We investigate the stationary states of one-dimensional driven diffusive systems, coupled to boundary reservoirs with fixed particle densities. We argue that the generic phase diagram is governed by an extremal principle for the macroscopic current irrespective of the local dynamics. In particular, we predict a minimal current phase for systems with local minimum in the currentdensity relation. This phase is explained by a dynamical phenomenon, the branching and coalescence of shocks, Monte-Carlo simulations confirm the theoretical scenario.] A recurrent problem in the investigation of many-body systems far from equilibrium is posed by the coupling of a driven particle system with locally conserved particle number to external reservoirs with which the system can exchange particles at its boundaries. In the presence of a driving force a particle current will be maintained and hence the system will always remain in an nonequilibrium stationary state characterized by some bulk density and the corresponding particle current. While for periodic boundaries the density is a fixed quantity, the experimentally more relevant scenario of open boundaries naturally leads to the question of steady-state selection, i.e. the question which stationary bulk density the system will assume as a function of the boundary densities [1]. In the topologically simplest case of quasi one-dimensional systems this is of importance for the understanding of many-body systems in which the dynamic degrees of freedom reduce to effectively one dimension as e.g. in traffic flow [2], kinetics of protein synthesis [3], or diffusion in narrow channels [4].Within a phenomenological approach this problem was first addressed in general terms by Krug [5] who postulated a maximal-current principle for the specific case where the density ρ + at the right boundary to which particles are driven is kept at zero. The exact solution of the totally asymmetric simple exclusion process (TASEP) [6] for arbitrary left and right boundary densities ρ − , ρ + confirms and extends the results by Krug. The complete phase diagram comprises boundary-induced phase transitions of first order between a low-density phase and a high density phase, and a second-order transition from these phases to a maximal current phase [8][9][10]. Analysis of the density profile [9] provides insight into the dynamical mechanisms that lead to these phase transitions and shows that the phase diagram is generic for systems with a single maximum in the current-density relation [11]. Experimental evidence for the first-order transition is found in the process of biopolymerization for which the TASEP with open boundaries was originally invented as a simple model [12], and more directly also in recent measurements of highway traffic close to on-ramps [13,14]. Renormalization group studies [15] indicate universality of the second order phase transition.In this Letter we develop a dynamical approach to generic driven one-component systems with several maxima in the current-density relation, and we show...

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...

We study the formation of localized shocks in one-dimensional driven diffusive systems with spacially homogeneous creation and annihilation of particles (Langmuir kinetics).We show how to obtain hydrodynamic equations which describe the density profile in systems with uncorrelated steady state as well as in those exhibiting correlations. As a special example of the latter case the Katz-Lebowitz-Spohn model is considered. The existence of a localized double density shock is demonstrated for the first time in one-dimensional driven diffusive systems. This corresponds to phase separation into regimes of three distinct densities, separated by localized domain walls. Our analytical approach is supported by Monte-Carlo simulations.