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arXiv:2303.17084 · Journal of Statistical Mechanics Theory and Experiment

Exact steady states of the impurity-doped XXZ spin chain coupled to dissipators

Chihiro Matsui, Naoto Tsuji

Published Mar 18, 2024• 2 citations
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Abstract

Abstract We give an exact matrix product steady state and matrix product forms of local observables for the bulk impurity-doped XXZ spin model coupled to dissipators at both ends, whose dynamics is described by the Lindblad quantum master equation. We find that local magnetization is induced at the impurity site when the spin current flows, which is contrary to the usual situation where the current suppresses magnetization due to heating. It is proved that this current-induced magnetization survives in the thermodynamic limit, and the spin current does not depend on the impurity strength. We discuss the role of bulk integrability by comparing the results with those of non-integrable models solved numerically by the quantum trajectory method.

DOI
10.1088/1742-5468/ad2b5c

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Abstract We give an exact matrix product steady state and matrix product forms of local observables for the bulk impurity-doped XXZ spin model coupled to dissipators at both ends, whose dynamics is described by the Lindblad quantum master equation. We find that local magnetization is induced at the impurity site when the spin current flows, which is contrary to the usual situation where the current suppresses magnetization due to heating. It is proved that this current-induced magnetization survives in the thermodynamic limit, and the spin current does not depend on the impurity strength. We discuss the role of bulk integrability by comparing the results with those of non-integrable models solved numerically by the quantum trajectory method.
We investigate a dissipatively driven XYZ spin-1/2 chain in the Zeno limit of strong dissipation, described by the Lindblad master equation. The nonequilibrium steady state is expressed in terms of a matrix product ansatz using novel site-dependent Lax operators. The components of Lax operators satisfy a simple set of linear recurrence equations that generalize the defining algebraic relations of the quantum group U_{q}(sl_{2}). We reveal connection between the nonequilibrium steady state of the nonunitary dynamics and the respective integrable model with edge magnetic fields, described by coherent unitary dynamics.
Current-carrying steady states of interacting spin chains exhibit rich structures generated through an interplay of constraints from the Hamiltonian dynamics and those induced by the current. The XXZ spin chain when coupled to maximally polarizing Lindblad terms (with opposite signs on either end) admits an exact solution for the steady state in a matrix product state (MPS) form. We use this exact solution to study the correlations and distributions of local spin observables in the nonequilibrium steady state. We present exact expressions for spin correlators, entropy per site, and scaled cumulant generating functions (SCGF) for distributions of local observables in the XX limit (Ising anisotropy Δ=0). Further, we use the exact MPS solution in the Δ>0 regime to calculate numerically exact entropy, correlations, as well as full distributions of spin observables in large systems. In systems where Δ is a cosine of rational multiple of π, we can numerically exactly estimate the large system limit of the SCGF and the large deviation/rate functions of local-z magnetization. For these, we show that the deviations of the SCGF, calculated in finite systems, from the asymptotic large system size limit decay exponentially with system size; however, the decay rate is a discontinuous function of Δ. The x magnetization density shows a double peak structure at Δ≲1, suggesting short-range ferromagnetic ordering in the x direction similar to what was reported for the ground state of the XXZ chain.
We will review some known exact solutions for the steady state of the open quantum Heisenberg XXZXXZXXZ spin chain coupled to a pair of baths [Phys. Rev. Lett. 107, 137201 (2011).]. The dynamics is modelled by the Lindblad master equation. We also review how to calculate some relevant physical observables and provide the statistics of spin current assuming the spin chain is weakly coupled to the baths [Phys. Rev. Lett. 112, 067201 (2014).].
We study the zero- and the finite-temperature behavior of the integrable spin-1/2 XXZ periodic chain with an impurity by the algebraic and thermal Bethe ansatz methods. We evaluate the local magnetization on the impurity site at zero temperature analytically and derive the impurity susceptibility exactly from it. In the graphs of the impurity specific heat versus temperature, we show how the impurity spin becomes more liberated from the bulk many-body effect as the exchange coupling between the impurity spin and other spins decreases and that at low temperature it couples strongly to them such as in the Kondo effect. Thus, we observe not only the crossover behavior from the high- to the low-temperature regime, but another from the N -site chain to the ( N − 1)-site chain with a free impurity spin. We also show that the estimate of the Wilson ratio at a given low temperature is independent of the impurity parameter if its absolute value is small enough with respect to the temperature and the universality class is described by the XXZ anisotropy in terms of the dressed charge.
Abstract. We show how to exploit algebraic relations among the operators (or matrices) which constitute the non-equilibrium matrix product steady state of a boundary driven quantum spin chain to find partial differential equations determining all the m -point correlation functions in the continuum (or thermodynamic) limit. These partial differential equations, the order of which is determined by scaling of the non-equilibrium partition function, are readily solved if we also know the boundary conditions. In this way we can avoid resorting to representation theory of the matrix product algebra. We apply our methods to study the distributions, or moments, of the magnetization and the spin current observables in boundary driven open XXZ spin chains, as well as some connected correlation functions. Interestingly, we find that the transverse connected correlation functions are thermodynamically non-decaying and long range at the isotropic point .
The effects of a local magnetic field on nonequilibrium stationary states (NESSs) of the open quantum XXZ spin chain are investigated with a Lindblad master equation approach in the limit of strong dissipation. The local magnetic field is applied to a single bulk spin of the chain while the ends are kept at fixed polarizations by dissipation. We show that suitable changes of the local magnetic field permit us to invert the spin current by switching pure NESS with opposite chiralities while preserving the state purity and achieving optimal transport.