Hidden and mirage collective modes in two dimensional Fermi liquids

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.


INTRODUCTION
Zero-sound (ZS) is a collective excitation of a Fermi liquid (FL) associated with a deformation of the Fermi surface (FS) [1][2][3][4] . The dispersion of the ZS mode ω = v zs q encodes important information about the strength of correlations, as was demonstrated in classical experiments on 3 He 5 . Conventional wisdom holds 6 that for a strong enough repulsive interaction in a given charge or spin channel, ZS excitations are anti-bound states which live outside the particle hole continuum (v zs > v F ) and appear as sharp peaks in spectroscopic probes, while for attractive interaction they are resonances buried inside the continuum. Possibly the best known example of a resonance is a Landau-overdamped mode near a Pomeranchuk transition 1-4,6-16 . These qualitative notions are consistent with rigorous results for a 3D FL [1][2][3][4]6 .
In this paper we report on two unconventional features of ZS excitations in a clean 2D FL. First, for relatively weak attraction, ZS modes with any angular momentum l are not the expected overdamped resonances but rather sharp propagating modes with v zs > v F . However, a spectroscopic probe will not show a peak at ω = v zs q. Second, for sufficiently strong repulsion, ZS modes with l ≥ 1 appear as peaks in a spectroscopic measurement with v zs > v F , but the modes are not the true poles of the dynamical susceptibility and, as a result, are not the longest lived excitations of the system. We argue that these two features come about because the charge (c) and spin (s) susceptibilities χ cðsÞ l ðq; ωÞ in the angular momentum channel l are nonanalytic functions of complex ω with branch points at ω = ±v F q, which arise from the threshold singularity at the edge of the particle hole continuum. Accordingly, χ cðsÞ l ðq; ωÞ is defined on the complex ω plane with branch cuts, located slightly below the real axis in the clean limit (see Fig. 1). In 3D, χ cðsÞ l ðq; ωÞ near a branch point has only a weak logarithmic non-analyticity. In 2D, however, the nonanalyticity is algebraic ( ffiffi ffi x p ). In this situation, the analytic structure of χ cðsÞ l ðq; ωÞ is encoded in a two-sheet genus 0 algebraic Riemann surface (a sphere) [17][18][19] . It has a physical sheet, on which χ cðsÞ l ðq; ωÞ is analytic in the upper half-plane by causality, and a nonphysical sheet. The ZS modes appear as poles of χ cðsÞ l ðq; ωÞ. Both the genus and the number of ZS poles are topological invariants of χ cðsÞ l ðq; ωÞ, which remain unchanged as the poles move on continuous trajectories over the complex plane. However, to pass smoothly through a branch cut, a ZS pole must move from the physical to unphysical sheet and vice versa. We show that, for relatively weak attractive interaction, the propagating pole is on the physical sheet, but below the branch cut. Consequently, it cannot be analytically extended to the real ω axis of the physical sheet and does not give rise to a sharp peak in Imχ cðsÞ l ðq; ωÞ above the continuum. We label such a mode as "hidden". It is similar to the "tachyon ghost" plasmon that appears in an ultra-clean 2D electron gas once retardation effects are taken into account 20,21 . For sufficiently weak repulsive interaction in channels with l ≥ 1, the pole is located above the branch cut but, when the interaction exceeds some critical value, the pole moves through the branch cut to the unphysical Riemann sheet. Although the pole is now below the branch cut, it does gives rise to a peak in χ l (q, ω) because the pole can be continued back through the branch cut to the physical real axis. We label such a mode as "mirage".
Hidden and mirage modes cannot be directly identified spectroscopically by probing Imχ cðsÞ l ðq; ωÞ, as hidden modes do not appear in such a measurement at all, while mirage modes do appear but cannot be distinguished from conventional modes. We argue, however, that they can be identified by studying the transient response of a 2D FL in real time, i.e., by analyzing χ cðsÞ l ðq; tÞ extracted from pump-probe measurements, which have recently emerged as a powerful technique for characterizing and controlling complex materials [22][23][24][25][26][27][28][29][30] . At long times, the response function χ cðsÞ l ðq; tÞ is the sum of contributions from the ZS poles and the branch points. One can readily distinguish a conventional ZS modes from a mirage one via χ cðsÞ l ðq; tÞ because a conventional ZS mode is located above the branch cut and decays slower than the branch point contribution, while a mirage mode decays faster. As a result, the response of a FL hosting a mirage mode undergoes 1 a crossover from oscillations at the ZS mode frequency to oscillations at the branch point frequency ω = v F q at some t = t cross (see Fig. 2). The detection of a hidden mode is a more subtle issue as this mode does not appear on the real frequency axis, and χ cðsÞ l ðq; tÞ at large t always oscillates at ω = v F q. However, we show that in the presence of the hidden pole the behavior of χ cðsÞ l ðq; tÞ changes from cosðv F qt þ π=4Þ=t 1=2 at intermediate t to cosðv F qt À π=4Þ=t 3=2 at the longest t, and the location of the hidden pole can be extracted from the crossover scale t cross between the two regimes (see Fig. 3a).

