Topological defects as relics of spontaneous symmetry breaking from black hole physics

Abstract

Formation and evolution of topological defects in course of non-equilibrium symmetry breaking phase transitions is of wide interest in many areas of physics, from cosmology through condensed matter to low temperature physics. Its study in strongly coupled systems, in absence of quasiparticles, is especially challenging. We investigate breaking of U(1) symmetry and the resulting spontaneous formation of vortices in a (2 + 1)-dimensional holographic superconductor employing gauge/gravity duality, a ‘first-principles’ approach to study strongly coupled systems. Magnetic fluxons with quantized fluxes are seen emerging in the post-transition superconducting phase. As expected in type II superconductors, they are trapped in the cores of the order parameter vortices. The dependence of the density of these topological defects on the quench time, the dispersion of the typical winding numbers, and the vortex-vortex correlations are consistent with predictions of the Kibble-Zurek mechanism.

Topological defects formation with momentum dissipation

“…The shaded regions correspond to the standard deviations from the mean values. In the fast quench regime (small τ Q ) the vortex number approximately saturates due to the finite size effect, which is consistent with previous results in condensed matter or holography [44][45][46][47]. However, in the slow quench regime (large τ Q ), the vortex number will decrease with respect to quench time satisfying the scaling relations as…”

“…The "freeze-out" timet can be reflected by studying the lag time t L that defined as order parameter begins to grow rapidly [15,44,46]. In numerics we operationally set t L as Ô ∼ 0.1 following [15,44,46]. On the right panel of figure 2 and (ν ≈ 0.510949, z ≈ 1.90386) for m = (0, 1.2, 2) respectively, in which the exponent matches the mean-field theory values with z = 2 and ν = 1/2.…”

“…It is convenient to work out the QNMs in the Schwarzschild-AdS metric of the spacetime [44,46]. Substituting the scalar field Ψ(t, z, x) into the EoMs (3.5), (3.6), and extracting the first order of the perturbations ψ(z), we arrive at the decoupled linear equation as…”

“…Vanishing boundary conditions are imposed at the boundary z = 0. We adopt the determinant method [46] to calculate the QNMs and fix the quench rate as τ Q = 20 (other quench rates will have similar results). More specifically, we employ the Chebyshev differential matrix to convert the differentiation with respect to spacetime coordinate into a matrix.…”

“…Applications of gauge/gravity duality on non-equilibrium dynamics can be found in [36][37][38][39][40][41][42][43]. Previous holographic work on KZM were carried out in [44][45][46][47][48][49][50], without any momentum dissipation in the background.…”