# Scaling behaviour of non-equilibrium planar $N$-atic spin systems under weak fluctuations

## TL;DR

This paper derives hydrodynamic equations for non-equilibrium $XY$ spin systems with $N$-atic symmetry, revealing two distinct scaling behaviors in two dimensions: long-range order for $N=1$ and quasi long-range order for others.

## Contribution

It provides a systematic derivation of hydrodynamic equations for non-equilibrium $XY$ systems with $N$-atic symmetry and characterizes their scaling behaviors in two dimensions.

## Key findings

- $N=1$ systems exhibit long-range order.
- $N
eq 1$ systems exhibit quasi long-range order.
- Two types of scaling behaviors are identified in 2D.

## Abstract

Starting from symmetry considerations, we derive the generic hydrodynamic equation of non-equilibrium $XY$ spin systems with $N$-atic symmetry under weak fluctuations. Through a systematic treatment we demonstrate that, in two dimensions, these systems exhibit two types of scaling behaviours. For $N=1$, they have long-range order and are described by the flocking phase of dry polar active fluids. For all other values of $N$, the systems exhibit quasi long-range order, as in the equilibrium $XY$ model at low temperature.