Exact Nonequilibrium Steady State of Open XXZ/XYZ Spin- 1/2 Chain with Dirichlet Boundar

Abstract

We investigate a dissipatively driven XYZ spin-1/2 chain in the Zeno limit of strong dissipation, described by 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 Uq(sl2). 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.Introduction.-One of the main current efforts of the condensed matter physics society is to understand quantum states of matter far from equilibrium [1,2]. Understandably, simple models with tractable but nontrivial exact solutions are of key importance in this game. The realm of driven dissipative quantum many-body systems [3] provides nice and rich examples of such models, capable of displaying genuinely out-of-equilibrium phenomena, for example, novel types of non-equilibrium phase transitions [4][5][6][7]. While exact treatment of the aforementioned class of models is essentially limited to quasi-free situations, it is remarkable that some exact solutions have been found even in the case of strong interactions, in particular, in quantum integrable spin chains with dissipation and incoherent driving localized at the chain's boundaries [8]. Despite the fact, that the exact matrix product form of these solutions has been found only for a very specific choice of the boundary jump operators [9, 10], this provided a fresh perspective on the effect of local and quasilocal conservation laws on quantum transport and relaxation [11]. It has, however, remained an open question of how these exact steady state solutions fit into the general framework of integrability. For example, except in the special case of dissipatively driven noninteracting models [12], the solvable dissipatively driven boundaries cannot be generated using the solutions of the ubiquitous reflection equations [13], which constitute the standard framework for generating integrable boundaries in the coherent (nondissipative, Hamiltonian) setting.In the present letter we make a step forward in the understanding of integrability of open XYZ spin-1/2 chain, dissipatively driven at the boundaries. We present an exact solution for the nonequilibrium steady state of the model with arbitrary boundary processes, as long as the edge spins are described by pure states in the large dissipation limit. The inhomogeneous matrix product ansatz for the steady state differs drastically from the previously known solutions that describe steady states of the spin chains driven along a particular axis at both edges.Firstly, we introduce the model and describe how the

Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains

“…These states are routinely created, and widely used in coherent experimental protocols [6][7][8]. SHS can also be generated as non-equilibrium steady states via a dissipative quantum protocol [9][10][11] or via controlled local boundary dissipation. Remarkably, the needed boundary dissipation is of the type which allows the system to retain, partly, its integrability [12].…”

“…These states are realized by non-standard string-type solutions of the Bethe ansatz equations with infinite rapidities. The Bethe Ansatz equations for singular roots are satisfied with a universal choice for their arrangement (11), which makes them effectively "disappear" from the set of Bethe Ansatz equations. For this reason we call the roots with infinite rapidities phantom Bethe roots and the respective excitations phantom excitations.…”

“…Phantom Bethe vectors for periodic chains. The Bethe vectors corresponding to the phantom Bethe roots (PBR) solution (11), under the conditions of Lemma 1, can be constructed as described below (10). The two signs ± in (11)…”

Ефект Потоку Енергії В Одновимірній Спін-1/2 XX Моделі Магнетоелектрика. Метод Множника Лагранжа

“…In modern statistical physics, much attention is paid to the theoretical study of nonequilibrium stationary processes of various nature that take place in various systems and models. In particular, the specific features of nonequilibrium steady states (NESSs) were studied in the framework of the nonequilibrium spinboson model [1,2], the simplified model of a system of noninteracting electrons (a one-dimensional chain of fermions consisting of a central part and two metal thermostats) [2], the electron-hole-photon system [3], the finite quantum system of interacting particles connected to electrodes that are simultaneously thermostats [4], the quantum wire [5], the system in which a quantum dot is placed between a metal and a superconductor or a ferromagnetic contact with opposite polarizations [6], the 𝑋𝑋 chain located in a transverse field and connected to quantum reservoirs with noninteracting spins and different temperatures [7], and the spin models with interactions between the nearest neighbors and the energy [8][9][10][11][12][13][14][15][16][17][18][19][20] or spin [9,[21][22][23][24] current. It should be noted that the main specific feature of NESSs is the presence of a permanent flux of some physical quantity (energy, magnetic moment, charge, and so on).…”

Full Spectrum of the Liouvillian of Open Dissipative Quantum Systems in the Zeno Limit

“…The solutions are given in terms of spectral data of a dissipation-projected Hamiltonian and other similar Hamiltonians, these being much simpler objects than the original Liouvillian. Our method is straightforwardly applicable to the XYZ model with dissipation acting on both boundaries, thus creating boundary gradients [31,32], which play a prominent role in studies of quantum transport [33]. All the auxiliary Hamiltonians have the form of an open XYZ spin chain with boundary fields and are integrable [34].…”