We present computational data on the thermal conductivity of nonlinear waves in disordered chains. Disorder induces Anderson localization for linear waves and results in a vanishing conductivity. Cubic nonlinearity restores normal conductivity, but with a strongly temperature-dependent conductivity $κ(T)$. We find indications for an asymptotic low-temperature $κ\sim T^4$ and intermediate temperature $κ\sim T^2$ laws. These findings are in accord with theoretical studies of wave packet spreading, where a regime of strong chaos is found to be intermediate, followed by an asymptotic regime of weak chaos (EPL 91 (2010) 30001).
We study the thermal conductivity, at fixed positive temperature, of a disordered lattice of harmonic oscillators, weakly coupled to each other through anharmonic potentials. The interaction is controlled by a small parameter $ε> 0$. We rigorously show, in two slightly different setups, that the conductivity has a non-perturbative origin. This means that it decays to zero faster than any polynomial in $ε$ as $ε\rightarrow 0$. It is then argued that this result extends to a disordered chain studied by Dhar and Lebowitz, and to a classical spins chain recently investigated by Oganesyan, Pal and …
The amorphous solids can be theoretically modeled by anharmonic disordered lattices. However, most of theoretical studies on thermal conductivity in anharmonic disordered lattices only focus on the potentials of hard-type (HT) anharmonicity. Here we study the thermal conductivity $κ$ of one-dimensional (1D) disordered lattices with both hard- and soft-type (ST) anharmonic on-site potentials. It is found, via both direct molecular dynamic simulations and theoretical method, that the anharmonicity dependence of $κ$ in the HT model is nonmonotonous, while in the ST model is monotonously increased…
We study heat conduction in one dimensional mass disordered harmonic and anharmonic lattices. It is found that the thermal conductivity $κ$ of the disordered anharmonic lattice is finite at low temperature, whereas it diverges as $κ\sim N^{0.43}$ at high temperature. Moreover, we demonstrate that a unique nonequilibrium stationary state in the disordered harmonic lattice does not exist at all.
In this paper, thermal transport in bond-disordered harmonic chains is revisited in detail using a nonequilibrium Green's function formalism. For strong bond disorder, thermal conductivity is independent of the system size. However, kinetic temperatures described by the local number of states coupling to external heat reservoirs are anomalous since they form a nonlinear profile in the interior of the system. Both results are accounted for in a unified manner in terms of the frequency-dependent localization length. From this argument, we derive a generic formula describing the asymptotic profil…
Anisotropic disordered system are studied in this work within the random barrier model. In such systems the transition probabilities in different directions have different probability density functions. The frequency-dependent conductivity at low temperatures is obtained using an effective medium approximation. It is shown that the isotropic universal ac-conduction law, $σ\ln σ=u$, is recovered if properly scaled conductivity ($σ$) and frequency ($u$) variables are used.
We consider a $d$-dimensional disordered harmonic chain (DHC) perturbed by an energy conservative noise. We obtain uniform in the volume upper and lower bounds for the thermal conductivity defined through the Green-Kubo formula. These bounds indicate a positive finite conductivity. We prove also that the infinite volume homogenized Green-Kubo formula converges.
Heat conduction in three types of 1D channels are studied. The channels consist of two parallel walls, right triangles as scattering obstacles, and noninteracting particles. The triangles are placed along the walls in three different ways: (a) periodic, (b) disordered in height, and (c) disordered in position. The Lyapunov exponents in all three models are zero because of the flatness of triangle sides. It is found numerically that the temperature gradient can be formed in all three channels, but the Fourier heat law is observed only in two disordered ones. The results show that there might be…
The escape of energy injected into one site in a disordered chain of nonlinear oscillators is examined numerically. When the disorder has a `fractal' pattern, the decay of the residual energy at the injection site can be fit to a stretched exponential with an exponent that varies continuously with the control parameter. At low temperature, we see evidence that energy can be trapped for an infinte time at the original site, i.e. classical many body localization.
We study the energy absorption in real time of a disordered quantum spin chain subjected to coherent monochromatic periodic driving. We determine characteristic fingerprints of the well-known ergodic (Floquet-ETH for slow driving/weak disorder) and many-body localized (Floquet-MBL for fast driving/strong disorder) phases. In addition, we identify an intermediate regime, where the energy density of the system -- unlike the entanglement entropy a local and bounded observable -- grows logarithmically slowly over a very large time window.
