Most of the investigations to date on tight-binding, quantum percolation models focused on the quantum percolation threshold, i.e., the analog to the Anderson transition. It appears to occur if roughly 30% of the hopping terms are actually present. Thus, models in the delocalized regime may still be substantially disordered, hence analyzing their transport properties is a nontrivial task which we pursue in the paper at hand. Using a method based on quantum typicality to numerically perform linear response theory we find that conductivity and mean free paths are in good accord with results from very simple heuristic considerations. Furthermore we find that depending on the percentage of actually present hopping terms, the transport properties may or may not be described by a Drude model. An investigation of the Einstein relation is also presented.
We consider sequences of measurements implemented by positive operator valued measures (POVMs). Starting from the assumption that these sequences may be described as consistent and Markovian, even and especially for closed quantum systems, we identify properties of the equilibrium state that coincide with the properties of typical pure quantum states. We define a physical entropy that converges against the standard entropies in the approach to equilibrium. Furthermore, strict limits to its possible decrease are derived on the basis of Renyi entropies. It is demonstrated that Landauer's principle follows directly from these limits. Since the above assumptions are rather strong, we exemplify the fact that they may nevertheless apply by checking them numerically for some transition paths in a concrete model.
We demonstrate that the concept of quantum typicality allows for significant progress in the study of real-time spin dynamics and transport in quantum magnets. To this end, we present a numerical analysis of the spin-current autocorrelation function of the antiferromagnetic and anisotropic spin-1/2 Heisenberg chain as inferred from propagating only a single pure state, randomly chosen as a "typical" representative of the statistical ensemble. Comparing with existing time-dependent density-matrix renormalization group data, we show that typicality is fulfilled extremely well, consistent with an error of our approach, which is perfectly under control and vanishes in the thermodynamic limit. In the long-time limit, our results provide for a new benchmark for the enigmatic spin Drude weight, which we obtain from chains as long as L=33 sites, i.e., from Hilbert spaces of dimensions almost O(104) larger than in existing exact-diagonalization studies.
A key feature of nonequilibrium thermodynamics is the Markovian, deterministic relaxation of coarse observables such as, for example, the temperature difference between two macroscopic objects which evolves independently of almost all details of the initial state. We demonstrate that the unitary dynamics for moderately sized spin-1/2 systems may yield the same type of relaxation dynamics for a given magnetization difference. This observation might contribute to the understanding of the emergence of thermodynamics within closed quantum systems.
We investigate transport properties of topologically disordered three-dimensional one-particle tight-binding models, featuring site-distance-dependent hopping terms. We start from entirely disordered systems into which we gradually introduce some short-range order by numerically performing a pertinent structural relaxation using local site-pair interactions. Transport properties of the resulting models within the delocalized regime are analyzed numerically using linear response theory. We find that even though the generated order is very short ranged, transport properties such as conductivity or mean free path scale significantly with the degree of order. Mean free paths may exceed the site-pair correlation length. It is furthermore demonstrated that while the totally disordered model is not in accord with a Drude- or Boltzmann-type description, moderate degrees of order suffice to render such a picture valid.
We aim at quantitatively determining transport parameters like conductivity, mean free path, etc., for simple models of spatially completely disordered quantum systems, comparable to the systems which are sometimes referred to as Lifshitz models. While some low-energy eigenstates in such models always show Anderson localization, we focus on models for which most states of the full spectrum are delocalized, i.e., on the metallic regime. For the latter we determine transport parameters in the limit of high temperatures and low fillings using linear response theory. The Einstein relation (proportionality of conductivity and diffusion coefficient) is addressed numerically and analytically and found to hold. Furthermore, we find the transport behavior for some models to be in accord with a Boltzmann equation, i.e., rather long mean free paths, exponentially decaying currents, while this does not apply to other models even though they are also almost completely delocalized.
We aim at deriving an equation of motion for specific sums of momentum mode occupation numbers from models for electrons in periodic lattices experiencing elastic scattering, electron-phonon scattering, or electron-electron scattering. These sums correspond to "grains" in momentum space. This equation of motion is supposed to involve only a moderate number of dynamical variables and/or exhibit a sufficiently simple structure such that neither its construction nor its analyzation or solution requires substantial numerical effort. To this end we compute, by means of a projection operator technique, a linear(ized) collision term which determines the dynamics of the above grain sums. This collision term results as nonsingular finite-dimensional rate matrix and may thus be inverted regardless of any symmetry of the underlying model. This facilitates calculations of, e.g., transport coefficients, as we demonstrate for a three-dimensional Anderson model featuring weak disorder.
We show that the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time. This is meant to apply to Schrödinger type dynamics in high dimensional Hilbert spaces. As a consequence individual dynamics of expectation values are then typically well described by the ensemble average. Our approach is based on the Hilbert space average method. We support the analytical investigations with numerics obtained by exact diagonalization of the full time-dependent Schrödinger equation for some pertinent, abstract Hamiltonian model. Furthermore, we discuss the implications for the applicability of projection operator methods with respect to initial states, as well as for irreversibility in general.
We investigate transport properties of one-dimensional fermionic tight binding models featuring nearest and next-nearest neighbor hopping, where the fermions are additionally subject to a weak short range mutual interaction. To this end we employ a pertinent approach which allows for a mapping of the underlying Schrödinger dynamics onto an adequate linear quantum Boltzmann equation. This approach is based on a suitable projection operator method. From this Boltzmann equation we are able to numerically obtain diffusion coefficients in the case of nonvanishing next-nearest neighbor hopping, i.e., the nonintegrable case, whereas the diffusion coefficient diverges without next-nearest neighbor hopping. For the latter case we analytically investigate the decay behavior of the current with the result that arbitrarily small parts of the current relax arbitrarily slowly which suggests anomalous diffusive transport behavior within the scope of our approach.
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