Nonextensive Statistical Mechanics

3.3 Correlations, Occupancy of Phase-Space, and Extensivity of S q 79where we assume periodic boundary conditions, i.e., we have a ring with N spins,and (σj x,σy j ,σz j) are the Pauli matrices. For |γ |=1wehavetheIsing ferromagnet,for 0 < |γ | < 1wehavetheanisotropic XY ferromagnet, and for γ = 0wehavetheisotropic XY ferromagnet (or, simply, the XY ferromagnet). This model, being onedimensional,has no phase transition at T > 0. But it does have a second order oneat T = 0. More precisely, it is critical at λ = 1ifγ ≠ 0, and at 0 ≤ λ ≤ 1ifγ = 0.See [201] for the details of the numerical and analytical calculations. The resultsare presented in Figs. 3.16, 3.17, 3.18, and 3.19. The numbers are consistent with themain present relation, namely q ent as a function of the central charge c. This conceptis since long known in quantum field theory (see [204] and references therein). Thecentral charge characterizes the critical universality class of vast sets of systems(more precisely, various critical exponents are shared between the systems that havethe same value of c).Reference [205] enables us to analytically confirm, at the critical point, thenumerical results exhibited in the above figures. The continuum limit of a (1+1)-dimensional critical system is a conformal field theory with central charge c.Inthisquite different context, the authors re-derive the resultS 1 (L) ∼ (c/3) ln L (3.144)for a finite block of length L in an infinite critical system. To obtain this (clearlynonextensive) expression of the von Neumann entropy S 1 (L), they first find an analyticalexpression, namely Tr ˆρ q L ∼ L−c/6(q−1/q) . Here, this expression is used quite15001250q = 0.05150100S q50q = 0.08100000 5 10 15 20S Lq750500250q = 0.07q = 0.09q = 0.1100 50 100 150 200 250 300LFig. 3.16 Block q-entropy S q (ˆρ L ) as a function of the block size L in a critical Ising chain (γ =1, λ= 1), for typical values of q. Only for q = q ent ≃ 0.0828, s q is finite (i.e., S q is extensive);for q < q ent (q > q ent ) it diverges (vanishes).