergodic behaviour of temporal correlation functions rendering anomalous transport properties. It turns out that such conservation laws can be generated by means of boundary universal quantum transfer operators of the fundamental integrable models.
Charge and energy are expected to diffuse in interacting systems of fermions at finite temperatures, even in the absence of disorder, with the interactions inducing a crossover from the coherent and ballistic streaming of quasi-particles at early times, to incoherent diffusive behavior at late times. The relevant crossover timescales and the transport coefficients are both controlled by the strength of interactions. In this work we develop a numerical method to simulate such systems at high temperatures, applicable in a wide range of interaction strengths, by adapting Dissipation-assisted Operator Evolution (DAOE) to fermions. Our fermion DAOE, which approximates the exact dynamics by systematically discarding information from high nnn-point functions, is tailored to capture non-interacting dynamics exactly, thus providing a good starting point for the weakly interacting problem. Applying our method to a microscopic model of weakly interacting fermions, we numerically demonstrate that the crossover from ballistic to diffusive transport happens at a time tD∼1/Δ2t_D\sim1/Δ^{2}tD​∼1/Δ2 and that the diffusion constant similarly scales as D∼1/Δ2D \sim 1/Δ^2D∼1/Δ2, where ΔΔΔ is the interaction strength. We substantiate this scaling with a Fermi's golden rule calculation in the operator spreading picture, interpreting tDt_DtD​ as the fermion-fermion scattering time and lifetime of the single-particle Green's function.