Phonon-Roton Dispersion Relation

• Phonon-Roton Dispersion Relation is a nonmonotonic energy-momentum profile featuring a linear phonon regime, maxon peak, and roton minimum.
• It arises from strong interparticle correlations and hybridization effects in systems like superfluid helium-4 and dipolar Bose fluids.
• Experimental and numerical methods, including molecular dynamics and quantum Monte Carlo, unveil its impact on critical velocity, density of states, and interface phenomena.

The phonon-roton dispersion relation describes the energy spectrum of collective excitations in various quantum and classical many-body systems, most notably superfluid helium-4, dipolar Bose fluids, and certain solids and metamaterials. It is distinguished by a nonmonotonic curve in energy-momentum space: a linear phonon regime at low momentum, a local maximum called the "maxon," and a pronounced local minimum—the "roton"—at finite momentum, before recovering monotonic behavior at large wave number. This structure emerges from strong interparticle correlations, hybridization phenomena, or long-range and nonlocal interactions, and plays a critical role in phenomena such as superfluidity, critical velocity, anomalous density of states, and mode conversion at interfaces.

1. Microscopic Origin and Correlational Mechanisms

The formation of the phonon-roton structure in quantum fluids such as superfluid helium-4 is tightly linked to strong correlations between constituent particles. Classical molecular dynamics (MD) studies, for example in two-dimensional bosonic dipole systems interacting via ϕ(r) = μ²/r³, directly demonstrate that nonmonotonicity (the maxon-roton structure) arises at sufficiently high coupling strengths, quantified either via Γ_D = (β μ²)/a³ (finite temperature) or r_D = (m μ²)/(ℏ²a) (zero temperature), with the roton minimum deepening as coupling surpasses thresholds near crystallization (Kalman et al., 2010). The dispersions extracted from MD-generated dynamical structure functions S(k, ω), and compared to quantum Monte Carlo calculations, consistently show that the roton is a manifestation of strong, short-range particle correlations enforcing a peak in static structure factor S(k), thereby "forcing" the dispersion relation into a maxon-roton profile. The Feynman relation ω(k) = (ℏ k²)/(2 m S(k)) similarly links the dispersion directly to many-body correlation effects.

In superfluid helium-4, ab initio quantum Monte Carlo simulations of 2D ^4He (Arrigoni et al., 2013) as well as path integral Monte Carlo (PIMC) studies of electron liquids (Chuna et al., 16 May 2025) consistently reveal that as density and interparticle coupling increase, the dispersion spectrum transitions from almost featureless to one with pronounced maxon-roton features; the roton wave vector itself shifts with density, indicating evolving microscopic structure and favoring crystallization. A second roton feature appears at higher harmonics in strongly coupled electron liquids as an incipient precursor to crystalline order.

2. Dispersion Relation, Mathematical Structure, and Density of States

The canonical form of the phonon-roton dispersion is given in Landau theory as:

• Phonon regime: $E(k) ≈ \hbar u_1 k$ (with $u_1 =$ speed of sound)
• Roton regime: $$E(k) ≈ Δ_{rot} + \frac{\hbar^2}{2m_{rot}} (k-k_{rot})^2$$

At intermediate and high densities, the dispersion for many Bose systems splits into two branches—a gapless sound (phonon) mode and a gapped optical mode—both of which may develop roton-like minima (Poluektov et al., 2022). The quartic equation $(\hbar \omega)^4 - L_k (\hbar \omega)^2 + N_k = 0$ yields two solutions with distinct long-wavelength behaviors: $\omega_k^{(-)} \approx c k$ (phonon) and $\omega_k^{(+)} \approx \omega_0 + \alpha k^2$ (optical), where $\omega_0$ denotes the gap (Poluektov et al., 2022).

A critical consequence of the roton minimum is the vanishing of group velocity $v_g = \partial E(k)/\partial k$ at the extremum. This produces a singularity in the density of states:

$$\rho(E) = \frac{4\pi k^2}{| \partial E/\partial k | }$$

yielding a strong enhancement at the roton or maxon point—a Van Hove singularity.

In finite systems, such as SHF resonators, the quantization of levels near the roton minimum produces a narrow absorption line, with the probability for photon-to-roton transitions (c-photon $\to$ c-roton) diverging at the minimum group velocity (Loktev et al., 2010). Similarly, engineered acoustic metamaterials can realize multiple coexisting roton minima through controllable long-range interactions, displaying features such as broadband backward wave propagation and mode coexistence (Zhu et al., 2021).

3. Interface Effects, Mode Conversion, and Nonlocal Hydrodynamics

At interfaces—such as the boundary between superfluid helium and solids—the full nonmonotonic dispersion relation controls transmission, reflection, and mode change of quasiparticles (Tanatarov et al., 2010, Adamenko et al., 2012). Nonlocal hydrodynamic theories, constructed with kernels $h(k) = k^2/\Omega^2(k)$ directly determined by experimental dispersion data, lead to rich behavior:

• Multiple propagating branches: phonon, R⁻ roton, R⁺ roton, each with characteristic $k_i(\omega)$
• Mode conversion at the interface: incident waves in the solid or helium can couple to all three branches
• Generalized Snell’s law: $\sin\theta_{i}/s_i(\omega) = \text{const}$ links incidence and transmission angles

A distinctive feature is the negative group velocity of the R⁻ roton branch ($d\Omega/dk < 0$), driving backwards reflection/refraction (Andreev-like process) so that energy flux direction reverses relative to momentum. The creation (D₂) and detection probability for R⁻ rotons is suppressed compared to phonons (D₁) and R⁺ rotons (D₃), explaining historical experimental challenges in generating and measuring these modes. Carefully engineered setups using h-phonons (high energy) can enhance R⁻ roton production (Adamenko et al., 2012).

