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arXiv 0711.2641cond-mat.stat-mechquant-ph
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量子自旋链中的广义非加性熵

Extensive nonadditive entropy in quantum spin chains

Filippo Caruso, Constantino Tsallis

备注：
9 pages, 4 figures, Invited Paper presented at the international conference CTNEXT07, satellite of STATPHYS23, 1-5 July 2007, Catania, Italy
发表信息：
AIP conference Proceedings 965 (2007) p. 51
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摘要：

我们提出了一个物理实现的细节，在多体哈密顿系统中，最近展示的抽象概率结构的盖尔曼，佐藤和我们之一（C.T.），非加性熵$S_q=k [1- Tr \hatρ^q]/[q-1]$（$\hatρ\equiv$密度矩阵; $S_1=-k Tr \hatρ \ln \hatρ$）可以符合，对于q的异常值（即，q不等于1），满足经典热力学对熵是广泛的要求。此外，我们发现熵指数q提供了一个工具来表征量子相变中的普适和非普适方面（例如，对于T=0临界横场的Ising铁磁链的L尺寸块，我们得到了$\lim_{L\to\infty}S_{\sqrt{37}-6}(L)/L=3.56 \pm 0.03$）。这些结果为量子多体系统纠缠度的测量提供了一种新的方法。根据这些结果，以及我们在别处讨论的d=2玻色子系统的类似结果，我们推测，对于一大类具有短程相互作用的费米和玻色子d维多体哈密顿量在T=0时的线性大小为L的块，我们得到：对于$d=1$为$ \ln L$，对于d>1为$ L^{d-1}$），因此它不是广泛的，而对于指数q的异常值，我们有非加性熵$S_q(L)\propto L^d$（$\forall d$），即，它是广泛的。本文的讨论清楚地表明，熵的可加性和熵的可拓性是完全不同的性质，即使它们在存在短程相关性的情况下基本上是一致的。

英文摘要：

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 $S_q=k [1- Tr \hatρ^q]/[q-1]$ ($\hatρ\equiv$ density matrix; $S_1=-k Tr \hatρ \ln \hatρ$) can conform, for an anomalous value of q (i.e., q not equal to 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 $\lim_{L\to\infty}S_{\sqrt{37}-6}(L)/L=3.56 \pm 0.03$). 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 $S_1(L) \propto [L^{d-1}-1]/(d-1)$ (i.e., $ \ln L$ for $d=1$, and $ L^{d-1}$ for d>1), hence it is not extensive, whereas, for anomalous values of the index q, we have that the nonadditive entropy $S_q(L)\propto L^d$ ($\forall d$), 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.

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