The fate of the flexural phonon dispersion and the classification of drifting and driftless second sound in two-dimensional materials

The unified theory of second sound in two-dimensional materials by Shang, Mao, Yang, Li and Lü (Phys. Rev. B 105, 165423 (2022)) predicts two distinct damped thermal-wave modes — drifting second sound, riding on a collective phonon drift, and driftless second sound, a temperature wave with no net drift — but it rests on an unresolved input: the asymptotic long-wavelength dispersion of the flexural (ZA) phonon mode in real suspended membranes, where symmetry-protected quadratic behavior and anharmonic power-law renormalization remain conflicting first-principles predictions. This problem asks for the low-wave-vector fate of the ZA dispersion and its downstream consequences: the existence, velocity, strain and finite-size dependence of the two second-sound modes in monolayer two-dimensional materials; the quantitative corrections to the relaxation times from energy-exchange channels the theory omitted, such as electron–phonon coupling; the causal relation between hydrodynamic phonon drift and the logarithmically size-divergent thermal conductivity; and the experimental conditions under which the two modes can be observed and told apart. As of 2026 both theory branches remain active, second sound has not been observed in a suspended monolayer, and neither the dispersion nor the drift–divergence question is resolved.

Background

In an insulating crystal, heat is carried by phonons — quantized lattice vibrations. When collisions that conserve total phonon momentum are much more frequent than processes that relax it, phonons flow collectively like a fluid; this regime is called phonon hydrodynamics. There, heat no longer diffuses but propagates as a damped wave, second sound, the thermal analogue of an ordinary sound wave. Momentum-conserving hydrodynamic heat transport was predicted for graphene and other two-dimensional crystals by Cepellotti et al. (Nat. Commun. 6, 6400 (2015)) and Lee et al. (Nat. Commun. 6, 6290 (2015)), and second sound has since been observed in bulk and layered crystals — in graphite above 100 K (Huberman et al., Science 364 (2019)), at room temperature in isotopically pure graphite (Xie et al., Nat. Commun. 17, 4806 (2026)), and in rapidly varying temperature fields in germanium (Beardo et al., Sci. Adv. 7, eabg4677 (2021)) — but never in a suspended two-dimensional monolayer.

A free two-dimensional crystal such as graphene behaves mechanically as a membrane, and its out-of-plane acoustic vibration branch — the flexural, or ZA, mode — is special: rotational symmetry forces its long-wavelength dispersion to be quadratic, $\omega_k\propto k^2$, rather than linear as for ordinary sound branches. The quadratic law makes low-frequency flexural phonons enormously populated and is the origin of the flagship prediction that the thermal conductivity of an ideal (infinite, unstrained, suspended) two-dimensional crystal grows logarithmically with system size, $\kappa\propto\ln L$. Whether this divergence survives in real materials depends on what anharmonicity — the coupling between out-of-plane bending and in-plane stretching — does at the smallest wave vectors. Mariani and von Oppen (Phys. Rev. Lett. 100, 076801 (2008)) showed that symmetry protects the quadratic law against bending–stretching coupling up to logarithmic corrections, while the membrane renormalization-group tradition expects a renormalized power law $\omega_k\propto k^{2-\eta/2}$ with $\eta$ a small anomalous exponent, i.e. an exponent strictly between 1 and 2, which would remove the $\ln L$ divergence. Non-perturbative self-consistent calculations by Aseginolaza et al. (Nat. Phys. 20, 1288–1293 (2024)) support the protected quadratic law, whereas elasticity-based treatments building on bending-rigidity renormalization, such as Ravichandran (arXiv:2604.03910 (2026)), support the renormalizing branch; the conflict is unresolved.

Shang, Mao, Yang, Li and Lü (Phys. Rev. B 105, 165423 (2022)) built a unified theory of second sound for such membranes from Callaway-type balance equations — a two-relaxation-time treatment of the phonon Boltzmann equation that separates momentum-conserving from momentum-relaxing collisions, with characteristic times $\langle\tau_c\rangle_q$ and $\langle\tau_R^{-1}\rangle_p$. The theory exhibits two second-sound modes: a drifting mode in which the temperature wave is advected by a net phonon drift, and a driftless mode in which no net drift occurs. Its quantitative predictions are sensitive to the assumed low-$k$ flexural dispersion, to the numerical protocol (force constants — the harmonic interatomic interaction matrices — must be symmetrized to enforce the membrane's exact symmetries and the quadratic ZA law, as in Carrete et al. (Mater. Res. Lett. 4, 204–211 (2016)), and the Brillouin-zone sampling needs a low-$k$ cutoff of order $2\pi/L$), and it deliberately neglects energy exchange with electrons and other quasiparticles. The authors themselves stress that hydrodynamic drift is not a necessary condition for a divergent thermal conductivity, since divergence also arises in plain relaxation-time approximations; the causal links among the ZA dispersion, finite-size behavior, drift, and the measured divergence are therefore not settled.

Problem Statement

Determine the asymptotic long-wavelength fate of the flexural (ZA) phonon dispersion in real suspended two-dimensional materials — does anharmonic renormalization produce $\omega_k\propto k^{\gamma}$ with $1<\gamma<2$, or does the quadratic law $\omega_k\propto k^2$ survive down to $k\to 0$ — and establish the consequences of that fate for monolayer two-dimensional materials, jointly addressing all four of the following, each a required part: (i) the existence, velocities, and strain and finite-size dependence of the drifting and driftless second-sound modes, including the relaxation times $\langle\tau_c\rangle_q$ and $\langle\tau_R^{-1}\rangle_p$ that control them; (ii) the quantitative modification of those relaxation times and of the visibility of the two second-sound modes — the prominence of each mode's peak in the temperature response — by the energy-exchange channels the theory omitted, such as electron–phonon coupling; (iii) the causal relation between hydrodynamic phonon drift and the logarithmic, size-dependent divergence of the thermal conductivity, resolved for whichever dispersion fate is determined, including the case in which that divergence is removed; and (iv) the experimental conditions, across excitation frequency, sample size and strain state, under which the second-sound modes that part (i) finds to exist can be observed and told apart by a stated measurable signature in a suspended two-dimensional monolayer.

References

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