Nonextensive Statistical Mechanics

78 3 Generalizing What We Learnt1000010000(a)(b)8000q = – 0.1q = 0.08000q = 1/2–0.1S q6000S q6000q = 1/24000q = + 0.1400020002000q = 1/2+0.10 2000 4000 6000 8000 10000N0 2000 4000 6000 8000 10000N10000(c)8000q = 2/3 – 0.16000S q4000q = 2/320000 2000 4000 6000 8000 10000Nq = 2/3+ 0.1Fig. 3.15 S q (N) for anomalous systems: (a) d = 1, (b) d = 2, and (c) d = 3. Only for q =1 − (1/d) wehaveafinite value for lim N→∞ S q (N)/N; itvanishes (diverges) forq > 1 + (1/d)(q < 1 + (1/d). From [199].tracing over all but L of the N elements), we will have (in the case of our quantumsystems) TrρL 2 < 1, i.e., a mixed state. Therefore, the block entropy S q(L, N) > 0.This fact is due to the nontrivial entanglement associated with quantum nonlocality.Our goal is to calculate for what value of the index q (noted q ent if such value exists,where ent stands for entropy) the block entropy S qent (L) ≡ lim N→∞ S qent (L, N) isextensive. In other words, S qent (L) ∼ s qent L (L →∞, after we have taken N →∞),with the slope s qent ∈ (0, ∞).Our first system [201] consists in the well-known linear chain of spin 1/2 XYferromagnet with transverse magnetic field λ. The Hamiltonian is given by∑N−1NH =− [(1 + γ )ˆσ j x ˆσ j+1 x + (1 − γ )ˆσ y jˆσ y j+1 ] − 2λ ∑ˆσ z j , (3.143)j=1j=1