# Condensation of Fluctuations in the Ising Model: a Transition without Spontaneous Symmetry Breaking

## TL;DR

This paper reveals a new type of transition in the Ising model under exotic boundary conditions, where fluctuations condense without symmetry breaking, challenging traditional understanding of ferromagnetic phase transitions.

## Contribution

It demonstrates that under cylindrical antiperiodic boundary conditions, the Ising model exhibits a fluctuation condensation transition below Tc without symmetry breaking.

## Key findings

- Transition occurs without ergodicity or symmetry breaking.
- Fluctuations of magnetization condense below Tc.
- Critical correlations persist for all T below Tc.

## Abstract

The ferromagnetic transition in the Ising model is the paradigmatic example of ergodicity breaking accompanied by symmetry breaking. It is routinely assumed that the thermodynamic limit is taken with free or periodic boundary conditions. More exotic symmetry-preserving boundary conditions, like cylindrical antiperiodic, are less frequently used for special tasks, such as the study of phase coexistence or the roughening of an interface. Here we show, instead, that when the thermodynamic limit is taken with these boundary conditions, a novel type of transition takes place below $T_c$ (the usual Ising transition temperature) without breaking neither ergodicity nor symmetry. Then, the low temperature phase is characterized by a regime (condensation) of strong magnetization's fluctuations which replaces the usual ferromagnetic ordering. This is due to critical correlations perduring for all T below Tc. The argument is developed exactly in the $d=1$ case and numerically in the d=2 case.