Energy flows through the interface can be quantified by

$$Q(T) = \int (d^3k_{sol}/(2\pi)^3)~\hbar\omega~n_T(\omega)~s_{sol}\cos\theta_{sol} D(\omega, k_\tau)$$

partitioned according to the probabilities for phonons and rotons.

4. Hybridization, Chirality, and Universality Across Media

In noncentrosymmetric micropolar crystals, the hybridization of translational ($u$) and micro-rotational ($\varphi$) degrees of freedom via chiral couplings gives rise to roton-like minima in the transverse acoustic branch (Kishine et al., 2020). The equations of motion, when expressed in the circular polarization basis, yield a polarization-dependent band splitting; the general dispersion form is

$$[\omega_\alpha^{O/A}]^2 = \frac{1}{2\rho j_\alpha} [ b_\alpha + j_\alpha a_\alpha \pm \sqrt{(b_\alpha - j_\alpha a_\alpha)^2 + 4j_\alpha\Delta_\alpha^2} ]$$

with hybridization terms producing roton minima at finite $k_m$. This phenomenon directly parallels the hybridization scenario in superfluid helium-4, where Bogoliubov quasiparticles and gapful density fluctuations couple (Nozières' mechanism), giving rise to the roton minimum as an incipient instability toward crystallization.

Chiral elastic media therefore realize nonreciprocal phonon dispersion and "acoustic activity," with potential applications in controlling elastic wave propagation, polarization rotation, and designing nonreciprocal or resonance-enhanced metamaterials.

5. Temperature Dependence, Interactions, and Critical Velocity

In superfluid ⁴He, temperature-dependent studies reveal that the roton energy and linewidth are governed by multi-particle scattering:

• Roton–roton: $$N_r(T) \propto \sqrt{T}[1+\alpha\sqrt{\mu T}] \exp(-\Delta/T)$$
• Roton–phonon (4 particle): linewidth $\Gamma_p(T) \sim T^7$
• Hartree and 3-particle (due to condensate’s broken gauge symmetry): energy shift $\delta(T) \propto T^4$ (Fak et al., 2012)

A crucial observation is that the net roton–phonon interaction is repulsive, implying an initial upward shift in the roton gap with temperature before decreasing at higher $T$ due to roton–roton scattering. The interplay of these processes governs the dynamical properties of the phonon-roton spectrum.

The Landau criterion for superfluid critical velocity is set by

$$v_{crit} = \min_{p} \left( \frac{\omega(p)}{|p|} \right)$$

With the roton minimum lowering this ratio below the speed of sound, the critical velocity is controlled by the position and depth of the roton.

6. Engineering Phonon-Roton Dispersion in Classical and Quantum Systems

The principles underlying phonon-roton dispersion have been successfully extended beyond quantum fluids. Acoustic metamaterials with programmable beyond-nearest neighbor interactions synthesize nonmonotonic, roton-like dispersions at ambient temperature (Zhu et al., 2021). The fundamental equations governing these systems are derived from spring-mass chains:

$$m\,\ddot{u}_n = K_1(u_{n+1} - 2u_n + u_{n-1}) + K_N(u_{n+N} - 2u_n + u_{n-N})$$

The corresponding dispersion is

$$\omega(k) = 2\sqrt{K_1/m}\sin\left(\frac{ka}{2}\right) + 2\sqrt{K_N/m}\sin\left(\frac{Nka}{2}\right)$$

By varying $N$, multiple roton minima—never observed in quantum systems—can be engineered, enabling broadband backward wave propagation, mode coexistence, and tunable density of states.

In superfluid vacuum theory, a Landau-like "roton" dispersion guarantees suppression of dissipative fluctuations, with photon-like modes acquiring effective mass corrections at small $p$ (Zloshchastiev, 2020):

$E_p^2 = c_0^2 p^2 + \mu^2(p) c_0^4$

with higher-order terms controlling deviations from linear behavior and ensuring the physical vacuum remains robustly non-dissipative.

7. Outlook and Broader Implications

The robust ubiquity of the phonon-roton dispersion relation—arising from many-body correlation, nonlocal interactions, or hybridization effects—illustrates its foundational role across quantum fluids, solids, electron liquids, and engineered metamaterials. A plausible implication is that the emergence of roton minima signals incipient crystallization instability or resonance between propagating and internal degrees of freedom. Enhanced density of states at roton points underpins narrow absorption phenomena, sets critical velocities, and constrains energy transport at interfaces.

The synthesis of classical analogues, the observation of higher-harmonic rotons in electron liquids (Chuna et al., 16 May 2025), and controlled mode conversions at interfaces highlight its utility for benchmarking many-body theory, quantum simulation, and engineering novel wave transport. Open directions include further probing of nonlocal and hybridization mechanisms, exploiting backward propagation effects for advanced device applications, and exploring universality of roton phenomena in new regimes such as straintronics and topological matter.