Zero-sound modes in 2D
A generic bosonic excitation of a FL with angular momentum l and dispersion ω(q) is the solution of χ cðsÞ l ðq; ωÞ À1 ¼ 0. ZS excitations are the modes with linear dispersion ω = v zs q in the limit q ≪ k F , where k F is the Fermi momentum. The quasiparticle susceptibility at small ω and q but fixed ω/v F q = s is expressed solely in terms of Landau parameters F cðsÞ l in the charge or spin sectors [1][2][3][4]6,7,[14][15][16] . An explicit form of χ cðsÞ l ðq; ωÞ is rather cumbersome but becomes much simpler if one of the Landau parameters, F cðsÞ l , is much larger than the others. Up to an irrelevant overall factor, for this case we have where χ l (s) is the quasiparticle contribution from states near the FS, normalized to χ l (0) = 1. The general structure of χ l (s) can be inferred from the particle-hole bubble of free fermions with propagators G 0 ðk; ωÞ ¼ ω þ iγ=2 À v F ðjkj À k F Þ ð Þ À1 and formfactors f l (θ) at the vertices, where θ is the angle between k and q, f 0 = 1, and f l ðθÞ ¼ ffiffi ffi 2 p cos lθ ð ffiffi ffi 2 p sin lθÞ for the longitudinal (transverse) channels with l ≥ 1. (The longitudinal/transverse modes correspond to oscillations of the FS that conserve/do not conserve its area.) However, to properly specify the position of the pole with respect to the branch cut one must include vertex corrections due to the same scattering processes that give rise to the iγ term in G 0 (refs 15,31 ). This is true even in the clean limit γ ! 0. To be specific, we assume that extrinsic damping is provided by short-range impurities, and account for the corresponding vertex corrections in all subsequent calculations. We study the case l = 0 as an example of a hidden mode, and the case l = 1, with f l ðθÞ ¼ ffiffi ffi 2 p cos θ, as an example of a mirage mode. (The l = 1 transverse mode has recently been discussed in refs 15,16 ).
For l = 0, χ 0 (s) with vertex corrections due to impurity scattering included is given by 15,31 where γ ¼γ=v F q. Observe that (i) χ 0 (s) vanishes at q → 0 and finite ω and γ, as required by charge/spin conservation, and (ii) χ 0 (s) has branch cuts at s = ±x − iγ, x > 1, see Fig. 1 q > 1 and γ zs ¼ γð1 þ F cðsÞ 0 Þ=ð1 þ 2F cðsÞ 0 Þ < γ. These are conventional ZS poles above the branch cut, which give rise to a peak in Imχ cðsÞ 0 ðq; ωÞ at ω = v F s zs q. For À1 < F cðsÞ 0 < À 1=2, Fig. 1 Trajectories of the poles of χ cðsÞ l ðq; ωÞ on the two-sheeted Riemann surface of complex s = ω/v F q. a l = 0 surface. Blue circles: overdamped ZS mode; magenta circles: hidden mode; orange circles: propagating ZS mode. b l = 1 surface. Blue circles: damped ZS mode; magenta circles: hidden mode; orange circles: propagating ZS mode; green circles: mirage mode. For clarity, additional poles on the unphysical sheet are not shown (see the "Methods" section). In both figures, solid (dashed) circles denote the poles on the physical (unphysical) Riemann sheet. Solid (dashed) blue arrows denote the direction of poles' motion on the physical (unphysical) sheet with increasing F cðsÞ l . ¼ 0:2 (orange) and a mirage mode at F cðsÞ 1 ¼ 8:0 (green). The modes correspond to the orange and green circles in Fig. 1b. The conventional mode displays an underdamped behavior with decay constant γ zs < γ and oscillation period T* = 2π/s zs < 2π at all times. The mirage mode decays with γ zs > γ and crosses over to oscillations with period T* = 2π at a crossover time t cross % ðγ zs À γÞ À1 . Inset: a zoomed-in view showing the crossover at t*~t cross . χ cðsÞ 1 ðt Ã Þ is multiplied by e γt Ã to enhance visibility. The solid line is added to the data points for clarity. The disorder strength is γ = 0.2.
