We study the nonequilibrium dynamics of one-dimensional adsorption-desorption processes with diffusional relaxation. According to the relative values of the transition probability rates, these systems can be mapped onto a soluble problem of many fermions. The time development of pair correlations, interparticle distribution functions, macroscopic density, structure factor, and density fluctuations is obtained by combining and comparing both exact results with Monte Carlo simulations. Depending on the spectrum gap of the evolution operator associated with the stochastic dynamics, these quantities display either exponential decay or power-law relaxation. For vanishing gaps, the pair correlations exhibit a faster decay as opposed to the standard diffusive relaxation of the macroscopic density. (c) 1995 The American Physical Society
We study the nonequilibrium dynamics of one-dimensional adsorption-desorption processes with diffusional relaxation. According to the relative values of the transition probability rates, these systems can be mapped onto a soluble problem of many fermions. The time development of pair correlations, interparticle distribution functions, macroscopic density, structure factor, and density fluctuations is obtained by combining and comparing both exact results with Monte Carlo simulations. Depending on the spectrum gap of the evolution operator associated with the stochastic dynamics, these quantities display either exponential decay or power-law relaxation. For vanishing gaps, the pair correlations exhibit a faster decay as opposed to the standard diffusive relaxation of the macroscopic density. (c) 1995 The American Physical Society
In this Thesis we discussed several issues related to the dynamics of isolated quantum many-body systems. The brief overview of Chapter 1 highlighted a series of open fundamental theoretical questions which are crucial for the investigation of nonequilibrium phenomena. Among them, we mainly focused on the emergence of thermal behaviors and on the characterization of dynamical phase transitions and of the associated universal dynamics. To this end, we analyzed some paradigmatic models, such as the Fermi-Hubbard model, the quantum Ising chain, and the O(N) model. -Chapter 2: Absence of thermalization in a Fermi liquid. In this Chapter we report two results that question the common expectation of thermalization in a Landau-Fermi liquid after an interaction quench. We first consider the perturbative expansion in the interaction strength of the long-wavelength structure factor S(q) and show that it does not satisfy the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) has an analytical component ∼ const. + O(q^2) for q → 0, compatible with thermalization, but it retains also a nonanalytic term ∼ |q| characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity turns in a power-law decay of the density-density correlations, in contrast with the exponential decay associated to thermalization. We next analyze the low-density case, where one can obtain results valid at any order in interaction but at leading in the density, and find that in the steady state the momentum distribution jump at the Fermi surface is strictly finite, though smaller than at equilibrium. -Chapter 3: Nonadiabatic stationary behavior in a driven Ising chain. In this Chapter we discuss the emergence of a nonadiabatic behavior in the dynamics of the order parameter in a low-dimensional system subject to a linear ramp of one of its parameters within a gapped phase, which should be the most favorable situation for an adiabatic evolution. We study in details the dynamics of the spontaneous magnetization m_x(t) in a quantum Ising chain after a linear variation in time of the transverse field within the ordered phase. In particular, focusing on the value of the order parameter at the end of the ramp m_x(τ), we show that the smaller the switching rate of the transverse field is the closer m_x(τ) gets to its ground state value. Nonetheless, this apparently small correction to adiabaticity eventually leads to the disruption of the order exponentially fast in the subsequent time evolution, no matter how slowly the ramp is performed. -Chapter 4: Aging and coarsening in isolated quantum systems after a quench. This Chapter analyzes the nonequilibrium dynamics of an isolated quantum system after a sudden quench to the dynamical critical point, where the emergence of scaling and universal exponents is expected, due to the absence of time and length scales. We explore these features for a bosonic interacting scalar field theory with O(N) symmetry in the large-N limit, where the model is exactly solvable and the exponents and scaling forms of the relevant two-time correlation functions can be analytically calculated. Moreover, we provide evidence of the emergence of a dynamical scaling behavior also for quenches below the dynamical critical point, associated with coarsening. We find that the latter case is characterized by the same scaling functions as those describing the critical case, yet with different exponents. -Chapter 5: Dynamical transitions and statistics of excitations. In this Chapter we study the dynamics of the O(N) model (introduced in the previous Chapter) resulting from a different protocol: a linear ramp of one of its parameters. We find that the presence of a dynamical phase transition, as well as its critical properties, are robust against the change of the protocol. We show that a characterization based on the critical dimensions and exponents would suggest that the dynamical phase transition is analogous to the equilibrium thermal one. However, its nonequilibrium nature becomes evident analyzing the statistics of excitations produced in the ramp process. In particular, the critical properties are encoded in the fluctuations in the number of excitations, which display qualitatively different behaviors depending on the ramp being performed above, at, or below the dynamical critical point. These behaviors bear no dependence on the duration of the protocol.
