We use a semiclassical approach to study out-of-equilibrium dynamics and transport in quantum systems with massive quasiparticle excitations having internal quantum numbers. In the universal limit of low-energy quasiparticles, the system is described in terms of a classical gas of colored hard-core particles. Starting from an inhomogeneous initial state, in this limit we give analytic expressions for the space- and time-dependent spin density and spin current profiles. Depending on the initial state, the spin transport is found to be ballistic or diffusive. In the ballistic case we identify a ``second front'' that moves more slowly than the maximal quasiparticle velocity. Our analytic results also capture the diffusive broadening of this ballistically propagating front. To go beyond the universal limit, we study the effect of nontrivial scattering processes in the $O(3)$ nonlinear sigma model by performing Monte Carlo simulations, and we observe local equilibration around the second front in terms of the densities of the particle species.
We use a semiclassical approach to study out-of-equilibrium dynamics and transport in quantum systems with massive quasiparticle excitations having internal quantum numbers. In the universal limit of low-energy quasiparticles, the system is described in terms of a classical gas of colored hard-core particles. Starting from an inhomogeneous initial state, in this limit we give analytic expressions for the space- and time-dependent spin density and spin current profiles. Depending on the initial state, the spin transport is found to be ballistic or diffusive. In the ballistic case we identify a ``second front'' that moves more slowly than the maximal quasiparticle velocity. Our analytic results also capture the diffusive broadening of this ballistically propagating front. To go beyond the universal limit, we study the effect of nontrivial scattering processes in the O(3) nonlinear sigma model by performing Monte Carlo simulations, and we observe local equilibration around the second front in terms of the densities of the particle species.
We investigate the non-equilibrium dynamics of the transverse field quantum Ising chain evolving from an inhomogeneous initial state given by joining two macroscopically different semi-infinite chains. We obtain integral expressions for all two-point correlation functions of the Jordan--Wigner Majorana fermions at any time and for any value of the transverse field. Using this result, we analytically compute the profiles of various physical observables in the space-time scaling limit and show that they can be obtained from a hydrodynamic picture based on ballistically propagating quasiparticles. Going beyond the hydrodynamic limit, we analyze the approach to the non-equilibrium steady state and find that the leading late time corrections display a lattice effect. We also study the fine structure of the propagating fronts which are found to be described by the Airy kernel and its derivatives. Near the front we observe the phenomenon of energy back-flow where the energy locally flows from the colder to the hotter region.
We consider two semi-infinite quantum Ising chains initially at thermal equilibrium at two different temperatures and subsequently joined by an interaction between their end points. Transport properties such as the heat current are determined by the dynamics of the left- and right-moving fermionic quasiparticles which characterize the ensuing unitary dynamics. Within the so-called semiclassical space-time scaling limit we extend known results by determining the full space and time dependence of the density and current of energy and of fermionic quasiparticles. Upon approaching the edge of the propagating front, these quantities as well as the two-point correlation function display qualitatively different behaviors depending on the transverse field of the chain being critical or not. While in the latter case corrections to the leading behavior are described, as expected, by the Airy kernel, in the former a novel scaling form emerges with universal features.
Recent experimental advances on ultracold atomic gases and trapped ions have made it possible to simulate exactly solvable quantum systems of interacting particles. In particular, the feasibility of making rapid changes, so-called quantum quenches, to such set-ups has allowed experimentalists to probe non-equilibrium phenomena in closed interacting quantum systems. This, in turn, has spurred a considerable theoretical interest in quantum many-body systems out of equilibrium. In this thesis, we study non-equilibrium properties of quantum many-body systems in the framework of exactly solvable quantum field theory in one spatial dimension. Specific systems include interacting fermions described by the Luttinger model and effective descriptions of spin chains using conformal field theory (CFT). Special emphasis is placed on heat and charge transport, studied from the point of view of quench dynamics, and, in particular, the effects of breaking conformal symmetries on transport properties. Examples include the Luttinger model with non-local interactions, breaking Lorentz and scale invariance, and inhomogeneous CFT, which generalizes standard CFT in that the usual propagation velocity v is replaced by a function v(x) that depends smoothly on the position x, breaking translation invariance. The quench dynamics studied here is for quantum quenches between, in general, different smooth inhomogeneous systems. An example of this is the so-called smooth-profile protocol, in which the initial state is defined by, e.g., smooth inhomogeneous profiles of inverse temperature and chemical potential, and the time evolution is governed by a homogeneous Hamiltonian. Using this protocol, we compute exact analytical results for the full time evolution of the systems mentioned above. In particular, we derive finite-time results that are universal in the sense that the same relations between the non-equilibrium dynamics and the initial profiles hold for any unitary CFT. These results also make clear that heat and charge transport in standard CFT are purely ballistic. Finally, we propose and study an inhomogeneous CFT model with v(x) given by a random function. We argue that this model naturally emerges as an effective description of one-dimensional quantum many-body systems with certain static random impurities. Using tools from wave propagation in random media, we show that such impurities lead to normal and anomalous diffusive contributions to heat transport on top of the ballistic one known from standard CFT.
