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 $\sim 1/E^2$. Our work establishes that random networks provide a yet-unexplored platform to host intriguing phases of topological quantum matter.
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. Our work establishes that random networks provide a yet-unexplored platform to host intriguing phases of topological quantum matter.
The question of whether the topological insulators can host a stable nontrivial phase in the presence of spatially correlated disorder is a fundamental question of interest. Based on theoretical investigations, we analyze the effect of random-dimer disorder on the quantum phase transitions of the Su-Schrieffer-Heeger model. We explicitly demonstrate that, due to the absence of symmetry, there are no in-gap edge states in certain disordered topological nontrivial gapped phases and the bulk-boundary correspondence breaks down. However, the fractionalized end charges still appear at the ends of the chain. The energy distribution possibility of the in-gap edge states and the possible values of fractionalized end charges are dependent on the concentration of random-dimer disorder. On the other hand, random-dimer disorder and dimer hopping intertwine in an interesting manner. Reentrant localization transition behavior appears, which is evidenced by the fingerprint of the inverse participation ratio, normalized participation ratio, and tunneling conductivity.
The conductance at the band edges of one-dimensional fermionic wires, with N sites, has been shown to have subdiffusive (1/N2) 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 H0 and H1, 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 Nx. 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 Nx. Our numerical results indicate a different subdiffusive behavior (1/Nx3) of the conductance at the transition point. We find the numerical results to be in good agreement with the theoretical predictions.
We introduce and study a toy model for anomalous transport and Griffiths effects in one-dimensional (1D) quantum disordered isolated systems near the many-body localization (MBL) transition. The model is constituted by a collection of 1D tight-binding chains with on-site random energies, locally coupled to a weak perturbation, corresponding to a random matrix which belongs to the Gaussian orthogonal ensemble (GOE) and mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in the absence of such a coupling the model is localized as expected for the one-dimensional Anderson model, increasing the coupling with the GOE perturbation we find a transition to a conducting phase driven by the proliferation of quantum avalanches, which does not fit the standard paradigm of the Anderson transition. In particular an intermediate Griffiths region emerges, where exponentially distributed insulating segments coexist with a few rare resonances. Typical correlations decay exponentially fast, while average correlations decay as stretched exponentials and diverge with the length of the chain, indicating that the conducting inclusions have a fractal structure and that the localization length is broadly distributed at the critical point. This behavior is consistent with a Kosterlitz-Thouless-like criticality of the MBL transition. Transport and relaxation are dominated by rare resonances and rare strong insulating regions, and show anomalous behaviors strikingly similar to those observed in recent simulations and experiments in the bad metal delocalized phase preceding MBL. In particular, we find subdiffusive transport and power-law decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region. Concomitantly, the ac conductivity vanishes near zero frequency with an anomalous power law.
In strictly one-dimensional lattices, free fermions and hard-core bosons exhibit equivalent physical properties in many-particle systems. However, this statistical equivalence generally breaks down upon moving to higher dimensions. Our work investigates the significant effects of this difference on the topological phase transition in a minimal system lying away from one dimension: the two-leg Su-Schrieffer-Heeger (SSH) ladder. Starting in the limit of two decoupled SSH chains having opposite hopping dimerization patterns (staggered dimerization), we demonstrate that introducing inter-leg coupling drives a transition from a topological to a trivial phase for both spinless fermions and hard-core bosons. However, the topological to trivial phase transition boundary is strongly sensitive to the underlying particle statistics. We find that this intriguing effect favors a trivial phase sandwiched between the two topological phases with opposite characters for the case of hard-core bosons, resulting in a reentrant topological phase transition. In contrast, this reentrant phenomenon is absent for spinless fermions, with the topological phases being separated by a gapless line instead of an intervening trivial phase. This study demonstrates the interplay between particle statistics and topological features, revealing fundamentally different properties in an experimentally accessible set-up.
Protected zero modes are a hallmark of topological phases of matter and are exponentially localized at sharp interfaces between distinct gapped phases. We investigate how this picture changes for smooth interfaces in a broad class of inhomogeneous Su-Schrieffer-Heeger (SSH) models. Combining an exact lattice solution with an inhomogeneous Dirac description, we show that the associated Jackiw-Rebbi zero mode becomes spatially extended. For arbitrary smooth hopping profiles, its lattice extension universally scales as the square root of the system size, independently of the microscopic details of the interface. This emergent length defines a mesoscopic critical region separating two gapped phases, within which correlations decay algebraically before crossing over to exponential decay. In addition, the entanglement entropy scales as the logarithm of the emergent length near the interface, confirming the interpretation of a mesoscopic critical region. Our results establish a universal critical length governing the low-energy physics of smooth topological interfaces.
The long-range Su-Schrieffer-Heeger (SSH) model, in which the second nearest-neighbor hopping is taken into account, exhibits a topological phase diagram that contains winding numbers w=0,1, and 2. In the clean system, the change in winding number stems from the band-touching phenomenon. In the presence of disorder, the renormalization of energy band and Fermi level results in the nonzero density of states in the energy gap. These midgap states cause the crossover phenomenon and the divergence of localization length at a critical disorder strength Uc in the finite SSH system. In this study, we numerically computed the mean winding number and localization length for the disordered SSH system. We find that the disorder is able to drive phase transitions between different mean winding numbers: $w=0\ensuremath{\rightarrow}1$, $0\ensuremath{\rightarrow}2$, $1\ensuremath{\rightarrow}2$, and $2\ensuremath{\rightarrow}1$ in the weak disorder regime. By investigating the wave function distribution and the self-energy, the nonzero mean winding numbers correspond to the so-called topological Anderson insulating (TAI) phases. The finite size scaling for the mean winding number in the TAI phase is shown. For describing the phase transitions in the thermodynamic limit, we apply the criterion of band gap closure resulting from the broadening of energy band and Fermi level to determine the critical disorder strength. The critical disorder strength for self-consistent Born approximation (SCBA) UcSCBA and that for first Born approximation (FBA) UcFBA are numerically calculated. UcSCBA is found to match with Uc qualitatively. Nonetheless, SCBA indicates the different roles of band shifts and Fermi level broadening near the topological phase transitions. Band shift/Fermi level broadening is more dominant for the transitions from low-to-high/high-to-low winding number. Interestingly, for the transition from bulk insulator to TAI, UcFBA is quantitatively closer to Uc than UcSCBA as long as the renormalized band gap is zero within FBA.