# Spontaneous isotropy breaking for vortices in nonlinear left-handed metamaterials

## TL;DR

This paper investigates how vortex beams in nonlinear left-handed metamaterials spontaneously break isotropy, forming polygonal patterns with symmetries depending on the vortex charge, explained through an effective Landau-Ginzburg model.

## Contribution

It introduces a novel theoretical framework linking symmetry breaking in vortex beams to nonlinear metamaterials using an effective field theory approach.

## Key findings

- Vortex beams induce polygonal symmetry patterns in metamaterials.
- The symmetry depends on the vortex charge and material response.
- Loop corrections improve the accuracy of theoretical predictions.

## Abstract

We explore numerically and analytically the pattern formation and symmetry breaking of beams propagating through left-handed (negative) nonlinear metamaterials. When the input beam is a vortex with topological charge (winding number) $Q$, the initially circular (isotropic) beam acquires the symmetry of a polygon with $Q$, $2Q$ or $3Q$ sides, depending on the details of the response functions of the material. Within an effective field-theory model, this phenomenon turns out to be a case of spontaneous dynamical symmetry breaking described by a Landau-Ginzburg functional. Complex nonlinear dependence of the magnetic permittivity on the magnetic field of the beam plays a central role, as it introduces branch cuts in the mean-field solution, and permutations among different branches give rise to discrete symmetries of the patterns. By considering loop corrections in the effective Landau-Ginzburg field theory we obtain reasonably accurate predictions of the numerical results.