Quantum Phase Transition in a Far-from-Equilibrium Steady State of anXYSpin Chain

Abstract

Using quantization in the Fock space of operators we compute the non-equilibrium steady state in an open Heisenberg XY spin 1/2 chain of finite but large size coupled to Markovian baths at its ends. Numerical and theoretical evidence is given for a far from equilibrium quantum phase transition with spontaneous emergence of long-range order in spin-spin correlation functions, characterized by a transition from saturation to linear growth with the size of the entanglement entropy in operator space.PACS numbers: 02.30. Ik, 05.70.Fh, 75.10.Pq, 03.67.Mn Non-perturbative physics of many-body open quantum systems far from equilibrium is largely an unexplored field. In one-dimensional locally interacting quantum systems equilibrium phase transitions -quantum phase transitions (QPT) -can occur at zero temperature only and are by now well understood [1]. QPT are typically characterized by vanishing of the Hamiltonian's spectral gap in the thermodynamic limit at the critical point, and (logarithmic) enhancement of the entanglement entropy and other measures of quantum correlations in the ground state [2]. Much less is known about the physics of QPT out of equilibrium, studies of which have been usually limited to near equilibrium regimes or using involved and approximate analytical techniques (e.g. [3,4]).There exist two general theoretical approaches to a description of non-equilibrium open quantum systems, namely the non-equilibrium Green's function method [5], and the quantum master equation [6,7]. In this Letter we adpot the latter and present a quasi-exactly solvable example of an open Heisenberg XY spin 1/2 chain exhibiting a novel type of phase transition far from equilibrium; characterized by a sudden appearance of long-range magnetic order in non-equilibrium steady state (NESS) as the magnetic field is reduced, and the transition from saturation to linear growth with size of the operator space entanglement entropy (OSEE) of NESS.The Hamiltonian of the quantum XY chain reads(1) where σ x,y,z m , m = 1, . . . , n are Pauli operators acting on a string of n spins. We may assume that parameters γ (anisotropy) and h (magnetic field) are non-negative. It is known that XY model (1) exhibits (equilibrium) critical behavior in the thermodynamic limit n → ∞ along the lines: γ = 0, h ≤ 1, and h = 1. Here we consider an open XY chain whose density matrix evolution ρ(t) is governed by the Lindblad master equation [6] (we set = 1)and study a phase transition in NESS. The simplest nontrivial bath (Lindblad) operators acting only on the first and the last spin are chosen (M = 4)where σ Note that Lindblad equation (2) can be rigorously derived within the so-called Markov approximation [7] which is justified for macroscopic baths with fast internal relaxation times. As shown in [9], Eq. (2) with (1,3) can be solved exactly in terms of normal master modes (NMM) which are obtained from diagonalization of 4n × 4n matrix A written in terms of 4 × 4 blockswhere Following [9], the key concept is 4 n dimensional Fock space of operators...

Relaxation times of dissipative many-body quantum systems

“…Similarly, one can show that the gap can be constant for some other systems with bulk dissipation [17,19], while in other systems it can also scale as ∼ 1/L 2 [14,19,24]. On the other hand, for open systems with boundary dissipation the observed gaps have so-far all been scaling as ∼ 1/L 3 , or smaller, examples being the XY [20,25,26] or the XXX [27,28] model. For scaling of gaps in the Redfield equation see Ref.…”

“…On the level of perturbative matrix R this can be understood as being due to the breaking of the underlying symmetry of two domain-wall states. Because the location of a symmetry-breaking perturbation is important (25), having σ ± dissipation at sites other than the boundary ones could still result in an exponentially small gap, see also recent Ref. [55] for a study of stability of edge modes to Markovian dissipation.…”

“…As mentioned, for boundary-driven open systems the gaps observed in the literature are all ∼ 1/L 3 or smaller. Examples are ∼ 1/L 3 in the XY chain with boundary magnetization driving [18,20,26] (or ∼ 1/L 5 at nonequilibrium phase transition points [25]; or even ∼ 1/L 7 on the socalled resonances [50]); ∼ 1/L 3 is the scaling also for the magnetization-driven XXX model [27], or for the XXZ model with an incoherent hopping as a driving [28].…”

Random phase approximation study of one-dimensional fermions after a quantum quench

“…The XY model has been extensively studied, [28][29][30] and its equilibrium properties are well understood. It is also a popular model 16,31,32 for studying non-equilibrium situations…”

Nonequilibrium Phase Transition in a Periodically DrivenXYSpin Chain

“…[8]), so many effects can be analyzed exactly or in great detail. For example, quantum phase transitions in nonequilibrium steady states have been observed either in quasi-free [5,9], or strongly interacting [6], or even dissipative [7,10] quantum systems in one dimension. …”