Microscopic Origin of the Shannon-Rényi Boundary Phase in the Critical Ising Chain
Abstract
Shannon-R\'enyi entropies extract universal information from the full measurement distribution of a quantum many-body state and have recently become experimentally accessible probes of conformal data. Yet their microscopic evaluation remains difficult even in exactly solvable free-fermion systems. We show that the Born distribution of the critical transverse-field Ising chain is exactly the parity boundary of a weak inverse-square classical Ising model, a structure revealed by the Cauchy form of its probability kernel. The construction is exact at finite size for every real n>0 and isolates the long-distance sectors controlling the n>1 thermodynamics. Cutting a finite interval generates the universal c n/[8(n−1)] logarithm directly from the missing 1/r2 interactions, with c=1/2 the Ising central charge. On the ring, the explicit finite-size factor has no constant term, while any additional noninteger singularity is confined to a residual cycle-only free energy. All explicit long-distance contributions become marginal only at n=1.
Related
- Analytical results for the Shannon entropy of the critical transverse-field Ising chain
- Microscopic Origin of the Equation of Motion for Open Thermodynamic Systems
- The Critical Cauchy Boundary for Gibbs Partitions
- Boundary phenomena for slow recurrences:finite and infinite critical sets
- The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II
BibTeX
@misc{rajabpour2026microscopic,
title = {{Microscopic Origin of the Shannon-Rényi Boundary Phase in the Critical Ising Chain}},
author = {Mohammad Ali Rajabpour},
howpublished = {HAL},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2609.27909},
eprint = {2609.27909},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05733389v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Microscopic Origin of the Shannon-Rényi Boundary Phase in the Critical Ising Chain
AU - Mohammad Ali Rajabpour
PY - 2026
DA - 2026/08/31
PB - HAL
DO - 10.48550/arxiv.2609.27909
UR - https://hal.science/hal-05733389v1
AB - Shannon-R\'enyi entropies extract universal information from the full measurement distribution of a quantum many-body state and have recently become experimentally accessible probes of conformal data. Yet their microscopic evaluation remains difficult even in exactly solvable free-fermion systems. We show that the Born distribution of the critical transverse-field Ising chain is exactly the parity boundary of a weak inverse-square classical Ising model, a structure revealed by the Cauchy form of its probability kernel. The construction is exact at finite size for every real n>0 and isolates the long-distance sectors controlling the n>1 thermodynamics. Cutting a finite interval generates the universal c n/[8(n−1)] logarithm directly from the missing 1/r^2 interactions, with c=1/2 the Ising central charge. On the ring, the explicit finite-size factor has no constant term, while any additional noninteger singularity is confined to a residual cycle-only free energy. All explicit long-distance contributions become marginal only at n=1.
ER -