Nonextensive Statistical Mechanics

82 3 Generalizing What We Learntspin-glasses (within the replica-trick and related approaches). Indeed, BG statisticalmechanics is essentially based on the ergodic hypothesis. It is firmly known thatglassy systems (e.g., spin-glasses) precisely violate ergodicity, thus leading to anintriguing and fundamental question. Consequently, a mathematical justification forthe use of BG entropy and energy distribution for such complex mean-field systemswould be more than welcome.The Hamiltonian (3.143) can be generalized into the following quantum Heisenbergone:∑N−1NH =− [(1 + γ )ˆσ j x ˆσ j+1 x + (1 − γ )ˆσ y jˆσ y j+1 + σ z j σ z j+1 ] − 2λ ∑j=1j=1ˆσ z j., (3.146)For = 1 and λ = 0 there also occurs a critical phenomenon. Its associated valueof c also is 1, hence q ent = √ 10 − 3 ≃ 0.16. If we include in this Hamiltonian saysecond-neighbor coupling (or, in fact, any short-range coupling which does not alterthe ferromagnetic order parameter), the value of c, hence that of q ent , remains thesame. Not so with the slope s qent , which depends on the details and not only on thesymmetry which is being broken at criticality.Let us address now our second system, the bosonic one [202]. It is the bidimensionalsystem of infinite coupled harmonic oscillators studied in Ref. [206], withHamiltonianH = 1 ∑ ( 2 x,y2+ ω2 0 2 x,yx,y+ ( x,y − x+1,y ) 2 + ( x,y − x,y+1 ) 2) (3.147)where x,y , x,y , and ω 0 are coordinate, momentum, and self-frequency of theoscillator at site r = (x, y). The system has the dispersion relationE(k) =√ω 2 0 + 4sin2 k x /2 + 4sin 2 k y /2 , (3.148)hence, a gap ω 0 at k = 0. Applying the canonical transformation b i = √ ω2 ( i +√iω i) with ω = ω0 2 + 4, the Hamiltonian (3.147) is mapped into the quadraticcanonical formH = ∑ ij[a † i A ija j + 1 ]2 (a† i B ija † j + h.c.) , (3.149)where a i are bosonic operators. It is found [206] that, for typical values of ω 0 ,S 1 (L) ∝ L (L >> 1) (3.150)