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Anomalous heat transport in harmonic long-range chains

Francesco Andreucci

Published Dec 11, 2023• 0 citations
OpenAlex

Abstract

In recent years there has been a remarkable rise of interest in the study of heat transport in long-range systems, i.e, systems in which the interaction between constituents scales as a power-law of their distance. The results present in the literature suggest that heat transport in long-range systems can be anomalous, but the understanding of transport properties in these systems remains a difficult and open problem, especially if one wants to obtain analytical results. We thus decided to analyze heat transport properties in linear long-range chains in one dimension, and this work contains the results of this analysis. We begin by considering the simplest long-range system: the fully-connected chain coupled to two external baths. In this case we are able to analytically extract the scaling of the heat-flux with the system's size: this system thus proved a reference result for more complex interactions. We then consider the case of a power-law long-range chain coupled to two baths: in this case we show numerically that the heat flux scales as non-trivial power of system's size and we provide an analytical estimate of the exponent. Finally, we consider a power-law long-range ring with stochastic collisions: in this case we are able to exactly compute the scaling exponent of the thermal conductivity using the Green-Kubo formula, and we find good agreement between our results and numerical simulations.

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In recent years there has been a remarkable rise of interest in the study of heat transport in long-range systems, i.e, systems in which the interaction between constituents scales as a power-law of their distance. The results present in the literature suggest that heat transport in long-range systems can be anomalous, but the understanding of transport properties in these systems remains a difficult and open problem, especially if one wants to obtain analytical results. We thus decided to analyze heat transport properties in linear long-range chains in one dimension, and this work contains the results of this analysis. We begin by considering the simplest long-range system: the fully-connected chain coupled to two external baths. In this case we are able to analytically extract the scaling of the heat-flux with the system's size: this system thus proved a reference result for more complex interactions. We then consider the case of a power-law long-range chain coupled to two baths: in this case we show numerically that the heat flux scales as non-trivial power of system's size and we provide an analytical estimate of the exponent. Finally, we consider a power-law long-range ring with stochastic collisions: in this case we are able to exactly compute the scaling exponent of the thermal conductivity using the Green-Kubo formula, and we find good agreement between our results and numerical simulations.
We consider heat transfer in one-dimensional systems with long-range interactions. It is known that typical short-range interacting systems shows anomalous behavior in heat transport when total momentum is conserved, whereas momentum-nonconserving systems do not exhibit anomaly. In this study, we focus on the effect of long-range interaction. We propose an exactly solvable model that reduces to the so-called momentum-exchange model in the short-range interaction limit. We exactly calculate the asymptotic time decay in the energy current correlation function, which is related to the thermal conductivity via the Green-Kubo formula. From the time decay of the current correlation, we show three qualitatively crucial results. First, the anomalous exponent in the time-decay continuously changes as a function of the index of the long-range interaction. Second, there is a regime where the current correlation diverges with increasing the system size with fixed time, and hence, the exponent of the time decay cannot be defined. Third, even momentum-nonconserving systems can show the anomalous exponent indicating anomalous heat transport. Higher dimensions are also considered, and we found that long-range interaction can induce the anomalous exponent even in three-dimensional systems.
