Topology versus Anderson localization: Nonperturbative solutions in one dimension

We present an analytic theory of quantum criticality in quasi-one-dimensional topological Anderson insulators. We describe these systems in terms of two parameters $(g,\chi )$ representing localization and topological properties, respectively. Certain critical values of $\chi$ (half-integer for $\mathbb{Z}$ classes, or zero for ${\mathbb{Z}}_2$ classes) define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two-parameter flow of the integer quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given in terms of the transfer-matrix solution of corresponding supersymmetric nonlinear sigma models. In ${\mathbb{Z}}_2$ classes we uncover a hidden supersymmetry, present at the quantum critical point.

Figure 1

Schematic phase diagram of topological insulators. The $\mathbb{Z}$ (top) or ${\mathbb{Z}}_2$ (bottom) valued topological number of a clean topological insulator can be controlled by a parameter $\mu$ (e.g., chemical potential, magnetic field, etc.). At the transition points separating distinct phases, band gaps close. Disorder, characterized by its strength $w$, induces a crossover from a clean band to an Anderson insulator (shaded lines). The amount of disorder required to close the band gap vanishes at the transition points. The transition points themselves become end points of transition lines in the $(\mu ,w)$ phase plane. Driving the system through one of these lines via a parameter change implies IQH-type transition with a divergent localization length. The right panels show the flow of the average topological number $\chi$ and the conductance $g$ upon increasing the system size $L$, starting from some nonuniversal bare values $(\tilde{g},\tilde{\chi })$, defined at a scale of the order of the mean-free path. In the $L\to \infty$ limit, the insulating $g=0$ and self-averaging $\chi =n$ Anderson topological insulator configurations are generically approached. The critical surfaces separating these regions are characterized by half-integer $\chi =n+\frac{1}{2}$ ($\mathbb{Z}$) or vanishing $\chi =0$ (${\mathbb{Z}}_2$) values of the average topological number.

Figure 2

Schematic of a multichannel AIII quantum wire with staggered hopping of strength $t,\mu$, respectively, and random interchain hopping described by matrix elements $R_{ss+1}^{k{k}^{\prime }}$.

Figure 3

Flow of the conductance $g$ and the topological parameter $\chi$ as a function of system size for class AIII system. Dots are for values, $L/\tilde{\xi }=1,2,4,\cdots ,32$.

Figure 4

Phase diagram of the $\mathrm{AIII}$ class three-channel disordered wire. Dashed lines show crossover regions between band insulator (bI) and Anderson insulator (AI) or tI and tAI phases, derived from the SCBA. Solid lines correspond to half-integer values of the SCBA computed topological number $\tilde{\chi }$ and mark boundaries between phases of different $n$. bI and AI have $n=0$, while for tI and tAI, $n\ne 0$. Squares: data points, representing phase boundaries found from a numerical analysis of Lyapunov exponents (Sec. 6).

Figure 5

Phase diagram of the $\mathrm{BDI}$ class three-channel disordered $p$-wave superconducting wire. Dashed lines show crossover regions between band insulator (bI) and Anderson insulator (AI) or tI and tAI phases, derived from the SCBA. Solid lines correspond to half-integer values of the SCBA computed topological number $\tilde{\chi }$ and mark boundaries between phases of different $n$. bI and AI have $n=0$, while for tI and tAI, $n\ne 0$.

Figure 6

Flow of the conductance $g$ and the kink's fugacity $\chi$ as a function of system size for class D system. Dots are for values, $L/\tilde{\xi }=1,2,4,\cdots ,32$.

Figure 7

Average density of Lyapunov exponents $\rho \left(\varphi \right)$ in the class $\mathrm{AIII}$ disordered wire in the case of $\tilde{\chi }=0$ shown for $L/\tilde{\xi }=1$ (weak localization, gray line), $L/\tilde{\xi }=4$ (black line), and $L/\tilde{\xi }=32$ (strong localization, solid red line) and in the case of $\tilde{\chi }=\frac{1}{2}$ for $L/\tilde{\xi }=32$ (dashed red line).

Figure 8

Average density of two minimal Lyapunov exponents $\rho \left(\varphi \right)$ in the class $\mathrm{DIII}$ wire shown for the bare value of the renormalized fugacity $v=0.2$ (see the text) and $L/\tilde{\xi }=v,1,8$ (gray, black, and red lines, respectively). In the limit $L\gg \xi \gg \tilde{\xi }$ (strong localization, red line) the minimal exponents “crystallize” around values $\pm L/2{\xi }_{*}$.

Figure 9

Disordered wire of symmetry class AIII connected to two normal terminals which are described by the Eilenberger functions $Q_{\pm }$. The $Q$ matrix at the boundaries between the wire and terminals is denoted by $Q_{\mathrm{L}}$ and $Q_{\mathrm{R}}$. The counter $\eta \left(x\right)$ which “measures” the parity current $j^P$ is located close to the left end of the wire.