A state of an open quantum system is described by a density matrix, whose dynamics is governed by a Liouvillian superoperator. Within a general framework, we explore fundamental properties of both first-order dissipative phase transitions and second-order dissipative phase transitions associated with a symmetry breaking. In the critical region, we determine the general form of the steady-state density matrix and of the Liouvillian eigenmatrix whose eigenvalue defines the Liouvillian spectral gap. We illustrate our exact results by studying some paradigmatic quantum optical models exhibiting critical behavior.

We show that short-range correlations have a dramatic impact on the steady-state phase diagram of quantum driven-dissipative systems. This effect, never observed in equilibrium, follows from the fact that ordering in the steady state is of dynamical origin, and is established only at very long times, whereas in thermodynamic equilibrium it arises from the properties of the (free) energy. To this end, by combining the cluster methods extensively used in equilibrium phase transitions to quantum trajectories and tensor-network techniques, we extend them to nonequilibrium phase transitions in dissipative many-body systems. We analyze in detail a model of spin-1=2 on a lattice interacting through an XYZ Hamiltonian, each of them coupled to an independent environment that induces incoherent spin flips. In the steady-state phase diagram derived from our cluster approach, the location of the phase boundaries and even its topology radically change, introducing reentrance of the paramagnetic phase as compared to the single-site mean field where correlations are neglected. Furthermore, a stability analysis of the cluster mean field indicates a susceptibility towards a possible incommensurate ordering, not present if short-range correlations are ignored.

We present a general variational approach to determine the steady state of open quantum lattice systems via a neural network approach. The steady-state density matrix of the lattice system is constructed via a purified neural network ansatz in an extended Hilbert space with ancillary degrees of freedom. The variational minimization of cost functions associated to the master equation can be performed using a Markov chain Monte Carlo sampling. As a first application and proof-of-principle, we apply the method to the dissipative quantum transverse Ising model.In spite of the tremendous experimental progress in the isolation of quantum systems, a finite coupling to the environment [1] is unavoidable and certainly plays a crucial role in the practical implementation of quantum information and quantum simulation protocols [2]. Moreover, through an active control of the environment via the so-called reservoir engineering, an open quantum manybody system can be prepared in non-trivial phases [3][4][5] with also possible quantum applications [6, 7]. The theoretical description of open quantum manybody systems is in general out-of-the equilibrium and much less developed than for equilibrium systems. A mixed state with a finite entropy can be described by a density matrix, whose evolution is described by a master equation. Recently, a few theoretical methods have been developed to solve the master equation of open quantum manybody systems, including analytical approaches based on the Keldysh formalism [8,9], numerical algorithms based on matrix product operator and tensor-network techniques [10][11][12][13][14], cluster mean-field methods [15,16], corner-space renormalization [17][18][19], Gutzwiller mean-field [20], full configuration-interaction Monte Carlo [21], permutationinvariant solvers [22] or efficient stochastic unravelings for disordered systems [23]. The research in the field is very active, since the different methods are optimal for different specific regimes. For example, the corner-space renormalization method is best suited for systems with moderate entropy, while matrix product operator techniques to systems with short-range quantum correlations.In the last decade, the field of artificial neural networks has enjoyed a dramatic expansion and success thanks to remarkable applications in the recognition of complex patterns such as visual images or human speech (for a recent review see, e.g., [24]). The optimization (supervised learning) of the network is obtained by tuning the weights quantifying the connections between neural units via a variational minimization of a properly defined cost function. The wavefunction of a manybody system is in general a complex quantity, which is hard to be recognized. Recent works have proposed to exploit arti-ficial neural networks to construct trial wavefunctions, where the connection weights in the network play the role of variational parameters [25,26]. Neural network approaches have already been succesffuly applied to a wide number (see e.g. [27][28][29][30][31]) of clos...

We study the energy transport in a system of two half-infinite XXZ chains initially kept separated at different temperatures, and later connected and let free to evolve unitarily. By changing independently the parameters of the two halves, we highlight, through bosonisation and time-dependent matrix-product-state simulations, the different contributions of low-lying bosonic modes and of fermionic quasi-particles to the energy transport. In the simulations we also observe that the energy current reaches a finite value which only slowly decays to zero. The general pictures that emerges is the following. Since integrability is only locally broken in this model, a pre-equilibration behaviour may appear. In particular, when the sound velocities of the bosonic modes of the two halves match, the low-temperature energy current is almost stationary and described by a formula with a nonuniversal prefactor interpreted as a transmission coefficient. Thermalisation, characterized by the absence of any energy flow, occurs only on longer time-scales which are not accessible with our numerics.

Using a non-Hermitian Hamiltonian approach to open systems, we study the interplay of disorder and superradiance in a one-dimensional Anderson model. Analyzing the complex eigenvalues of the non-Hermitian Hamiltonian, a transition to a superradiant regime is shown to occur. As an effect of openness the structure of eigenstates undergoes a strong change in the superradiant regime: we show that the sensitivity to disorder of the superradiant and the subradiant subspaces is very different; superradiant states remain delocalized as disorder increases, while subradiant states are sensitive to the degree of disorder.Comment: 7 pages, submitted to the special issue on "Physics with non-Hermitian operators: Theory and Experiment" of the journal "Fortschritte der Physik - Progress of Physics

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We explore theoretically the nonequilibrium photonic phases of an array of coupled cavities in presence of incoherent driving and dissipation. In particular, we consider a Hubbard model system where each site is a Kerr nonlinear resonator coupled to a two-level emitter, which is pumped incoherently. Within a Gutzwiller mean-field approach, we determine the steady-state phase diagram of such a system. We find that, at a critical value of the intercavity photon hopping rate, a second-order nonequilibrium phase transition associated with the spontaneous breaking of the U(1) symmetry occurs. The transition from an incompressible Mott-like photon fluid to a coherent delocalized phase is driven by commensurability effects and not by the competition between photon hopping and optical nonlinearity. The essence of the mean-field predictions is corroborated by finite-size simulations obtained with matrix product operators and corner-space renormalization methods

The competition between interactions and dissipative processes in a quantum many-body system can drive phase transitions of different order. Exploiting a combination of cluster methods and quantum trajectories, we show how the systematic inclusion of (classical and quantum) nonlocal correlations at increasing distances is crucial to determine the structure of the phase diagram, as well as the nature of the transitions in strongly interacting spin systems. In practice, we focus on the paradigmatic dissipative quantum Ising model: in contrast to the non-dissipative case, its phase diagram is still a matter of debate in the literature. When dissipation acts along the interaction direction, we predict important quantitative modifications of the position of the first-order transition boundary. In the case of incoherent relaxation in the field direction, our approach confirms the presence of a second-order transition, while does not support the possible existence of multicritical points. Potentially, these results can be tested in up-to-date quantum simulators of Rydberg atoms. arXiv:1810.08112v2 [cond-mat.stat-mech]