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Physical Review E

Universality classes for phonon relaxation and thermal conduction in one-dimensional vibrational systems

Santhosh G., Deepak Kumar

Published Oct 14, 2011• 3 citations
Publisher

Abstract

We study phonon relaxation in chains of particles coupled through polynomial-type pair-interaction potentials and obeying quantum dynamics. We present detailed calculations for the sixth-order potential and find that the wave-vector-dependent relaxation rate follows a power-law behavior, $\ensuremath{\Gamma}(q)\ensuremath{\sim}{q}^{\ensuremath{\delta}}$, with $\ensuremath{\delta}=5/3$, which is identical to that of the fourth-order potential. We argue through diagrammatic analysis that this is a generic feature of even-power potentials. Our earlier analysis has shown that $\ensuremath{\delta}=3/2$ when the leading-order term in the nonlinear potential is odd, suggesting that there are two universality classes for the phonon relaxation rates dependent on a simple property of the potential. This implies that the thermal conductivity $\ensuremath{\kappa}$ which diverges as a function of chain size $N$ as $\ensuremath{\kappa}\ensuremath{\propto}{N}^{\ensuremath{\alpha}}$ also has two universal behaviors, in that $\ensuremath{\alpha}=1\ensuremath{-}1/\ensuremath{\delta}$ as follows from a finite-size argument. We support these arguments by numerical calculations of conductivity for chains obeying classical dynamics for polynomial potentials of some even and odd powers.

DOI
10.1103/physreve.84.041119

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We study the low-temperature transport properties of clean one-dimensional spin-12\frac{1}{2}21​ chains coupled to phonons. Due to the presence of approximate conservation laws, the heat current decays very slowly giving rise to an exponentially large heat conductivity, $\ensuremath{\kappa}\ensuremath{\sim}{e}^{{T}^{*}/T}.$ As a result of an interplay of umklapp scattering and spinon-phonon coupling, the characteristic energy scale T∗{T}^{*}T∗ turns out to be of order ${\ensuremath{\Theta}}_{D}/2,$ where ${\ensuremath{\Theta}}_{D}$ is the Debye energy, rather than the magnetic exchange interaction JJJ---in agreement with recent measurements in SrCuO compounds. A large magnetic field h strongly affects the heat transport by two distinct mechanisms. First, it induces a linear spinon-phonon coupling, which alters the nature of the $\stackrel{\ensuremath{\rightarrow}}{T}0$ fixed point: the elementary excitations of the system are composite spinon-phonon objects. Second, the change of the magnetization and the corresponding change of the wave vector of the spinons strongly affects the way in which various umklapp processes can relax the heat current, leading to a characteristic fractal-like spiky behavior of $\ensuremath{\kappa}(T,h)$ as a function of h.
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(ln⁡ω)2\omega ^{2}(\ln \omega )^{2}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
Quasicritical exponents of one-dimensional models displaying a quasitransition at finite temperatures are examined in detail. The quasitransition is characterized by intense sharp peaks in physical quantities such as specific heat and magnetic susceptibility, which are reminiscent of divergences accompanying a continuous (second-order) phase transition. The question whether these robust finite peaks follow some power law around the quasicritical temperature is addressed. Although there is no actual divergence of these quantities at a quasicritical temperature, a power-law behavior fits precisely both ascending as well as descending parts of the peaks in the vicinity but not too close to a quasicritical temperature. The specific values of the quasicritical exponents are rigorously calculated for a class of one-dimensional models (e.g., Ising-XYZ diamond chain, coupled spin-electron double-tetrahedral chain, Ising-XXZ two-leg ladder, and Ising-XXZ three-leg tube), whereas the same set of quasicritical exponents implies a certain "universality" of quasitransitions of one-dimensional models. Specifically, the values of the quasicritical exponents for one-dimensional models are: α=α^{'}=3 for the specific heat, γ=γ^{'}=3 for the susceptibility and ν=ν^{'}=1 for the correlation length.
We investigate the mechanism of heat conduction in one-dimensional (1D) quantum FPU chain with quantum fluctuation in the framework of semiquantal molecular dynamics. In the semiquantal many-body simulations, a Hartree-type many-body wave function is adopted for a whole chain and a single-particle state of a particle is represented by a trial wave function with Jackiw–Kerman (JK) form. With the help of the Dirac's time-dependent variational principle (TDVP), a set of equations of the variational parameters contained in the JK wave packet is obtained, and it describes the quantum dynamics of the nonlinear lattices approximately. These equations not only prove highly efficient in recovering dynamics of classical heat conduction, but also allow exploring the case with quantum mechanical characteristics. As a consequence, we find the thermal conductivity diverges with system size as well as in the classical version when considering quantum fluctuation. Furthermore, in three different temperature regions it is observed that the enhancement of the quantum fluctuation increases the heat flux through the 1D quantum FPU chain.
This thesis consists in a theoretical study of some one-dimensional models. Special attention is given to simple but non-trivial soluble models. The study starts with a resume of the Green function method which is the formalism used throughout the work. A discussion of coupled electron-phonon and spin-phonon systems is presented and a decoupling scheme introduced. The approximation gives the transition temperature for one-dimensional systems exactly, namely T = 0, and consequently represents a great improvement on the mean-field approximation. The impure XY-chain in a transverse field is studied and the formal solution presented. The static and time-dependent correlations are discussed, and the effect of the boundary term reviewed in details. The one-impurity case is solved for the isotropic chain, and it is shown how to derive from it the open chain result in the thermodynamic limit. The heat-flux operator is obtained for the impure isotropic chain by solving the continuity equation for energy density. The dynamics of the one-dimensional transverse Ising model is discussed within several approximations. The limitations of each approximation are examined in detail and the time-dependent correlations are calculated using the most successful approximation. It is also shown that this approximation is only valid in the high temperature limit in which case no critical behaviour is to be observed. The one-impurity solution of the isotropic impure XY-chain is extended to many impurities in two different cases. In the first case the specific heat of the dilute chain is calculated in the framework of the average t-matrix approximation, and the results compared with exact numerical calculations for finite chains. In the second case the specific heat and thermal conductivity of the isotropic XY-chain in a random transverse field are calculated using the coherent-potential approximation (CPA). The results are used to explain the low temperature thermal properties of praseodymium ethyl sulphate. Finally, two interacting systems, an electron-phonon system and a spin-phonon system, are studied after a brief discussion of the so-called Peierls instability. The first model is solved exactly in the framework of the approximation discussed at the beginning of the work. It presents a giant Kohn anomaly at zero temperature which drives a Peierls transition. The renormalized modes are discussed, and the real and imaginary parts of the phonon propagator are presented for wave vector π⁄a and several coupling constants and temperatures. The spin-phonon system is also studied in the framework of the approximation discussed initially in the work, and it is chosen in such a way that the relevant properties are given in terms of Green functions calculated in the transverse Ising model. The results obtained for the dynamics of this model are used, and since they are valid only in the high temperature limit, the discussion is restricted to this temperature region. Since this region is far away from the critical one, no critical behaviour is observed, and the study is restricted to a simple discussion of modes. The real and imaginary parts of the renormalized phonon propagator are presented for the wave vector π⁄a, high temperature and various coupling constants.