Anderson localization and the topology of classifying spaces

We construct the generic phase diagrams encoding the topologically distinct localized and delocalized phases of noninteracting fermionic quasiparticles for any symmetry class from the tenfold way in one, two, and three dimensions. To this end, we start from a massive Dirac Hamiltonian perturbed by a generic disorder for any dimension of space and for any one of the ten symmetry classes from the tenfold way. The physics of Anderson localization is then encoded by a two-dimensional phase diagram that we deduce from the topology of the space of normalized Dirac masses. This approach agrees with previously known results and gives an alternative explanation for the even-odd effect in the one-dimensional chiral symmetry classes. We also give a qualitative explanation for the Gade singularity and Griffiths effects in the density of states using the first homotopy group of the normalized Dirac masses in two dimensions. Finally, this approach is used to analyze the stability of massless Dirac fermions on the surface of three-dimensional topological crystalline insulators.

Figure 1

Connectedness of the compact topological space $V_{d,r}^{}$ parameterized by the normalized Dirac masses. The zeroth homotopy groups ${\pi }_0^{}\left(V_{d,r}^{}\right)$ index the disconnected parts of the compact topological space $V_{d,r}^{}$. The zeroth homotopy group ${\pi }_0^{}\left(V_{d,r_{\min }^{}N}^{}\right)$ is either (a) $\mathbb{Z}$ (in the limit $N\to \infty$), (b) ${\mathbb{Z}}_2^{}$, or (c) $\left\{0\right\}\equiv 0$ according to Table 2. (a) When ${\pi }_0^{}\left({lim}_{N\to \infty }{V}_{d,r_{\min }^{}N}^{}\right)=\mathbb{Z}, V_{d,r_{\min }^{}N}^{}$ is disconnected and given by the union of path-connected and compact topological subspaces indexed by the half-integers or integers defined in Eq. (3.8c). For odd $N, V_{d,r_{\min }^{}N}^{}$ is the union of path-connected and compact topological subspaces labeled by negative and positive half integers. The total “volume” of the union of all subspaces with negative half-integer labels equals that of the union of all subspaces with positive half-integer labels when the mean values for the Dirac masses vanish. For even $N, V_{d,r_{\min }^{}N}^{}$ is the union of path-connected and compact topological subspaces labeled by the integers (3.8c). The total “volume” of the union of all subspaces with strictly negative integer labels equals that of the union of all subspaces with strictly positive integer labels when the mean values for the Dirac masses vanish. The subspace labeled by the integer 0 has a distinct (larger) “volume” when the mean values for the Dirac masses vanish. (b) When ${\pi }_0^{}\left(V_{d,r}^{}\right)={\mathbb{Z}}_2^{}, V_{d,r}^{}$ is the union of two path-connected and compact topological subspaces that are indexed by the integers (3.15d) and are of equal “volume” when the mean values for the Dirac masses vanish. (c) When ${\pi }_0^{}\left(V_{d,r}^{}\right)=\left\{0\right\}\equiv 0, V_{d,r}^{}$ is a path-connected and compact topological space.

Figure 2

For each realization of a random potential perturbing the massless Dirac Hamiltonian defined in $d$-dimensional Euclidean space ${\mathbb{R}}^d$, we may decompose ${\mathbb{R}}^d$ into open sets (domains) of linear size ${\xi }_{\mathrm{dis}}^{}$. In each of these domains, the normalized Dirac masses correspond to a unique value of their zeroth homotopy group. At the boundary between domains differing by the values taken by their zeroth homotopy group, the Dirac masses must vanish. Such boundaries support quasi-zero-energy boundary states. When it is possible to classify the elements of the zeroth homotopy group by the pair of indices A and B, we may assign the letters A or B to any one of these domains as is illustrated. When the typical “volume” of a domain of type A equals that of type B, quasizero modes undergo quantum percolation through the sample and thus establish either a critical or a metallic phase of quantum matter in $d$-dimensional space.

Figure 3

Qualitative quantum phase diagrams for 1D disordered wires with the quasiparticle energy fixed at $\varepsilon =0$. The horizontal axis is parameterized by the characteristic value $\mathsf{m}\in \mathbb{R}$ of the disorder-averaged Dirac masses allowed by the symmetry class. The vertical axis is parameterized by the characteristic value $\mathsf{g}\ge 0$ taken by the strength of the generic static and local random mass disorder allowed by the symmetry class. Arrows on the phase boundaries and on the horizontal axis indicate flows under renormalization group transformations.

