Umklapp-Free Lattice Models

• Umklapp-Free Lattice (UFL) models are engineered lattices that remove umklapp scattering to enforce strictly momentum-conserving phonon interactions.
• They utilize tailored long-range nonlinear (cubic and quartic) interactions to cancel umklapp terms, resulting in unique thermal transport properties.
• Numerical studies reveal that proper interaction truncation leads to near-ballistic transport, validating Peierls’s hypothesis on thermal resistance.

The Umklapp-Free Lattice (UFL) is a class of engineered lattice models in which all umklapp (U) scattering processes are explicitly removed from the phonon interaction dynamics, leaving only crystal momentum-conserving (normal) processes. The UFL paradigm enables the mathematical realization and computational exploration of Peierls's hypothesis that umklapp scattering is the sole phononic mechanism responsible for finite thermal resistance in crystals. UFL constructions have been developed for both long-range quartic nonlinearities and (more recently) for cubic nonlinear interactions, with significant consequences for thermal transport and nonequilibrium dynamics.

1. Hamiltonian Formulation of Umklapp-Free Lattices

For a chain of $N$ identical particles under periodic boundary conditions, the cubic UFL Hamiltonian proposed in (Ono et al., 5 Nov 2025) is

$$H(\{q_n,p_n\}) = \sum_{n=1}^N \frac{1}{2}p_n^2 + \sum_{n=1}^N \frac{1}{2}(q_{n+1}-q_n)^2 + \alpha\sum_{n=1}^N\sum_{l=1}^{L}\frac{a_l}{3}(q_{n+l}-q_n)^3,$$

where $q_n, p_n$ are the displacements and conjugate momenta, $\alpha$ sets the cubic nonlinearity scale, $a_l$ is the coupling constant for particles separated by $l$, and $L\le N/2$ defines the truncation range (with $L=N/2$ for full range). This form augments standard nearest-neighbor harmonic coupling with tailored, long-range, anharmonic (cubic) interactions.

The quartic UFL, as introduced in (Yoshimura et al., 2021), employs a similar construction: $$H = \sum_{n=-\infty}^{\infty}\frac{1}{2}p_{n}^2 + \sum_{n=-\infty}^{\infty}\left[\frac{\mu_0}{2}q_n^2 + \frac{\mu_1}{2}(q_{n+1}-q_n)^2\right] + \beta\sum_{n=-\infty}^\infty\sum_{r=1}^\infty\frac{1}{4r^2}\left[q_{n+r}-(-1)^r q_n\right]^4,$$ where $\mu_0$ controls an optional on-site potential, $\mu_1$ is the harmonic spring constant, and $\beta$ sets the strength of the quartic nonlinearity. The quartic interactions decay as $1/r^2$ and are modulated by a $(-1)^r$ factor, enforcing a specific form of "potential symmetry."

2. Analytical Construction and Removal of Umklapp Terms

In normal-mode (Fourier) coordinates, the cubic part of the UFL produces both "normal" ($i+j-k=0$) and umklapp ($i+j-k = \pm N$) three-phonon processes. The umklapp contributions are given by sums involving $$(-1)^l a_l \sin(\pi l i/N)\sin(\pi l j/N)\sin(\pi l k/N)$$ for $i+j+k=N$. To nullify all possible umklapp terms, $a_l$ must satisfy

$a_l = \frac{2+(-1)^l}{l}, \quad l=1,2,\ldots$

in the thermodynamic limit ($N\to\infty$), leading to an exact cancellation of all umklapp-channel matrix elements for cubic interactions. For finite $N$, the precise periodic-chain solution is

$a_l = \frac{\pi[2+(-1)^l]}{N\tan(\pi l/N)},$

which asymptotically behaves as $1/l$ for $l\ll N$. Quartic UFLs (Yoshimura et al., 2021) achieve the same effect via the $(-1)^r/r^2$ dependence, for which all umklapp resonances in the nonlinear mode coupling tensor vanish identically.

These constructions enforce the absence of momentum-destroying phonon-phonon scattering at the level of the lattice Hamiltonian, restricting the nonlinear dynamics to strictly normal, momentum-conserving processes.

