Fugacity-Resolved Stabilizer Entropy in Critical Quantum Chains: Discrete Selberg Sums and Exactly Solvable Rényi Indices

Abstract

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.