A. Klein et al. the two poles are located along the imaginary s axis, one on the physical Riemann sheet, at s zs ¼ Àið1 À jF , and the other on the unphysical Riemann sheet. This is another conventional behaviorthe ZS is Landau overdamped, and at F cðsÞ 0 ! À1 its frequency vanishes, signaling a Pomeranchuk instability 6,15 . The hidden ZS mode emerges at À1=2 < F cðsÞ l < 0. Here the two modes are again located near the real axis, at cðsÞ 0 jÞ > γ. Since s h > 1, the ZS mode is formally outside the continuum, i.e., it is an anti-bound state, even though the interaction is attractive (F cðsÞ 0 < 0). However, because γ h > γ, the pole is located below the branch cut. Since a pole cannot pass smoothly through the cut without moving to a different Riemann sheet, a hidden pole does not give rise to a peak in Imχ cðsÞ ðq; ωÞ at ω = v F s h q. The evolution of the poles with F cðsÞ 0 is depicted in Fig. 1a. For l = 1 one finds: In this case too, a hidden pole exists for attractive interaction, in the interval À1=9 < F cðsÞ 1 < 0. In addition, a new type of behavior emerges for F cðsÞ 1 > 0. Namely, χ cðsÞ 1 has a conventional ZS pole above the branch cut only for a finite range 0 < F cðsÞ 1 < F m 1 , where F m 1 ¼ 3=5 in the clean limit. At F cðsÞ 1 ¼ F m 1 the pole merges with the branch cut and, for larger F cðsÞ 1 , it moves below the branch cut and, simultaneously, to the unphysical Riemann sheet. We call this pole a "mirage" one because although it is located on the unphysical Riemann sheet, it can be connected to the physical real axis through the branch cut. As a result, the pole gives rise to a sharp peak in Imχ cðsÞ 1 ðq; ωÞ; however, the width of the mirage mode, γ m , is larger than γ.

Detection of hidden and mirage modes
We argue that hidden and mirage modes can be observed experimentally by analyzing the transient response of a FL which, for an instantaneous initial perturbation, is described by the susceptibility in the time domain, χ cðsÞ l ðq; tÞ. At first glance, it seems redundant to study χ cðsÞ l ðq; tÞ, which is just a Fourier transform of χ cðsÞ l ðq; ωÞ for real ω, expressed via Imχ cðsÞ l ðq; ωÞ as χ cðsÞ l ðq; t > 0Þ ¼ ð2=πÞ 0 sinðωtÞImχ cðsÞ l ðq; ωÞ by causality. A hidden mode does not give rise to a peak in Imχ cðsÞ l ðq; ωÞ for real ω, while the peak due to a mirage mode is essentially indistinguishable from that due to a conventional ZS mode. However, we will show below that there are subtle features in Imχ cðsÞ l ðq; ωÞ for hidden and mirage modes that manifest themselves in the time evolution of χ cðsÞ l ðq; tÞ. Our reasoning is based on the argument that χ cðsÞ l ðq; tÞ can be obtained by closing the contour of integration over ω on the Riemann surface. A choice of the particular contour is a matter of convenience, but a contour can always be decomposed into a part enclosing the poles in the lower half-plane (either on the physical or unphysical sheet) and a part connecting the branch points on the Riemann sphere. For both conventional and mirage modes the second contribution at long times comes from the vicinity of the branch points and behaves as χ cðsÞ l ðq; tÞ / cosðt Ã À π=4Þe Àγt Ã t À3=2 , where t* = v F qt. The pole contribution behaves as χ cðsÞ l ðq; tÞ / sinðs a t Ã Þe Àγ a t Ã , where a = zs, h, m. For a conventional ZS mode γ zs < γ, and the long-t behavior of χ cðsÞ l ðq; tÞ is dominated by oscillations at the ZS frequency. For a mirage mode γ m > γ, and the oscillations associated with the mirage mode decay faster than the ones associated with the branch points. We illustrate this behavior in Fig. 2 ¼ 8, which correspond to the cases of a conventional and mirage ZS mode, respectively. Alternatively, of course, the mirage mode may be identified from the width of the ZS peak if an independent measurement of γ is available.
For a hidden mode, the situation is more tricky as the pole contribution is cancelled out by a portion of the branch cut contribution and so a hidden pole does not contribute directly to χ cðsÞ 0 ðq; tÞ. The only oscillations in χ cðsÞ 0 ðq; tÞ are due to the branch points, with a period T = 2π/v F q. However, a more careful study shows (see "Methods") that in the presence of a hidden pole the branch point contribution undergoes a crossover between two types of oscillations with the same period: at intermediate t,  Fig. 1a). The gray lines show the characteristic power-law decays ∝ t −1/2 , t −3/2 . b A damped l = 1 mode at F cðsÞ 1 ¼ À0:9 (blue circles in Fig. 1b). At even longer times (not shown), the period of oscillations approaches 2π. c A hidden l = 1 mode at F cðsÞ 1 ¼ À0:121 (magenta circles in Fig. 1b). d The numerically extracted variation of the phase shift between the two regimes of the hidden mode described in the text (solid), and the analytic prediction (dashed), for F  ðq; tÞ / cosðt Ã À π=4Þ=ðt Ã Þ 3=2 . We illustrate this behavior in Fig. 3a. Note that both the t-dependence of the envelope changes and the phase is shifted by π/2. The crossover scale t Ã cross is determined by the position of a hidden pole in relation to the branch point. For small F cðsÞ 0 it is just t Ã cross ¼ js h À ð1 À iγÞj À1 ; this relation is verified numerically in the Methods section. Hence, a hidden pole can be extracted from time-dependent measurements even though it does not show up in spectroscopic probes.
