正温度下的量子自旋概率是霍德尔吉布斯概率
Quantum Spin probabilities at positive temperature are Hölder Gibbs probabilities
摘要：
我们考虑量子自旋晶格$\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes ...$上与哈密顿量$H= σ^x \otimes σ^x$相关的KMS态。对于形式为$L \otimes L \otimes L \otimes ...$的固定可观测量，其中$L:\mathbb{C}^2 \to \mathbb{C}^2 $是自伴的，并且对于正温度$T$，可以得到伯努利空间$\{1,2\}^\mathbb{N}$上自然定义的平稳概率$μ_T$。$μ_T$的雅可比矩阵可以通过某种连分数展开式来表示。我们将证明这个概率是霍德尔势的吉布斯概率。因此，该概率对于移位映射是混合的。对于这样的概率$μ_T$，我们将显示某类函数的显式偏差函数。当降低温度时，我们将能够展示显式转变值$T_c$，其中吉布斯概率的雅可比矩阵的值的集合$μ_T$从康托尔集变为区间。我们也给出了零温度下量子自旋几率的一些性质（例如，熵的显式值）。
英文摘要：
We consider the KMS state associated to the Hamiltonian $H= σ^x \otimes σ^x$ over the quantum spin lattice $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes ...$. For a fixed observable of the form $L \otimes L \otimes L \otimes ...$, where $L:\mathbb{C}^2 \to \mathbb{C}^2 $ is self adjoint, and for positive temperature $T$ one can get a naturally defined stationary probability $μ_T$ on the Bernoulli space $\{1,2\}^\mathbb{N}$. The Jacobian of $μ_T$ can be expressed via a certain continued fraction expansion. We will show that this probability is a Gibbs probability for a Hölder potential. Therefore, this probability is mixing for the shift map. For such probability $μ_T$ we will show the explicit deviation function for a certain class of functions. When decreasing temperature we will be able to exhibit the explicit transition value $T_c$ where the set of values of the Jacobian of the Gibbs probability $μ_T$ changes from being a Cantor set to being an interval. We also present some properties for quantum spin probabilities at zero temperature (for instance, the explicit value of the entropy).
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