Hidden and mirage collective modes in two dimensional Fermi liquids

Zero sound is a collective excitation of a Fermi liquid linked to Fermi-surface deformations. Conventional wisdom (well established in 3D) states that for repulsive interactions, ZS is a sharp anti-bound state outside the particle-hole continuum, whereas for attractive interactions it is a Landau-overdamped resonance within the continuum. This work asks whether, in 2D Fermi liquids, additional qualitatively distinct zero-sound behaviors can occur due to the different analytic structure of susceptibilities near the particle-hole threshold. The purpose is to reveal and characterize new ZS regimes (hidden and mirage), explain their analytic origin on a two-sheet Riemann surface, and propose experimental strategies (time-domain pump-probe) to detect them when frequency-domain probes may fail. The study is significant because it revises the standard picture of collective modes in 2D Fermi liquids and connects response dynamics to topological/analytic properties of susceptibilities.

Prior theory of Landau Fermi liquids (Abrikosov-Gorkov-Dzyaloshinski; Lifshitz-Pitaevskii; Baym-Pethick; Nozières-Pines) establishes ZS behavior in 3D and criteria for Pomeranchuk instabilities. For attractive interactions, Landau damping near Pomeranchuk transitions leads to overdamped modes. In 2D, previous work discussed collective modes near Pomeranchuk instabilities and transverse l=1 modes, but the standard dichotomy of sharp (repulsive) vs overdamped (attractive) ZS was retained. The authors emphasize a key 2D difference: the particle-hole threshold produces algebraic (square-root) branch points (versus logarithmic in 3D), implying χ(q,ω) lives on a two-sheet genus-0 Riemann surface. Related “tachyon ghost” behavior was noted for 2D plasmons with retardation. These insights motivate re-examining pole trajectories and observable consequences in 2D.

- Framework: Landau Fermi-liquid theory in 2D with charge/spin susceptibilities χ^(c,s)(q,ω) decomposed into angular momentum channels l. Focus on regimes where one Landau parameter F_C^(s) (or F_l^(s)) dominates. - Impurities and vertex corrections: Include weak disorder through a single-particle damping γ (from short-range impurities) in Green’s functions G_0 and incorporate the corresponding vertex corrections to correctly place poles relative to branch cuts, even in the clean limit γ→0. - Analytic structure: Work with dimensionless s = ω/(v_F q). The susceptibility χ^(s)_l(s) has branch points at s = ±(1 − iγ) with branch cuts slightly below the real axis in the clean limit. The nonanalyticity in 2D is algebraic, producing a two-sheet Riemann surface. ZS modes are poles of χ on this surface; their locations relative to branch cuts determine visibility in spectroscopic probes. - Representative channels: Derive explicit forms for l=0 and l=1 longitudinal channels as exemplars. - l=0 with vertex corrections: χ_0^(s)(s) = 1 + i s / [1 − (s + iγ)^2]. Solve 1 + F_C^(s) χ_0^(s)(s) = 0 for pole positions. - l=1 longitudinal: χ_1(s) = 1 + 2 s^2 + i γ_1 sqrt[1 − (s + iγ)^2] / [1 − (s + iγ)^2]. Analyze pole evolution with F_C^(s). - Time-domain analysis: Compute χ(q,t) via Fourier transform of χ(q,ω) for real ω using contour integration on the Riemann surface. Decompose χ(q,t) into: (i) pole contributions (on physical or unphysical sheets depending on contour choice) and (ii) branch-cut contributions from near the threshold points. Evaluate long-time asymptotics and identify crossovers. - Regime classification and detection: - Conventional ZS: pole above branch cut; underdamped with intrinsic decay γ_ZS < γ; dominates long-time dynamics. - Hidden ZS: pole outside continuum (s_h > 1) for attractive interactions but located below the branch cut; no peak in Im χ(ω); cancels in time-domain pole contribution; detectable via altered branch-point asymptotics (t^(-1/2) to t^(-3/2) crossover with phase shift). - Mirage ZS (l ≥ 1): for sufficiently strong repulsion, the pole crosses the cut to the unphysical sheet; despite not being a true pole on the physical sheet, it yields a peak in Im χ(ω) with width larger than γ; time-domain dynamics crosses over to branch-point controlled oscillations at long times. - Also treat cases with comparable F_0^(s) and F_1^(s); extend to charged systems (plasmon in l=0 charge channel; coexisting plasmons and acoustic ZS in l ≥ 1).

