Integer quantum Hall transition on a tight-binding lattice

Even though the integer quantum Hall transition has been investigated for nearly four decades its critical behavior remains a puzzle. The best theoretical and experimental results for the localization length exponent $\nu$ differ significantly from each other, casting doubt on our fundamental understanding. While this discrepancy is often attributed to long-range Coulomb interactions, Gruzberg et al. [Phys. Rev. B 95, 125414 (2017)] recently suggested that the semiclassical Chalker-Coddington model, widely employed in numerical simulations, is incomplete, questioning the established central theoretical results. To shed light on the controversy, we perform a high-accuracy study of the integer quantum Hall transition for a microscopic model of disordered electrons. We find a localization length exponent $\nu =2.58\left(3\right)$ validating the result of the Chalker-Coddington network.

Figure 1

Critical state (left, $E=E_{\mathrm{c}}=3.4221$) and edge state (right, $E=3.0056$) for a square system of linear size $L=128$, magnetic flux $\mathrm{\Phi }=1/10$, and disorder strength $W=0.5$. Coloring represents wave-function intensities $|{\psi }_j{|}^2$ at site $j$ via ${log}_{L^2}(L^2|{\psi }_j{|}^2)$. Periodic and open boundary conditions are applied in the $y$ and $x$ directions, respectively.

Figure 2

Left: Hofstadter butterfly. Energy spectrum of Hamiltonian (1) in the clean case $(W=0)$. For $\mathrm{\Phi }=0$ and 1, there is a single band from $E=-4$ to 4. Landau levels are shown for $\mathrm{\Phi }=p/q$ with coprime numbers $p,q\le 20$. The free-electron approximation (solid red lines), $E_n^{+}=4-4\pi \mathrm{\Phi }(n+1/2)$, is shown for $n=0$, 1, and 2. Right: LL spacing (midband distance $\mathrm{\Delta }$ between the LLs with $n=0$ and $n=1$) in multiples of the intrinsic LL width $\delta$ as a function of $\mathrm{\Phi }$.

Figure 3

(a) Dimensionless Lyapunov exponent $\mathrm{\Gamma }(E,L)$ for $\mathrm{\Phi }=1/10$ and $W=0.5$ as a function of $E$ for several $L$. The statistical errors are well below the symbol size. The blue area shows the density of states in arbitrary units. Lines are guides to the eye only. (b) $\mathrm{\Gamma }(E,L)$ close to the lowest IQH transition. The solid lines are fits (see main text for details). The inset shows the immediate vicinity of the transition. The tiny red bar at the bottom shows the energy range of the LL in absence of disorder (compare Fig S1).

Figure 4

System-size dependence of $\mathrm{\Gamma }$ at $E_{\text{min}}$ for several $\mathrm{\Phi }$ under assumption of an irrelevant exponent $y=0.38$. The fits of Eq. (4) for $\mathrm{\Phi }\le 1/10$ (black dashed line) and $\mathrm{\Phi }=1/5$ (black dotted line) are based on the data with $L\ge 128$. The statistical errors are below the symbol size. The insets show the deviations $\mathrm{\Delta }\mathrm{\Gamma }$ of the data points from the fit functions. Solid lines are guides to the eye only.