We address the hydrodynamics of operator spreading in interacting integrable lattice models. In these models, operators spread through the ballistic propagation of quasiparticles, with an operator front whose velocity is locally set by the fastest quasiparticle velocity. In interacting integrable systems, this velocity depends on the density of the other quasiparticles, so equilibrium density fluctuations cause the front to follow a biased random walk, and therefore to broaden diffusively. Ballistic front propagation and diffusive front broadening are also generically present in nonintegrable systems in one dimension; thus, although the mechanisms for operator spreading are distinct in the two cases, these coarse-grained measures of the operator front do not distinguish between the two cases. We present an expression for the front-broadening rate; we explicitly derive this for a particular integrable model (the ``Floquet-Fredrickson-Andersen'' model), and argue on kinetic grounds that it should apply generally. Our results elucidate the microscopic mechanism for diffusive corrections to ballistic transport in interacting integrable models.
The primary area, on which we focus in the thesis, are the out-of-equilibrium properties of quantum and classical systems. The playground we use to study those properties, are interacting integrable and solitonic systems, whose structure enables us to perform analytical calculations.
In the context of the out-of-equilibrium physics we focus on two questions. The first problem is related to the set of stationary states in quantum systems. The question is especially important in the context of the homogeneous quantum quenches, since it makes possible the calculations of expectation values of local observables. The building blocks of stationary states are quasilocal conserved quantities which constitute the information that is preserved under the time evolution. In the first part of the thesis we deal with the construction of quasilocal conserved quantities in one dimensional anisotropic Heisenberg model and bosonic models. Quasilocal charges are put in the context of the standard theory of integrability, which in turn allows for the calculation of physical quantities.
The second question we deal with are the transport properties of one dimensional systems. The connection between the local integrals of motion and ideal transport was established in 90's. Here we extend the connection to the diffusive transport. This enables us to obtain explicit lower bound on diffusion constant in Heisenberg model, which proves that the Heisenberg model is not an insulator at finite temperatures.
In the last chapter we deal with the transport properties of classical celular automata. The first example we consider is the gas of charged hard core interacting particles, and the second one the gas of interacting solitons. Explicit solution of the dynamics of local observables allows for the explicit calculation of transport coeffcients, and the solution of the inhomogeneous quench problem. Despite their simplicity, the models exhibit wide range of transport phenomena ranging from insulating, through diffusive to ballistic transport.
The semiclassical approach introduced by Sachdev and collaborators proved to be extremely successful in the study of quantum quenches in massive field theories, both in homogeneous and inhomogeneous settings. While conceptually very simple, this method allows one to obtain analytic predictions for several observables when the density of excitations produced by the quench is small. At the same time, a novel generalized hydrodynamic (GHD) approach, which captures exactly many asymptotic features of the integrable dynamics, has recently been introduced. Interestingly, also this theory has a natural interpretation in terms of semiclassical particles and it is then natural to compare the two approaches. This is the objective of this work: we carry out a systematic comparison between the two methods in the prototypical example of the sine-Gordon field theory. In particular, we study the ``bipartitioning protocol'' where the two halves of a system initially prepared at different temperatures are joined together and then left to evolve unitarily with the same Hamiltonian. We identify two different limits in which the semiclassical predictions are analytically recovered from GHD: a particular nonrelativistic limit and the low-temperature regime. Interestingly, the transport of topological charge becomes subballistic in these cases. Away from these limits we find that the semiclassical predictions are only approximate and, in contrast to the latter, the transport is always ballistic. This statement seems to hold true even for the so-called ``hybrid'' semiclassical approach, where finite time DMRG simulations are used to describe the evolution in the internal space.