For completeness, we also briefly discuss the behavior of χ cðsÞ 0 ðq; tÞ in the range À1 < F cðsÞ 0 < À1=2, where the pole is Landau overdamped even in the absence of disorder, i.e., ω = −iv F qγ zs 15 . In this situation, dynamics at intermediate t is dominated by a non-oscillatory, exponentially decaying pole contribution, while dynamics at longer t is dominated by algebraically decaying oscillations arising from the branch points, with the period T = 2π/ (v F q). The crossover time is ðt Ã cross Þ À1 ¼ ðγ zs À γÞ À1 to logarithmic accuracy. We also present the results for χ crosses the critical value of −1/9, the poles transform into hidden ones, and oscillations are now controlled by the branch points (Fig. 3c). As a final remark, we also verified that the behavior does not change qualitatively for a more realistic case when two Landau parameters, F cðsÞ 0 and F cðsÞ 1 , have comparable magnitudes.

DISCUSSION
In this work we argued that ZS collective excitations in a 2D FL have two unexpected features. First, for any angular momentum l and for the Landau parameter F cðsÞ l in some negative range, a ZS mode is not a damped resonance inside a particle-hole continuum, as is the case in 3D, but a propagating mode with velocity larger than v F . In the clean limit, a ZS pole of χ cðsÞ l is located arbitrary close to the real axis, but still below the branch cut, which hides the pole. Such a "hidden" mode does not manifest itself in spectroscopic probes but can be identified by transient, pump-probe techniques. Second, for l ≥ 1 and positive F cðsÞ l above some critical value, a ZS pole moves from the physical Riemann surface to the unphysical one and becomes a "mirage" one. In this situation, Imχ cðsÞ l ðq; ωÞ still has a peak at the pole frequency in the clean limit. However, the long-time behavior of χ cðsÞ l ðq; tÞ is now determined by the branch points rather than by the pole.
The existence of hidden modes in 2D can be traced to the fact that in 2D the branch points associated with the particle-hole threshold are algebraic. The consequence of this is that the poles move continuously on the Riemann surface as F cðsÞ l is varied. This feature is best seen for the case of weak interaction (jF cðsÞ l j ( 1) and vanishingly small damping. In this case, the poles of χ cðsÞ l ðq; ωÞ are near the branch points: ω = v zs q(±1 − iγ) with v zs ≈ v F and γ ≪ 1. Then the form of branch point singularity determines the trajectory of the pole as F cðsÞ l is varied. For the square-root branch point, the pole's trajectory is described by ðω=v F qÞ À ð1 À iγÞ cðsÞ l , which gives rise to hidden modes. (To see this for l = 0 mode, note that the equation for the pole, following from Eqs. (1) and (2), is reduced for small jF . For small z, this gives the required trajectory.) In contrast, in 3D the cut is logarithmic and poles move discontinuously 15 . For example, in the l = 0 channel in 3D the pole position moves from above the branch cut for F cðsÞ 0 > 0 to the imaginary axis for F cðsÞ 0 < 0 (ref. 6 ). We also stress that in our calculations we always assumed ω ≫ v F qγ, which corresponds to the collisionless regime. In the opposite limit of ω ≪ γv F q, there is no hidden mode.
The existence of mirage modes for l ≥ 1 but not for l = 0 is a consequence of the fact that the l = 0 channel represents the response function of a conserved quantity (total particle number or spin), while the l ≥ 1 channels represent the response functions of the quantities which are not conserved in the presence of even infinitesimally weak disorder (for example, l = 1 corresponds to the charge or spin current). As a result, the free susceptibility χ 0 ≡ χ l = 0 in the long wavelength limit (γ ≫ 1) must have a diffusion pole with small magnitude, s = 1/(2iγ). Because of this constraint, the pole in χ 0 (s) remains above the branch cut for all values of F cðsÞ 0 . For l ≥ 1, there are no constraints limiting the damping term. The result of this is that the imaginary part of the ZS frequency grows with increasing repulsion F cðsÞ l , and at some critical F cðsÞ l the pole frequency crosses the branch cut. We note in passing that the difference between the l = 0 and l ≥ 1 channels is not special to 2D, although 2D is a more natural setting to search for a mirage mode, since the pole positions move continuously on the Riemann surface as a function of F cðsÞ l . Indeed, it can be shown that there is a mirage mode in the 3D l = 1 longitudinal channel as well. (The calculation is analogous to the one for the 2D case. The pole equation is 1 þ F cðsÞ 1 χ 1 ðsÞ ¼ 0, where χ 1 (s) is the particle-hole bubble with vertex corrections from impurities, with a form factor We find that the crossover to a mirage mode occurs for vanishing γ at F m 1 ¼ 0:44.) In more general terms, our work establishes that dynamics of a 2D FL, even of an isotropic and Galilean-invariant one, is determined not just by the poles of its response functions, but also by topological properties encoded in the Riemann surfaces defined by those functions. Here we studied the simplest case, where the Riemann surface is a closed sphere. There exist more complex cases, e.g., for two bands with different Fermi velocities, v F,1 and v F,2 , there are four branch points in the complex plane, at ω = ±v F,1 q, ±v F,2 q, and the associated Riemann surface is a torus.
