The impact of dimensionality on universality of 2D quantum Hall transitions

TL;DR

The paper tackles the apparent non-universality of 2D quantum Hall transition exponents by examining quasi-2D Weyl semimetal slabs with finite thickness $L_z$, disorder, and a perpendicular magnetic field. It combines transfer-matrix scaling and recursive Green's function analysis to extract localization-length exponents and multifractal dimensions, revealing that increasing $L_z$ drives a crossover from 2D universal behavior toward 3D Gaussian Unitary Ensemble physics, with asymmetries and a bifurcated scaling structure emerging near the critical point. The results highlight that auxiliary degrees of freedom such as thickness substantially affect universal critical properties, offering a concrete mechanism to reconcile disparities among experiments, 2D models, and 3D theories, and suggesting broader implications for quasi-2D quantum Hall systems. This work implies that thickness and similar hidden dimensions should be carefully accounted for when interpreting critical transport and multifractal statistics in disordered topological systems.

Abstract

Regardless of model and platform details, the critical phenomena exhibit universal behaviors that are remarkably consistent across various experiments and theories, resulting in a significant scientific success of condensed matter physics. One widely known and commonly used example is the 2D quantum Hall transition; yet, its universal exponents still somewhat conflict between experiments, theoretical models, and numerical ansatzes. We study critical behaviors of quasi-2D Weyl semimetal systems with a finite thickness $L_z>1$, disorder, and external magnetic field $B_z$. By analyzing the scaling behaviors of the localization lengths and local density of states using recursive methods, we find that the finite thickness yields a deviation from the 2D quantum Hall universality ($L_z=1$ case) and a crossover toward the 3D Gaussian Unitary Ensemble ($L_z\rightarrow \infty$ limit), potentially offering another cause of the discrepancy. Our work demonstrates the often-overlooked importance of auxiliary degrees of freedom, such as thickness, and that 3D quantum Hall physics is not merely a trivial finite-thickness extension of its 2D counterpart.

Figures (8)

• Figure 1: (a) Illustration of the semiclassical electron orbit in a slab-shaped Weyl semimetal subject to a perpendicular magnetic field. The electron slides along the Fermi arc on the top surface, tunnels through the chiral Landau band to the opposite surface, and repeats this process on the bottom surface to form a closed cyclotron orbit. The circles denote the projections of the bulk Weyl points onto the surfaces, which also serve as the endpoints of the Fermi arcs; the $\pm$ symbols denote their respective chirality. (b) Both the averaged density of states $D(E=0)$ and the quasi-1D localization length $\lambda$ on disordered ($W=0.6$) systems of $L_x=20000$, $L_y=20$, and $L_z=11$ versus the inverse of the perpendicular magnetic field $1/B_z$ show clear signatures of quantum oscillations - a manifestation of the 3D quantum Hall effect zhang2016quantum.
• Figure 2: The (inverse) dimensionless localization length $\Gamma$ for critical states in disordered 2D quantum Hall systems ($L_z=1$). (a) Results of $\Gamma$ around a critical magnetic field from the transfer matrix method (dots) display satisfactory fitting to the finite-size scaling (solid lines). (b) With the dimensionless quantity $L_y/\xi$, the data approximately collapses into a single curve; see further discussions in the main text and Appendix \ref{['sec:appC']}. Here, we estimate $\xi=|1/B-1/B_c|^{-\nu}$. We set the system width $L_y\in [20, 48]$, disorder strength $W=0.6$, and Fermi energy $E=0$ (at the original Weyl nodes).
• Figure 3: The multifractality of wavefunctions at critical points in disordered 2D quantum Hall systems ($L_z=1$). (a) The LDOS of a single critical state, obtained from the recursive Green's function method, exhibits multifractal behaviors. Here, $\eta=0.01/L_xL_y$. (b) The scaling of the IPR $P_2$ with the system size $L\in [30,120]$ determines the fractal dimensions $\tau_2=1.45$ of the wavefunctions at the critical point.
• Figure 4: The (inverse) dimensionless localization length $\Gamma$ versus the Fermi energy $E$ for quasi-2D systems in Eq. \ref{['eq:ham']} with magnetic field and disorder in slab geometry with thickness (a) $L_z=7$ and (c) $L_z=11$. $L_y\in [20, 48]$. The solid curves represent fits from the $L_y$ and $E-E_c$ scalings, as given by Eq. \ref{['eq:fit']}, with the resulting parameters listed in Table \ref{['tab1']}. Moreover, (b) and (d) are the corresponding data collapses near $E_c$, which increasingly reflect two separate branches, one for $E<E_c$ and one for $E>E_c$, as $L_z$ increases.
• Figure 5: The evolution of the IPR scaling exponent, i.e., the fractal dimension $\tau_2$ in the 2D $x-y$ plane, increases with the layer thickness $L_z>1$, indicating the advent of crossover from 2D to 3D universality in 3D quantum Hall systems. $L_x=L_y=L$, $L\in [30, 120]$. The main panel presents the IPR scaling of the entire system, while the inset shows the IPR scaling of the $z=(L_z+1)/2$ central layer. Clearly, all data displays excellent linearity (in the log-log plot), as indicated by the high $R^2$ value, as well as a change in slope as $L_z$ increases. Quantitatively, the resulting $\tau_2$ are listed in Table \ref{['tab3']}.