Normal Heat Conduction in a Chain with a Weak Interparticle Anharmonic Potential

Pereira, Emmanuel; Falcao, Ricardo

doi:10.1103/physrevlett.96.100601

Cited by 58 publications

(46 citation statements)

References 19 publications

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“…The classic solution for the problem suggested by Peierls [3] was questioned after seminal numeric experiment of Fermi, Pasta and Ulam [4]; the latter has disproved common belief concerning fast thermalization and mixing in nonintegrable systems with weak nonlinearity. Despite large amount of work done, necessary and sufficient conditions for microscopic model of solid to obey macroscopically the Fourier law with finite and size -independent heat conduction coefficient [4][5][6][7][8][9][10][11][12][13][14][15][16] are not known yet. Numerous anomalies in the heat transfer in microscopic models of dielectrics were revealed by means of direct numeric simulation over recent years, including qualitatively different behavior of models of different types (with and without on-site potential) and dimensionality [1].…”

Section: Introductionmentioning

confidence: 99%

“…The only known exception with convergent heat conduction coefficient is the chain of coupled rotators [8,9]. As for the models without the moment conservation, their heat conduction is convergent [10,14], with exception of integrable models [11]. In two dimensions, the divergence still exists [1], although is reported to be of logarithmic rather than power-like type.…”

Section: Introductionmentioning

confidence: 99%

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Nonstationary heat conduction in one-dimensional models with substrate potential

et al. 2012

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The paper investigates non-stationary heat conduction in one-dimensional models with substrate potential. In order to establish universal characteristic properties of the process, we explore three different models -Frenkel-Kontorova (FK), phi4+ (φ 4 +) and phi4-(φ 4 −). Direct numeric simulations reveal in all these models a crossover from oscillatory decay of short-wave perturbations of the temperature field to smooth diffusive decay of the long-wave perturbations. Such behavior is inconsistent with parabolic Fourier equation of the heat conduction and clearly demonstrates the necessity of hyperbolic models. The crossover wavelength decreases with increase of average temperature. The decay patterns of the temperature field almost do not depend on the amplitude of the perturbations, so the use of linear evolution equations for temperature field is justified. In all model investigated, the relaxation of thermal perturbations is exponential -contrary to linear chain, where it follows a power law. However, the most popular lowest-order hyperbolic generalization of the Fourier law, known as Cattaneo-Vernotte (CV) or telegraph equation (TE) is not valid for description of the observed behavior of the models with on-site potential. In part of the models a characteristic relaxation times exhibit peculiar scaling with respect to the temperature perturbation wavelength. Quite surprisingly, such behavior is similar to that of well-known model with divergent heat conduction (Fermi-Pasta-Ulam chain) and rather different from the model with normal heat conduction (chain of rotators). Thus, the data on the non-stationary heat conduction in the systems with on-site potential do not fit commonly accepted concept of universality classes for heat conduction in one-dimensional models.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonstationary heat conduction in one-dimensional models with substrate potential

et al. 2012

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“…However, it is still largely an open problem in the gapless case, and, with few exceptions (see, e.g., [8,9,10,11]), the threshold exponents µ are not known. For the model (1), the most complete results so far were obtained by combining numerics with the algebraic Bethe ansatz [12,13]. The main limitation of this technique is very slow convergence towards the thermodynamic limit, which makes it very difficult to evaluate the exponent.…”

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confidence: 99%

Threshold Singularities in the Dynamic Response of Gapless Integrable Models

Cheianov

Pustilnik

2008

Phys. Rev. Lett.

