Chalker scaling, level repulsion, and conformal invariance in critically delocalized quantum matter: Disordered topological superconductors and artificial graphene

We numerically investigate critically delocalized wave functions in models of two-dimensional Dirac fermions, subject to vector potential disorder. These describe the surface states of three-dimensional topological superconductors, and can also be realized through long-range correlated bond randomness in artificial materials like molecular graphene. A “frozen” regime can occur for strong disorder in these systems, wherein a single wave function presents a few localized peaks separated by macroscopic distances. Despite this rarefied spatial structure, we find robust correlations between eigenstates at different energies, at both weak and strong disorder. The associated level statistics are always approximately Wigner-Dyson. The system shows generalized Chalker (quantum critical) scaling, even when individual states are quasilocalized in space. We confirm analytical predictions for the density of states and multifractal spectra. For a single Dirac valley, we establish that finite energy states show universal multifractal spectra consistent with the integer quantum Hall plateau transition. A single Dirac fermion at finite energy can therefore behave as a “Quantum Hall critical metal.” For the case of two valleys and non-Abelian disorder, we verify predictions of conformal field theory. Our results for the non-Abelian case imply that both delocalization and conformal invariance are topologically protected for multivalley topological superconductor surface states.

Figure 1

Probability density of the exact zero-energy wave functions in the single-valley Dirac fermion model [23]. (a) Represents the critically delocalized states in the weak disorder region (${\Delta }_A=0.4\pi <2\pi$). The distribution is spatially inhomogeneous and multifractal. (b) Shows the spatial signature of the “frozen” states in the strong disorder region (${\Delta }_A=3\pi >2\pi$). The wave function is characterized by rarefied peaks. For both cases, we generate the analytical wave function [23] with a 64-by-64 spatial resolution.

Figure 2

Sketch of the DoS in the MFD, in the presence of disorder. The red dashed lines mark the position of the energy cutoff $\Lambda =\mathcal{N}(2\pi /L)$, which is fixed to a constant. The chiral region is circled by a blue dashed line, wherein the dynamic critical exponent is modified by the disorder strength ${\Delta }_A$ [23, 24, 32, 33]. The states away from the chiral region but inside the cutoff typically show a DoS linear in energy, which is also the result for clean 2D Dirac fermions.

Figure 3

Bipartite lattices with low-energy Dirac fermions: square lattice (a) with $\pi$ flux and the honeycomb lattice (b). Labels $A$ and $B$ indicate the sublattices. In the clean limit, the homogeneous hopping amplitudes $t_{{\mathrm{r}}_A,{\mathrm{r}}_B}^{\left(0\right)}$ [Eq. (2.9)] are equal to $+1$ for solid lines and $-1$ for dashed lines.

Figure 4

The DoS in MFD near zero energy with $\mathcal{N}=40$, $\xi =0.25r$, and $r=L/\mathcal{N}$. The results were obtained by averaging over 80 realizations of the disorder. Dots are the numerical results from 200 energy levels, and the solid lines are the analytical formula. (From bottom to top) ${\Delta }_A=$ $0.2\pi$, $0.4\pi$, $0.6\pi$, $0.8\pi$, $\pi$, and $1.2\pi$. The data are rescaled so that the rightmost points are placed at the same position. As described in the text, for ${\Delta }_A\ge \pi$ we extract an effective disorder strength from the data, which is later employed in the study of level statistics and Chalker scaling. ${\Delta }_{A,\text{eff}}=0.96\pi$ for ${\Delta }_A=\pi$ and ${\Delta }_{A,\text{eff}}=1.125\pi$ for ${\Delta }_A=1.2\pi$.

Figure 5

The $\xi$ dependence of the DoS in MFD. $\mathcal{N}=40$, ${\Delta }_A=1.8\pi$, $r=L/\mathcal{N}$, and we average over 20 disorder realizations. (Inset) The dynamical exponent $z$ computed with different $\xi$ values. The dots are extracted from the numerical DoS. The blue solid line is the analytical dynamical exponent with ${\Delta }_A=1.8\pi$. We choose $\xi =0.35r$ as the proper parameter in MFD, because this is where $z$ is least sensitive to the value of $\xi$. We extract ${\Delta }_{A,\text{eff}}\approx 1.42\pi$.

Figure 6

The DoS near-zero energy for the MDH model on (a) the $\pi$-flux lattice with $L=256$ and (b) honeycomb lattice with the same size. Results are averaged over 40 disorder realizations. Dots are the numerical results from 149 energy levels (excluding the first positive level), and the dashed lines are the analytical formula. The data are rescaled such that the rightmost points are placed at the same position.

Figure 7

The $f\left(\alpha \right)$ spectra of low-energy states in MFD with $\mathcal{N}=40$. For each value of ${\Delta }_A$, results are averaged over 40 disorder realizations. Here $r=L/\mathcal{N}$. (a) ${\Delta }_A=0.4\pi$; (b) ${\Delta }_A=0.6\pi$; (c) ${\Delta }_A=0.8\pi$; (d) ${\Delta }_A=1.0\pi$. The data are extracted from the numerical derivative of the IPR with respect to $b$, the size of the binning cell. Data A is extracted from $b=1$ and $b=2$. Data B is extracted from $b=2$ and $b=4$. The solid lines are the analytical prediction from Eqs. (4.6) and (4.7).

