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arXiv:1612.07265 · Physical review. B./Physical review. B

Nonequilibrium spin transport in integrable spin chains: Persistent currents and emergence of magnetic domains

Andrea De Luca, Mario Collura, Jacopo De Nardis

Published Jul 6, 2017• 142 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.96.020403

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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 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.
Anomalous transport in low dimensional spin chains is an intriguing topic that can offer key insights into the interplay of integrability and symmetry in many-body dynamics. Recent studies have shown that spin-spin correlations in spin chains, where integrability is either perfectly preserved or broken by symmetry-preserving interactions, fall in the Kardar-Parisi-Zhang (KPZ) universality class. Similarly, energy transport can show ballistic or diffusive-like behavior. Although such behavior has been studied under equilibrium conditions, no results on nonequilibrium spin transport in classical spin chains has been reported so far. In this work, we investigate both spin and energy transport in classical spin chains (integrable and nonintegrable) when coupled to two reservoirs at two different temperatures/magnetizations. In both the integrable case and the broken-integrability (but spin-symmetry preserving) case, we report anomalous scaling of spin current with system size (J^{s}∝L^{-μ}), with an exponent value that, within error bars, is close to the KPZ universality class value μ≈2/3. On the other hand, it is noteworthy that the energy current remains ballistic (J^{e}∝L^{-η} with η≈0) in the purely integrable case while there is departure from ballistic behavior (η>0) when integrability is broken regardless of spin-symmetry. We also present results on other interesting observables in the nonequilibrium steady state such as the spatial profiles of magnetization and energy, and spin-spin correlations.
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.
We present two directions of research in statistical mechanics of nonequilibrium onedimensional quantum systems. One is related to isolated integrable models, the other one to exactly solvable dissipatively driven spin chains. The device enabling their analysis is identified as the quantum group symmetry. In the framework of isolated systems we focus on the concept of integrable Floquet driven systems. In particular, we show how to build such systems out of the basic constituents of integrability structure. This allows substantiated and conclusive statements,for instance, about the transport phenomena. Despite their inherent relation, the presented integrable periodically driven models originate in two different ideas. On the one hand we have Trotterisations of integrable spin chains, concieved in an effort to better understand the dynamics of spin models. Then, there are systems originating in the attempts to solve quantum field theory in an algebraically closed form, for example, the quantum Hirota equation. The bulk of our consideration of isolated quantum systems consists of the construction of extensive conservation laws that either (i) constitute the generalised Gibbs ensemble and the hydrodynamic description of thermalisation after a quantum quench, or (ii) prevent the decay of current autocorrelations and thus characterise ideal transport in the system. The exact way in which these charges determine the dynamics stems from the symmetries of the quantum group representations. The second part of the exposition is dedicated to open quantum systems. Again we distinguish two formally related settings. The first one is that of a boundary driven quantum cellular automaton, our particular example being the integrable Trotterisation of the Heisenberg magnet. Starting from a repeated interaction protocol, in which the system under scrutiny is repeatedly coupled to the environment, we introduce the Kraus map as a general form of a dissipative time evolution of the density matrix. We then solve for its unique nonequilibrium steady state, using integrability structure of the model. The other setting is that of a dissipatively boundary driven spin chain in the continuous time. Here we present the recently developed formalism of inhomogeneous Lax structure. Using it we demonstrate the solvability of the XXZ and XYZ spin chains, acted upon by the Lindblad operators that polarise the boundary spins in arbitrary directions. The ansatz for the steady state is particularly interesting, since it exhibits a previously unknown integrability structure, differing from site to site in the spin chain. This structure can independently produce nontrivial conservation laws in an isolated spin chain with arbitrary boundary fields.
Abstract Recently found spin-flip non-invariant (SFNI) conserved quantities play important roles in discussing nonequilibrium physics of the XXZ model. The representative examples are the generalized Gibbs ensemble (GGE) and the ballistic transport of the spin current. In spite of big progress in understanding nonequilibrium physics of integrable systems, the general framework to determine a minimal complete set of conserved quantities which describes the long-time steady state has not yet been found. This paper shows that the GGE of the gapless XXZ model consists of functionally independent conserved quantities rather than linearly independent. At the same time, the physical meaning of SFNI conserved quantities is provided. We also discuss that there exist ballistic channels of the spin current supported by non-quasilocal conserved quantities. The saturation of the lower bound for the Drude weight by quasilocal conserved quantities reads the linear dependence of non-quasilocal conserved quantities on quasilocal ones. We show that their (generalized) linearly dependence relation is consistent with the statement that the GGE consists of functionally independent conserved quantities without containing all linearly independent conserved quantities.
Abstract We review the non-equilibrium dynamics of many-body quantum systems after a quantum quench with spatial inhomogeneities, either in the Hamiltonian or in the initial state. We focus on integrable and many-body localized systems that fail to self-thermalize in isolation and for which the standard hydrodynamical picture breaks down. The emphasis is on universal dynamics, non-equilibrium steady states and new dynamical phases of matter, and on phase transitions far from thermal equilibrium. We describe how the infinite number of conservation laws of integrable and many-body localized systems lead to complex non-equilibrium states beyond the traditional dogma of statistical mechanics.