Dispersion and absorption in one-dimensional nonlinear lattices: A resonance phonon approach

Based on the linear response theory, we propose a resonance phonon (r-ph) approach to study the renormalized phonons in a few one-dimensional nonlinear lattices. Compared with the existing anharmonic phonon (a-ph) approach, the dispersion relations derived from this approach agree with the expectations of the effective phonon (e-ph) theory much better. The application is also largely extended, i.e., it is applicable in many extreme situations, e.g., high frequency, high temperature, etc., where the existing one can hardly work. Furthermore, two separated phonon branches (one acoustic and one optical) with a clear gap in between can be observed by the r-ph approach in a diatomic anharmonic lattice. While only one combined branch can be detected in the same lattice with both the a-ph approach and the e-ph theory.

Figure 1

Time evolution of space modes in the momentum-conserving lattices [(a) the $\mathrm{FPU}-\beta$ lattice and (b) the $\mathrm{FPU}-\alpha 2\beta$ lattice] and a momentum-nonconserving lattice [(c) the ${\varphi }^4$ lattice]. The curves from far to near correspond to $k=\frac{\pi }{16},\frac{\pi }{8},\frac{2\pi }{8},\frac{3\pi }{8},\frac{4\pi }{8},\frac{5\pi }{8},\frac{6\pi }{8}$, and $\frac{7\pi }{8}$, respectively. $N=1024$ and $T=0.2$.

Figure 2

The $\mathrm{FPU}-\beta$ lattice. $N=2048, T=0.2$. The wave number $k$ increases from 0 to $\pi$ along the arrow. (a) and (b) are the real and the imaginary parts of $\chi (k,\omega )$, and (c) their values vary in the $\omega$ space for each value of $k$. (d) The 2D projection of (c) in $({\chi }^{\prime },{\chi }^{\prime \prime })$ space, which are just the thermodynamic counterparts of the Cole-Cole diagram.

Figure 3

Columns from left to right correspond to the $\mathrm{FPU}-\beta$ lattice, the $\mathrm{FPU}-\alpha 2\beta$ lattice, and the ${\varphi }^4$ lattice, respectively. Upper row: the dispersion relations for those different lattices derived from various approaches. Open triangles, solid circles, and dashed lines are for the a-phs, r-phs, and e-phs, respectively. The symbol groups from the top down correspond to $T=10$, 1, and 0.2, respectively. Apparent disagreement between a-phs and e-phs is observed. In contrast, the r-phs agree with the e-phs much better. Only very slight discrepancies can be observed for very large $k$ and high temperature $T$. Lower row: ${\chi }^{\prime }\left({\omega }_0\right)$ vs $k$. The curves of ${\chi }^{\prime }\left({\omega }_0\right)$ in (d) and (e) display a power-law divergence as $k\to 0$, while those in (f) approach to constants. The slopes $k^{-1.5}$ and $k^{-1.3}$ in (d) and (e) are plotted for reference only.

Figure 4

${\chi }^{\prime }\left(\omega \right)$ in (a) the $\mathrm{FPU}-\alpha 2\beta$ lattice and (b) the ${\varphi }^4$ lattice. $N=1024, T=0.2$. The wave number $k$ increases from 0 to $\pi$ along the arrow. The resonance peaks for the $\mathrm{FPU}-\alpha 2\beta$ lattice are much wider than those for the $\mathrm{FPU}-\beta$ lattice plotted in Fig. 2. As for the ${\varphi }^4$ lattice, the height of the peaks increases with $k$.

Figure 5

The diatomic $\mathrm{FPU}-\beta$ lattice. $N=1024, T=0.2$. The first and the second rows are for the cases that the driving force is applied to a heavy particle with mass $M=1.618$ and a light particle with mass $m=1$, respectively. The left and the right columns are for $k\in (0,\pi /2]$ and $k\in [\pi /2,\pi )$, respectively. (e) The dispersion relations derived from different approaches. The r-ph displays two separated branches with a clear gap in between, while both the e-ph and a-ph display a combined branch only. The blue dashed lines are for the corresponding diatomic harmonic lattice obtained using Eq. (12).