  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.
