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SUMMARY:Shannon mutual information of critical quantum chains - Francisco 
  Alcaraz (Universidade de São Paulo )
DTSTART:20160111T163000Z
DTEND:20160111T173000Z
UID:TALK64619@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Associated to the equilibrium Gibbs state of a given critical 
 classical  system in d dimensions we can associate a special quantum mecha
 nical  eigenfunction defined in a Hilbert space with the dimension given b
 y the number  of configurations of the classical system and components giv
 en by the Boltzmann  weights of the equilibrium probabilities of the criti
 cal system. This class of  eigenfunctions are generalizations of the Rokhs
 ar-Kivelson\, initially proposed  for the dimer problem in 2 dimensions. I
 n particular in two dimensions\, where  most of the critical systems are c
 onformal invariant\, such functions exhibit  quite interesting universal f
 eatures. The entanglement entropy of a line of  contiguous variables (clas
 sical spins)\, is given by the shannon entropy of d=1  quantum chains\, an
 d the entanglement spectrum of the two dimensional system are  given by th
 e amplitudes of the ground-state eigenfunction of the quantum chain.  We p
 resent a conjecture showing that the Shannon mutual information of the  qu
 antum chains in some appropriate basis (we called conformal basis) show a 
  universal behavior with the size of the line of the entangled spins (subs
 ystem  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 f
 or  integrable and non integrable quantum chains will be presented. We als
 o consider  numerical results for two distinct generalizations of the Shan
 non mutual  information: the one based in the concept of the R\\&#39\;enyi
  entropy and the one  based on the R\\&#39\;enyi divergence. A numerical t
 est of the extension of this  conjecture for critical random chains (not c
 onformal invariant) is also  presented.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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