Non-universal dynamics is shown to occur in a one-dimensional non-equilibrium system of hard-core particles. The stochastic processes included are pair creation and annihilation (with rates and ) and symmetric hopping rates which alternate from one bond to the next ( , ). A dynamical scaling relation between the relaxation time and the correlation length in the steady state is derived in a simple way for the case . We find that the dynamical exponent takes the non-universal value . For the special condition , where the stochastic system is in principle soluble by reduction to a free fermion system, the model is mapped to the Glauber dynamics of an Ising chain with alternating ferromagnetic bonds of values and , in contact with a quantum thermal bath. The full time dependence of the space-dependent magnetization and of the equal time spin-spin correlation function are studied by writing the master equation for this system in the quantum Hamiltonian formalism. In particular, we obtain the dispersion relations and rigorously confirm the results obtained for the correlation length and for the dynamical exponent.
Non-universal dynamics is shown to occur in a one-dimensional non-equilibrium system of hard-core particles. The stochastic processes included are pair creation and annihilation (with rates e and e') and symmetric hopping rates which alternate from one bond to the next (Pa, Pb). A dynamical scaling relation between the relaxation time and the correlation length in the steady state is derived in a simple way for the case e' > Pa >> Pb >> e. We find that the dynamical exponent takes the non-universal value z = 2 ln(e'/e) / ln[(Pb e')/(Pa e)]. For the special condition e + e'= Pa + Pb, where the stochastic system is in principle soluble by reduction to a free fermion system, the model is mapped to the Glauber dynamics of an Ising chain with alternating ferromagnetic bonds of values J1 and J2, in contact with a quantum thermal bath. The full time dependence of the space-dependent magnetization and of the equal time spin-spin correlation function are studied by writing the master equation for this system in the quantum Hamiltonian formalism. In particular we obtain the dispersion relations and rigorously confirm the results obtained for the correlation length and for the dynamical exponent.
We investigate the quantum reaction-diffusion dynamics of fermionic particles which coherently hop in a one-dimensional lattice and undergo annihilation reactions. The latter are modelled as dissipative processes which involve losses of pairs 2A→∅, triplets 3A→∅, and quadruplets 4A→∅ of neighboring particles. When considering classical particles, the corresponding decay of their density in time follows an asymptotic power-law behavior. The associated exponent in one dimension is different from the mean-field prediction whenever diffusive mixing is not too strong and spatial correlations are relevant. This specifically applies to 2A→∅, while the mean-field power-law prediction just acquires a logarithmic correction for 3A→∅ and is exact for 4A→∅. A mean-field approach is also valid, for all the three processes, when the diffusive mixing is strong, i.e., in the so-called reaction-limited regime. Here we show that the picture is different for quantum systems. We consider the quantum reaction-limited regime and we show that for all the three processes power-law behavior beyond mean field is present as a consequence of quantum coherences, which are not related to space dimensionality. The decay in 3A→∅ is further, highly intricate, since the power-law behavior therein only appears within an intermediate time window, while at long times the density decay is not power law. Our results show that emergent critical behavior in quantum dynamics has a markedly different origin, based on quantum coherences, to that applying to classical critical phenomena, which is, instead, solely determined by the relevance of spatial correlations.
The dynamics of a one-dimensional many-particle Fermi system containing one particle spin down (the d particle) in a sea of spin-up particles (u particles) is discussed in the context of an exactly solvable model problem. It is assumed that random variables associated with the d particle are relevant, and the marginal distributions of these random variables are calculated. The marginal distribution of the u-d relative coordinate is found to be constant in time and therefore to provide a connection of dynamics to equilibrium. This constant probability distribution affects the temperature-volume dependence of the partition function and, in the case of an attractive interaction, leads to a phase transition from an unpaired to a paired state at a well-defined critical temperature.
The raise and peel model is a one-dimensional nonlocal stochastic model where adsortion happens locally and desorption is nonlocal.The model depends on the single parameter u that is the ratio among the desorption and adsorption rates.At a special value of this parameter (u = 1) the model has interesting features.The spectrum is described by a conformal field theory (central charge c = 0), and its stationary probability density is related to the equilibrium distribution of a two dimensional system.The phase diagram of the model, as a function of the parameter u, has a massive phase (gapped phase) and a massless (gapless phase) whose critical exponents vary continuously with u.In this monography we study a one-parameter extension of the raise and peel model at u = 1, that depends on the additional parameter p.The new model exhibits conformal invariance for the whole range of values of its parameter p, and it is in the same universality class as the usual raise and peel model.The single difference between the models is the value of the sound velocity v s (p) which is a function of p.The methods used in this monography are the exact diagonalization of the evolution operator of the stochastic model (Hamiltonian), for small lattice sizes and Monte Carlo simulations.