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arXiv:1006.5368 · Journal of Physics A Mathematical and Theoretical

A matrix product solution for a nonequilibrium steady state of an XX chain

Marko Žnidarič

Published Sep 27, 2010• 66 citations
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Abstract

A one-dimensional XX spin chain of finite length coupled to reservoirs at both ends is solved exactly in terms of a matrix product state ansatz. An explicit representation of matrices of fixed dimension 4 independent of the chain length is found. Expectations of all observables are evaluated, showing that all connected correlations, apart from the nearest neighbor z – z , are zero.

DOI
10.1088/1751-8113/43/41/415004

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A one-dimensional XX spin chain of finite length coupled to reservoirs at both ends is solved exactly in terms of a matrix product state ansatz. An explicit representation of matrices of fixed dimension 4 independent of the chain length is found. Expectations of all observables are evaluated, showing that all connected correlations, apart from the nearest neighbor z – z , are zero.
We study a one-dimensional XX chain under nonequilibrium driving and local dephasing described by the Lindblad master equation. The analytical solution for the nonequilibrium steady state found for particular parameters in a previous study [M. Žnidarič J. Stat. Mech. (2010) L05002] is extended to arbitrary coupling constants, driving, and a homogeneous magnetic field. All one-, two-, and three-point correlation functions are explicitly evaluated. It is shown that the nonequilibrium stationary state is not Gaussian. Nevertheless, in the thermodynamic and weak-driving limit it is only weakly correlated and can be described by a matrix product operator ansatz with matrices of fixed dimension 4. A nonequilibrium phase transition at zero dephasing is also discussed. It is suggested that the scaling of the relaxation time with the system size can serve as a signature of a nonequilibrium phase transition.
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 study the steady state of a finite XX chain coupled at its boundaries to quantum reservoirs made of free spins that interact one after the other with the chain. The two-point correlations are calculated exactly, and it is shown that the steady state is completely characterized by the magnetization profile and the associated current. Except at the boundary sites, the magnetization is given by the average of the reservoirs' magnetizations. The steady-state current, proportional to the difference in the reservoirs' magnetizations, shows a nonmonotonic behavior with respect to the system-reservoir coupling strength, with an optimal current state for a finite value of the coupling. Moreover, we show that the steady state can be described by a generalized Gibbs state.
We calculate a nonequilibrium steady state of a quantum XX chain in the presence of dephasing and driving due to baths at chain ends. The obtained state is exact in the limit of weak driving while the expressions for one- and two-point correlations are exact for an arbitrary driving strength. In the steady state the magnetization profile and the spin current display diffusive behavior. The spin–spin correlation function, on the other hand, has long-range correlations which, though, decay to zero in either the thermodynamical limit or for equilibrium driving. At zero dephasing a nonequilibrium phase transition occurs from a ballistic transport having short-range correlations to a diffusive transport with long-range correlations.
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.
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)].