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arXiv:2109.08390 · Physical review. B./Physical review. B

Modified Matthiessen's rule: More scattering leads to less resistance

Marko Žnidarič

Published Jan 28, 2022• 18 citations
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Abstract

We study the breaking of integrability by a finite density of dilute impurities, specifically the emerging diffusive transport. Provided the distance between impurities (localized perturbations) is large, one would expect the scattering rates to be additive, and therefore, the resistivity to be proportional to the number of impurities (the so-called Matthiessen's rule). We show that this is, in general, not the case. If transport is anomalous in the original integrable system without impurities, the diffusion constant in the nonintegrable system at low impurity density has a nontrivial power-law dependence on the impurity density, with the power being determined by the dynamical scaling exponent of anomalous transport. We also find a regime at high impurity density in which, counterintuitively, adding more impurities to an already diffusive system increases transport rather than decreases it.

DOI
10.1103/physrevb.105.045140

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We study the breaking of integrability by a finite density of dilute impurities, specifically the emerging diffusive transport. Provided the distance between impurities (localized perturbations) is large, one would expect the scattering rates to be additive, and therefore, the resistivity to be proportional to the number of impurities (the so-called Matthiessen's rule). We show that this is, in general, not the case. If transport is anomalous in the original integrable system without impurities, the diffusion constant in the nonintegrable system at low impurity density has a nontrivial power-law dependence on the impurity density, with the power being determined by the dynamical scaling exponent of anomalous transport. We also find a regime at high impurity density in which, counterintuitively, adding more impurities to an already diffusive system increases transport rather than decreases it.
We study how perturbations affect dynamics of integrable many-body quantum systems, causing transition from integrability to chaos. Looking at spin transport in the Heisenberg chain with impurities we find that in the thermodynamic limit transport gets diffusive already at an infinitesimal perturbation. Small extensive perturbations therefore cause an immediate transition from integrability to chaos. Nevertheless, there is a remnant of integrability encoded in the dependence of the diffusion constant on the impurity density, namely, at small densities it is proportional to the square root of the inverse density, instead of to the inverse density as would follow from Matthiessen's rule. We show that Matthiessen's rule has to be modified in nonballistic systems. Results also highlight a nontrivial role of interacting scattering on a single impurity, and that there is a regime where adding more impurities can actually increase transport.
Abstract We study anomalous transport arising in disordered one-dimensional spin chains, specifically focusing on the subdiffusive transport typically found in a phase preceding the many-body localization transition. Different types of transport can be distinguished by the scaling of the average resistance with system’s length. We address the following question: what is the distribution of resistance over different disorder realizations, and how does it differ between transport types? In particular, an often evoked so-called Griffiths picture, that aims to explain slow transport as being due to rare regions of high disorder, would predict that the diverging resistivity is due to fat power-law tails in the resistance distribution. Studying many-particle systems with and without interactions we do not find any clear signs of fat tails. The data is compatible with distributions that decay faster than any power law required by the fat tails scenario. Among the distributions compatible with the data, a simple additivity argument suggests a Gaussian distribution for a fractional power of the resistance.
The anomalous spin diffusion of the integrable easy-axis Heisenberg chain originates in the ballistic transport of symmetry sectors with nonzero magnetization. Ballistic transport is replaced by normal dissipative transport in all magnetization sectors upon introducing the integrability-breaking perturbations, including external driving. Such behavior implies that the diffusion constant obtained for the integrable model is relevant for the spread of spin excitations but not for the spin conductivity. We present numerical results for closed systems and driven open systems, indicating that the diffusion constant shows a discontinuous variation as the function of perturbation strength.
Integrable and nonintegrable systems have very different transport properties. In this paper, we highlight these differences for specific one-dimensional models of interacting lattice fermions. The technique used is a finite temperature numerical calculation of the adiabatic stiffness (also called the Drude weight or charge stiffness) and isothermal stiffness (also called the ``Meissner'' stiffness) in electrical and energy transport, and the momentum dependent dynamical conductivities $\ensuremath{\sigma}(q,\ensuremath{\omega})$ and $\ensuremath{\kappa}(q,\ensuremath{\omega})$. We apply a flux twist to break the Kramers degeneracy; thus, allowing us to focus on the effect of the dynamical degeneracies in the integrable system. In this situation, we show that the isothermal stiffness goes to zero rapidly with the system size for both types of systems even at high temperatures; while the adiabatic stiffness appears to go to zero in the nonintegrable system and to a finite value in the integrable one. We analyze this difference in terms of the statistics of the current matrix elements and the degeneracies of the systems, and show that in the integrable system, despite the presence of degeneracies, the dominant contribution to the adiabatic stiffness comes from large-current-carrying nondegenerate states. We also show that energy transport at nonzero $\ensuremath{\omega}$ and qqq occurs within a banded continuum in the integrable system indicative of ballistic transport while the nonintegrable system shows diffusion but with the existence of overdamped excitations at large values of momentum.
We consider an impurity in a one-dimensional weakly interacting Bose gas and analytically calculate the density profile of the Bose gas. Within the mean-field approximation, by increasing the distance from the impurity, the Bose gas density saturates exponentially fast to its mean thermodynamic-limit value at distances beyond the healing length. The effect of quantum fluctuations drastically changes this behavior, leading to a power-law decay of the density deviation from the mean density. At distances longer than the healing length and shorter than a new length scale proportional to the impurity coupling strength, the power-law exponent is 2, while at the longest distances the corresponding exponent becomes 3. The latter crossover does not exist in two special cases. The first one is realized for infinitely strongly coupled impurity; then the density deviation always decays with the exponent 2. The second special case occurs when the new length scale is smaller than the healing length, i.e., at weak impurity coupling; then the density deviation always decays with the exponent 3. The obtained results are exact in the impurity coupling strength and account for the leading order in the interaction between the particles of the Bose gas.
Strange metals are highly entangled gapless states of matter that exhibit anomalous transport, such as linear in temperature resistivity, over more than a decade of temperature. Why a single power law should be so robust is an open question. We propose a scenario in which interactions enhance the domain of certain scattering regimes, effectively suppressing other ``would-be regimes." We test this proposal in a one-dimensional Luttinger liquid coupled to a one-dimensional acoustic phonon. We use the memory matrix formalism to calculate the dc electrical and thermal conductivities at low and high temperatures, relative to the Debye cutoff on phonon frequencies, in both the ``clean" (umklapp scattering) and ``dirty" (disorder scattering) limits. We find the crossover temperature separating the low and high temperature regimes to be interaction-dependent, with repulsive interactions substantially increasing it, generally by more than an order of magnitude. This provides a concrete illustration for how interactions can extend a single transport regime over a wider temperature range.