In such cases, one should expect new topological features of ZS excitations.
A few remarks about real systems. First, our results apply to both neutral and charged FLs, with a caveat that for charged FLs the l = 0 charge mode becomes a plasmon 32 . Second, to observe a ZS mode, one needs to either employ finite-q versions of the pump-probe techniques, e.g., time resolved RIXS 33 and neutron scattering 34 , or spatially modulate/laterally confine 2D electrons. The most readily verifiable prediction is the hidden mode in the spin channel, which occurs for 0 < F s 0 < À1=2. Previous measurements on a GaAs/AlGaAs quantum well 35,36 indicate that F s 0 for this system is exactly in the required range.

METHODS
In this section we present the details of our calculations of the charge/spin susceptibility in the time domain, χ cðsÞ l ðq; tÞ, and discuss the analytic structure of the Riemann surface of χ cðsÞ l ðq; ωÞ. In Section A we discuss the framework to calculate χ cðsÞ l ðq; tÞ for a generic l in the charge or spin channel. In Sections B and C we give detailed derivations of χ cðsÞ l ðq; tÞ in the l = 0 and the l = 1 longitudinal channels and briefly discuss how these calculations can be extended to arbitrary l. In Section E we show that the results, discussed in the main text, i.e. the existence of conventional, hidden, and mirage poles, also hold when two Landau parameters, F Throughout this section, we assume an isotropic system, such that at low enough momenta and frequency the fermionic dispersion can be approximated as F m=m Ã and m* is the FL effective mass. We assume that single-particle states are damped by impurity scattering and that the damping rate,γ, is small compared to Fermi energy. We also assume that the temperature T is low enough such that the quasiparticle damping rate can be neglected, but still higher than the critical temperature of a superconducting (Kohn-Luttinger) instability.
Dynamical susceptibliities χ cðsÞ l ðq; ωÞ and χ cðsÞ l ðq; tÞ In this section we provide details of our calculations of the response functions in the frequency and time domains, χ cðsÞ l ðq; ωÞ and χ cðsÞ l ðq; tÞ. We assume that typical frequencies and momentum transfers are small, i.e., q ≪ k F and ω ≪ E F . In this limit the response of a FL to a weak external perturbation comes predominantly from quasiparticles near the FS. The quasiparticle contribution to the dynamical susceptibility was obtained by Leggett back in 1965 (ref. 37 ). To get it diagrammatically, one needs to sum up series of bubble diagrams coupled by quasiparticle interactions. For the case when one Landau parameter dominates, the quasiparticle contribution to χ cðsÞ l ðq; ωÞ has the form Here the Landau parameter F l is the properly normalized l'th moment of the antisymmetrized four-fermion vertex, ν F is the (renormalized) thermodynamic density of states, and χ l (s) is the retarded free-fermion susceptibility in the l'th channel. The subscript qp makes explicit the fact that this is only the quasiparticle response. The full χ cðsÞ l ðq; ωÞ differs from (4) by an overall factor, which accounts for renormalizations by fermions with higher energies, and also contains (for a non-conserved order parameter) an additional term, which comes solely from high-energy fermions 37 . These additional terms are relevant for the full form of the susceptibility near Pomeranchuk instabilities towards states with special order parameter 13,15,38,39 but not for collective modes studied in this paper. The expression for the free-fermion susceptibility χ l (s) in the presence of impurity scattering is obtained by (a) evaluating a particle-hole bubble using propagators of free fermions with fermionic frequency ω shifted to ω þ iγ and (b) summing up the ladder diagrams for the vertex renormalizations due to impurity scattering. The detailed form of χ l (s) depends both on the channel angular momentum l and its polarization (longitudinal/transverse). For a detailed derivation of Eq. (4) and explicit forms of χ l (s) we refer the reader to refs 14,15,31 . Here we just state the final results for χ cðsÞ qp;l ðsÞ and focus on calculating its time-domain form. To shorten the notations, henceforth we skip the subindex "qp" in χ cðsÞ qp;l q; ω ð Þ, as we did in the main text.
The retarded time-dependent susceptibility is a Fourier transform of χ cðsÞ l ðq; ωÞ: where t* = v F qt. In physical terms, χ cðsÞ l ðq; tÞ describes a response of the order parameter in the l'th charge or spin channel to a pulse-like excitation of the form h l e −iq⋅r δ(t).
