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Non-equilibrium steady states of the boundary-driven Heisenberg quantum spin chain

Gunter Markus Schutz

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

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 $N$ sites. The magnetization profiles are harmonic functions with a frequency proportional to the twisting angle $\theta$. 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 $1/N^{2}$ for fixed and sufficiently large boundary coupling, and vanish as the coupling increases, while the transversal current increases with the coupling and saturates to $2\theta/N$.

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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.
We consider an open isotropic Heisenberg quantum spin chain, coupled at the ends to boundary reservoirs polarized in different directions, which sets up a twisting gradient across the chain. Using a matrix product ansatz, we calculate the exact magnetization profiles and magnetization currents in the nonequilibrium steady state of a chain with N sites. The magnetization profiles are harmonic functions with a frequency proportional to the twisting angle θ. 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/N^{2} 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/N.
We consider the open isotropic spin-1/2 Heisenberg quantum spin chain with a finite number N of sites coupled at the ends to a dissipative environment that favors polarization of the boundary spins in different directions. We review the matrix product ansatz (MPA) that yields the exact reduced density matrix of the Heisenberg chain. We develop the matrix algebra coming from the MPA in more detail than in previous work. We hence obtain exact results for the nonequilibrium partition function, about the impact of quantum fluctuations on the targeted boundary states, and for current–magnetization correlations in the steady state. The boundary states turn out to be pure to the order o(N −2 ). We show that the local magnetization and the local current perpendicular to the plane spanned by the boundary polarizations exhibit long-range correlations while the local magnetization correlations with the local in-plane currents are strongly suppressed.
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.
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 .
We investigate an open XXZ spin 1/2 chain driven out of equilibrium by coupling with boundary reservoirs targeting different spin orientations in the XY-plane. Symmetries of the model are revealed which appear to be different for spin chains of odd and even sizes. As a result, the spin current is found to alternate with chain length, ruling out the possibility of ballistic transport. Heat transport is switched off completely by virtue of another global symmetry. Further, we investigate the model numerically and analytically. At strong coupling, we find an exact nonequilibrium steady state using perturbation theory. The state is determined by solving secular conditions which guarantee self-consistency of the perturbative expansion. We find nontrivial dependence of the magnetization current on the spin chain anisotropy Δ in the critical region |Δ| < 1, and a phenomenon of tripling of the twisting angle along the chain for narrow lacunae of Δ.
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.