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arXiv:2310.08474 · Physical review. B./Physical review. B

Topologically protected subdiffusive transport in two-dimensional fermionic wires

Junaid Majeed Bhat

Published Mar 14, 2024• 2 citations
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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 ${N}_{x}\ifmmode\times\else\texttimes\fi{}{N}_{y}$ rectangular lattice comprised of vertical chains with a Hermitian intrachain and interchain hopping matrices given by ${H}_{0}$ and ${H}_{1}$, respectively. We study particle transport using the nonequilibrium Green's function formalism, and we show that the asymptotic behavior of the conductance, $T(\ensuremath{\omega})$, at the Fermi level $\ensuremath{\omega}$, is controlled by the spectrum of a dimensionless matrix $A(\ensuremath{\omega})=(\ensuremath{-}\ensuremath{\omega}+{H}_{0}){H}_{1}^{\ensuremath{-}1}$. This gives three simple conditions on the spectrum of $A(\ensuremath{\omega})$ for observing ballistic, subdiffusive, and exponentially decaying $T(\ensuremath{\omega})$ with respect to ${N}_{x}$. We show that certain eigenvalues of $A(\ensuremath{\omega})$ 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 $A(\ensuremath{\omega})$ 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 $A(\ensuremath{\omega})$ 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 to be in good agreement with the theoretical predictions.

DOI
10.1103/physrevb.109.125415

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We show that a one-dimensional ordered fermionic lattice system with power-law-decaying hopping, when connected to two baths at its two ends with different chemical potentials at zero temperature, features two phases showing subdiffusive scaling of conductance with system size. These phases have no analogues in the isolated system (i.e., in absence of the baths) where the transport is perfectly ballistic. In the open system scenario, interestingly, there occurs two chemical-potential-driven subdiffusive to ballistic phase transitions at zero temperature. We discuss how these phase transitions, to our knowledge, are different from all the known nonequilibrium quantum phase transitions. We provide a clear understanding of the microscopic origin of these phases and argue that the subdiffusive phases are robust against the presence of arbitrary number-conserving many-body interactions in the system. These phases showing subdiffusive scaling of conductance with system size in a two-terminal setup are therefore universal properties of all ordered one-dimensional number-conserving fermionic systems with power-law-decaying hopping at zero temperature.
Advances in the next generation of mesoscopic electronics require an understanding of topological phases in inhomogeneous media and the principles that govern them. Motivated by the nature of motifs available in printable conducting inks, we introduce and study quantum transport in a minimal model that describes a bundle of one-dimensional metallic wires that are randomly interconnected by semiconducting chains. Each of these interconnects is represented by a Su-Schrieffer-Heeger chain, which can reside in either a trivial or a topological phase. Using a tight-binding approach, we show that such a system can transit from an insulating phase to a robust metallic phase as the interconnects undergo a transition from a trivial to a topological phase. In the latter, despite the random interconnectedness, the metal evades Anderson localization and exhibits a ballistic conductance that scales linearly with the number of wires. We show that this behavior originates from hopping renormalization in the wire network. The zero-energy modes of the topological interconnects act as effective random dimers, giving rise to an energy-dependent localization length that diverges as ∼1/E2\sim 1/E^2∼1/E2. Our work establishes that random networks provide a yet-unexplored platform to host intriguing phases of topological quantum matter.
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