To evaluate Eq. (5), it is convenient to close the integration contour in the complex plane. As discussed in the main text, χ cðsÞ l ðsÞ has two types of singularities in complex s plane, both of which contribute to the result of integration. First, it has a set of poles s j , which can be either on the physical or unphysical sheet. To be concrete, in the subsequent calculations for l = 0, 1 we will label by s 1 the pole in the lower-right quadrant of a complex plane of frequency, where Res ≥ 0, Ims < 0. We express the coordinates of the pole s 1 as where a = zs, h, m, and the notations are for three different types of the poles corresponding to a "conventional" ZS mode (either a propagating one, or a resonance within the particle-hole continuum), a hidden mode, and a mirage mode, respectively. These are the same notations that we used in the main text. To make the text less cumbersome, we will refer to each pole according to the mode it gives rise to, i.e. we will call them a "conventional pole", a "hidden pole", and a "mirage pole". Second, χ cðsÞ l ðsÞ has branch points at s = ±1 − iγ, where γ ¼γ=v F q, and we chose the branch cuts to run along the lines ±x − iγ, 1 < x < ∞. Because of the sign of the argument of the exponential function in Eq. (5), the contour must be closed in the lower half-plane for t > 0, so it traces over the branch cuts in the manner shown in Here χ cðsÞ l;pole ðt Ã Þ is a contribution from the residues of the poles of χ cðsÞ l ðsÞ on the physical sheet: e Àisj t Ã Res s!sj χ l ðsÞ: Since the sum over s j is restricted to the poles on the physical sheet, it includes conventional ZS and and hidden poles, but not mirage poles.
The second term in (7) is the branch-cut contribution  Another way to define the integration contour over complex s. We added to the integral over real s the integration segments over s immediately above the branch cuts on the physical sheet and immediately below the branch cuts on the unphysical sheet. These additional integrals then cancel out between the two Riemann sheets. We then added the integral over an infinite semi-circle to the unphysical sheet, and for both sheets added and subtracted the integrals over the range of s between the branch points. The resulting integration contour in each Riemann sheet consists of the closed contour (the solid line) and an additional piece (the dashed line).
where Δ c(s) χ l (x) is the discontinuity of χ cðsÞ l ðsÞ at the branch cut: It is also possible to re-arrange the contour integral into the one depicted in Fig. 5. This is done by (a) closing the integration contour in complex s on the physical sheet along the line x − iγ + iε, where ε is infinitesimal and x =−∞…∞, i.e. along the line which is located right above the branch cuts, (b) adding an integration contour on the unphysical sheet along the line x − iγ + iε, x = −∞…∞, i.e., right below the branch cut, (c) closing this second contour via an infinite half-circle in the unphysical lower half plane, and (d) adding two compensating integration segments along the lines x − iγ − iε, where −1 ≤ x ≤ 1, on the physical sheet, and along x − iγ + iε, −1 ≤ x ≤ 1 on the unphysical sheet (dashed lines in Fig. 5). Because χ cðsÞ l ðsÞ varies smoothly through the branch cuts if one simultaneously move between physical and unphysical Riemann sheets, the integration segments running above and below the branch cuts cancel out.
The evaluation of the integrals again yields an expression of the form of Eq. (7), but now the sum in Eq. (8) is over the poles on the physical sheet above the branch cut (i.e., conventional poles with damping rate γ zs < γ), and over mirage poles: χ cðsÞ l;pole ðt Ã Þ ¼ Ài X sj 2conv:;mirage e Àisj t Ã Res s!sj χ l ðsÞ: In addition, the second contribution in Eq. (7) now comes from the difference between the values of χ cðsÞ l ðsÞ on the two Riemann sheets rather than from a discontinuity at the branch cut: It can be verified that the integration contour of Fig. 5 is equivalent to a contour on the physical sheet, when the branch cut is chosen to run along the line x − iγ, − 1 < x < 1, see Fig. 6. In this case, the integral for χ branch can be understood as running around the circumference of the contour glueing the two Riemann sheets together into a single sphere.
In what follows, we will present calculations using both integration contours, the one in Fig. 4 and the one in Fig. 5. Although the result, of course, does not depend on the choice of a contour, some details of the calculation are more transparent when using one contour and some are clearer when using the other. In this section we provide detailed calculations for the case of l = 0. First, we use the integration contour in Fig. 4 and then the one in Fig. 5.
The free-fermion susceptibility is given by Eq.
(2) of the main text The quasiparticle susceptibility is obtained by plugging χ 0 into Eq. (4). The two poles of χ cðsÞ 0 ðsÞ are located at In Fig. 7 we show a 3D depiction of the poles' trajectories on the Riemann surface. In what follows, we assume that γ ≪ 1, as we did in the main text.
The discontinuity of χ 0 (s) at the branch cut is where s 1,2 are given by (14), see Eq. (10). We obtain χ 0 (q, t*) for the three cases shown in Fig. 1a of the main text, i.e., for a ZS resonance (an overdamped l = 0 mode), hidden mode, and weakly damped ZS mode.
