Anomalous thermal conductivity and local temperature distribution on harmonic Fibonacci chains

The harmonic Fibonacci chain, which is one of a quasiperiodic chain constructed with a recursion relation, has a singular continuous frequencyspectrum and critical eigenstates. The validity of the Fourier law is examined for the harmonic Fibonacci chain with stochastic heat baths at both ends by investigating the system size N dependence of the heat current J and the local temperature distribution. It is shown that J depends on N as J ∼ (lnN) and the local temperature strongly oscillates along the chain. These results indicate that the Fourier law does not hold on the harmonic Fibonacci chain. Furthermore the local temperature exhibits two different distribution according to the generation of the Fibonacci chain, i.e., the local temperature distribution does not have a definite form in the thermodynamic limit. The relations between Ndependence of J and the frequency-spectrum, and between the local temperature and critical eigenstates are discussed. PACS numbers: 44.10.+i, 61.44.Br, 05.40.-a, 05.60.-k Many studies in these decades have shown that arbitrarily defined one-dimensional (1D) systems of interacting particles do not exhibit normal thermal transport properties, i.e., the Fourier law does not hold on such systems [1, 2, 3, 4, 5]. For the steady-state of the homogeneous 1D chain of system size N , the Fourier law, J = −κ∇T with thermal conductivity κ, indicates that the heat current depends on the system size as J ∼ 1/N and that the temperature gradient ∇T is constant along the chain. Rieder et al [1] have shown for a 1D chain of equal mass particles interacting with identical harmonic potential that the heat current is independent of the system size. They also have obtained the local temperature distribution. The local temperature behaves in unphysical way: the temperature takes a constant value in the bulk. Furthermore, near the end of the chain, the temperature decreases as we move in the direction of the hotter heat bath, and rise only at the end particle in contact with the heat bath; the temperature exhibits corresponding behavior at the other end of the chain. Casher and Lebowitz [2] have shown for the same model but with random mass distribution that J ∼ N. For the same random mass distribution model but with different type of boundary conditions, Rubin and Greer [3] have obtained the result J ∼ N. These system size dependence of J for periodic or disordered chains may be attributed to the localization property of eigenstates on these chains. Since the eigenstates are extended in periodic chains, the ballistic energy transport of extended eigenstates results in the constant heat current of the periodic chains. Letter to the Editor 2 In contrast, for the disordered chains, the decrease in heat current with increasing system size is caused from the localized eigenstates, which can not transport energy over the length of the system. The extended and localized eigenstates correspond to continuous and pure-point spectra, respectively. It is shown that, for the Casher and Lebowitz type chain and heat bath, the thermal conductivity κ diverges as the system size increases if the spectrum contains absolutely continuous part [2]. Some of quasiperiodic systems have a Cantor set-like spectrum, i.e., a singular continuous spectrum [6, 7, 8, 9, 10]. Then, on such systems, eigenstates show power law decay; such neither extended nor exponentially localized states are called critical states [6]. For example, a harmonic chain with quasiperiodically arranged spring constants and/or mass of particles has a singular continuous spectrum and critical eigenstates [7, 8, 9]. We may thus expect that such quasiperiodic systems show exotic heat transport properties compared with periodic or disordered systems. In the present paper we investigate the anomaly of heat transport phenomena on harmonic Fibonacci chains, which is one of discrete quasiperiodic and self-similar 1D lattices. We focus the anomaly resulting from the spectral properties of the Fibonacci chain. In order to check the validity of the Fourier law on the Fibonacci chain, we investigate the system size dependence of the heat current J . Although Maciá [11] have already studied the thermal conductivity κ of the harmonic Fibonacci chain, he has not checked the system size dependence of κ and thus the validity of the Fourier law is not clear yet. Our results show that the heat current behaves as J ∼ (lnN), which is in contrast with that of periodic or disordered chain. We discuss the fact that the total bandwidth of the phonon spectrum of the Fibonacci chain has similar N dependence to J . We also calculate the local temperature distribution on the Fibonacci chain; it seems not to converge to a definite form even in the thermodynamic limit. We relate {Ti} with the critical eigenstates of the Fibonacci chain. The harmonic Fibonacci chain which we consider is a 1D chain of N particles; each particle interact with its neighbouring particles with equal spring constant k. We make the sequence of mass of particles {mi|i = 1, ..., N,mi = mα or mβ} according to the Fibonacci sequence. The Fibonacci sequence of the n-th generation Ln, which consists of two kinds of components mα and mβ , is constructed by the recursion relation: Ln = Ln−1Ln−2, L0 = mβ , and L1 = mα. Then the system size of the n-th generation Fibonacci sequence is the Fibonacci number Fn, which obeys the recursion relation Fn = Fn−1 + Fn−2, F0 = 1, and F1 = 1. The Fibonacci number Fn is asymptotically behaves as Fn ∼ τ with golden ratio τ = ( √ 5 + 1)/2. We can obtain the asymptotic properties of the Fibonacci chain in the limit of N → ∞ by concerning the infinite-generation limit n → ∞. We set both the spring constant k and the mass mβ unity; we calculate the heat current and the temperature distribution varying with the mass mα. We consider the following Langevin equations for particles of the chain with stochastic heat baths at both ends: m1ẍ1 = −2x1 + x2 − γẋ1 + ηL(t), miẍi = −2xi + xi−1 + xi+1, mN ẍN = −2xN + xN−1 − γẋN + ηR(t), (1) where xi are the displacements of the particles from their equilibrium positions; γ is the friction constant; ηL and ηR are the random forces caused from left and right heat baths, respectively. We choose the random forces the white noises, i.e., the Fourier Letter to the Editor 3 transforms of the random force obey 〈ηL(ω)ηL(ω)〉 = 4πγTLδ(ω + ω′), 〈ηR(ω)ηR(ω)〉 = 4πγTRδ(ω + ω′), (2) where angular bracket is the average over the random force; TL and TR are the temperatures of the left and right heat baths, respectively. The energy current Ji from the site i to the site i+ 1 is Ji = 1 2 〈(ẋi+1 + ẋi)(xi+1 − xi)〉. (3) In the steady state, the energy current does not depend on the site; then from the Langevin equation (1),