# Polarization Plateaus in Hexagonal Water Ice $I_h$

## TL;DR

This paper investigates how external electric fields induce polarization plateaus and phase transitions in hexagonal water ice I$_h$, revealing complex degeneracy reduction and algebraic correlations through analytical and numerical methods.

## Contribution

It introduces a minimal model to study polarization plateaus and order-disorder transitions in water ice I$_h$ under external fields, connecting ice rules to dimer models and Kasteleyn theory.

## Key findings

- Identification of polarization plateaus at specific field directions.
- Observation of an order-disorder transition of Kasteleyn type.
- Protons exhibit algebraically decaying correlations and characteristic structure factors.

## Abstract

The protons in water ice are subject to so called \emph{ice rules} resulting in an extensive ground state degeneracy. We study how an external electric field reduces this ground state degeneracy in hexagonal water ice I$_h$ within a minimal model. We observe polarization plateaus when the field is aligned along the $[001]$ and $[010]$ directions. In each case, one plateau occurs at intermediate polarization with reduced but still extensive degeneracy. The remaining ground states can be mapped to dimer models on the honeycomb and the square lattice, respectively. Upon tilting the external field, we observe an order-disorder transition of Kasteleyn type into a plateau at saturated polarization and vanishing entropy. This transition is investigated analytically using the Kasteleyn matrix and numerically using a modified directed-loop Monte Carlo simulation. The protons in both cases exhibit algebraically decaying correlations. Moreover, the features of the static structure factor are discussed.