Topologically protected subdiffusive transport in two-dimensional fermionic wires

Abstract

The conductance at the band edges of one-dimensional fermionic wires, with $N$ sites, has been shown to have subdiffusive $(1/N^2)$ behavior. We investigate this issue in two-dimensional fermionic wires described by a hopping model on an $N_x\times N_y$ rectangular lattice comprised of vertical chains with a Hermitian intra-chain and inter-chain hopping matrices given by $H_0$ and $H_1$, respectively. We study particle transport using the non-equilibrium Green's function formalism, and show that the asymptotic behavior of the conductance, $T(\omega)$, at the Fermi level $\omega$, is controlled by the spectrum of a dimensionless matrix $A(\omega)=(-\omega+H_0)H_1^{-1}$. This gives three simple conditions on the spectrum of $A(\omega)$ for observing ballistic, subdiffusive, and exponentially decaying $T(\omega)$ with respect to $N_x$. We show that certain eigenvalues of $A(\omega)$ give rise to subdiffusive contributions in the conductance, and correspond to the band edges of the isolated wire. We demonstrate that the condition for observing the subdiffusive behavior can be satisfied if $A(\omega)$ has nontrivial topology. In that case, a transition from ballistic behavior to subdiffusive behavior of the conductance is observed as the hopping parameters are tuned within the topological regime. We argue that at the transition point, different behaviors of the conductance can arise as the trivial bulk bands of $A(\omega)$ also contribute subdiffusively. We illustrate our findings in a simple model by numerically computing the variation of the conductance with $N_x$. Our numerical results indicate a different subdiffusive behavior ($1/N_x^3$) of the conductance at the transition point. We find the numerical results in good agreement with the theoretical predictions.