Geometric Mutual Information at Classical Critical Points

Abstract

A practical use of the entanglement entropy in a 1d quantum system is to identify the conformal field theory describing its critical behavior. It is exactly (c/3) ln for an interval of length in an infinite system, where c is the central charge of the conformal field theory. Here we define the geometric mutual information, an analogous quantity for classical critical points. We compute this for 2d conformal field theories in an arbitrary geometry, and show in particular that for a rectangle cut into two rectangles, it is proportional to c. This makes it possible to extract c in classical simulations, which we demonstrate for the critical Ising and 3-state Potts models. Introduction.-In studies of new and exotic phases of quantum matter, the entanglement entropy has established itself as an important resource [1]. It is particularly useful in the many 1d quantum critical systems governed by a conformal field theory (CFT) [2] in the large-distance limit. Here the Rényi entanglement entropy S n of the ground-state is universal [3][4][5][6], and the leading piece is proportional to the central charge c of the CFT characterizing the universality class. Namely, for a periodic system of length L cut into two open segments of respective sizes L A and L B = (L − L A ),

Shannon and Rényi mutual information in quantum critical spin chains

“…40,41 ), but in slightly different limits. Let us finally mention that universal scaling forms proportional to n/(n − 1) have been found in the 2d classical Rényi mutual information 24 for Ising, as well as in the Rényi entropy of the 2d quantum transverse field Ising model 27 . In both cases the underlying ordering assumption is easier to justify: for the former the critical system is coupled to a bulk in the ordered phase 42 , while for the latter the higher dimensionality makes an extraordinary transition more likely.…”

“…Such terms should also impact related quantities in classical systems. For example, the Shannon mutual information of a infinite cylinder (strip) in a 2d classical system is exactly (twice) the Shannon entropy of the dominant eigenvector of the corresponding transfer matrix 24 . Hence by universality the (ln L) 2 contribution should also appear in the 2d MI, most probably also in a finite system.…”

Classical mutual information in mean-field spin glass models

“…Here G n is the so-called geometric mutual information 29 . Interestingly, for critical systems G n depends only on the geometry of A and B, and it is universal.…”

“…(C2) can be sampled efficiently in Monte Carlo 29 . Notice that the same trick has also been used in Ref.…”

“…For instance, at a conformally invariant critical point entanglement measures contain universal information about the underlying conformal field theory (CFT), such as the central charge [24][25][26][27] . For classical spin models a lot of attention has been focused on the classical Rényi entropy 28,29 . Given a bipartition of the system into two complementary subregions A and B, the classical Rényi entropy S n (A) (with n ∈ N) is defined as…”