ZS resonance, À1 < F cðsÞ 0 < À1=2. An overdamped ZS resonance occurs for À1 < F cðsÞ 0 < À1=2. The pole contribution can be found directly from Eq. (8). As follows from Eq. (14), there is only one pole in the lower half-plane, at s 1 = −iγ zs , where Note γ zs ≫ γ everywhere but in the narrow vicinity of the Pomeranchuk instability at F Now we turn to χ cðsÞ 0;branch ðt Ã Þ, Eq. (9). One can readily verify that at large t*, the leading contribution to the integral in (9) comes from the vicinity of the branch point s = 1 − iγ. Accordingly, we shift the integration variable in Eq. (9) to y = 1 + x and expand the integrand to leading order in y. We obtain χ cðsÞ 0;branch ðt Ã Þ % À 2 ffiffi π p e Àγt Ã where σ 1;2 ¼ s 1;2 À ð1 À iγÞ; are the pole coordinates measured from the branch point at s = 1 − iγ Fig. 6 Integration contour over an alternatively defined Riemann surface. Contour of integration over complex s with a branch cut (dashed line) chosen to run horizontally between the branch points at ∓1 − iγ. cosðt Ã À π=4Þ Comparing χ cðsÞ 0;pole and χ cðsÞ 0;branch , we see that at F cðsÞ 0 \ À1, where γ zs ≪ 1 (but still γ zs > γ), the pole contribution dominates up to t*~t cross , where t cross ¼ 3 2ðγ zs À γÞ log F cðsÞ 0 2 ð2jF cðsÞ 0 j À 1Þðγ zs À γÞ For t* ≫ t cross , the branch-cut contribution becomes the dominant one. At F cðsÞ 0 not close to −1, t cross~1 . In this situation, the branch-cut contribution dominates over the pole one for all t* ≫ 1.
Weakly damped ZS mode, F Hidden mode, À1=2 < F cðsÞ 0 < 0. We next consider the range À1=2 < F cðsÞ 0 < 0, where the ZS pole is a hidden one: Note that to get the prefactor right, one has to keep γ finite, otherwise the pole and the branch cut would be at the same depth below the real axis, and the prefactor in (23) would be smaller by a factor of two because the angle integration around the pole would be only over a half-circle rather than over a full circle.
The branch cut contribution in Eq. (9) reduces to where now s 1,2 = ±s h − iγ h . Evaluating the integral, we find two dominant contributions: one from x ≈ 1, i.e., from the vicinity of the branch point, and another one from x ≈ s h , i.e., from the vicinity of the hidden pole (there is only one such term because Re s 2 < 0). Accordingly, we write To obtain χ cðsÞ 0;branch;a , we expand near x = s h as x = s h + ϵ and keep the leading terms in ϵ. We obtain χ cðsÞ 0;branch;a ðt Ã Þ ¼ where γ ¼ γ h À γ > 0. The integral in (26) yields, by Cauchy theorem Observe that the exponential factor in (25) is e Àγ h t Ã , despite that the overall factor in (24) is e Àγt Ã . The extra factor e Àðγ h ÀγÞt Ã appears after the integration in (27).
The second term in Eq. (25) is the contribution from the vicinity of the branch point. At the largest t*, this contribution has the same form as in Eq. (18): However, the full form of χ cðsÞ 0;branch;b ðt Ã Þ is more involved, and the 1=ðt Ã Þ 3=2 behavior sets in only after some characteristic time t cross,1 , which becomes progressively larger as jF cðsÞ 0 j decreases and s h approaches 1. To see this, we expand the integrand of (24) in y = x − 1, but do not assume that y is small compared to σ h = s h − 1. We obtain, at t* ≫ 1 where z = −iyt* and where ffiffiffiffiffi Ài p in (32) stands for ð1 À iÞ= ffiffi ffi 2 p . Note that both σ h and ðF Accordingly, in the two limits χ cðsÞ 0;branch;b ðt Ã Þ behaves as We see that both the exponent of the power law decay and the phase of oscillations vary between the two regimes. In particular, the phase changes by π/2 between the regimes of σ h t* ≪ 1 and σ h t* ≫ 1 (up to corrections O (γ). The crossover between the two regimes occurs at t*~t cross,1 , where is related to the coordinate of the hidden pole. This relation provides a way to detect the hidden mode experimentally, particularly for small F where s h − 1 ≪ 1 and t cross,1 ≫ 1, by either by looking at the crossover in the power-law decay of χ cðsÞ 0 ðt Ã Þ or by studying a variation of the phase shift.
In the intermediate regime of t*~t cross,1 (assuming that t cross,1 ≫ 1) the susceptibility behaves as χ cðsÞ 0 ðt Ã Þ $ Aðσ h t Ã Þ cosðt Ã À ϕðt Ã ÞÞ=ðt Ã Þ 1=2 . In Fig. 8 we depict ϕ(t*) extracted from numerical evaluation of χ cðsÞ 0 ðt Ã Þ for different F cðsÞ 0 . To obtain the data in the figure, we fit segments of the data at different t* onto a trial function A cosðt Ã À ϕÞ=ðt Ã Þ α , where A, ϕ, α are fitting parameters. We then fit ϕ(t*/t cross ) to the prediction of Eq. (37). The data shows a good collapse of the phase evolution onto a universal function of σ h t* = t*/t cross,1 , given by Eqs. (31) and (32), even for not-toosmall F cðsÞ 0 , and a very good agreement between the numerical value of t cross,1 and the asymptotic expression in Eq. (35).