Figure 8

The $f\left(\alpha \right)$ spectra of low-energy states of the MDH model with $L=256$. For each value of ${\Delta }_A$, results are averaged over 40 disorder realizations. (a) ${\Delta }_A=0.4\pi$; (b) ${\Delta }_A=0.8\pi$; (c) ${\Delta }_A=1.2\pi$; (d) ${\Delta }_A=1.6\pi$; (e) ${\Delta }_A=2.0\pi$; (f) ${\Delta }_A=2.4\pi$. The data are extracted from the numerical derivative of the IPR with respect to $b$, the size of the binning cell. Data B is extracted from $b=2$ and $b=4$. Data C is extracted from $b=4$ and $b=8$. The solid lines are the analytical prediction from Eqs. (4.6), (4.7), and (4.8).

Figure 9

The level spacing distribution in the chiral region of the Dirac fermion in MFD. Here the system size is $\mathcal{N}=32$. We keep 100 energy levels per realization, and we have averaged over 400 disorder realizations; $r=L/\mathcal{N}$. The effective disorder strength ${\Delta }_{A,\text{eff}}$ for the presented data is $0.4\pi$, $1.55\pi$, $1.9\pi$, and $2.55\pi$. (Inset) The tail distribution. The black solid line is the Wigner surmise with $\beta =2$; the blue solid line is the Poisson distribution.

Figure 10

The level spacing distribution in the chiral region of the MDH model. The system size is $L=256$. We keep 149 energy levels per realization, and we have averaged over 400 disorder realizations. (Inset) The tail distribution. The black solid line is the Wigner surmise with $\beta =1$; the blue solid line is the Poisson distribution.

Figure 11

The Chalker scaling exponent $\mu$ [defined via Eq. (5.5)] as a function of the disorder strength in the MFD and MDH approaches. $\mathcal{N}=40$ and we average over 80 realizations of the disorder for MFD. We show data for two system sizes of the MDH model. For $L=256$, we average over 200 realizations of the disorder. For $L=512$, we average over 40 disorder realizations. For MFD, the effective disorder strengths are presented when ${\Delta }_A>\pi$. For the MDH model, the wave functions are coarse grained with binning size $b=2$. The solid curve is the analytical prediction that includes termination and freezing effects [Eq. (5.7)].

Figure 12

The $f\left(\alpha \right)$ spectra of finite energy states in the single-valley Dirac model using the momentum space formalism with $\mathcal{N}=40$. (a) Weak disorder; (b) intermediate disorder. We perform an average over 80 realizations of the disorder; $r=L/\mathcal{N}$. Finite energy states with ${\Delta }_A=0.4\pi$ and ${\Delta }_A=0.6\pi$ show deviations from the parabolic spectrum in Eq. (6.1) with $\theta =0.26$. The latter is a good approximation of the integer quantum Hall plateau transition spectrum [59, 60, 61]. For ${\Delta }_A=0.8\pi$, $\pi$, and $1.2\pi$, the $f\left(\alpha \right)$ spectra are consistent with the $\theta =0.26$ curve. ${\Delta }_{A,\text{eff}}=0.96\pi$ for ${\Delta }_A=\pi$, and ${\Delta }_{A,\text{eff}}=1.125\pi$ for ${\Delta }_A=1.2\pi$. Data A is extracted from $b=1$ and $b=2$.

Figure 13

The $f\left(\alpha \right)$ spectra of low-energy states for a single-valley Dirac fermion with two kinds of disorder in MFD; $\mathcal{N}=40$. We perform an average over 80 realizations of the disorder. (a) ${\Delta }_A={\Delta }_M=0.8\pi$; (b) ${\Delta }_V={\Delta }_A=0.8\pi$; (c) ${\Delta }_V={\Delta }_M=0.8\pi$. ${\Delta }_V$ and ${\Delta }_M$ correspond to the disorder variance of scalar and mass potentials, respectively. $\xi =0.25r$ for all the cases, where $r=L/\mathcal{N}$. Data A is extracted from binning sizes $b=1$ and $b=2$. Data B is extracted from $b=2$ and $b=4$. The dashed line is the same as stated in Fig. 12.

Figure 14

The DoS near zero energy for two-valley class CI and AIII Dirac models, with $\mathcal{N}=40$. Dots are the numerical results from 400 energy levels, averaged over 40 disorder realizations. The strength of the non-Abelian SU(2) vector potential disorder ${\Delta }_N$ is fixed to $0.8\pi$ for all three cases; $r=L/\mathcal{N}$. The case(s) with ${\Delta }_A=0$ (${\Delta }_A>0$) correspond to class CI (AIII). The solid lines are the analytical result implied by Eqs. (3.2) and (7.3). The data are rescaled so that the rightmost points are placed at the same position.

Figure 15

The $f\left(\alpha \right)$ spectra of Dirac fermions with non-Abelian SU(2) vector potential in MFD. Here $\mathcal{N}=40$, $\xi =0.25r$ ($r=L/\mathcal{N}$), and we average over 40 disorder realizations for all three cases: (a) ${\Delta }_N=0.8\pi$; (b) ${\Delta }_A=0.1\pi$; (c) ${\Delta }_A=0.2\pi$. The data are extracted from the numerical derivatives of the IPR. Data A is extracted from binning sizes $b=1$ and $b=2$. Data B is extracted from $b=2$ and $b=4$. The solid lines are the analytical prediction in Eqs. (6.1) and (7.4).