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arXiv:2606.03426

Criticality and Quench Dynamics at the Anderson Transition of a Chern Insulator

Lara Ulčakar, Jernej Mravlje, G. Lemut, Tomaž Rejec

Published Jun 2, 2026• 0 citations
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Abstract

We study the critical properties of the topological Anderson phase transition in a strongly disordered Chern insulator, separating the topological phase from a trivial Anderson insulator. We show that the transition is characterized by a non-zero electrical conductance and by the emergence of a critical length scale in the real-space profile of the local Chern marker. From this, we extract the correlation-length and the dynamical critical exponents, which are consistent with those of non-interacting models of the integer quantum Hall effect. We then ramp the disorder strength across the transition and study the ensuing dynamics. In contrast to clean topological systems, we find that the excitation density does not follow the Kibble-Zurek scaling. The non-equilibrium length scale associated with the local Chern marker is decoupled from the generation of excitations. For studied system sizes, we find it to be close to the Kibble-Zurek prediction for the topological-to-trivial quench, while it deviates from it for the reverse direction.

DOI
10.48550/arxiv.2606.03426

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We study the critical properties of the topological Anderson phase transition in a strongly disordered Chern insulator, separating the topological phase from a trivial Anderson insulator. We show that the transition is characterized by a non-zero electrical conductance and by the emergence of a critical length scale in the real-space profile of the local Chern marker. From this, we extract the correlation-length and the dynamical critical exponents, which are consistent with those of non-interacting models of the integer quantum Hall effect. We then ramp the disorder strength across the transition and study the ensuing dynamics. In contrast to clean topological systems, we find that the excitation density does not follow the Kibble-Zurek scaling. The non-equilibrium length scale associated with the local Chern marker is decoupled from the generation of excitations. For studied system sizes, we find it to be close to the Kibble-Zurek prediction for the topological-to-trivial quench, while it deviates from it for the reverse direction.
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 study the quantum quench of a finite rate across topological quantum transitions in two-dimensional Chern and Z2{Z}_{2}Z2​ topological insulators. We choose the representative Haldane model and the Kane-Mele model to investigate the behavior of excitation density generated by the quench and the impact of disorder on it. For the Haldane model, as long as the spectral gap is not closed by disorder, we find the excitation density at the end of the quench obeys the power-law decay with decreasing quench rate, and the power is consistent with the prediction of the Kibble-Zurek mechanism. By contrast, the Kibble-Zurek scaling of excitation density is absent in the Kane-Mele model once disorder is switched on, which we attribute to the emergence of a disorder-induced gapless region. In particular, the anti-Kibble-Zurek behavior of excitation density, namely, larger excitation density at slower quench, is observed at suitable model parameters. Moreover, we demonstrate that particle's onsite occupation can be used as a local measurable quantity to probe the breakdown of adiabatic evolution. The difference of onsite occupation between the time-evolved state and the instantaneous ground state at the end of the quench can successfully capture the key features of excitation density for both the Haldane and Kane-Mele models under periodic and more realistic open boundary conditions, thus facilitating the experimental characterization of the quench dynamics in these models.
We investigate disorder-driven transitions between trivial and topological insulator (TI) phases in two-dimensional (2D) systems. Our study primarily focuses on the Bernevig-Hughes-Zhang (BHZ) model with Anderson disorder, while other standard 2D TI models exhibit equivalent features. The analysis is based on the local Chern marker (LCM), a local quantity that allows for the characterization of topological transitions in finite and disordered systems. Our simulations indicate that disorder-driven trivial to topological insulator transitions are nicely characterized by C0{\mathcal{C}}_{0}C0​, the disorder-averaged LCM near the central cell of the system. We show that C0{\mathcal{C}}_{0}C0​ is characterized by a single-parameter scaling, namely, ${\mathcal{C}}_{0}(M,W,L)\ensuremath{\equiv}{\mathcal{C}}_{0}(z)$, with $z=[{W}^{\ensuremath{\mu}}\ensuremath{-}{W}_{c}^{\ensuremath{\mu}}(M)]L$, where MMM is the Dirac mass, WWW is the disorder strength, and LLL is the system size, while ${W}_{c}(M)\ensuremath{\propto}\sqrt{M}$ and $\ensuremath{\mu}\ensuremath{\approx}2$ stand for the critical disorder strength and the critical exponent, respectively. Our numerical results are in agreement with a theoretical prediction based on a first-order Born approximation analysis. These observations lead us to speculate that the universal scaling function we have found is rather general for amorphous and disorder-driven topological phase transitions.
We study topological insulators under dephasing noise. With examples of both a 2d2d2d Chern insulator and a 3d3d3d topological insulator protected by time-reversal symmetry, we demonstrate that there is a phase transition at finite dephasing strength between phases with nontrivial and trivial topological indices. Here, the topological index is defined through the correlation matrix. The transition can be diagnosed through the spectrum of the whole correlation matrix or of a local subsystem. Interestingly, even if the topological insulator is very close to the topological-trivial critical point in its Hamiltonian, it still takes finite strength of dephasing to change the topological index, suggesting the robustness of topological insulators under dephasing. We further consider Chern insulators in the presence of real-space disorder, which exhibit a ground-state transition between topological and Anderson insulating phases. We find that even strongly disordered Chern insulators, close to the critical disorder strength, exhibit robustness with respect to dephasing.
We investigate the phases and phase transitions of the disordered Haldane model in the presence of on-site disorder. We use the real-space Chern marker and transfer matrices to extract critical exponents over a broad range of parameters. The disorder-driven transitions are consistent with the plateau transitions in the Integer Quantum Hall Effect (IQHE), in conformity with recent simulations of disordered Dirac fermions. Our numerical findings are compatible with an additional line of mass-driven transitions with a continuously varying correlation length exponent. The values interpolate between free Dirac fermions and the IQHE with increasing disorder strength. We also show that the fluctuations of the Chern marker exhibit a power-law divergence in the vicinity of both sets of transitions, yielding another varying exponent. We discuss the interpretation of these results.
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.