AI 中文总结
本文针对临界横场伊辛链引入逸度分辨配分函数，将其逸度分辨全子式求和映射为离散塞尔伯格系综，得到多类雷尼指数的精确结果，揭示了稳定子熵的统计演化规律。
AI 中文摘要
稳定子雷尼熵通过量子态的泡利期望值分布的雷尼矩来量化其非稳定子性，其标准形式对泡利串自由度求和，仅保留总雷尼权重。本文针对临界横场伊辛链引入逸度分辨的配分函数，该配分函数在平衡马约拉纳表象中分辨上述自由度并生成其全计数统计。此伊辛问题的适用范围超出单一模型：精确约化恒等式及与半填充XX链的对应关系，使其成为稳定子与计算基香农-雷尼熵的通用有限尺寸构建块。我们将所有正雷尼指数对应的逸度分辨全子式求和，映射为半填充双根格上棋盘加权的离散塞尔伯格系综；对于正整数指数，其简化为有限混叠狄森常数项。在α=1/2、1、2处，行列式与 pfaffian（普菲夫）压缩给出乘积公式；在α=4处，一般逸度问题存在精确的逆杰克-科斯特卡表示，而在单位逸度下，互补的中间子式恒等式将其与α=2结果的平方关联。这些生成函数决定了平衡自由度统计：互补对称性使分布关于k=L/2对称，且具有指数依赖的宽度；在α=1时其为精确二项分布，在α=1/2时方差与L成正比，在α=2时出现LlogL增强；经方差重标度后，三个指数对应的中心化分布均收敛于高斯极限。超出这些可解情形，有限尺寸数值计算显示，随着雷尼指数增大，分布从中心峰演变为对称双峰，最终变为端点主导。
英文摘要
Stabilizer Rényi entropy quantifies the nonstabilizerness of a quantum state through Rényi moments of its Pauli expectation-value distribution. Its standard form sums over Pauli-string degrees and retains only the total Rényi weight. We introduce a fugacity-resolved partition function for the critical transverse-field Ising chain that resolves this degree in the balanced Majorana representation and generates its full counting statistics. This Ising problem extends beyond a single model: exact decimation identities and the correspondence with the half-filled \(XX\) chain establish it as a common finite-size building block for stabilizer and computational-basis Shannon--Rényi entropies. We map the fugacity-resolved all-minors sum, for every positive Rényi index, to a checkerboard-weighted discrete Selberg ensemble on a half-filled doubled root lattice. For positive integer indices, it reduces to a finite aliased Dyson constant term. At \(α=\tfrac12,1,2\), determinant and Pfaffian compressions yield product formulas. At \(α=4\), the generic-fugacity problem admits an exact inverse Jack--Kostka representation, while at unit fugacity a complementary middle-minor identity relates it to the square of the \(α=2\) result. These generating functions determine the balanced-degree statistics. Complement symmetry makes the distribution symmetric about \(k=L/2\), with an index-dependent width. It is exactly binomial at \(α=1\), has variance proportional to \(L\) at \(α=\tfrac12\), and develops an \(L\log L\) enhancement at \(α=2\). After variance rescaling, the centered distributions converge to Gaussian limits at all three indices. Beyond these solvable cases, finite-size numerics reveal an evolution from a central peak to a symmetric bimodal profile and eventually to endpoint dominance as the Rényi index increases.
Comments46 pages + 6 Figures
