Construction of cubic nonlinear lattice free from umklapp processes

TL;DR

The paper addresses suppressing umklapp scattering in a 1D nonlinear lattice by enforcing a symmetry under a phase-shifting transform on complex normal modes, enabling a cubic long-range UFL. It derives analytic coupling constants $a_l$, providing finite- and infinite-$N$ forms that ensure umklapp-free dynamics, and validates these findings through mode-excitation analyses. Non-equilibrium MD confirms ballistic energy transport emerging as the cubic-range truncation grows, with truncation effects more pronounced than in quartic-only UFLs. Overall, the work demonstrates how lattice symmetry can be harnessed to eliminate umklapp processes and promote ballistic heat transport in cubic nonlinear lattices.

Abstract

We propose a novel type of umklapp-free lattice (UFL), where umklapp processes are completely absent. The proposed UFL incorporates cubic long-range nonlinearity, a feature not addressed in previous studies. In this paper, we derive an analytical expression for the cubic nonlinear coupling constants by imposing mathematical conditions such that the nonlinear coupling strength between particle pairs decays inversely with their separation distance. The absence of umklapp processes in the proposed lattice is confirmed through numerical comparisons with the Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. Furthermore, molecular dynamics simulations are performed to investigate the thermal conductivity of the proposed lattice in the non-equilibrium steady state. Compared to the original FPUT lattice, the proposed UFL is closer to ballistic transport. Our results demonstrate that the umklapp processes induced by cubic nonlinearity are suppressed in the proposed UFL. Moreover, compared to the UFL with only quartic nonlinearity, truncation of long-range interactions plays a significant role in the proposed lattice.

Figures (6)

• Figure 1: The nonlinear lattice model for considering the UFL. The cubic long-range interaction is included. Only the connections related to the blue particles are shown.
• Figure 2: Change in $\color{black} a_l \color{black}$ based on Eq. \ref{['eq:finiteN']} for $N = 512, 1024, 2048$. $\color{black} b_l \color{black}$ is also plotted for comparison. Solid and dashdot lines correspond to the $\color{black} a_l \color{black}$ and $\color{black} b_l \color{black}$ in the limit $N \rightarrow \infty$, respectively.
• Figure 3: Excited phonon modes ($i$, $x$-axis) from externally perturbed phonon modes ($j$, $y$-axis) on the UFL (top) and the FPUT lattice (bottom). Red, blue, and black dots are separated by the energy spectrum threshold of phonon modes: energy spectra of red dots are greater than $10^{-3}$, and that of blue and black dots are less than $10^{-3}$.
• Figure 4: Illustration of numerical simulation model. Black dotted lines correspond to long-range interactions. Long-range interactions extend to thermostats. Fixed boundary conditions are applied to the left ends in the high temperature thermostat, and the right end in the low temperature thermostat, respectively.
• Figure 5: Relation between thermal conductivity $\kappa$ and lattice size $N$. Dashed line represents the linear case: $\alpha, \beta = 0$, and shows the ballistic energy transport.

Theorems & Definitions (2)

• Theorem 1
• proof