We study the finite-temperature stabilizer Rényi entropy of the open critical quantum spin chains. At Rényi index one half, this observable probes the distribution of thermal Pauli-string expectation values and can be written as a sum over absolute values of all square minors of a finite-temperature correlation matrix for the transverse-field Ising chain. We show that this exponentially large sum is exactly reducible to a single Pfaffian. The Pfaffian representation reveals a block Toeplitz--Hankel structure and allows us to extract the large-size scaling in several thermal regimes. In the crossover window where the inverse temperature is proportional to the system size, the stabilizer entropy factorizes into a saturated extensive contribution and a universal finite-size scaling function. We find that this scaling function is a level-eight eta quotient, rather than the ordinary free-boundary Majorana thermal factor. The deviation is exponentially hidden at low temperature but controls the high-temperature crossover, where it gives a Cardy-like asymptotic for the Pauli-string expectation-weight spectrum. These results show that finite-temperature stabilizer entropy reveals hidden defect-like conformal data invisible to ordinary thermodynamic probes.
We study the finite-temperature stabilizer Rényi entropy of the open critical quantum spin chains. At Rényi index one half, this observable probes the distribution of thermal Pauli-string expectation values and can be written as a sum over absolute values of all square minors of a finite-temperature correlation matrix for the transverse-field Ising chain. We show that this exponentially large sum is exactly reducible to a single Pfaffian. The Pfaffian representation reveals a block Toeplitz--Hankel structure and allows us to extract the large-size scaling in several thermal regimes. In the crossover window where the inverse temperature is proportional to the system size, the stabilizer entropy factorizes into a saturated extensive contribution and a universal finite-size scaling function. We find that this scaling function is a level-eight eta quotient, rather than the ordinary free-boundary Majorana thermal factor. The deviation is exponentially hidden at low temperature but controls the high-temperature crossover, where it gives a Cardy-like asymptotic for the Pauli-string expectation-weight spectrum. These results show that finite-temperature stabilizer entropy reveals hidden defect-like conformal data invisible to ordinary thermodynamic probes.
Stabilizer Rényi entropy has emerged as a probe of nonstabilizerness in quantum many-body systems, but its scaling structure beyond critical points remains poorly understood compared with entanglement entropy. Recent field-theory approaches indicate that stabilizer entropy contains universal critical data and boundary-sensitive terms, raising the question of how these structures extend into massive and crossover regimes. We address this problem for a broad class of finite-range spin chains at Rényi index one-half. We derive exact finite-size formulas for both full periodic chains and finite intervals of the infinite chain, making the universal crossover from critical to noncritical behavior analytically accessible. In periodic geometry, the entropy obeys a volume law away from criticality and exhibits a universal finite-size crossover controlled by the competition between system size and correlation length. We also show that the large-scale SRE density develops a cusp across the field-tuned critical line, while the XX endpoint is governed by a distinct scaling regime associated with the saturation point. In the subsystem geometry, the interval entropy separates bulk critical behavior from boundary contributions generated by the way the finite region cuts the infinite chain. The crossover from critical to massive behavior is then encoded in boundary constants and universal functions controlled by the correlation length. Through exact stabilizer-entropy correspondences, the scaling theory extends to internal XY reductions, Finite-range spin chains, and Cluster--Ising representatives. Our results provide an exact lattice benchmark for the emerging QFT description of stabilizer entropy beyond isolated conformal points.
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=21,1,2, 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=21, and develops an LlogL 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.
Quantum magic---the resource that separates universal quantum computation from efficiently simulable Clifford circuits---has emerged as a diagnostic of many-body quantum states, yet the stabilizer Rényi entropy (SRE) that quantifies it remains largely inaccessible in interacting fermion systems. Estimating the global SRE requires a sum over exponentially many Majorana strings, which in determinant quantum Monte Carlo can be sampled without a sign problem only for the interacting density matrix obtained after averaging the auxiliary fields: sampling strings and fields simultaneously incurs a sign problem even for sign-problem-free models, so that the known sign-free alternative is a nested Monte Carlo that does not scale. We propose instead the two-point SRE---a practical stabilizer-Rényi correlator built from rank-2 SREs of one- and two-site reduced density matrices, which are fixed exactly by single-particle Green functions and density correlations already measured in standard simulations---and show that it obeys universal finite-size scaling at fermionic quantum critical points. In the one-dimensional half-filled spinless t-V chain, the correlator distinguishes algebraic and exponential decay regimes and tracks the inverse-logarithmic finite-size drift characteristic of the Berezinskii--Kosterlitz--Thouless transition. On the honeycomb lattice, sign-problem-free quantum Monte Carlo yields Gross--Neveu--Ising scaling with finite anomalous dimension at zero temperature and two-dimensional Ising collapse at the thermal transition. Our results suggest stabilizer-Rényi microscopy as a spatially resolved, quantitatively universal probe of critical correlations in interacting fermionic matter, on par with conventional order-parameter correlators and accessible to quantum-simulator measurements.
Shannon-Rényi and stabilizer entropies are key diagnostics of structure, "nonstabilizerness," phase transitions, and universality in quantum many-body states. We establish an exact correspondence for quadratic fermions: for any Gaussian eigenstate, the stabilizer Rényi entropy equals the Shannon-Rényi entropy of a number-conserving free-fermion eigenstate on a doubled system, evaluated in the computational basis. Specializing to the transverse-field Ising (TFI) chain, the TFI ground-state stabilizer entropies map to the Shannon-Rényi entropies of the XX-chain ground state of length 2L. Building on this correspondence, together with other exact identities we prove, we derive closed expressions for the stabilizer entropy at indices α=1/2,2, 4 for a broad class of critical closed free-fermion systems. Each of these can be written with respect to the universal functions of the TFI chain. We further derive conformal-field-theory scaling laws for the stabilizer entropy at arbitrary Rényi index under both periodic and open boundary conditions. At α=4, these scaling forms display a discontinuity for both open and periodic boundary conditions.
Universal information extracted from a critical quantum state depends on how the state is probed. Spatial entanglement, participation statistics, and stabilizer Rényi entropy correspond to different replica geometries and need not isolate the same boundary data. We show that the complete distribution of Pauli magnitudes in critical Ising chains carries an exact geometric fingerprint of boundary termination, distinguishing lattice realizations that flow to the same infrared boundary fixed point. This structure is organized by discrete Selberg ensembles associated with the classical root systems A, B, C, and D: three open chains sharing the free--free Ising boundary condition retain distinct BL, CL, and DL trigonometric wall geometries. The standard symmetry-breaking representatives instead produce rectangular, pinned, and character-inserted C ensembles. Their four boundary moments contain exactly two independent multiplicative combinations invariant under local endpoint normalizations. One of these reduces exactly at finite rank to a moment of the fundamental symplectic character and provides a lattice target for a defect--measurement-boundary junction amplitude. A conditional Gaussian-character law predicts its values throughout the unlocked phase, while exact-arithmetic finite-rank results at the α=4 transition depart sharply from the Gaussian continuation. These results show that the full Pauli spectrum resolves boundary information invisible to the infrared boundary fixed point alone and provide concrete targets for replicated boundary conformal field theory.
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∣ln2, 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.