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arXiv:2407.19780 · Physical review. B./Physical review. B

Kibble-Zurek behavior in a topological phase transition with a quadratic band crossing

Huan Yuan, Jinyi Zhang, Shuai Chen, Xiaotian Nie

Published Oct 15, 2024• 5 citations
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Abstract

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 ${t}_{\mathrm{f}}$ 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.

DOI
10.1103/physrevb.110.165130

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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 \ensuremathν\ensuremath≃1\ensuremath{\nu}\ensuremath{\simeq}1\ensuremathν\ensuremath≃1 and z\ensuremath≃1z\ensuremath{\simeq}1z\ensuremath≃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 \ensuremathν\ensuremath≃1/2\ensuremath{\nu}\ensuremath{\simeq}1/2\ensuremathν\ensuremath≃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 \ensuremathξ(tf)\ensuremath{\xi}({t}_{\mathrm{f}})\ensuremathξ(tf​) both satisfy the KZ scaling, verifying z\ensuremath≃2z\ensuremath{\simeq}2z\ensuremath≃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.
The discovery of nonlocal order parameters in real space provides a feasible scheme for studying dynamical critical behavior in topological systems. We study the critical phenomena in the one-dimensional Su-Schrieffer-Heeger (SSH) model by investigating the inhomogeneities in the local winding number in real space. By slowly quenching the system across the topological phase transition during a finite time interval, we find that the length scale defined through the local winding number satisfies the Kibble-Zurek mechanism. In contrast to the density of excitation, the scaling of this length scale is in full analog to the behavior of traditional continuous phase transitions with local order parameter and spontaneous symmetry breaking. In addition, the critical behavior and Kibble-Zurek mechanism in the generalized SSH with next-nearest-neighbor hopping are also studied. These results extend our understanding to the Kibble-Zurek mechanism and topological phase transition in nonequilibrium.
The Kibble-Zurek (KZ) mechanism renders a theoretical framework for elucidating the formation of topological defects across continuous phase transitions. Nevertheless, it is not immediately clear whether the KZ mechanism applies to topological phase transitions. The direct experimental study for such a topic is hindered by quenching a certain parameter over orders of magnitude in topological materials. Instead, we investigate the KZ behavior across topological transitions of a Chern band in two-dimensional (2D) optical Raman lattices with quantum gases. Defined as the defects, excitation density is reconstructed via measuring the spin wave functions, with which the power-law scaling of total excitation density is extracted and such scaling could be interpreted within the KZ framework. Our work has heralded the commencement of experimentally exploring the KZ mechanism of the topological phase transitions.
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.
Even though no local order parameter in the sense of the Landau theory exists for topological quantum phase transitions in Chern insulators, the highly nonlocal Berry curvature exhibits critical behavior near a quantum critical point. We investigate the critical properties of its real space analog, the local Chern marker, in weakly disordered Chern insulators. Because of disorder, inhomogeneities appear in the spatial distribution of the local Chern marker. Their size exhibits power-law scaling with the critical exponent matching the one extracted from the Berry curvature of a clean system. We drive the system slowly through such a quantum phase transition. The characteristic size of inhomogeneities in the nonequilibrium postquench state obeys the Kibble-Zurek scaling. In this setting, the local Chern marker thus does behave in a similar way as a local order parameter for a symmetry breaking second order phase transition. The Kibble-Zurek scaling also holds for the inhomogeneities in the spatial distribution of excitations and of the orbital polarization.
We present a formulation for investigating quench dynamics across quantum phase transitions in the presence of decoherence. We formulate decoherent dynamics induced by continuous quantum non-demolition measurements of the instantaneous Hamiltonian. We generalize the well-studied universal Kibble-Zurek behavior for linear temporal drive across the critical point. We identify a strong decoherence regime wherein the decoherence time is shorter than the standard correlation time, which varies as the inverse gap above the groundstate. In this regime, we find that the freeze-out time \bar{t}\sim\tau^{{2\nu z}/({1+2\nu z})} t - ∼ τ 2 ν z / ( 1 + 2 ν z ) for when the system falls out of equilibrium and the associated freeze-out length \bar{\xi}\sim\tau^{\nu/({1+2\nu z})} ξ ‾ ∼ τ ν / ( 1 + 2 ν z ) show power-law scaling with respect to the quench rate 1/\tau 1 / τ , where the exponents depend on the correlation length exponent \nu ν and the dynamical exponent z z associated with the transition. The universal exponents differ from those of standard Kibble-Zurek scaling. We explicitly demonstrate this scaling behavior in the instance of a topological transition in a Chern insulator system. We show that the freeze-out time scale can be probed from the relaxation of the Hall conductivity. Furthermore, on introducing disorder to break translational invariance, we demonstrate how quenching results in regions of imbalanced excitation density characterized by an emergent length scale which also shows universal scaling. We perform numerical simulations to confirm our analytical predictions and corroborate the scaling arguments that we postulate as universal to a host of systems.
We present a formulation for investigating quench dynamics across quantum phase transitions in the presence of decoherence. We formulate decoherent dynamics induced by continuous quantum non-demolition measurements of the instantaneous Hamiltonian. We generalize the well-studied universal Kibble-Zurek behavior for linear temporal drive across the critical point. We identify a strong decoherence regime wherein the decoherence time is shorter than the standard correlation time, which varies as the inverse gap above the groundstate. In this regime, we find that the freeze-out time tˉ∼τ2νz/(1+2νz)\bar{t}\simτ^{{2νz}/({1+2νz})}tˉ∼τ2νz/(1+2νz) for when the system falls out of equilibrium and the associated freeze-out length ξˉ∼τν/(1+2νz)\barξ\simτ^{ν/({1+2νz})}ξˉ​∼τν/(1+2νz) show power-law scaling with respect to the quench rate 1/τ1/τ1/τ, where the exponents depend on the correlation length exponent ννν and the dynamical exponent zzz associated with the transition. The universal exponents differ from those of standard Kibble-Zurek scaling. We explicitly demonstrate this scaling behavior in the instance of a topological transition in a Chern insulator system. We show that the freeze-out time scale can be probed from the relaxation of the Hall conductivity. Furthermore, on introducing disorder to break translational invariance, we demonstrate how quenching results in regions of imbalanced excitation density characterized by an emergent length scale which also shows universal scaling. We perform numerical simulations to confirm our analytical predictions and corroborate the scaling arguments that we postulate as universal to a host of systems.