Localized mode and nonergodicity of a harmonic oscillator chain

We present a simple and microscopic physical model that breaks the ergodicity. Our model consists of coupled classical harmonic oscillators, and the motion of the tagged particle obeys the generalized Langevin equation satisfying the second fluctuation dissipation theorem. It is found that although the nonergodicity strength, which is expected to detect the ergodicity breaking, for this model vanishes, the velocity autocorrelation function of the tagged particle asymptotically oscillates. We analyze the model by using the molecular dynamics and the exact diagonalization as well as the rigorous mapping to the generalized Langevin equation. Our analysis reveals that the asymptotic oscillation is caused by a localized mode with an isolated frequency from the continuous phonon spectrum.

Figure 1

Schematic picture of our model. The harmonic oscillator chain in the orange dashed box is regarded as the bath, and the tagged particle is connected to the center of harmonic chain with the harmonic interaction. The motion of all the particles is restricted in one dimensional, and they are connected by ideal springs. See Eq. (11) for the detailed definition of the model.

Figure 2

VACF and its spectrum (inset) obtained by the molecular dynamics simulation $N+1={10}^4$. The spectrum is obtained by the discrete Fourier transformation. In the inset, the divergence around $\omega \simeq {\omega }_{\mathrm{iso}}$ (indicated by the vertical dashed line) manifests the existence of a $\delta$ function. The analytic solution for $\tilde{a}\left(\omega \right)$ [Eq. (50)] is plotted by the dashed curve in the inset.

Figure 3

Density of states of the normal modes obtained by the exact diagonalization $N+1={10}^4$ (filled boxes). The phonon mode of a simple harmonic chain is represented by the solid line. The arrow (labeled as ${\omega }_0$) represents the position of the isolated normal mode with the largest frequency, ${\omega }_0$. The inset shows the size dependence of ${\omega }_i-{\omega }_{i+1}$ for $i=0$ (blue circles), 1 (purple squares), 2 (green downward triangles), 3 (red upward triangles). The gap between ${\omega }_0$ and ${\omega }_1$ in the $N=\infty$ limit is estimated as 0.09355570(4) by the least squares method.

Figure 4

Amplitude of the isolated normal mode, $A_0\left(n\right)$, near $n=0$ for $N+1={10}^4$, $m=1.0$, and $k=0.2,0.4,0.8,1.0$. The horizontal axis, $n$, denotes the site index of the bath particles. The inset shows a semi-log plot of the amplitude squared.