Fugacity-Resolved Stabilizer Entropy in Critical Quantum Chains: Discrete Selberg Sums and Exactly Solvable R\'enyi Indices

Reyhaneh Khasseh; M. A. Rajabpour

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arXiv · 2608.06995

Fugacity-Resolved Stabilizer Entropy in Critical Quantum Chains: Discrete Selberg Sums and Exactly Solvable R\'enyi Indices

Reyhaneh Khasseh·M. A. Rajabpour

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Abstract

Stabilizer R\'enyi entropy quantifies the nonstabilizerness of a quantum state through R\'enyi moments of its Pauli expectation-value distribution. Its standard form sums over Pauli-string degrees and retains only the total R\'enyi 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--R\'enyi entropies. We map the fugacity-resolved all-minors sum, for every positive R\'enyi 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 $\alpha=\tfrac12,1,2$, determinant and Pfaffian compressions yield product formulas. At $\alpha=4$, 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 $\alpha=2$ 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 $\alpha=1$, has variance proportional to $L$ at $\alpha=\tfrac12$, and develops an $L\log L$ enhancement at $\alpha=2$. 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 R\'enyi index increases.

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Reyhaneh Khasseh, M. A. Rajabpour. 2026-08-07. Fugacity-Resolved Stabilizer Entropy in Critical Quantum Chains: Discrete Selberg Sums and Exactly Solvable R\'enyi Indices. https://arxiv.org/abs/2608.06995

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