Thermodynamic limit of the first Lee‐Yang zero
Thermodynamic limit of the first Lee‐Yang zero
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DOI：
10.1002/cpa.22159
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发表时间：
2022-10
影响因子：
3
通讯作者：
Jianping Jiang;Charles M. Newman
中科院分区：
文献类型：
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作者：
Jianping Jiang;Charles M. Newman
我们完成了1952年Yang和Lee提出的关于热力学奇点正是C ${\mathbb {C}}$中有限体积奇点在R ${\mathbb {R}}$中的极限的验证。对于在逆温度β≥0 $\beta \ge 0$和外场h下的有限的Λ∧Zd $\Lambda \subset \mathbb {Z}^d$上定义的Ising模型，设α1（Λ，β） $\alpha _1(\Lambda ,\beta )$是其配分函数（在变量h中）的第一个零（最接近原点的零）的模。我们证明了α1（Λ，β） $\alpha _1(\Lambda ,\beta )$随着Λ增大到Zd $\mathbb {Z}^d$而减小到α1(Zd,β) $\alpha _1(\mathbb {Z}^d,\beta )$，其中α1(Zd,β)∈[0,∞)$\alpha _1(\mathbb {Z}^d,\beta )\in [0,\infty )$是以原点为中心的最大圆盘的半径，该圆盘的热力学极限自由能是解析的。我们还注意到α1(Zd,β) $\alpha _1(\mathbb {Z}^d,\beta )$是严格正的当且仅当β严格小于临界逆温度。
We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in R${\mathbb {R}}$ of finite‐volume singularities in C${\mathbb {C}}$ . For the Ising model defined on a finite Λ⊂Zd$\Lambda \subset \mathbb {Z}^d$ at inverse temperature β≥0$\beta \ge 0$ and external field h, let α1(Λ,β)$\alpha _1(\Lambda ,\beta )$ be the modulus of the first zero (that closest to the origin) of its partition function (in the variable h). We prove that α1(Λ,β)$\alpha _1(\Lambda ,\beta )$ decreases to α1(Zd,β)$\alpha _1(\mathbb {Z}^d,\beta )$ as Λ increases to Zd$\mathbb {Z}^d$ where α1(Zd,β)∈[0,∞)$\alpha _1(\mathbb {Z}^d,\beta )\in [0,\infty )$ is the radius of the largest disk centered at the origin in which the free energy in the thermodynamic limit is analytic. We also note that α1(Zd,β)$\alpha _1(\mathbb {Z}^d,\beta )$ is strictly positive if and only if β is strictly less than the critical inverse temperature.