We find that, in the presence of weak incoherent effects from surrounding environments, the low-temperature conductance of nearest-neighbor tight-binding fermionic chains exhibits a counter-intuitive monotonic growth with system length when Fermi energy is near the band edges, indicating a superballistic scaling. This fascinating environment-assisted superballistic scaling of conductance occurs over a finite but extended regime of system lengths. This regime can be systematically expanded by decreasing the coupling to the surrounding environments and by reducing temperature. This behavior is robust against weak disorder and slight shifts from band edge, although the extent of the superballistic scaling regime is affected by them. We give precise predictions of how the superballistic scaling regime depends on coupling to surrounding environments, disorder strength, shifts from band edge, and temperature. There is no corresponding analog of this behavior in isolated systems. The superballistic scaling stems from an intricate interplay of incoherent effects from surrounding environments and exceptional points of the system's transfer matrix that occur at every band edge.
We find that, in the presence of weak incoherent effects from surrounding environments, the low-temperature conductance of nearest-neighbor tight-binding fermionic chains exhibits a counter-intuitive monotonic growth with system length when Fermi energy is near the band edges, indicating a superballistic scaling. This fascinating environment-assisted superballistic scaling of conductance occurs over a finite but extended regime of system lengths. This regime can be systematically expanded by decreasing the coupling to the surrounding environments and by reducing temperature. This behavior is robust against weak disorder and slight shifts from band edge, although the extent of the superballistic scaling regime is affected by them. We give precise predictions of how the superballistic scaling regime depends on coupling to surrounding environments, disorder strength, shifts from band edge, and temperature. There is no corresponding analog of this behavior in isolated systems. The superballistic scaling stems from an intricate interplay of incoherent effects from surrounding environments and exceptional points of the system's transfer matrix that occur at every band edge.
Recently, it has been shown that, in one-dimensional fermionic systems, close to band edges, the zero temperature conductance scales as 1/N2, where N is the system length. This universal subdiffusive scaling of conductance at band edges has been tied to an exceptional point (EP) of the transfer matrix of the system that occurs at every band edge. Further, in the presence of bulk dephasing probes, this EP has been shown to lead to a counterintuitive superballistic scaling of conductance, where the conductance increases with N over a finite but large regime of system lengths. In this work, we explore how these behaviors are affected by the order of the transfer matrix EP at the band edge. We consider a one-dimensional fermionic lattice chain with a finite range of hopping. Depending on the range of hopping and the hopping parameters, this system can feature band edges which correspond to arbitrarily higher order EPs of the associated transfer matrix. Using this system we establish in generality that, in the absence of bulk dephasing, surprisingly, the universal 1/N2 scaling of conductance is completely unaffected by the order of the EP. This is despite the fact that existence of transfer matrix EP is crucial for such behavior. In the presence of bulk dephasing, however, the phase coherence length, the extent of the superballisitic scaling regime, and the exponent of superballistic scaling all encode the order of the transfer matrix EP.
Dynamical conductivity in a disordered one-dimensional model of interacting fermions is studied numerically at high temperatures and in the weak-interaction regime in order to find a signature of many-body localization and vanishing dc transport coefficients. On the contrary, we find in the regime of moderately strong local disorder that the dc conductivity ${\ensuremath{\sigma}}_{0}$ scales linearly with the interaction strength while being exponentially dependent on the disorder. According to the behavior of the charge stiffness evaluated at the fixed number of particles, the absence of the many-body localization seems related to an increase in the effective localization length with the interaction.
Abstract We investigate two one-dimensional tight-binding models with disorder that have extended states at zero energy. We use exact and partial diagonalisation of the Hamiltonian to obtain the eigenmodes and the associated participation ratios, and the transfer-matrix method to determine the localisation length. The first model has no on-site disorder, but random couplings. While the participation ratio remains finite at zero energy, the localisation length diverges logarithmically as the energy goes to zero. We provide an intuitive derivation of this logarithmic divergence based on the weak coupling of the two sublattices. The second model has a conserved quantity as the row sums of the Hamiltonian are zero. This model can be represented as a harmonic chain with random couplings, or as a diffusion model on a lattice with random links. We find, in agreement with existing analytical calculations, that the number of system-spanning eigenmodes increases proportionally to the square root of the system size, and we related this power law to other power laws that characterise the scaling behaviour of the eigenmodes, the participation ratio, the localisation length, and their dependence on energy and system size. When disorder is so strong that the smallest hopping terms can be arbitrarily close to zero, all these power laws change, and we show a crossover between the two scaling regimes. All these results are explained by intuitive arguments based on scaling.
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.
Charge and energy are expected to diffuse in interacting systems of fermions at finite temperatures, even in the absence of disorder, with the interactions inducing a crossover from the coherent and ballistic streaming of quasi-particles at early times, to incoherent diffusive behavior at late times. The relevant crossover timescales and the transport coefficients are both controlled by the strength of interactions. In this work we develop a numerical method to simulate such systems at high temperatures, applicable in a wide range of interaction strengths, by adapting Dissipation-assisted Operator Evolution (DAOE) to fermions. Our fermion DAOE, which approximates the exact dynamics by systematically discarding information from high n-point functions, is tailored to capture non-interacting dynamics exactly, thus providing a good starting point for the weakly interacting problem. Applying our method to a microscopic model of weakly interacting fermions, we numerically demonstrate that the crossover from ballistic to diffusive transport happens at a time tD∼1/Δ2 and that the diffusion constant similarly scales as D∼1/Δ2, where Δ is the interaction strength. We substantiate this scaling with a Fermi's golden rule calculation in the operator spreading picture, interpreting tD as the fermion-fermion scattering time and lifetime of the single-particle Green's function.
Abstract We study the transport properties of a one-dimensional quantum system with disorder. We numerically compute the frequency dependence of the conductivity of a fermionic chain with nearest-neighbor interaction and a random chemical potential by using the Chebyshev matrix product state (CheMPS) method. As a benchmark, we investigate the noninteracting case first. Comparison with exact diagonalization and analytical solutions demonstrates that the results of CheMPS are reliable over a wide range of frequencies. We then calculate the dynamical conductivity spectra of the interacting system for various values of the interaction and disorder strengths. In the high-frequency regime, the conductivity decays as a power law, with an interaction-dependent exponent. This behavior is qualitatively consistent with the bosonized field theory predictions, although the numerical evaluation of the exponent shows deviations from the analytically expected values. We also compute the characteristic pinning frequency at which a peak in the conductivity appears. We confirm that it is directly related to the inverse of the localization length, even in the interacting case. We demonstrate that the localization length follows a power law of the disorder strength with an exponent dependent on the interaction, and find good quantitative agreement with the field theory predictions. In the low-frequency regime, we find a behavior consistent with the one of the noninteracting system ω2(lnω)2 ω 2 ( ln ω ) 2 independently of the interaction. We discuss the consequences of our finding for experiments in cold atomic gases. Graphical abstract