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arXiv:2608.09749

Quantized Stabilizer-Rényi Boundary Response across Fermionic SPT Transitions

M. A. Rajabpour

quant-phcond-mat.str-elhep-thSubmitted Aug 10, 2026• Announced Aug 11, 2026• 0 citations• Revised Aug 21, 2026 (v2)
v1 · Aug 10, 2026v2 · Aug 21, 2026
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Abstract

Open boundaries host symmetry-protected Majorana modes, yet their imprint on the stabilizer Rényi entropy is obscured by a nonuniversal volume law. At Rényi index $α=1/2$, we isolate a bulk-subtracted boundary response in free-fermion BDI chains using exact finite-open-chain relations, large-$L$ Pfaffian evaluations, and direct positive-weight checks. Across a mass inversion, this response approaches $|Δω|\ln 2$, where $Δω$ is the change in winding number. The response survives primitive bulk deformations, the tested local boundary perturbations, and broken bulk duality. Thus, for the families studied here, the boundary response counts the Majorana channels that change the fermionic SPT index.

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6+29 pages and 7 figures

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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 XXXXXX 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=12,1,2α=\tfrac12,1,2α=21​,1,2, determinant and Pfaffian compressions yield product formulas. At α=4α=4α=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=2α=2α=2 result. These generating functions determine the balanced-degree statistics. Complement symmetry makes the distribution symmetric about k=L/2k=L/2k=L/2, with an index-dependent width. It is exactly binomial at α=1α=1α=1, has variance proportional to LLL at α=12α=\tfrac12α=21​, and develops an Llog⁡LL\log LLlogL enhancement at α=2α=2α=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.
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