- Hidden zero sound in 2D: - l=0: For attractive interaction −1/2 < F_C^(s) < 0, poles are at ω = v_F q (± s_h − i γ_h) with s_h = (1 − F_C^(s)) / sqrt(1 − 2 F_C^(s)) > 1 and γ_h ≈ γ (1 − F_C^(s)) / (1 − 2 F_C^(s)) > γ. The pole lies below the branch cut and does not produce a sharp peak in Im χ(ω). Nevertheless, it controls transient response via a characteristic crossover in χ(q,t). - l=1 longitudinal: A hidden pole exists for −1/9 < F_C^(s) < 0 with analogous properties. - Mirage zero sound (l ≥ 1): - For repulsive interaction above a critical value, the conventional ZS pole moves through the branch cut to the unphysical sheet. For l=1, the crossover occurs at F_C^(s) = F_m ≈ 3/5 (clean limit). The resulting mirage pole yields a peak in Im χ(ω) despite not being a true pole on the physical sheet. The peak width γ_m exceeds the single-particle scattering rate γ. - Conventional regimes and thresholds: - l=0, F_C^(s) > 0: conventional propagating ZS with s_ZS = (1 + F_C^(s)) / sqrt(1 + 2 F_C^(s)) > 1 and γ_ZS = γ (1 + F_C^(s)) / (1 + 2 F_C^(s)) < γ. - l=0, −1 < F_C^(s) < −1/2: Landau-overdamped resonance with purely imaginary frequency approaching zero at F_C^(s) → −1 (Pomeranchuk instability). - Time-domain signatures and crossovers: - Branch-point asymptotics in 2D: χ(q,t) ∝ cos(v_F q t − π/4) e^(−γ t) t^(−3/2) at long times. - Hidden mode (l=0): χ(q,t) exhibits a crossover from cos(v_F q t + π/4) t^(−1/2) to cos(v_F q t − π/4) t^(−3/2), with a phase shift of π/2. The crossover time scales as t_cross,1 ≈ 1/(s_h − 1); for small |F_0^(s)|, t_cross ≈ ℏ (1 − iγ)^(−1) (in dimensionless units t* = v_F q t, t* ≈ 1/(s_h − 1)). - Conventional ZS: χ(q,t) dominated by underdamped oscillations at ω_ZS, decay rate γ_ZS < γ. - Mirage ZS: χ(q,t) initially shows oscillations at ω_m (set by the mirage pole) with decay rate γ_m > γ, but at longer times crosses over to branch-point oscillations at ω = v_F q. Typical crossover scale t_cross,2 ~ (γ_ZS − γ)^(−1) log(...), mirage analog t_cross,4 ~ (γ_m − γ)^(−1) log(...). - Spectroscopy vs time-domain: - Hidden modes do not produce peaks in Im χ(ω) and are invisible to standard spectroscopy but can be revealed by pump-probe time-domain measurements via their distinctive crossover in decay law and phase. - Mirage modes do produce peaks in Im χ(ω) but can be distinguished from conventional ZS because their linewidth exceeds γ and by observing time-domain crossover to ω = v_F q oscillations. - Generalization and robustness: - Results persist when two Landau parameters F_0^(s) and F_1^(s) are comparable; mirage modes can then occur in both l=0 and l=1 and at smaller interaction strengths (e.g., for F_1^(s) > 0.15 if F_0^(s) = 1). - In charged Fermi liquids: l=0 charge mode becomes a plasmon; for l ≥ 1 longitudinal channels, both acoustic ZS and plasmon modes exist. - There is also a mirage mode in the 3D l=1 longitudinal channel (crossover at F_0 ≈ 0.44 as γ→0), but 2D is the natural setting due to continuous pole trajectories on the Riemann surface. - Experimental relevance: - Detection requires finite-q pump-probe (e.g., time-resolved RIXS, neutron scattering) or spatial modulation/confinement in 2D. Hidden spin-mode most accessible for systems with 0 < F_0 < 1/2; GaAs/AlGaAs 2DEGs have measured F_0 in the requisite range.

The research demonstrates that 2D Fermi liquids possess richer zero-sound phenomenology than the standard picture suggests. The algebraic threshold singularity in 2D creates a two-sheet Riemann surface for χ(q,ω), enabling pole trajectories that cross branch cuts while preserving topological invariants (number of poles and genus). As a result, for weak attractive interactions, ZS poles reside outside the continuum yet below the branch cut (hidden modes) and thus are not visible in Im χ(ω) but shape long-time dynamics. For sufficiently strong repulsion in l ≥ 1, poles migrate to the unphysical sheet (mirage modes), still producing spectroscopic peaks but not dominating long-time dynamics, which is governed by branch points. These findings resolve how 2D analytic structure modifies the conventional 3D expectations and provide concrete time-domain signatures to distinguish conventional and mirage ZS. The work emphasizes that topological/analytic properties of response functions (branch points, Riemann sheets) are crucial for interpreting observed collective modes. It also delineates experimental conditions and materials regimes where hidden or mirage modes should be observable.

This paper identifies and characterizes two unconventional zero-sound excitations in 2D Fermi liquids: hidden modes (outside the continuum for attractive interactions but below the branch cut, invisible in spectroscopy) and mirage modes (poles on the unphysical sheet for sufficiently strong repulsion in l ≥ 1, visible as broadened peaks). The authors connect these modes to the square-root branch points at the particle-hole threshold and the resulting two-sheet Riemann surface, and show how time-resolved measurements can distinguish them via long-time asymptotics and crossovers in χ(q,t). The results generalize to cases with multiple comparable Landau parameters and to charged systems; more complex band structures (multiple Fermi velocities) lead to higher-genus Riemann surfaces and potentially new topological features of ZS. Future directions include experimental pump-probe studies at finite q (e.g., time-resolved RIXS or neutron scattering) to detect hidden/mirage modes, exploration of multiband systems with toroidal Riemann surfaces, and systematic analysis of hydrodynamic-to-collisionless crossovers where hidden modes are absent.

- Assumes isotropic, Galilean-invariant 2D Fermi liquids with one dominant Landau parameter (extensions to comparable F_0 and F_1 were analyzed but more complex interactions may add structure). - Disorder included as weak, short-range impurity scattering; analysis focused on the collisionless regime (ω ≫ ν_eq γ). Hidden modes do not exist in the hydrodynamic limit (ω ≪ ν_eq γ). - Temperature assumed low enough to neglect inelastic quasiparticle damping and above any superconducting instability; finite-T or strong inelastic scattering may modify linewidths and crossovers. - Experimental detection requires finite-q time-domain techniques or engineered spatial modulation; spectral identification of mirage modes relies on independent knowledge of γ to separate linewidth contributions. - Most explicit analytics presented for l=0 and l=1 longitudinal channels; while qualitative generalizations are argued, detailed quantitative thresholds may vary for higher l and in anisotropic or multiband systems.