A resonance in phonons scattering off a kink in the absence of a Peierls-Nabarro potential

Technical Summary: Resonance in Phonons Scattering off a Kink in the Absence of a Peierls–Nabarro Potential

Problem Statement
The interaction between topological solitons (kinks) and small-amplitude linear waves (phonons) is a fundamental problem in nonlinear physics, relevant to systems ranging from crystal dislocations to DNA dynamics. In continuum models like the $\phi^4$ theory, kinks are often reflectionless, and phonon scattering is well-understood. However, in discrete systems, the breaking of continuous translational invariance typically introduces a static Peierls–Nabarro potential (PNP). This potential pins moving kinks, induces energy radiation, and fundamentally alters scattering outcomes. While "exceptional discretizations" have been developed to eliminate the static PNP, allowing for mobile kinks, the specific scattering dynamics of phonons off these PNP-free kinks—particularly the dependence on lattice spacing and the emergence of resonant effects—remains an open area of investigation. This study addresses the scattering of phonons from a static kink in a discretized $\phi^4$ model that is free of the static PNP, focusing on how lattice discreteness modifies transmission, reflection, and momentum transfer.

Methodology
The authors investigate a real scalar field theory in $(1+1)$ dimensions using a specific discretization of the $\phi^4$ model that preserves the Bogomolny–Prasad–Sommerfield (BPS) structure, thereby ensuring the absence of a static PNP. The model is defined by a Hamiltonian where the discrete field equations are derived to satisfy $u_n = 0$, a discrete analogue of the continuum BPS equation.

1. Model Construction: The authors utilize a discretization scheme based on energy conservation (as introduced in prior works [28, 39]) to generate static kink profiles ($\phi^{(0)}_n$) that are free of the PNP. They analyze the kink profiles for varying lattice spacings $h$, noting a qualitative transition in the asymptotic tails of the kink from monotonic (for $h < 1$) to oscillatory (for $h > 1$).
2. Phonon Generation: To study scattering, a single lattice site ($n^* = -100$) is driven by a localized harmonic source $\epsilon_{n^*}(t) = A \sin(\Omega t)$ with a small amplitude ($A=0.05$) to ensure the system remains in the linear regime. This generates phonon wave packets that propagate toward the kink, initially centered at $n=0$.
3. Numerical Simulation: The system is simulated with $N=800$ lattice sites. The authors compute the time evolution of the energy density to track the incident, reflected, and transmitted energy components.
4. Analysis: Transmission ($T$) and reflection ($R$) coefficients are calculated across the entire phonon band for different lattice spacings. The velocity of the kink post-scattering is determined by tracking its position in space-time. Theoretical analysis includes the derivation of the phonon dispersion relation, group velocity, and the application of Doppler shift corrections to understand resonance conditions.

Key Contributions and Results

• Dependence on Lattice Spacing ($h$): The study reveals a stark contrast in scattering behavior based on the lattice spacing relative to a critical value ($h=1$).

  • Weak Discreteness ($h < 1$): The kink acts as a nearly transparent scattering center. Phonons are transmitted through the kink with high efficiency over most of the phonon spectrum. Reflection is only significant in narrow frequency intervals near the upper edge of the phonon band.
  • Strong Discreteness ($h > 1$): Despite the underlying continuum kink being reflectionless, the discrete kink exhibits significant reflection over a broad range of frequencies. As $h$ increases, the kink behaves as a more effective barrier, with a substantial fraction of incident energy being reflected.

• Negative Radiation Pressure and Kink Acceleration: A central finding is that the kink experiences negative radiation pressure, accelerating toward the incoming phonon source.

  • In the weak-discreteness regime, the kink often remains at rest or moves only slightly, except near the band edge.
  • In the strong-discreteness regime ($h > 1$), the kink is consistently accelerated toward the source for all driving frequencies within the phonon band. The magnitude of this acceleration is significantly stronger than in the weak-discreteness case.

• Resonance Mechanisms: The paper identifies two primary mechanisms governing the kink's velocity and energy transfer:

  1. Group Velocity Extrema: The kink velocity reaches maxima when the driving frequency corresponds to the extrema of the phonon group velocity.
  2. Doppler-Shifted Resonance: The authors demonstrate that the maxima in kink velocity coincide with a resonance condition where the Doppler-shifted frequency of the incoming phonon (in the kink's moving frame) matches the frequency of the kink's internal (shape) mode. Specifically, the condition $n\omega_s = \omega(q) - q V_k$ is satisfied, where $\omega_s$ is the internal mode frequency. This resonance leads to enhanced momentum transfer.

• Energy Dynamics: In the weak-discreteness regime, the kinetic energy of the kink shows damped oscillations, indicating resonant coupling with the phonon band and energy radiation. Conversely, in the strong-discreteness regime, the kinetic energy exhibits persistent oscillations with constant amplitude, suggesting the internal mode frequency lies outside the phonon band, preventing efficient energy radiation and leading to long-lived internal excitations.

Significance
The paper demonstrates that strong lattice discreteness can qualitatively modify phonon-kink interactions even in systems where the static Peierls–Nabarro potential is absent. The findings challenge the intuition derived from continuum models, showing that a "reflectionless" kink in the continuum limit can become a strong reflector in a discrete setting. Furthermore, the work elucidates the mechanism of negative radiation pressure, showing that kinks in the $\phi^4$ model are never repelled by phonons but are instead accelerated toward the source. The identification of resonance between Doppler-shifted phonon frequencies and kink internal modes provides a detailed explanation for the observed acceleration peaks. These results highlight the critical role of lattice effects in the controllability and dynamics of topological solitons, distinct from the pinning effects of the PNP.