Figure 4

Qualitative quantum phase diagrams for 2D random Dirac Hamiltonians with the quasiparticle energy fixed at $\varepsilon =0$. The horizontal axis is parameterized by the characteristic value $\mathsf{m}\in \mathbb{R}$ of the disorder-averaged Dirac masses allowed by the symmetry class. We choose the probability distribution (3.10) for the case of $N=2$ in the symmetry class D. The vertical axis is parameterized by the characteristic value $\mathsf{g}\ge 0$ taken by the strength of the generic local disorder allowed by the 2D AZ symmetry class. The zeroth homotopy group associated with the space of normalized Dirac mass fixes the indexing of the topologically distinct insulating phases. The first homotopy group of the normalized Dirac masses determines if the density of states is regular or singular at the band center. The existence of a metallic phase, the flows along phase boundaries, and the dimensionality of the phase boundaries all follow from perturbative renormalization-group calculations.

Figure 5

Qualitative quantum phase diagrams for 3D random Dirac Hamiltonians with the quasiparticle energy fixed at $\varepsilon =0$. The horizontal axis is parameterized by the characteristic value $\mathsf{m}\in \mathbb{R}$ of the disorder-averaged Dirac masses allowed by the symmetry class. We choose the probability distribution (3.10) for the case of $N=2$ in the symmetry classes AIII, DIII, and CI. The vertical axis is parameterized by the characteristic value $\mathsf{g}\ge 0$ taken by the strength of the generic local disorder allowed by the 3D AZ symmetry class. The zeroth homotopy group of the topological space associated with the normalized Dirac mass fixes the indexing of the topologically distinct insulating phases. The existence of a metallic phase, the flows along phase boundaries, and the dimensionality of the phase boundaries all follow from perturbative renormalization-group calculations.

Figure 6

(a) Phase diagram in the parameter space (2.7). There are three gapped phases all meeting at the origin, the clean massless Dirac Hamiltonian of rank $r=4$ in two-dimensional space. The origin of parameter space is a critical point that is unstable to both $\mathsf{m}$ or ${\mathsf{m}}_{+}^{}$ as is indicated by the arrows. The gray region with the mass $\mathsf{m}>{\mathsf{m}}_{\pm }^{}>0$ supports the dimensionless quantized thermal Hall conductivity $\nu =+1$. The gray region with the mass $\mathsf{m}<-{\mathsf{m}}_{+}^{}<0$ supports the dimensionless quantized thermal Hall conductivity $\nu =-1$. The white region with ${\mathsf{m}}_{+}^{}>\left|\mathsf{m}\right|>0$ support the dimensionless quantized thermal Hall conductivity $\nu =0$. The critical lines $\mathsf{m}=\pm {\mathsf{m}}_{+}^{}$ separates two insulating phases whose Chern numbers differ by the number 1 in magnitude. The line parallel to the horizontal axis at fixed value of ${\mathsf{m}}_{+}^{}$ intersects the phase boundaries at two critical points depicted by two filled circles. (b) Phase diagram in the parameter space $g_M^{},g_V^{},g_A^{}\ge 0$ spanned by the variances of a random mass, a random scalar potential, and a random vector potential (all of vanishing means) in the 2D symmetry class A for a random Dirac Hamiltonian of rank $r=2$ [15]. The bullet represents the stable flows to the critical point that describes the plateau transition in the integer quantum Hall effect. The unstable flow away from this critical point corresponds to a nonvanishing mean value of the random mass and is thus not present in this three-dimensional parameter space. (c) Projected phase diagram with the vanishing mean masses ${\mathsf{m}}_{+}^{}=0$ onto a two-dimensional cut with the same horizontal axis as in (a) but with a vertical axis encoding multiples sources of disorder including the variance $\mathsf{g}$ associated with the random mass of mean $\mathsf{m}$. The direction $\mathsf{g}$ can be thought of as a cross section of all the directions associated with the variances of the disorder coming out of the plane in (a).