These are notes for lectures delivered at the NATO ASI on Dynamics in Leiden, The Netherlands, in July 1998. The quantum kinetic theory for noninteracting electrons in a disordered solid is introduced and discussed. We first use many-body theory to derive the quantum Boltzmann equation that describes transport and time correlation functions in this system. Particular attention is paid to the calculation of the electrical conductivity and the density response function. We then consider corrections to the Boltzmann equation due to wave interference effects. The disorder expansion of the conducti…
Transport quantities of the classical spin chain with the quenched disorder in the antiferromagnetic coupling $J_i$ are evaluated using the dynamical simulation at finite temperatures $T>0$ . Since the classical model is nonintegrable, spin and thermal conductivities remain finite even in the pure case. On the other hand, the role of disorder becomes crucial at low $T $ leading to a vanishing transport due to the Anderson localization within the linearized regime. The crossover from the insulator to the conductor appears both for the spin and thermal transport at quite low $T^* \ll J$. Still t…
In one-dimensional electronic systems with strong repulsive interactions, charge excitations propagate much faster than spin excitations. Such systems therefore have an intermediate temperature range [termed the "spin-incoherent Luttinger liquid'" (SILL) regime] where charge excitations are "cold" (i.e., have low entropy) whereas spin excitations are "hot." We explore the effects of charge-sector disorder in the SILL regime in the absence of external sources of equilibration. We argue that the disorder localizes all charge-sector excitations; however, spin excitations are protected against ful…
We consider $d$-dimensional chains of (an)harmonic oscillators we perturb by a noise conserving energy or energy and momentum. We review the thermal conduction properties we obtained for these systems and conclude by several open questions.
We investigate the mechanism of heat conduction in ordered and disordered harmonic onedimensional chains within the quantum mechanical Langevin method. In the case of the disordered chains we find indications for normal heat conduction which means that there is a finite temperature gradient but we cannot clearly decide whether the heat resistance increases linearly with the chain length. Furthermore we observe characteristic quantum mechanical features like Bose-Einstein statistics of the occupation numbers of the normal modes, freezing of the heat conductivity and the influence of the entangl…
We study a model of nonequilibrium quantum transport of particles and energy in a many-body system connected to mesoscopic Fermi reservoirs (the so-called meso-reservoirs). We discuss the conservation laws of particles and energy within our setup as well as the transport properties of quasi-periodic and disordered chains.
We study a frustrated mixed spin chain with side chains, where the spin species and the exchange interactions are spatially varied. A nonlinear sigma model method is formulated for this model, and a phase diagram with two disordered spin-gap phases is obtained for typical cases. Among them we examine the case with a main chain consisting of an alternating array of spin-1 and spin-1/2 sites and side chains each of a single spin-1/2 site in great detail. Based on numerical, perturbational, and variational approaches, we propose a singlet cluster solid picture for each phase, where the ground sta…
We uncover a novel mechanism for inducing a gapful phase in interacting many-body quantum chains. The mechanism is nonperturbative, being triggered only in the presence of both strong interactions and strong aperiodic (disordered) modulation. In the context of the critical antiferromagnetic spin-1/2 XXZ chain, we identify an emerging dimerization which removes the system from criticality and stabilizes the novel phase. This mechanism is shown to be quite general in strongly interacting quantum chains in the presence of strongly modulated quasiperiodic disorder which is, surprisingly, perturbat…
The multifractal characterization of the distribution over disorder of the mean first-passage time in a finite chain is revisited. Both, absorbing-absorbing and reflecting-absorbing boundaries are considered. Two models of dichotomic disorder are compared and our analysis clarifies the origin of the multifractality. The phenomenon is only present when the diffusion is anomalous.
Negative differential conductivity is reported for the far from equilibrium quantum spin transport in the insulating regime (J_x < J_z) of finite Heisenberg XXZ spin 1/2 chains. The phenomenon is reproduced analytically for small chains of N=4 spins and further analyzed numerically, for up to N=16, using an efficient pure-state simulation with stochastic spin baths.
