Nonextensive Statistical Mechanics

3.3 Correlations, Occupancy of Phase-Space, and Extensivity of S q 8110.90.80.70.6q ent 0.50.150.1250.40.100.30.0750.20.050.0250.100 0.2 0.4 0.6 0.8 100 1 2 3 4 5 6 7 8 9 10cFig. 3.19 q ent vs. c with the q-entropy, S q (ˆρ L ), being extensive, i.e.,lim L→∞ S √ (ˆρ9+c 2 −3 L )/L < ∞.cWhen c increases from 0 to infinity, q ent increases from 0 to unity (von Neumann entropy); forc = 4, q = 1/2andforc ≫ 1, see Ref. [203]. Inset: for the critical quantum Ising and XY modelsc = 1/2 andq ent = √ 37 − 6 ≃ 0.0828, while for the critical isotropic XX model c = 1andq ent = √ 10 − 3 ≃ 0.16.differently. We impose the extensivity of S q (L) finding the value of q for which−c/6(q ent − 1/q ent ) = 1, i.e.,√9 + c2 − 3q ent =. (3.145)cConsequently, lim L→∞ S √ 9+c 2 −3c(L)/L < ∞. When c increases from 0 to infinity(see Fig. 3.19), q ent increases from 0 to unity (von Neumann entropy). For c = 4(dimensionof physical space-time), q = 1/2; c = 26 corresponds to a 26-dimensionalbosonic string theory, see [203]. It is well-known that for critical quantum Ising andanisotropic XY models the central charge is equal to c = 1/2 (indeed they are inthe same universality class and can be mapped to a free (nonlocal) fermionic fieldtheory). For these models, at λ = 1, the value of q for which S q (L) isextensive isgiven by q ent = √ 37 − 6 ≃ 0.0828, in perfect agreement with our numerical resultsin Fig. 3.17. The critical isotropic XX model (γ = 0 and |λ| ≤1) is, instead, inanother universality class, the central charge is c = 1 (free bosonic field theory) andS q (L)isextensive for q ent = √ 10−3 ≃ 0.16, as found numerically also. We finallynotice that, in the c →∞limit, q ent → 1. The physical interpretation of this fact isnot clearly understood. However, since c in some sense plays the role of a dimension(see [203]), this limit could correspond to some sort of mean field approximation. Ifso, it is along a line such as this one that a mathematical justification could emergefor the widely spread use of BG concepts in the discussion of mean-field theories of