Quench dynamics in integrable systems

These notes cover in some detail lectures I gave at the Les Houches Summer School 2012. I describe here work done with Deepak Iyer with important contributions from Hujie Guan. I discuss some aspects of the physics revealed by quantum quenches and present a formalism for studying the quench dynamics of integrable systems. The formalism presented generalizes an approach by Yudson and is applied to Lieb-Liniger model which describes a gas of $N$ interacting bosons moving on the continuous infinite line while interacting via a short range potential. We carry out the quench from several initial states and for any number of particles and compute the evolution of the density and the noise correlations. In the long time limit the system dilutes and we find that for any value of repulsive coupling independently of the initial state the system asymptotes towards astrongly repulsive gas, while for any value of attractive coupling, the system forms a maximal bound state that dominates at longer times. In either case the system equilibrates but does not thermalize, an effect that is consistent with prethermalization. These results can be confronted with experiments. For much more detail see: Phys. Rev. A 87, 053628 (2013) on which these notes are based. Further applications of the approach to the Heisenberg model and to the Anderson model will be presented elsewhere.Comment: Lecture Notes of the 2012 Les Houches Summer School of Physics "Strongly Interacting Quantum Systems Out of Equilibrium", Oxford University Press (to appear

Reply to the Comment on "Bound States in the One-dimensional Hubbard Model"

We reply to the comment (cond-mat/9806125) by Essler, Goehmann and Korepin, and show that their points are unfounded.Comment: 2 pages, revte

Time Evolution of Superradiance

The superradiant behaviour of the Dicke model is examined using the Yudson representation. This is achieved by computing the time evolution of the mean photon current density and photon number. Extensions of this model including energy splitting and spatial separation are then investigated using this technique.Comment: 7 pages, 3 Figure

Quantum impurity in a Luttinger liquid: Exact solution of the Kane-Fisher model

A Luttinger Liquid coupled to a quantum impurity describes a large number of physical systems. The Hamiltonian consists of left- and right-moving fermions interacting among themselves via a density-density coupling and scattering off a localised transmitting and reflecting impurity. We solve exactly the Hamiltonian by means of an incoming-outgoing scattering Bethe basis which properly incorporates all scattering processes. A related model, the Weak-Tunnelling model, wherein the impurity is replaced by a tunnel junction, is solved by the same method. The consistency of the construction is established through a generalised Yang-Baxter relation. Periodic boundary conditions are imposed and the resulting Bethe Ansatz equations are derived by means of the Off Diagonal Bethe Ansatz approach. We derive the spectrum of the model for all coupling constant regimes and calculate the impurity free energy. We discuss the low energy behaviour of the systems for both repulsive and attractive interactions.Comment: 15 page

Quantum Dot in Interacting Environments

A quantum impurity attached to an interacting quantum wire gives rise to an array of of new phenomena. Using Bethe Ansatz we solve exactly models describing two geometries of a quantum dot coupled to an interacting quantum wire: a quantum dot that is (i) side-coupled and (ii) embedded in a Luttinger liquid. We find the eigenstates and determine the spectrum through the Bethe Ansatz equations. Using this we derive exact expressions for the ground state dot occupation. The thermodynamics are then studied using the thermodynamics Bethe Ansatz equations. It is shown that at low energies the dot becomes fully hybridized and acts as a backscattering impurity or tunnel junction depending on the geometry and furthermore that the two geometries are related by changing the sign of the interactions. Although remaining strongly coupled for all values of the interaction in the wire, there exists competition between the tunneling and backscattering leading to a suppression or enhancement of the dot occupation depending on the sign of the bulk interactions.Comment: 13 pages, v2 expanded and more calculations adde

Equilibration and Generalized GGE in the Lieb Liniger gas

We study the nonequilibrium properties of the one dimensional Lieb Liniger model in the finite repulsion regime. Introducing a new version of the Yudson representation applicable to finite size systems and appropriately taking the infinite volume limit we are able to study equilibration of the Lieb Liniger gas in the thermodynamic limit. We provide a formalism to compute various correlation functions for highly non equilibrium initial states. We are able to find explicit analytic expressions for the long time limit of the expectation of the density, density density and related correlation functions. We show that the gas equilibriates to a diagonal ensemble which we show is equivalent to a generalized version of the GGE for sufficiently simple correlation functions, which in particular include density correlations.Comment: 4+epsilon pages, 2 figur

How to experimentally detect a GGE? - Universal Spectroscopic Signatures of the GGE in the Tonks gas

In this work we study the properties of the density density correlation function of the 1-D Lieb-Liniger model with infinite repulsion in the GGE regime. The GGE describes the equilibrated system in the long time limit after a quench from a generic initial state. In the case that the initial and hence the final state has low entropy per particle we find that the density density correlation function has a universal form, in particular it depends on a few parameters corresponding to "key" momenta and has power law dependence on the distance. This provides an experimental signature of the GGE which may readily be identified through spectroscopy. These signatures are universal and robust to initial sate preparation.Comment: 6 pages, 4 figure

Quench Dynamics of the Anisotropic Heisenberg Model

We develop an analytic approach for the study of the quench dynamics of the anisotropic Heisenberg model (XXZ model) on the infinite line. We present the exact time-dependent wavefunctions after a quench in an integral form for any initial state and for any anisotropy $\Delta$ by means of a generalized Yudson contour representation. We calculate the evolution of several observables from two particular initial states: starting with a local N\`eel state we calculate the time evolution of the antiferromagnetic order parameter--staggered magnetization; starting with a state with consecutive flipped spins we calculate the propagation of magnons and bound state excitations, and the induced spin currents. We also show how the "string" solution of Bethe Ansatz equations emerge naturally from the contour approach. We confront our results with experiments and numerical methods where possible.Comment: 4 pages, 5 figure

Stron eigenstate thermalization hypothesis

We present a generalization of the ETH conjecture. Using this generalization we are able to derive the fact that an arbitrary eigenstate of a general many body system may be used to represent microcanonical ensemble in any many body experiment that involves only local operators and projectors onto eigenstates of the system Hamiltonian. In particular we extend the ETH to include some non-local operators. We present a derivation of this conjecture in the case of a many body model whose Hamiltonian is composed of two parts: an integrable Hamiltonian and a small but finite Gaussian perturbation.Comment: 4 pages, 1 figur

Equilibration and GGE for hard wall boundary conditions

In this work we present an analysis of a quench for the repulsive Lieb-Liniger gas confined to a large box with hard wall boundary conditions. We study the time average of local correlation functions and show that both the quench action approach and the GGE formalism are applicable for the long time average of local correlation functions. We find that the time average of the system corresponds to an eigenstate of the Lieb-Liniger Hamiltonian and that this eigenstate is related to an eigenstate of a Lieb-Liniger Hamiltonian with periodic boundary conditions on an interval of twice the length and with twice as many particles (a doubled system). We further show that local operators with support far away from the boundaries of the hard wall have the same expectation values with respect to this eigenstate as corresponding operators for the doubled system. We present an example of a quench where the gas is initially confined in several moving traps and then released into a bigger container, an approximate description of the Newton cradle experiment. We calculate the time average of various correlation functions for long times after the quench.Comment: 10 pages, 1 figur