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arXiv:2504.04436 · Physical Review Research

Intertwined geometries in collective modes of two-dimensional Dirac fermions

Ankan Biswas, Avraham Klein

Published Jul 7, 2025• 0 citations
arXivHTMLPDF

Abstract

It is well known that the time-dependent response of a correlated system can be inferred from its spectral correlation functions. As a textbook example, the zero-sound collective modes of a Fermi liquid appear as poles of its particle-hole susceptibilities. However, the Fermi liquid’s interactions endow these response functions with a complex analytic structure, so that this time-frequency relationship is no longer straightforward. We study how the geometry of this structure is modified by a nontrivial band geometry, via a calculation of the zero-sound spectrum of a Dirac cone in two dimensions. We find that the chiral wave functions, which encode the band geometry, fundamentally change the analytic structure of the response functions, which encode its Riemannian geometry. As a result, isotropic interactions can give rise to a variety of unconventional zero-sound modes, which, due to the geometry of the functions in frequency space, can only be identified via time-resolved probes. These modes are absent in a conventional Fermi liquid with similar interactions, so that these modes can be used as a sensitive probe for the existence of Dirac points in a band structure.

DOI
10.1103/bh3c-4sng

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It is well known that the time-dependent response of a correlated system can be inferred from its spectral correlation functions. As a textbook example, the zero-sound collective modes of a Fermi liquid appear as poles of its particle-hole susceptibilities. However, the Fermi liquid’s interactions endow these response functions with a complex analytic structure, so that this time-frequency relationship is no longer straightforward. We study how the geometry of this structure is modified by a nontrivial band geometry, via a calculation of the zero-sound spectrum of a Dirac cone in two dimensions. We find that the chiral wave functions, which encode the band geometry, fundamentally change the analytic structure of the response functions, which encode its Riemannian geometry. As a result, isotropic interactions can give rise to a variety of unconventional zero-sound modes, which, due to the geometry of the functions in frequency space, can only be identified via time-resolved probes. These modes are absent in a conventional Fermi liquid with similar interactions, so that these modes can be used as a sensitive probe for the existence of Dirac points in a band structure.
Abstract The longstanding view of the zero sound mode in a Fermi liquid is that for repulsive interaction it resides outside the particle-hole continuum and gives rise to a sharp peak in the corresponding susceptibility, while for attractive interaction it is a resonance inside the particle-hole continuum. We argue that in a two-dimensional Fermi liquid there exist two additional types of zero sound: “hidden” and “mirage” modes. A hidden mode resides outside the particle-hole continuum already for attractive interaction. It does not appear as a sharp peak in the susceptibility, but determines the long-time transient response of a Fermi liquid and can be identified in pump-probe experiments. A mirage mode emerges for strong enough repulsion. Unlike the conventional zero sound, it does not correspond to a true pole, yet it gives rise to a peak in the particle-hole susceptibility. It can be detected by measuring the width of the peak, which for a mirage mode is larger than the single-particle scattering rate.
We study the appearance of a sharp collective mode which features transverse current fluctuations within the bosonization approach to interacting two-dimensional Fermi liquids. This mode is analogous to the shear sound modes in elastic media, and, unlike the conventional zero sound mode, it is damped in weakly interacting Fermi liquids and only separates away from the particle-hole continuum when the quasiparticle mass becomes twice the transport mass ${m}^{*}\ensuremath{\gtrsim}2m$. The shear sound should be present in a large class of interacting charged and neutral Fermi liquids especially those proximate to critical points where the quasiparticle mass diverges. In metals this mode remains linearly dispersing in the presence of the long-ranged Coulomb force, unlike the conventional zero sound mode which becomes the plasma mode. We also detail a quick path between bosonization and classical Landau's Fermi liquid theory by constructing a mapping between the solutions of the classical kinetic equation and the quantized bosonic eigenmodes. By further mapping the kinetic equation into a one-dimensional tight-binding model we solve for the entire spectrum of collective and incoherent particle-hole excitations of Fermi liquids with nonzero F0{F}_{0}F0​ and F1{F}_{1}F1​ Landau parameters.
