Emptiness formation probability, Toeplitz determinants, and Conformal field theory
University of Virginia
Mon, Mar. 25th 2013, 11:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
This talk deals with various aspects of the emptiness formation probability (EFP), the probability of forming a sequence of N consecutive spins with the same orientation, in the ground-state of a quantum spin chain. I will argue that the study of this toy quantity provides an illuminating way of thinking about a variety of problems at criticality, including full counting statistics, information-theoretic aspects of 2d classical systems, and quantum quenches. I will mainly focus on two examples of critical systems for which the EFP exhibits strikingly different behavior, as the number of spins N goes to infinity. For the Ising chain in transverse field, the sequence of spins acts as an additional boundary in imaginary time, and the behavior of the EFP can be studied combining boundary conformal field theory (CFT) arguments with asymptotic results on Toeplitz and Toeplitz+Hankel determinants. This allows for a systematic derivation of all universal and non-universal terms in its scaling. The other example, the XXZ chain, is more challenging. The leading behavior can be understood in terms of an imaginary time ``arctic picture'', in which all degrees of freedom inside of a certain region are frozen. The CFT, whose exact properties need to be determined, then lives at the exterior of this region. par Finally, I will explain on a specific example how these ideas may be applied to quantum quenches problems.