Crossover from ballistic to normal heat transport in the ${\varphi }^4$ lattice: If nonconservation of momentum is the reason, what is the mechanism?

Anomalous (non-Fourier) heat transport is no longer just a theoretical issue since it has been observed experimentally in a number of low-dimensional nanomaterials, such as SiGe nanowires, carbon nanotubes, and others. To understand these anomalous behaviors, exploring the microscopic origin of normal (Fourier) heat transport is a fascinating theoretical topic. However, this issue has not yet been fully understood even for one-dimensional (1D) model chains, in spite of a great amount of thorough studies done to date. From those studies, it has been widely accepted that the conservation of momentum is a key ingredient to induce anomalous heat transport, while momentum-nonconserving systems usually support normal heat transport where Fourier's law is valid. But if the nonconservation of momentum is the reason, what is the underlying microscopic mechanism for the observed normal heat transport? Here we carefully revisit a typical 1D momentum-nonconserving ${\varphi }^4$ model, and we present evidence that the mobile discrete breathers, or, in other words, the moving intrinsic localized modes with frequency components above the linear phonon band, can be responsible for that.

Figure 1

The phonon-dispersion relation for the ${\varphi }^4$ lattice, where lines 1 and 2 indicate the lower and upper edges of the linear phonon band. Note that for the considered hard-type anharmonicity, only DBs with frequencies above line 2 can exist.

Figure 2

${\rho }_Q(m,t)$ calculated for several $\beta$ values at three typical long times $t=500$ (dotted), $t=1000$ (dashed), and $t=1500$ (solid).

Figure 3

Parts (a), (c), and (e) show the height $H_c$ of the central peaks of ${\rho }_Q(m,t)$ vs $t$ for extracting the scaling exponent $\gamma$. Parts (b), (d), and (f) show the rescaled ${\rho }_Q(m,t)$ using formula (7): (a) and (b) for $\beta =0.01$, (c) and (d) for $\beta =0.2$, and (e) and (f) for $\beta =5$. In (b), (d), and (f), three typical long times of $t=500$ (dotted), $t=1000$ (dashed), and $t=1500$ (solid) are compared.

Figure 4

The scaling exponent $\gamma$ vs $\beta$ for indicating the crossover process from ballistic to normal heat transport, where the horizontal lines, from bottom to top, denote $\gamma =1$ and 2, while the vertical lines, from left to right, represent $\beta =0.1$ and 0.4, respectively.

Figure 5

The power spectrum $P_0\left(\omega \right)$ of the equilibrium states at $T=0.5$ for the chains with different strengths of nonlinearity: (a) $\beta =0.01$, (b) $\beta =0.1$, (c) $\beta =0.2$, (d) $\beta =0.5$, (e) $\beta =1$, and (f) $\beta =5$. The lower and upper boundaries of the linear phonon band are shown by the vertical dashed lines, and they are indicated as 1 and 2, respectively.

Figure 6

Several typical snapshots of the residual thermal fluctuations after a long-time absorption for three $\beta$ values: (a),(b) for $\beta =0.01$; (c),(d) for $\beta =0.2$; and (e),(f) for $\beta =5$. The insets in (e) and (f) indicate the Sievers-Takeno DB modes.

Figure 7

The power spectrum $P\left(\omega \right)$ of the residual thermal fluctuations after a long-time absorption: (a) $\beta =0.01$, (b) $\beta =0.1$, (c) $\beta =0.2$, (d) $\beta =0.5$, (e) $\beta =1$, and (f) $\beta =5$, where the vertical dashed lines 1 and 2 show the lower and upper edges of the linear phonon band, respectively.

Figure 8

Velocity $v_{\mathrm{DB}}$ of DBs excited with the help of the ansatz Eq. (9) as a function of $\delta$: (a) $\beta =1$ and (b) $\beta =5$. The DB's amplitude is indicated in the legends. Other DB parameters are taken from Table 1.

Figure 9

(a) Counter plot showing the evolution of total (kinetic plus potential) energies of particles with $t$ in the chain with $\beta =1$. Initially ($t=0$), in the center $k_c$, a standing DB is excited by using ansatz (9) with $A_{\mathrm{DB}}=0.5$, $\delta =0$, ${\omega }_{\mathrm{DB}}=2.257$, and $\theta =0.309$, and an ac driving phonon source at $k^{*}=k_c-500$ is operated until $t=1500$. One can see that the DB is accelerated toward the phonon source during the scattering with the phonon wave packet. (b) Time evolution of DB's velocity during this process.

Figure 10

The momentum correlation function ${\rho }_p(m,t)$ for several small $\beta$ values for a long time $t=1500$: (a) $\beta =0.01$, (b) $\beta =0.05$, (c) $\beta =0.1$, and (d) $\beta =0.2$.