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Shannon mutual information of critical quantum chains

Presented by: 
Francisco Alcaraz
Monday 11th January 2016 - 16:30 to 17:30
INI Seminar Room 1
Associated to the equilibrium Gibbs state of a given critical classical system in d dimensions we can associate a special quantum mechanical eigenfunction defined in a Hilbert space with the dimension given by the number of configurations of the classical system and components given by the Boltzmann weights of the equilibrium probabilities of the critical system. This class of eigenfunctions are generalizations of the Rokhsar-Kivelson, initially proposed for the dimer problem in 2 dimensions. In particular in two dimensions, where most of the critical systems are conformal invariant, such functions exhibit quite interesting universal features. The entanglement entropy of a line of contiguous variables (classical spins), is given by the shannon entropy of d=1 quantum chains, and the entanglement spectrum of the two dimensional system are given by the amplitudes of the ground-state eigenfunction of the quantum chain. We present a conjecture showing that the Shannon mutual information of the quantum chains in some appropriate basis (we called conformal basis) show a universal behavior with the size of the line of the entangled spins (subsystem size). This dependence allow us to identify the conformal charge of the associated classical critical system (used to define the d=2 quantum eigenfunction) or the quantum critical chain. Tests of this conjecture for integrable and non integrable quantum chains will be presented. We also consider numerical results for two distinct generalizations of the Shannon mutual information: the one based in the concept of the R\'enyi entropy and the one based on the R\'enyi divergence. A numerical test of the extension of this conjecture for critical random chains (not conformal invariant) is also presented. 
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons