Zero sound (ZS), a collective excitation in Fermi liquids (FLs), involves Fermi surface (FS) deformation and reveals correlation strengths. In 3D FLs, ZS modes are typically anti-bound states (outside the particle-hole continuum) for repulsive interactions, manifesting as sharp peaks in spectroscopic measurements, and resonances (inside the continuum) for attractive interactions. This paper investigates ZS excitations in clean 2D FLs, revealing two unconventional behaviors. First, with weak attraction, ZS modes (any angular momentum *l*) are propagating modes (*V*_ZS > *V*_F) but lack sharp peaks in spectroscopic probes. Second, with strong repulsion (*l* ≥ 1), ZS modes appear as peaks with *V*_ZS > *V*_F but aren't true poles, resulting in broader peaks than the single-particle scattering rate would suggest. The non-analyticity of the charge (c) and spin (s) susceptibilities χ^(c,s)(*q*, ω) in 2D, with branch points at ω = ±*V*_F*q*, is central to these observations. This non-analyticity is described by a two-sheet genus 0 algebraic Riemann surface (a sphere). The ZS modes are poles of χ^(c,s)(*q*, ω) whose movement across branch cuts between physical and unphysical sheets explains the observed behavior. The paper explores these unusual modes for *l* = 0 (hidden mode example) and *l* = 1 (mirage mode example), using the susceptibility equation (1 + F_C(s)χ^(s)(s)) = 0, where F_C(s) is a Landau parameter.

The authors extensively review the existing literature on Fermi liquid theory and zero sound, referencing key works by Abrikosov, Gorkov, and Dzyaloshinski; Lifshitz and Pitaevskii; Baym and Pethick; and Nozières and Pines. They discuss previous research on the propagation of zero sound in liquid ³He and theoretical work concerning the stability of Fermi liquids and Pomeranchuk instabilities. The review also touches upon prior research on Fermi liquid instabilities, including work focusing on nematic Fermi fluids and spin channel instabilities. Furthermore, the authors cite studies on the limits of dynamically generated spin-orbit coupling and the impact of vertex corrections on impurity scattering. This substantial review establishes the context for their novel findings and clearly positions their work within the existing theoretical framework.

The authors employ a theoretical approach based on the analysis of the charge/spin susceptibility χ^(s)(q, ω) in the angular momentum channel *l*. They use the quasiparticle susceptibility at small ω and q, expressed in terms of Landau parameters, particularly focusing on cases where one Landau parameter, F_C(s), is significantly larger than others. The quasiparticle susceptibility is expressed using equation (1): χ^(s)(s) ∝ 1/(1 + F_C(s)χ^(s)), where χ^(s) is the quasiparticle contribution. To determine the pole positions relative to the branch cut, vertex corrections due to impurity scattering are incorporated. The analysis involves studying the analytic structure of χ^(c,s)(*q*, ω) on the complex ω-plane, focusing on branch points and branch cuts. The paper utilizes the concept of a two-sheeted Riemann surface to describe the analytic structure of the susceptibility. The locations of poles (representing ZS modes) on this surface determine whether a mode is conventional, hidden, or mirage. The authors perform calculations for *l* = 0 (hidden mode) and *l* = 1 (mirage mode) and examine the time-domain response function χ^(s)(q, t) through Fourier transformation, focusing on how poles and branch points contribute to the long-time behavior. The time evolution of χ^(s)(q, t) is then analyzed to identify distinctive features of hidden and mirage modes. The methodology involves analyzing the crossover time (t_cross) where the dominant contribution to the response shifts from pole to branch point contributions. Numerical calculations and simulations support the analytical findings. The paper provides detailed descriptions of calculation techniques, including the treatment of impurity scattering, vertex corrections, and branch cut integrations.

The primary findings revolve around two unconventional zero sound modes in 2D Fermi liquids: 1. **Hidden modes:** These modes exist even under attractive interactions (−1/2 < F^(s)_c < 0 for *l* = 0). Although formally outside the particle-hole continuum (anti-bound states), they are located below the branch cut and do not show up as sharp peaks in Imχ^(s)(*q*, ω). However, they influence the long-time transient response function χ^(s)(q, t), exhibiting a crossover from cos(v_Fqt + π/4)/t^1/2 to cos(v_Fqt − π/4)/t^3/2 behavior at long times. This crossover time (t_cross) is related to the hidden pole's position. 2. **Mirage modes:** These modes emerge in channels with *l* ≥ 1 and sufficiently strong repulsive interaction (F^(s)_c > F^(s)_m, where F^(s)_m ≈ 3/5). They initially reside above the branch cut as conventional ZS poles but, beyond a critical interaction strength, move to the unphysical Riemann sheet. Despite being on the unphysical sheet, they create peaks in Imχ^(s)(*q*, ω), distinguishable from conventional modes by their broader width (γ_m > γ). In the time domain, mirage modes display a crossover from oscillations at the ZS frequency to oscillations at the branch point frequency ω = v_Fq at time t = t_cross. The authors demonstrate that the time evolution of the susceptibility provides a crucial tool to distinguish between conventional ZS modes, hidden modes, and mirage modes. Specific analysis of the time-dependent susceptibility χ^(s)(q, t) reveals characteristic decay and oscillation patterns associated with each type of mode. Numerical results show excellent agreement with the theoretical predictions. The analysis is further extended to the case where multiple Landau parameters ($F_0^{(s)}$ and $F_l^{(s)}$) are comparable. The authors find that the existence of hidden and mirage modes remains largely unchanged, even under this more complex condition.

The discovery of hidden and mirage modes significantly expands our understanding of collective excitations in 2D Fermi liquids. The existence of these unconventional modes challenges the traditional understanding of zero sound, highlighting the importance of considering the topological features of the Riemann surface associated with the susceptibility. The findings demonstrate that the dynamics of 2D Fermi liquids are not solely determined by the poles of their response functions but also by the Riemann surface's topological properties. The algebraic nature of the branch points in 2D, in contrast to the logarithmic branch points in 3D, is identified as the origin of these novel modes. The distinction between the *l* = 0 (conserved quantity) and *l* ≥ 1 (non-conserved quantity) channels also plays a critical role in determining the appearance of mirage modes. The authors propose that experimental investigations using pump-probe techniques can directly detect hidden and mirage modes by analyzing the time-dependent response functions, providing valuable insights into the properties of 2D Fermi liquids. The identification of the hidden mode is especially promising in systems where the Landau parameter falls within the predicted range.

This research reveals the existence of hidden and mirage zero sound modes in 2D Fermi liquids, challenging conventional understanding and emphasizing the role of Riemann surface topology. Hidden modes, undetectable spectroscopically, manifest in long-time transient responses, while mirage modes, though appearing as peaks, show broader widths. These modes' existence is linked to the algebraic branch points characteristic of 2D systems. Future studies could explore similar phenomena in more complex systems, such as multi-band systems or those with anisotropic Fermi surfaces. Experimental verification using advanced pump-probe techniques is crucial to validate the theoretical predictions and open new avenues for understanding 2D quantum materials.

The study primarily focuses on clean systems with weak impurity scattering. The effects of stronger disorder or interactions beyond the Landau Fermi-liquid framework could alter the observed behavior of hidden and mirage modes. The analysis assumes isotropic systems; the impact of anisotropy on these modes requires further investigation. The study primarily focuses on theoretical aspects; experimental confirmation of the predicted behavior through sophisticated pump-probe techniques remains a critical next step.