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arXiv:0804.2431 · Journal of Statistical Mechanics Theory and Experiment

Time evolution of one-dimensional gapless models from a domain wall initial state: stochastic Loewner evolution continued?

Pasquale Calabrese, Christian Hagendorf, Pierre Le Doussal

Published Jul 14, 2008• 60 citations
arXivHTMLPDF

Abstract

We study the time evolution of quantum one-dimensional gapless systems evolving from initial states with a domain wall.We generalize the path integral imaginary time approach that together with boundary conformal field theory allows us to derive the time and space dependence of general correlation functions.The latter are explicitly obtained for the Ising universality class, and the typical behavior of one-and two-point functions is derived for the general case.Possible connections with the stochastic Loewner evolution are discussed and explicit results for one-point time dependent averages are obtained for generic κ for boundary conditions corresponding to stochastic Loewner evolution.We use this set of results to predict the time evolution of the entanglement entropy and obtain the universal constant shift due to the presence of a domain wall in the initial state.

DOI
10.1088/1742-5468/2008/07/p07013

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We study the time evolution of quantum one-dimensional gapless systems evolving from initial states with a domain wall.We generalize the path integral imaginary time approach that together with boundary conformal field theory allows us to derive the time and space dependence of general correlation functions.The latter are explicitly obtained for the Ising universality class, and the typical behavior of one-and two-point functions is derived for the general case.Possible connections with the stochastic Loewner evolution are discussed and explicit results for one-point time dependent averages are obtained for generic κ for boundary conditions corresponding to stochastic Loewner evolution.We use this set of results to predict the time evolution of the entanglement entropy and obtain the universal constant shift due to the presence of a domain wall in the initial state.
We show that the dynamics resulting from preparing a one-dimensional quantum system in the ground state of two decoupled parts, then joined together and left to evolve unitarily with a translational invariant Hamiltonian (a local quench), can be described by means of quantum field theory. In the case when the corresponding theory is conformal, we study the evolution of the entanglement entropy for different bipartitions of the line. We also consider the behavior of one- and two-point correlation functions. All our findings may be explained in terms of a picture, that we believe to be valid more generally, whereby quasiparticles emitted from the joining point at the initial time propagate semiclassically through the system.
Abstract We revisit the out-of-equilibrium physics arising during the unitary evolution of a one-dimensional XXZ spin chain initially prepared in a domain wall state ∣ψ0⟩=∣⋯↑↑↓↓… ⟩\vert \psi _0\rangle =\vert \dots \uparrow \uparrow \downarrow \downarrow \dots \rangle∣ψ0​⟩=∣⋯↑↑↓↓…⟩ | ψ 0 ⟩ = | ⋯ ↑ ↑ ↓ ↓ ⋯ ⟩ . In absence of interactions, we review the exact lattice calculation of several conserved quantities, including e.g. the magnetization and the spin current profiles. At large distances x and times t, we show how these quantities allow for a ballistic scaling behavior in terms of the scaling variable ζ=x/t\zeta = x/tζ=x/t ζ = x / t , with exactly computable scaling functions. In such a limit of large space-time scales, we show that the asymptotic behavior of the system is suitably captured by the local occupation function of spinless fermionic modes, whose semi-classical evolution in phase space is given by a Euler hydrodynamic equation. Similarly, analytical results for the asymptotic fronts dynamics are obtained for the interacting chain via Generalized Hydrodynamics. In the last part of the work, we include large-scale quantum fluctuations on top of the semi-classical hydrodynamic background in the form of a conformal field theory that lives along the evolving Fermi contour. With this procedure, dubbed quantum generalized hydrodynamics, it is possible to obtain exact asymptotic results for the entanglement spreading during the melting dynamics.
We study in general the time evolution of correlation functions in a extended quantum system after the quench of a parameter in the Hamiltonian. We show that correlation functions in d dimensions can be extracted using methods of boundary critical phenomena in d +1 dimensions. For d = 1 this allows us to use the powerful tools of conformal field theory in the case of critical evolution. Several results are obtained in generic dimension in the Gaussian (mean field) approximation. These predictions are checked against the real time evolution of some solvable models that allow us also to understand which features are valid beyond the critical evolution. All our findings may be explained in terms of a picture generally valid, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate with a finite speed through the system. Furthermore we show that the long time results can be interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible future developments.
We study the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length and its complement, starting from a pure state which is not an eigenstate of the Hamiltonian. We use path integral methods of quantum field theory as well as explicit computations for the transverse Ising spin chain. In both cases, there is a maximum speed v of propagation of signals. In general the entanglement entropy increases linearly with time t up to , after which it saturates at a value proportional to , the coefficient depending on the initial state. This behaviour may be understood as a consequence of causality.
We study analytically the role of initial conditions in nonequilibrium quantum dynamics considering the one-dimensional ferromagnets in the regime of spontaneously broken symmetry. We analyze the expectation value of local operators for the infinite-dimensional space of initial conditions of domain wall type, generally intended as initial conditions spatially interpolating between two different ground states. At large times the unitary time evolution takes place inside a light cone produced by the spatial inhomogeneity of the initial condition. In the innermost part of the light cone the form of the space-time dependence is universal, in the sense that it is specified by data of the equilibrium universality class. The global limit shape in the variable x/t changes with the initial condition. In systems with more than two ground states the tuning of an interaction parameter can induce a transition which is the nonequilibrium quantum analog of the interfacial wetting transition occurring in classical systems at equilibrium. We illustrate the general results through the examples of the Ising, Potts and Ashkin-Teller chains.
We investigate a class of local quantum quenches in which the conformal boundary condition of a (1+1)-dimensional conformal field theory is abruptly changed. We derive a remarkably simple and universal expression for the time evolution of one-point functions on the half-line. This result provides a direct description of the propagation of the disturbance generated by the quench and, in turn, allows us to determine the dynamics of bipartite entanglement for subsystems adjacent to the boundary. We show that, once the subsystem becomes fully causally connected to the quench event, the entanglement entropy undergoes a sharp finite jump whose magnitude is universally given by the logarithm of the ratio of the boundary g-factors associated with the initial and final boundary conditions. We benchmark these analytical predictions against Matrix Product State simulations of the critical Ising spin chain, finding excellent agreement. The numerical analysis further allows us to investigate the time evolution of the spin-flip entanglement asymmetry, revealing how the symmetry-breaking perturbation emitted from the boundary propagates through the system. Our results uncover universal dynamical signatures of boundary quenches and establish a direct connection between nonequilibrium entanglement dynamics and boundary critical phenomena.