We investigate the competition of coherent and dissipative dynamics in many-body systems at continuous quantum transitions. We consider dissipative mechanisms that can be effectively described by Lindblad equations for the density matrix of the system. The interplay between the critical coherent dynamics and dissipation is addressed within a dynamic finite-size scaling theory, which allows us to identify the regime where they develop a nontrivial competition. We put forward general scaling behaviors involving the Hamiltonian parameters and the coupling associated with the dissipation. This scaling scenario is supported by a numerical study of the dynamic behavior of a one-dimensional lattice fermion gas undergoing a quantum Ising transition, in the presence of dissipative mechanisms such as local pumping, decaying and dephasing.
The aims of this paper are two. The first is to give a brief review of the most relevant theoretical results concerning the uniqueness of the steady-state solution of the Lindblad-Gorini-Kossakowski-Sudarshan master equation and the criteria which guarantee relaxingness and irreducibility of dynamical semigroups. In particular, we test and discuss their physical meaning by considering their applicability to the characterisation of the simplest open quantum system i.e. a two-level system coupled to a bath of harmonic oscillators at zero temperature. The second aim is to provide a set of sufficient conditions which guarantees the uniqueness of the steady-state solution and its attractivity. Starting from simple assumptions, we derive simple criteria that can be efficiently exploited to characterise the behavior of dissipative systems of spins and bosons (with truncated Fock space), and a wide variety of other open quantum systems recently studied.
We study the scaling properties of the statistics of the work done on a generic many-body system at a quantum phase transition of any order and type, arising from quenches of a driving control parameter. For this purpose we exploit a dynamic finite-size scaling framework. Namely, we put forward the existence of a nontrivial finite-size scaling limit for the work distribution, defined as the large-size limit when appropriate scaling variables are kept fixed. The corresponding scaling behaviors are thoroughly verified by means of analytical and numerical calculations in two paradigmatic many-body systems as the quantum Ising model and the Bose-Hubbard model.
We give a prescription to perform the continuum limit of the d-dimensional Hubbard model in the presence of a harmonic trap at zero temperature. We perform the continuum limit at fixed number of particles. In d ≥ 3 the lattice system of spin-1/2 particles is mapped into a non-interacting two-component Fermi gas in a harmonic trap. In d = 1 and d = 2 the particles with opposite spin interact via a Dirac delta interaction. We show that the properties of this continuum limit can be put in correspondence with those derived applying the Trap-Size scaling (TSS) formalism to the confined Hubbard model in the so called Dilute Regime (fixed number of particles and weak confinement). The correspondence in d = 1 and d = 2 has been tested comparing the numerical results obtained for lattice system with those of the continuum limit in the case of two-particle and in absence of spin-polarization (N = 2,N ↑ = N ↓ = 1).
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