Professor Fabian Essler

Abstract:

We consider quantum quenches in integrable models. We argue that the behavior of local observables at late times after the quench is given by their expectation values with respect to a single representative Hamiltonian eigenstate. This can be viewed as a generalization of the eigenstate thermalization hypothesis to quantum integrable models. We present a method for constructing this representative state by means of a generalized thermodynamic Bethe ansatz (GTBA). Going further, we introduce a framework for calculating the time dependence of local observables as they evolve towards their stationary values. As an explicit example we consider quantum quenches in the transverse-field Ising chain and show that previously derived results are recovered efficiently within our framework. © 2013 American Physical Society.

Abstract:

We consider a quantum quench in the spin-1/2 Heisenberg XXZ chain. At late times after the quench it is believed that the expectation values of local operators approach time-independent values, that are described by a generalized Gibbs ensemble. Employing a quantum transfer matrix approach we show how to determine short-range correlation functions in such generalized Gibbs ensembles for a class of initial states.

Abstract:

We consider the von Neumann and Rényi entropies of the one-dimensional quarter-filled Hubbard model. We observe that for periodic boundary conditions the entropies exhibit an unexpected dependence on system size: for L=4 mod 8 the results are in agreement with expectations based on conformal field theory, while for L=0 mod 8 additional contributions arise. We explain this observation in terms of a shell-filling effect and develop a conformal field theory approach to calculate the extra term in the entropies. Similar shell-filling effects in entanglement entropies are expected to be present in higher dimensions and for other multicomponent systems. © 2013 American Physical Society.

Abstract:

We consider the reduced density matrix (RDM) \rho_A(t) for a finite subsystem A after a global quantum quench in the infinite transverse-field Ising chain. It has been recently shown that the infinite time limit of \rho_A(t) is described by the RDM \rho_{GGE,A} of a generalized Gibbs ensemble. Here we present some details on how to construct this ensemble in terms of local integrals of motion, and show its equivalence to the expression in terms of mode occupation numbers widely used in the literature. We then address the question, how \rho_A(t) approaches \rho_{GGE,A} as a function of time. To that end we introduce a distance on the space of density matrices and show that it approaches zero as a universal power-law t^{-3/2} in time. As the RDM completely determines all local observables within A, this provides information on the relaxation of correlation functions of local operators. We then address the issue, of how well a truncated generalized Gibbs ensemble with a finite number of local higher conservation laws describes a given subsystem at late times. We find that taking into account only local conservation laws with a range at most comparable to the subsystem size provides a good description. However, excluding even a single one of the most local conservation laws in general completely spoils this agreement.