3. Thermally Driven Transport: Numerical Validation and Scaling

To probe transport properties, non-equilibrium molecular dynamics (NEMD) simulations connect finite UFL chains to Langevin heat baths at both boundaries. For the cubic UFL, the equations of motion for bulk sites are

$$\begin{aligned} \ddot{q}_n &= q_{n+1} - 2q_n + q_{n-1} \ &+ \alpha\sum_{l=1}^{L_3} \frac{2 + (-1)^l}{l} \left[(q_{n+l}-q_n)^2 - (q_n-q_{n-l})^2\right] \ &+ \beta\sum_{l=1}^{L_4} \frac{1}{l^2} \left\{ [(-1)^l q_{n+l} - q_n]^3 - [q_n - (-1)^l q_{n-l}]^3 \right\} \end{aligned}$$

with standard Langevin friction and stochastic noise at the bath sites. The stationary energy flux and thermal conductivity $\kappa$ are extracted as

$\kappa = \frac{J}{(T_H - T_L)/N}.$

The empirical relationship between $\kappa$ and system size $N$ depends on the UFL parameters and truncation ranges:

• For the linear chain ($\alpha = \beta = 0$): $\kappa \propto N$ (ideal ballistic transport)
• Cubic UFL with $L_3=10$: $\kappa \propto N^{0.538}$ (sub-ballistic transport due to insufficient umklapp cancellation)
• Cubic UFL with $L_3=200$: $\kappa \propto N^{0.855}$ (approaches ballistic scaling)
• Quartic-only UFL ($L_4=200$): $\kappa \propto N$ (ballistic transport even for moderate truncation) (Ono et al., 5 Nov 2025, Yoshimura et al., 2021)

For the quartic UFL, the scaling law

$\kappa(N, M) = \frac{a N}{1 + b N/M^2}$

with $a, b$ fitted parameters, quantifies the effect of the cutoff $M$ in interaction range. For $M \sim N$, strictly ballistic scaling is retained up to very large $N$.

4. Microscopic Mechanism: Normal vs Umklapp Processes

Normal processes ($\mathbf{Q} = \mathbf{Q}' \pm \mathbf{Q}''$) conserve crystal momentum and cannot fully relax a heat current, as shown in the Peierls-Boltzmann formalism (Allen, 2013). Umklapp processes ($$\mathbf{Q} = \mathbf{Q}' \pm \mathbf{Q}'' + \mathbf{G}$$, with $\mathbf{G}$ nonzero reciprocal lattice vector) are required for momentum relaxation and finite thermal resistance.

The improved Callaway model formalizes this in the context of the Peierls-Boltzmann equation: $$\left[\frac{\partial N_Q}{\partial t}\right]_\text{coll} = -\frac{N_Q-n_Q}{\tau^U_Q} - \frac{N_Q-n_Q^*}{\tau^N_Q}$$ where the $U$-free limit ($\tau^U_Q\to\infty$) leads to divergence of the lattice thermal conductivity: $\kappa_{xx} \to \infty$ in alignment with Peierls’s result. Numerically, perfect UFL systems exhibit a flat temperature profile and ballistic energy transport, in contrast to standard FPUT lattices where umklapp peaks persist in the mode spectrum.

5. Effect of Interaction Truncation and Physical Design Principles

The decay rate of the nonlinear coupling with distance directly determines the sensitivity of umklapp cancellation to the interaction cutoff:

• Cubic UFL ($a_l \sim 1/l$): Slow decay necessitates very long-range interactions for perfect umklapp suppression; truncation at small $L_3$ results in residual momentum-nonconserving scattering, sub-ballistic scaling, and incomplete suppression.
• Quartic UFL ($a_l \sim 1/l^2$): Faster decay allows near-ballistic transport even for moderate truncation; the system is more robust to practical limitations on interaction range.

This dependence suggests a general design principle for the engineering of UFLs with arbitrary nonlinear order: the slower the decay of the required coupling constants, the more important it is to implement interactions at long range.

6. Physical Relevance, Implications, and Applications

UFLs, by rigorously removing all umklapp scattering at the model Hamiltonian level, provide both a theoretical platform for probing fundamental phonon and heat transport questions and a quantitative confirmation of Peierls's hypothesis. Only umklapp processes can produce intrinsic bulk thermal resistance; in their absence, even in nonlinear systems, energy transport remains ballistic.

Potential applications exist in the design of materials, optical-lattice simulators, and engineered low-dimensional nanostructures where thermal conductivity control is critical. Realizing (or approximating) UFL-like conditions could enable ultrahigh or tunable thermal transport regimes, limited only by extrinsic sources of momentum relaxation (e.g., impurities, boundary scattering) rather than anharmonic lattice dynamics themselves.

The direct correspondence between UFLs as constructed (Ono et al., 5 Nov 2025Yoshimura et al., 2021) and the $U$-free limit of the Peierls-Boltzmann-Callaway formalism (Allen, 2013) underscores their utility as testbeds for computational and theoretical exploration of nonequilibrium statistical mechanics, energy transport, and the emergent hydrodynamics of lattice systems. The mechanical realization of UFLs in experimental or synthetic systems remains a target for future research, with the decay law of the tailored nonlinear interaction serving as a key engineering constraint.