Classical Root Systems Reveal Defect-Junction Data in Stabilizer Renyi Entropy

Abstract

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 $B_L$, $C_L$, and $D_L$ 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.

The paper asks what boundary information the complete Pauli-magnitude distribution of a critical Ising ground state retains beyond its infrared boundary condition, and whether stabilizer Rényi moments yield normalization-independent boundary amplitudes. It maps Pauli expectations to balanced or augmented minors of free-fermion correlation matrices, then uses weighted Cauchy determinants to identify each magnitude with a classical Weyl alternant. Periodic chains give $A$-type discrete Selberg ensembles; three parity-preserving open terminations give $B_L,C_L,D_L$ ensembles despite sharing the free–free Ising boundary fixed point. Longitudinal boundary fields instead produce rectangular, pinned, and character-inserted $C$ ensembles. An exact rank-shift identity relates equal-fixed and free–free moments: $Z_{\alpha,L}^{++}=Z_{\alpha,L+1}^{ff}/2$. Two independent multiplicative ratios cancel local free/fixed endpoint normalizations. The opposite-fixed/equal-fixed ratio is exactly a $2\alpha$th absolute moment of the fundamental symplectic character. Assuming Gaussian character fluctuations with variance $1/\alpha$ and uniform moment control predicts $R_\alpha^{(\eta)}=2^\alpha\Gamma(\alpha+1/2)/(\sqrt\pi\alpha^\alpha)$ for $0<\alpha<4$. A separate conjecture gives $R_\alpha^{(\times)}=2^{\alpha-1}$; its value $R_2^{(\times)}=2$ is proved. At the previously identified transition $\alpha=4$, exact-arithmetic finite-rank data support $R_4^{(\eta)}\to143/240$, rather than the Gaussian continuation $105/256$. The lattice correspondences are exact, but Gaussian convergence, required tail control, several all-rank reconstructions, and generic infrared universality remain unproved. Boundary-field and termination comparisons support common ratio limits over accessible sizes. Their identification as replicated-BCFT junction amplitudes remains a continuum target; fixed-$\alpha$ ratios above four are undetermined.