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arXiv:1106.2978 · Physical Review Letters

Exact Nonequilibrium Steady State of a Strongly Driven Open XXZ Chain

Tomaž Prosen

Published Sep 19, 2011• 290 citations
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Abstract

An exact and explicit ladder-tensor-network ansatz is presented for the nonequilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is-apart from a normalization constant-a polynomial of degree 2n - 2 in the coupling constant. Efficient computation of physical observables is facilitated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cosine spin profiles, 1/n(2) scaling of the spin current, and long-range correlations in the steady state. This is a fully nonperturbative extension of a recent result [Phys. Rev. Lett. 106, 217206 (2011)].

DOI
10.1103/physrevlett.107.137201

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An exact and explicit ladder-tensor-network ansatz is presented for the nonequilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is-apart from a normalization constant-a polynomial of degree 2n - 2 in the coupling constant. Efficient computation of physical observables is facilitated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cosine spin profiles, 1/n(2) scaling of the spin current, and long-range correlations in the steady state. This is a fully nonperturbative extension of a recent result [Phys. Rev. Lett. 106, 217206 (2011)].
We demonstrate that the exact nonequilibrium steady state of the one-dimensional Heisenberg XXZ spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the nonequilibrium density matrix where the matrices satisfy a quadratic algebra. This algebra turns out to be related to the quantum algebra U(q)[SU(2)]. Coherent state techniques are introduced for the exact solution of the isotropic Heisenberg chain with and without quantum boundary fields and Lindblad terms that correspond to two different completely polarized boundary states. We show that this boundary twist leads to nonvanishing stationary currents of all spin components. Our results suggest that the matrix product ansatz can be extended to more general quantum systems kept far from equilibrium by Lindblad boundary terms.
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).].
An explicit matrix product ansatz is presented, in the first two orders in the (weak) coupling parameter, for the nonequilibrium steady state of the homogeneous, nearest neighbor Heisenberg XXZ spin 1/2 chain driven by Lindblad operators which act only at the edges of the chain. The first order of the density operator becomes, in the thermodynamic limit, an exact pseudolocal conservation law and yields-via the Mazur inequality-a rigorous lower bound on the high-temperature spin Drude weight. Such a Mazur bound is a nonvanishing fractal function of the anisotropy parameter Δ for |Δ|<1.
In this paper, we provide two extensions of recent explicit results on the matrix-product ansatz for the non-equilibrium steady state of a Markovian boundary-driven anisotropic Heisenberg XXZ spin-1/2 chain. We write a perturbative solution for the steady-state density matrix in the system–bath coupling for an arbitrary (asymmetric) set of four spin-flip rates at the two chain ends, generalizing the symmetric-driving ansatz of Prosen (2011 Phys. Rev. Lett. 106 217206). Furthermore, we generalize the exact (non-perturbative) form of the steady state for just two Lindblad channels (spin-up flipping on the left and spin-down flipping on the right) to an arbitrary (asymmetric) ratio of the spin-flipping rates (Prosen 2011 Phys. Rev. Lett. 107 137201). In addition, we also indicate a simple proof of the uniqueness of our steady states.
We consider an open Heisenberg quantum spin chain, coupled at the ends to boundary reservoirs polarized in different directions, which sets up a twisting gradient across the chain. We derive a matrix product ansatz for the non-equilibrium density matrix. For the isotropic chain we calculate the exact magnetization profiles and magnetization currents in the nonequilibrium steady steady state of a chain with NNN sites. The magnetization profiles are harmonic functions with a frequency proportional to the twisting angle θ\thetaθ. The currents of the magnetization components lying in the twisting plane and in the orthogonal direction behave qualitatively differently: In-plane steady state currents scale as 1/N21/N^{2}1/N2 for fixed and sufficiently large boundary coupling, and vanish as the coupling increases, while the transversal current increases with the coupling and saturates to 2θ/N2\theta/N2θ/N.
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.