Bayesian phase transition for the critical Ising model: Enlarged replica symmetry in the epsilon expansion and in 2D

TL;DR

This paper investigates measurement-induced phase transitions in the critical Ising model using replica field theory and simulations, revealing multiscaling, enlarged symmetries, and exact critical exponents.

Contribution

It introduces a novel analysis of measurement phases in the Ising model, highlighting enlarged replica symmetries and their implications for critical behavior.

Findings

Multiscaling of correlation functions at the critical point

Enlarged symmetry of the replica description analogous to Nishimori phenomenon

Exact value of the Edwards-Anderson correlator exponent in 2D and near critical dimension

Abstract

A process that images or measures bond energies in the critical Ising model can be in distinct measurement ``phases'', depending on the precision of measurement. We study the transition into the strong-measurement phase using replica field theory (an epsilon expansion around six dimensions) and numerical simulations in two dimensions. The results reveal multiscaling of correlation functions at the critical point, and a striking enlarged symmetry of the replica description. This is an analog of the Nishimori phenomenon in the Ising spin glass, in a distinct replica limit. The enlarged symmetry is present microscopically for certain measurement protocols, but more generally can emerge in the infrared, and it fixes the exact value of the exponent for the Edwards-Anderson correlator both in 2D and near the upper critical dimension. We also examine the epsilon expansion for models with power-law interactions and/or long-range measurement.