Exact Matrix Product Solution for the Boundary-Driven LindbladXXZChain

Abstract

We demonstrate that the exact non-equilibrium 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 non-equilibrium density matrix where the matrices satisfy a quadratic algebra. This algebra turns out to be related to the quantum algebra Uq[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 non-vanishing 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.PACS numbers: 03.65. Yz, 75.10.Pq, The non-equilibrium behaviour of open quantum systems has become accessible through recent advances in artificially assembled nanomagnets consisting of just a few atoms [1] or in the study of quasi one-dimensional spin chain materials like SrCuO 2 where many transport characteristics are measurable experimentally [2,3]. In particular, it is desirable to understand the interplay between many-body bulk properties (e.g. magnon excitations or magnetization currents in quantum spin systems) and local pumping (applied to the boundary of a system) driving the system constantly out of equilibrium. A good starting point is provided by the anisotropic Heisenberg modelof coupled spins. The pure quantum version of this model is exactly solvable by the Bethe ansatz. Interestingly, within linear response theory, i.e., close to equilibrium, it was found that at finite temperature a diffusive contribution to the Drude weight appears [5][6][7], which is at variance with the long-held belief that integrability protects the ballistic nature of transport phenomena. Unfortunately the Bethe-ansatz fails in the more relevant context of open far-from-equilibrium systems where these questions can be addressed directly in terms of the Lindblad Master equationfor the reduced density matrix ρ associated to the chain (here and below we set = 1). The dissipative termswith the Lindblad operators D L,R acting locally at the open ends of the quantum chain (see below) describe the coupling to external reservoirs that drive a current through the system and thus keep the system in a permanent non-equilibrium steady state. Indeed, using dissipative dynamics for the preparation of quantum states is becoming a promising field of research [9, 10].Significant progress has been achieved very recently in two remarkable papers by Prosen [11,12] who observed that the exact stationary density matrix for the XXZ chain with one specific pair of Lindblad boundary terms can be constructed explicitly in matrix product operator form [13] by a matrix product ansatz (MPA) somewhat reminiscent of the matrix product ansatz of Derrida et al. [14] for the stationary distribution of purely classical stochas...

Closed hierarchy of correlations in Markovian open quantum systems

“…In the classical setting steady states have been numerically and analytically investigated in stochastic lattice gas models [1] and in exactly solvable one-dimensional exclusion processes [2][3][4][5], where the steady state is framed in a matrix product form [6]. This ansatz has proven useful also in quantum setting, where it has been utilized to construct the nonequilibrium steady state of the boundary driven XXZ spin-1/2 chain [7][8][9]. Furthermore, the existing classes of Markovian many-body open quantum system models that can be exactly solved (by this we mean to analytically obtain the steady state) are strongly related to integrable closed quantum models, namely the quasi-free bosonic or fermionic models [10][11][12] and Yang-Baxter integrable models [13].…”

Quantum quench from a thermal tensor state: Boundary effects and generalized Gibbs ensemble

“…In other words, such infinitely extended system will always have, at each instant of time, two infinite portions (the far left and the far right playing the role of reservoirs) that remain equilibrated at the two initial temperatures. Between these two reservoirs, the (finite portion of the) system will show nonlocal properties reflecting the non-local structure of the non-equilibrium steady state (which is widely believed conjecture) [49][50][51][52][53][54][55][56] . Recently, by means of thermodynamic Bethe anstaz 57 , the energy current in non-equilibrium steady states of integrable models of relativistic quantum field theory has been evaluated 58 and interestingly, seems to be in disagreement with what has been numerically found in Ref.…”

Obtaining pure steady states in nonequilibrium quantum systems with strong dissipative couplings

“…Due to isotropy, this choice can be parametrized by a single angle Φ between the left and right boundary polarizations, i.e., we can put l L = (1, 0, 0), l R = (cos Φ, sin Φ, 0). Various properties of the XXZ model with boundary twisting in the XY plane for strong and weak driving have been investigated for Φ = π/2 and arbitrary ∆ in [32,33], while for the isotropic case ∆ = 1, the full analytic NESS for arbitrary Γ, Φ has been obtained in [29,30].…”

“…For specific boundary gradients, the NESS of this model has been calculated analytically at arbitrary dissipation strength [28][29][30][31]. The explicit form of L L , L R is given in Appendix A, where we also detail the content of Eq.…”