We discuss results on the dynamics of thermalization for a model with Gaussian interactions between two classical many-body systems trapped in external harmonic potentials. Previous work showed an approximate power-law scaling of the interaction energy with the number of particles, with particular focus on the dependence of the anomalous exponent on the interaction strength. Here we explore the role of the interaction range in determining anomalous exponents, showing that it is a more relevant parameter to differentiate distinct regimes of responses of the system. More specifically, on varying the interaction range from its largest values while keeping the interaction strength constant, we observe a crossover from an integrable system, approximating the Caldeira-Leggett interaction term in the long-range limit, to an intermediate interaction range in which the system manifests anomalous scaling, and finally to a regime of local interactions in which anomalous scaling disappears. A Fourier analysis of the interaction energy shows that nonlinearities give rise to an effective bath with a broad band of frequencies, even when starting with only two distinct trapping frequencies, yielding efficient thermalization in the intermediate regime of interaction range. We provide qualitative arguments, based on an analogous Fourier analysis of the standard map, supporting the view that anomalous scaling and features of the Fourier spectrum may be used as proxies to identify the role of chaotic dynamics. Our work, that encompasses models developed in different contexts and unifies them in a common framework, may be relevant to the general understanding of the role of nonlinearities in a variety of many-body classical systems, ranging from plasmas to trapped atoms and ions.
We study the heat conductivity in Anderson insulators in the presence of a power-law interaction. Particle-hole excitations built on localized electron states are viewed as two-level systems randomly distributed in space and energy and coupled due to electron-electron interaction. A small fraction of these states form resonant pairs that in turn build a complex network allowing for energy propagation. We identify the character of energy transport through this network and evaluate the thermal conductivity. For physically relevant cases of two-dimensional and three-dimensional spin systems with 1/r31/{r}^{3}1/r3 dipole-dipole interaction (originating from the conventional 1/r1/r1/r Coulomb interaction between electrons), the found thermal conductivity $\ensuremath{\kappa}$ scales with temperature as $\ensuremath{\kappa}\ensuremath{\propto}{T}^{3}$ and $\ensuremath{\kappa}\ensuremath{\propto}{T}^{4/3}$, respectively. Our results may be of relevance also to other realizations of random spin Hamiltonians with long-range interactions.
Recent experimental progress on the realization of quantum systems with highly controllable long-range interactions has impelled the study of quantum phase transitions in low-dimensional systems with power-law couplings. Long-range couplings mimic higher-dimensional effects in several physical contexts. Here, we provide the exact relation between the spectral dimension d at the band bottom and the exponent α that tunes the range of power-law hoppings of a one-dimensional ideal lattice Bose gas. We also develop a finite-size scaling analysis to obtain some relevant critical exponents and the critical temperature of the BEC transition. In particular, an irrelevant dangerous scaling field has to be taken into account when the hopping range is sufficiently large to make the effective dimensionality d>4.
There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance rrr as $1∕{r}^{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}0$. Here, we review old nonextensive scaling. In addition, we propose a Hamiltonian scaled by $2\frac{{(N∕2)}^{1\ensuremath{-}\ensuremath{\alpha}}\ensuremath{-}1}{1\ensuremath{-}\ensuremath{\alpha}}$ that explicitly includes symmetry of the lattice and dependence on the size NNN of the system. The approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.
We explore transport properties in a disordered nonlinear chain of classical harmonic oscillators, and thereby identify a regime exhibiting behavior analogous to that seen in quantum many-body-localized systems. Through extensive numerical simulations of this system connected at its ends to heat baths at different temperatures, we computed the heat current and the temperature profile in the nonequilibrium steady state as a function of system size N, disorder strength Δ, and temperature T. The conductivity κ_{N}, obtained for finite length (N), saturates to a value κ_{∞}>0 in the large N limit, for all values of disorder strength Δ and temperature T>0. We show evidence that for any Δ>0 the conductivity goes to zero faster than any power of T in the (T/Δ)→0 limit, and find that the form κ_{∞}∼e^{-B|ln(CΔ/T)|^{3}} fits our data. This form has earlier been suggested by a theory based on the dynamics of multioscillator chaotic islands. The finite-size effect can be κ_{N}<κ_{∞} due to boundary resistance when the bulk conductivity is high (the weak disorder case), or κ_{N}>κ_{∞} due to direct bath-to-bath coupling through bulk localized modes when the bulk is weakly conducting (the strong disorder case). We also present results on equilibrium dynamical correlation functions and on the role of chaos on transport properties. Finally, we explore the differences in the growth and propagation of chaos in the weak and strong chaos regimes by studying the classical version of the out-of-time-ordered commutator.