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Physical Review Letters

Defect Production across Higher-Order Phase Transitions beyond Kibble-Zurek Scaling

Menghua Deng, Zhoujian Sun, Fuxiang Li

Published Jan 7, 2025• 10 citations
Publisher

Abstract

The Kibble-Zurek mechanism predicts a universal scaling behavior of the defect production when a system is slowly quenched across a critical phase transition point. Here, we discover that the universal scaling behavior across a higher-order topological phase transition does not conform to the traditional Kibble-Zurek mechanism. To explain the anomalous scaling exponent, we develop a solvable Landau-Zener model that takes into account the role of topological edge band that dominates the phase transition. For a two-dimensional boundary-obstructed higher-order topological system, the Kibble-Zurek scaling must be modified to adopt the effective dimension of topological edge band instead of the real physical dimension. We also find that across the boundary-obstructed higher-order topological phase transition, boundary conditions can drastically modify the scaling behaviors. For comparison, we investigate the slow quench dynamics across the bulk-obstructed phase transitions and a single multicritical point, which obeys the Kibble-Zurek mechanism with physical dimension.

DOI
10.1103/physrevlett.134.010409

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The Kibble-Zurek mechanism predicts a universal scaling behavior of the defect production when a system is slowly quenched across a critical phase transition point. Here, we discover that the universal scaling behavior across a higher-order topological phase transition does not conform to the traditional Kibble-Zurek mechanism. To explain the anomalous scaling exponent, we develop a solvable Landau-Zener model that takes into account the role of topological edge band that dominates the phase transition. For a two-dimensional boundary-obstructed higher-order topological system, the Kibble-Zurek scaling must be modified to adopt the effective dimension of topological edge band instead of the real physical dimension. We also find that across the boundary-obstructed higher-order topological phase transition, boundary conditions can drastically modify the scaling behaviors. For comparison, we investigate the slow quench dynamics across the bulk-obstructed phase transitions and a single multicritical point, which obeys the Kibble-Zurek mechanism with physical dimension.
We study the nonadiabatic dynamics of a two-dimensional higher-order topological insulator when the system is slowly quenched across the boundary-obstructed phase transition, which is characterized by edge band gap closing. We find that the number of excitations produced after the quench exhibits power-law scaling behaviors with the quench rate. Boundary conditions can drastically modify the scaling behaviors: The scaling exponent is found to be α=1/2α=1/2α=1/2 for hybridized and fully open boundary conditions, and α=2α=2α=2 for periodic boundary condition. We argue that the exponent α=1/2α=1/2α=1/2 cannot be explained by the Kibble-Zurek mechanism unless we adopt an effective dimension deff=1d^{\rm eff}=1deff=1 instead of the real dimension d=2d=2d=2. For comparison, we also investigate the slow quench dynamics across the bulk-obstructed phase transitions and a single multicritical point, which obeys the Kibble-Zurek mechanism with dimension d=2d=2d=2.
When a system is swept through a quantum critical point, the Kibble-Zurek mechanism predicts that the average number of topological defects follows a universal power-law scaling with the ramp timescale. This scaling behavior is determined by the equilibrium critical exponents of the underlying phase transition. We show that the correspondence between Kibble-Zurek scaling and quantum criticality does not hold generally. In particular, the defect density can exhibit a suppression faster than the Kibble-Zurek prediction even when the quench crosses a critical point, while conventional Kibble-Zurek scaling may persist for quenches through a noncritical point. Our results, based on models representative of a broad class of quasi-one-dimensional Fermi systems, identify the dynamical conditions under which universal defect scaling emerges and clarify the relation between defect generation and equilibrium criticality. © 2026 authors. Published by the American Physical Society.
When a system is swept through a quantum critical point (QCP), the Kibble-Zurek mechanism predicts that the average number of topological defects follows a universal power-law scaling with the ramp time scale. This scaling behavior is determined by the equilibrium critical exponents of the underlying phase transition. We show that the correspondence between Kibble-Zurek scaling and quantum criticality does not hold generally. In particular, the defect density can exhibit a suppression faster than the Kibble-Zurek prediction even when the quench crosses a critical point, while conventional Kibble-Zurek scaling may persist for quenches through a non-critical point. Our results, based on models representative of a broad class of quasi-one-dimensional Fermi systems, identify the dynamical conditions under which universal defect scaling emerges and clarify the relation between defect generation and equilibrium criticality.
The Kibble-Zurek (KZ) mechanism describes the scaling behavior when driving a system across a continuous symmetry-breaking transition. Previous studies have shown that the KZ-like scaling behavior also lies in the topological transitions in the Qi-Wu-Zhang model (2D) and the Su-Schrieffer-Heeger model (1D), although symmetry breaking does not exist in these systems. Both models with linear band crossings yield $\ensuremath{\nu}\ensuremath{\simeq}1$ and $z\ensuremath{\simeq}1$. A natural question is whether different critical exponents can emerge in topological transitions beyond the linear band crossings. In this paper, we look into the KZ behavior in a topological 2D checkerboard lattice with a quadratic band crossing. We investigate from dual perspectives: momentum distribution of the Berry curvature in clean systems for simplicity, and real-space analysis of domain-like local Chern marker configurations in disordered systems, which is a more intuitive analog to conventional KZ description. In equilibrium, we find the correlation length diverges at the transition with a power $\ensuremath{\nu}\ensuremath{\simeq}1/2$. Then, by slowly quenching the system across the topological phase transition, we find that the freeze-out time tf{t}_{\mathrm{f}}tf​ and the unfrozen length scale $\ensuremath{\xi}({t}_{\mathrm{f}})$ both satisfy the KZ scaling, verifying $z\ensuremath{\simeq}2$. We subsequently explore KZ behavior in topological phase transitions with other higher-order band crossings and find the relationship between the critical exponents and the order. Our results extend the understanding of the KZ mechanism and nonequilibrium topological phase transitions.
We study the nonequilibrium driven dynamics at topologically nontrivial quantum critical points (QCPs), and find that topological edge modes at criticality give rise to anomalous dynamical scaling behavior. By analyzing the driven dynamics of bulk and boundary order parameters at topologically distinct QCPs in quantum spin chains, we demonstrate that, while the bulk dynamics remain indistinguishable and follow standard Kibble Zurek (KZ) scaling, the anomalous boundary dynamics are unique to topological criticality, obeying modified scaling relation beyond the traditional KZ framework. To elucidate the unified origin of this anomaly, we further study the dynamics of defect production at topologically distinct QCPs in free-fermion models and demonstrate similar anomalous scaling exclusive to topological criticality. These findings establish the existence of anomalous dynamical scaling arising from the interplay between topology and driven dynamics, challenging standard paradigms of quantum critical dynamics.
The crossing of a continuous phase transition gives rise to the formation of topological defects described by the Kibble-Zurek mechanism (KZM) in the limit of slow quenches. The KZM predicts a universal power-law scaling of the defect density as a function of the quench time. We focus on the deviations from KZM experimentally observed in rapid quenches and establish their universality. While KZM scaling holds below a critical quench rate, for faster quenches the defect density and the freeze-out time become independent of the quench rate and exhibit a universal power-law scaling with the final value of the control parameter. These predictions are verified in several paradigmatic scenarios in both the classical and quantum domains.