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We develop a method of an asymptotically exact treatment of threshold singularities in dynamic response functions of gapless integrable models. The method utilizes the integrability to recast the original problem in terms of the low-energy properties of a certain deformed Hamiltonian. The deformed Hamiltonian is local, hence it can be analysed using the conventional field theory methods. We apply the technique to spinless fermions on a lattice with nearest-neighbors repulsion, and evaluate the exponent characterizing the threshold singularity in the dynamic structure factor.PACS numbers: 02.30. Ik, 71.10.Pm, 75.10.Pq One-dimensional (1D) interacting systems [1, 2, 3] occupy a special place in quantum physics. Although interactions have much stronger effect in 1D than in higher dimensions, it is often possible to evaluate observable quantities exactly. Besides forming the basis of our understanding of strong correlations, 1D models have long served as a test bed for various approximate methods.Many properties of interacting 1D systems can be understood by considering integrable models [2]. The paradigmatic example is N spinless fermions on a lattice with L cites with periodic boundary conditions,Here n m = ψ † m ψ m and ∆ is the repulsion strength [4]. The experimentally relevant [5] response function for the model (1) is the dynamic structure factor|f is an eigenstate of Eq. (1) with energy ǫ f , and |0 is the ground state with ǫ 0 being the ground state energy.In any 1D system, conservation laws restrict the support of correlation functions in (ω, q) plane. For example, S(q, ω) > 0 only at ω > ω 0 (q), see Fig. 1. On general grounds, S(q, ω) is expected to exhibit a power-law singularity at the threshold [6],Although exact eigenstates of integrable models can be constructed using the Bethe ansatz [2], evaluation of dynamic correlation functions is very difficult. A considerable progress was achieved in understanding gapful models [7]. However, it is still largely an open problem in the gapless case, and, with few exceptions (see, e.g., [8,9,10,11]), the threshold exponents µ are not known. For the model (1), the most complete results so far were obtained by combining numerics with the algebraic Bethe ansatz [12,13]. The main limitation of this technique is very slow convergence towards the thermodynamic limit, which makes it very difficult to evaluate the exponent. In the alternative Luttinger liquid approach [1,3,14], a 1D system is described by two parameters, the sound velocity v = (dω 0 /dq) q→0 , and the Luttinger parameter κ characterizing the interaction strength. At q → 2k F , Luttinger liquid theory yields Eq. (3) with the exponentAt q → 0, however, one finds S ∝ δ(ω − vq). The discrepancy with Eq. (3) is due to the omission in the fixed-point Luttinger liquid Hamiltonian [3,14] of the irrelevant in the RG sense operators [13,14] representing the spectrum nonlinearity. Indeed, for small q, most of the spectral weight of S(q, ω) is confined to a narrow interval of ω of the width δω ∼ ω 1 − ω 0 about ...

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“…The above results have to be contrasted with recent exact results for the dynamic response of the spinless fermion model [26][27][28][29], which -unsurprisingly -are not correctly reproduced within CDFT-LDA. For example, a continuum of collective excitations is found in [27,28] for a certain frequency range, ω − < ω < ω + , where ω ± are q- and interaction-dependent.…”

Section: Numerical Results Inmentioning

confidence: 60%

Current density functional theory for one‐dimensional fermions

Dzierzawa

Eckern

Schenk

et al. 2009

Physica Status Solidi (b)

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The frequency-dependent response of a one-dimensional fermion system is investigated using Current Density Functional Theory (CDFT) within the local approximation (LDA). DFT-LDA, and in particular CDFT-LDA, reproduces very well the dispersion of the collective excitations. Unsurprisingly, however, the approximation fails for details of the dynamic response for large wavevectors.In particular, we introduce CDFT for the one-dimensional spinless fermion model with nearest-neighbor interaction, and use CDFT-LDA plus exact (Bethe ansatz) results for the groundstate energy as function of particle density and boundary phase to determine the linear response. The successes and failures of this approach are discussed in detail.1 Introduction Density Functional Theory (DFT) is an efficient and powerful tool for determining the electronic structure of solids. While originally developed for continuum electron systems with Coulomb interaction [1,2], DFT has also been applied to lattice models, such as the Hubbard model [3][4][5][6], in order to develop new approaches to correlated electron systems: lattice models often allow for exact solutions which hence can serve as benchmarks for assessing the quality of approximations.Very useful for applications is the Local Density Approximation (LDA) where the exchange-correlation energy of the inhomogeneous system under consideration is constructed via a local approximation from the homogeneous electron system. A lattice version of LDA has been suggested for one-dimensional systems [5] where the underlying homogeneous system can be solved using Bethe ansatz. For recent applications of Bethe ansatz LDA, see also, for example, Refs. [6][7][8][9][10][11][12].In addition, the time-dependent version of DFT has been developed and applied [13,14], in particular, the current density version [15]; a recent review [16] and book [17] provide excellent overviews, including the relation to standard many-body Green's function approaches.In this article, we focus on the one-dimensional spinless fermion model with nearest-neighbor interaction, which is exactly solvable in the homogeneous case [19,20]. We extend our recent DFT-LDA approach [18] to current den-

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Normal Heat Conduction in a Chain with a Weak Interparticle Anharmonic Potential

Cited by 58 publications

References 19 publications

Nonstationary heat conduction in one-dimensional models with substrate potential

Nonstationary heat conduction in one-dimensional models with substrate potential

Threshold Singularities in the Dynamic Response of Gapless Integrable Models

Current density functional theory for one‐dimensional fermions

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