Extensive nonadditive entropy in quantum spin chains

We present details on a physical realization, in a many-body Hamiltonian system, of the abstract probabilistic structure recently exhibited by Gell-Mann, Sato and one of us (C.T.), that the nonadditive entropy S-q - k[1 - Tr (p) over cap (q)[q - 1] ((p) over cap density matrix; S-1 = kTr (p) over cap 1n (p) over cap) can conform, for an anomalous value of q (i.e., q not equal 1), to the classical thermodynamical requirement for the entropy to be extensive. Moreover, we find that the entropic index q provides a tool to characterize both universal and nonuniversal aspects in quantum phase transitions (e.g., for a L-sized block of the Ising ferromagnetic chain at its T = 0 critical transverse field, we obtain lim(L-->infinity)S(root 37-6)(L)/L = 3.56 +/- 0.03). The present results suggest a new and powerful approach to measure entanglement in quantum many-body systems. At the light of these results, and similar ones for a d = 2 Bosonic system discussed by us elsewhere, we conjecture that, for blocks of linear size L of a large class of Fermionic and Bosonic d-dimensional many-body Hamiltonians with short-range interaction at T = 0, we have that the additive entropy S-1 (L) proportional to [Ld-1 - 1]/(d- 1) (i.e., 1nL for d= 1, and Ld-1 for d > 1), hence it is not extensive, whereas, for anomalous values of the index q, we have that the nonadditive entropy S-q(L) proportional to L-d (for all d), i.e., it is extensive. The present discussion neatly illustrates that entropic additivity, and entropic extensivity, are quite different properties, even if they essentially coincide in the presence of short-range correlations.