Quantum Hall criticality in an amorphous Chern insulator

TL;DR

This work investigates the quantum Hall critical point in a two-dimensional amorphous Chern insulator, testing its universality with the integer quantum Hall transition. It employs two-terminal conductance and wavefunction multifractality to extract critical exponents, finding $\nu \simeq 2.60(5)$ and $y \simeq 0.3(1)$ at $E=0$, and reveals a broad, scale-invariant conductance distribution at criticality. The multifractal analysis shows reciprocity $\Delta_q=\Delta_{1-q}$ consistent with conformal invariance, but also hints at deviations from the exact parabolic spectrum and possible quartic corrections. Overall, the amorphous QAH transition exhibits IQH-like universality in the unitary class, with finite-size corrections playing a significant role in the observed spectra and their corrections, offering insights for experiments in disordered topological systems.

Abstract

We explore the critical properties of a topological transition in a two-dimensional, amorphous lattice with randomly distributed points. The model intrinsically breaks the time-reversal symmetry without an external magnetic field, akin to a Chern insulator. Here, the topological transition is induced by varying the density of lattice points or adjusting the mass parameter. Using the two-terminal conductance and multifractality of the wavefunction, we found that the topological transition belongs to the same universality class as the integer quantum Hall transition. Regardless of the approach to the critical point across the phase boundary, the localization length exponent remains within $ν\approx 2.55 - 2.61$. The irrelevant scaling exponent for both the observables is $y \approx 0.3(1)$, comparable to the values obtained using transfer matrix analysis in the Chalker-Coddigton network. Additionally, the investigation of the entire distribution function of the inverse participation ratio at the critical point shows possible deviations from the parabolic multifractal spectrum at the anomalous quantum Hall transition.

Figures (12)

• Figure 1: Setup for two terminal conductance $\overline{g}$ calculations. Square lattice leads are connected to a scattering region with random lattice points. The system shown here has system size $L=8$ with a density of lattice points $\rho=0.7$ and with open boundary condition perpendicular to the transport direction. The gray lines connect the lattice sites within a hopping distance of $R=4$. One extra slice of regular lattice sites (indicated with red dots) is embedded into the scattering region on both sides for a smooth connection to the ideal square leads. Right panel: Shows the qualitative topological phase diagram of the model \ref{['eq:ham']}. The symbol is calculated using the Bott index (for system size $L=48$ and a single lattice configuration), and the dotted line is a guide to the eye.
• Figure 2: Finite size analysis of the mean conductance $\overline{g}$ and $\overline{\ln g}$. (a) Show the raw conductance data as a function of the mass parameter $M$ for different system sizes $L=\{32-768\}$ with density $\rho=0.7$. Shift in the crossing point with successive system sizes indicates a strong finite size effect. (b) The corrected conductance $\overline{g}_\mathrm{corr.}$ using Eq. \ref{['eq:gcorr_expand']} with $N_\mathrm{R}=2$. (c) Show the scaling collapse of the data with $\nu\simeq 2.58(2)$, and $M_\mathrm{c}\simeq-2.145(1)$ with a $\chi^2\approx 1.22$. For better visibility, only a few system sizes are shown. Inset shows the scaling collapse of the $\overline{\ln g}_\mathrm{corr.}$ with similar leading exponent $\nu\approx 2.55(5)$. The data collapse is obtained with an irrelevant scaling exponent $y\approx 0.36$.
• Figure 3: Finite-size scaling analysis of the log-conductance $\overline{\ln g}$ by varying the density $\rho$. The mass parameter is fixed at $M=-1$. The main panel shows the scaling collapse of $\overline{\ln g}_\mathrm{corr.}$ after subtracting the irrelevant correction (similar analysis as in Fig. (\ref{['fig:g_collapse']})). The critical parameters obtained are shown in the figure and table \ref{['tab:cond']}. The inset figure shows the raw data at various densities and the system sizes $L=\{64-512\}$.
• Figure 4: (a) Shows the scale-invariant conductance distribution at the estimated critical point $M_\mathrm{c}\simeq -2.146$. For small system sizes, the finite size effects are visible in the tail of the distribution. (b) Highlighting the scaling of the $\overline{g}$ close to the critical point. The solid line indicates the two correction terms mentioned in the legend with $y$ given in the legend. Depending on the precise value of the $M_\mathrm{c}$ the $y$ also changes as the $g_\text{c}$.
• Figure 5: Effective anomalous MF exponent across the IQH transition at $\rho=0.7$ for the moment $q=0.5$ for several system sizes (left) and approximate collapsed with $\nu_\mathrm{IQH}\sim 2.6$ (right). The horizontal line indicates the value of $\Delta_q/q(1-q) \rvert_{q=0.5}$ for parabolic prediction at the IQH transition \ref{['eq:deltaq']}. The finite size correction is taken into account with the $L$ dependent critical mass term $\tilde{M}_\mathrm{c}(L)$.