We study the Fermi liquid properties of the cold atomic dipolar Fermi gases with the explicit dipolar anisotropy using perturbative approaches. Due to the explicit dipolar anisotropy, Fermi surfaces exhibit distortions of the ${d}_{{r}^{2}\ensuremath{-}3{z}^{2}}$ type in three dimensions and of the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ type in two dimensions. The fermion self-energy, effective mass, and Fermi velocity develop the same anisotropy at the Hartree-Fock level proportional to the interaction strength. The Landau interaction parameters in the isotropic Fermi liquids become the tridiagonal Landau interaction matrices in the dipolar Fermi liquids which renormalize thermodynamic susceptibilities. With large dipolar interaction strength, the Fermi surface collapses along directions perpendicular to the dipole orientation. The dynamic collective zero sound modes exhibit an anisotropic dispersion with the largest sound velocity propagating along the polar directions. Similarly, the longitudinal ppp-wave channel spin mode becomes a propagating mode with an anisotropic dispersion in multicomponent dipolar systems.
Abstract Using the Landau kinetic equation to study the non-equilibrium behavior of interacting Fermi systems is one of the crowning achievements of Landau’s Fermi liquid theory. While thorough study of transport modes has been done for standard three-dimensional Fermi liquids, an equally in-depth analysis for two dimensional Fermi liquids is lacking. In applying the Landau kinetic equation (LKE) to a two-dimensional Fermi liquid, we obtain unconventional behavior of the zero sound mode c 0 . As a function of the usual dimensionless parameter s = ω / q v F , we find two peculiar results: first, for | s | > 1 we see the propagation of an undamped mode for weakly interacting systems. This differs from the three dimensional case where an undamped mode only propagates for repulsive interactions and the mode experiences Landau damping for any arbitrary attractive interaction. Second, we find that regardless of interaction strength, a propagating mode is forbidden for | s | < 1. This is profoundly different from the three-dimensional case where a mode can propagate, albeit damped. In addition, we present a revised Pomeranchuk instability condition for a two-dimensional Fermi liquid as well as equations of motion for the fluid that follow directly from the LKE. In two dimensions, we find a constant minimum for all Landau parameters for ℓ ⩾ 1 which differs from the three dimensional case. Finally we discuss the effect of a Coulomb interaction on the system resulting in the plasmon frequency ω p exhibiting a crossover to the zero sound mode.
We consider a system of interacting fermions in two dimensions beyond the second-order perturbation theory in the interaction. It is shown that the mass-shell singularities in the self-energy, arising already at the second order of the perturbation theory, manifest a nonperturbative effect: an interaction with the zero-sound mode. Resumming the perturbation theory for a weak, short-range interaction and accounting for a finite curvature of the fermion spectrum, we eliminate the singularities and obtain the results for the quasiparticle self-energy and the spectral function to all orders in the interaction with the zero-sound mode. A threshold for emission of zero-sound waves leads a nonmonotonic variation of the self-energy with energy (or momentum) near the mass shell. Consequently, the spectral function has a kinklike feature. We also study in detail a nonanalytic temperature dependence of the specific heat $C(T)\ensuremath{\propto}{T}^{2}$. It turns out that although the interaction with the collective mode results in an enhancement of the fermion self-energy, this interaction does not affect the nonanalytic term in C(T)C(T)C(T) due to a subtle cancellation between the contributions from the real and imaginary parts of the self-energy. For a short-range and weak interaction, this implies that the second-order perturbation theory suffices to determine the nonanalytic part of C(T)C(T)C(T). We also obtain a general form of the nonanalytic term in C(T)C(T)C(T), valid for the case of a generic Fermi liquid, i.e., beyond the perturbation theory.