Topological Defect Formation Beyond the Kibble-Zurek Mechanism in Crossover Transitions with Approximate Symmetries

TL;DR

The work addresses topological defect formation during quenches across crossover transitions with an approximate $U(1)$ symmetry, where explicit breaking by a small source $h$ smooths the transition and challenges the applicability of the Kibble-Zurek mechanism (KZM). It demonstrates, through both a weakly coupled Ginzburg-Landau model and a strongly coupled holographic superfluid, that in the slow-quench regime the conventional KZ power-law defect scaling breaks down and is replaced by an exponential suppression, with $\hat{N} \propto t_Q^{-1/2} e^{-\beta_h t_Q}$ and $\beta_h \propto h^2$. The authors formulate a generalized KZ framework that uses the non-equilibrium freeze-out correlation length $\hat{\xi}$, yielding $\hat{N} = N_0 L^d/\hat{\xi}^d$ and $\hat{\xi} \sim t_Q^{1/4} e^{\beta_h t_Q/2}$, which captures defect formation across all quench rates in both weakly and strongly coupled regimes. This universal behavior suggests applicability to diverse systems, including QCD chiral transitions, charge-density waves with impurities, and magnetism under external fields, and provides a robust way to account for explicit symmetry-breaking effects in defect dynamics beyond the traditional KZM.

Abstract

The formation of topological defects during continuous second-order phase transitions is well described by the Kibble-Zurek mechanism (KZM). However, when the spontaneously broken symmetry is only approximate, such transitions become smooth crossovers, and the applicability of KZM in these scenarios remains an open question. In this work, we address this problem by analyzing both a weakly coupled Ginzburg-Landau model and a strongly coupled holographic setup, each featuring pseudo-spontaneous breaking of a global U(1) symmetry. In the slow quench regime, we observe a breakdown of the universal power-law scaling predicted by the Kibble-Zurek Mechanism. Specifically, the defect density acquires an exponential correction dependent on the quench rate, following a universal form dictated by the source of explicit symmetry breaking. Although these dynamics extend beyond the scope of the traditional KZM, we demonstrate that a generalized framework, that incorporates the effects of explicit symmetry breaking into the dynamical correlation length, remains valid and accurately captures the non-equilibrium defect formation across the entire range of quench rates.

Figures (11)

• Figure 1: Quenching across a continuous crossover. The spatially averaged expectation value of the order parameter $\langle \bar{\psi}\rangle$ as a function of time upon quenching the system from an initial state near the critical point $T=T_c$ to a final state at $T=T_f$. After a freeze-out time $\hat{t}$, the system starts to evolve following non-equilibrium dynamics in which the order parameter grows incoherently creating various spatial sub-regions with different phase $\theta=\arg (\psi)$. Topological defects form at the interfaces between these clusters whose size is determined by the correlation length $\xi$ (blue and purple symbols are respectively defects with positive and negative winding number). The top left inset shows the equilibrium value of $\langle \psi \rangle$ as a function of temperature across a continuous second order phase transition (blue line) as well as a continuous crossover (green line).
• Figure 2: Ginzburg-Landau model.(a) Normalized number of vortices $\hat{N}$ as a function of the quench rate $t_Q$. Background colors indicate respectively the fast quench region, the crossover regime and the KZ scaling region. Different colors correspond to different values of $h$. Symbols are the numerical data, solid lines the theoretical prediction introduced in the text, Eq. \ref{['scaling_law']}. (b) Zoom in the KZ scaling region and fit of the numerical data using $\hat{N}\sim t_Q^{-0.5}e^{-\beta_h t_Q}$. (c) The behavior of $\beta_h$ as a function of the symmetry-breaking source $h$.
• Figure 3: Holographic model.(a) Normalized number of topological defects $\hat{N}$ formed during nonequilibrium quenches with inverse quench rate $t_Q$ in the strongly coupled holographic superfluid model. Symbols are the numerical data while solid lines are the theoretical predictions discussed in the text, Eq. \ref{['scaling_law']}. The dashed line guides the eyes towards the KZ scaling $\hat{N} \propto t_Q^{-0.5}$. (b) Verification of the scaling law $\beta_h\propto h^2$ by fitting the large $t_Q$ data using $\hat{N}\sim t_Q^{-0.5}e^{-\beta t_Q}$.
• Figure 4: Ginzburg-Landau model.(a) Freeze-out time $\hat{t}$ as a function of the quench rate $t_Q$ for different values of $h$. The dashed line indicates the KZ scaling in the slow-quench regime. As explained in detail in the text, $\hat{t}$ is defined as the time at which the order parameter reaches $10\%$ of its final value. (b) The freeze-out correlation length $\hat{\xi}$ for the same data. (c) A zoom of $\hat{\xi}$ in the slow-quench regime displaying the deviations from the KZ predictions and fits to a generalized power-law scaling with exponential correction. $\beta_h=0$ corresponds to the KZ prediction. (d) Scaling of the parameter $\beta_h$ with $h$, confirming the origin of this trend.
• Figure 5: The structure of the GL functional in Eq. \ref{['pote']} as a function of the real and imaginary parts of the order parameter $(\psi_R,\psi_I)$. In this figure, the external symmetry breaking field is fixed to a constant value $h=10^{-3}$ and $\beta=2$ for simplicity. The stationary points of the potential are indicated as $\psi^{(i)}$ with $i=1,2,3$. In panels (a)-(d) the parameter $\alpha$ is respectively fixed to $\alpha \approx [0.1,0,-0.0238,-0.1]$. The solid blue and red curves show respectively the cuts of the potential along the $\psi_R,\psi_I$ axes. Panel (c) corresponds to the critical value $\alpha_c$ below which three distinct stationary points appear. $\psi^{(1)}$ is always a minimum, $\psi^{(2)}$ is a saddle point and $\psi^{(3)}$ a maximum.