Calculations using the contour of Fig. 5. We now demonstrate how to evaluate χ cðsÞ 0 ðt Ã Þ in the case of a hidden pole, i.e., at À1=2 < F cðsÞ 0 < 0, using the contour of Fig. 5. The advantage of using this contour is that there is no need to account for a partial cancellation between the pole and brunchcut contributions. Inspecting the integration contours, we note that χ 0, pole (t*) = 0 because there are no poles either above the branch cuts on the physical sheet or below it on the unphysical sheet. We are left only with χ 0, branch , defined in Eq. (12). We shift the integration variable in (12) to y = 1 − x. At t* ≫ 1 only small y matter, and one can safely extend the limits of integration to ±∞. We then obtain It is easy to verify that Eq. (36) is the analog of Eq. (24), up to small corrections due to γ. The integral in Eq. (36) can be solved exactly with the result where Z(a) was defined in Eq. (32). This result is the same as in Eq. (31), but with corrections due to finite γ.
We also note in passing that at small t* < 1, χ cðsÞ 0 ðt Ã Þ is linear in t* for all values of F cðsÞ 0 . In the limit γ → 0 the dependence is given by: At small but finite γ, the slope at t* → 0 changes to  61)). It can be seen that the two traces begin in phase, then move out of phase, and finally become in-phase again. This is an indication that χ cðsÞ 1 ðq; tÞ oscillates at different frequencies that correspond to poles for different F cðsÞ 1 , until oscillations from the branch points take over at long times.  In this section we provide a detailed derivation of χ cðsÞ 1 ðt Ã Þ in the longitudinal channel. The free-fermion susceptibility is In the limit γ → 0, the pole coordinates are the solutions of This gives four poles, which are located on both physical and unphysical sheets. In Fig. 9 we present a 2D sketch of the evolution of the four poles on the Riemann surface. As before, we label the pole with Re s > 0, Ims > 0 as s 1 , We label the pole in the first quadrant of the unphysical sheet as s 3 and define s 2 ¼ Às Ã 1 ; s 4 ¼ Às Ã 3 . At finite γ, the expressions for the coordinates of the poles are much more involved, but the number of poles remains unchanged, as does their qualitative behavior.
The discontinuity at the branch cut is Before proceeding to a calculation of χ  (We recall, that on the Riemann surface the points ±∞, +i∞ on the unphysical sheet, and −i∞ on the physical sheet, are identical). The pole on the physical sheet moves up from −i∞ and the pole on the unphysical sheet moves down from +i∞. At finite γ, the trajectories are slightly deformed, so that, e.g., s 1,2 never quite reach the branch cut and s 3,4 are never true mirror images, but the qualitative behavior remains the same.
We now evaluate χ cðsÞ 1 ðt Ã Þ. As we did in the l = 0 case, we first use the contour of Fig. 4. The evaluation proceeds along similar lines as for l = 0, except for two differences related, first, to the existence of mirage poles, and second, to the fact that for some ranges of F cðsÞ 1 we need to take into account contributions from all four poles.
Weakly damped ZS mode, F cðsÞ 1 \ À1. Consider first the limiting case F cðsÞ 1 \ À1. Here s 1 = s zs − iγ zs , where s zs % ðð1 À jF cðsÞ 1 jÞ=2Þ 1=2 and γ zs % ð1 À jF cðsÞ 1 jÞ=4. The real part of s 1 is much larger than the imaginary one (γ zs ≪ s zs ≪ 1), i.e., the mode is underdamped. The pole and branch contributions to χ c(s) (t*) are given by The branch cut contribution has the same form as in the l = 0 case, cf. Eq. (30): For F cðsÞ 1 % À1, the pole contribution is larger than the branch-cut one over a wide range of t* because the pole contributions contains a large prefactor 1/s zs while the branch cut contribution is reduced by 1=ðt Ã Þ 3=2 at large t*. Still, at any jF cðsÞ 1 j< 1, intrinsic γ zs is finite and by our construction is larger than extrinsic γ. Then, at large enough t* > t cross,2 , the branch-cut contribution becomes larger than the contribution from the pole. The crossover scale is This t cross,2 is the l = 1 analog of t cross in the l = 0 channel, Eq. (21).
Damped ZS mode for F cðsÞ 1 À1=9. In this section we consider the range of À1 < F cðsÞ 1 < À1=9, excluding the immediate vicinity of −1, which has been already considered in Section 1. For F cðsÞ 1 t À 1=9 the pole is close to but somewhat below the branch cut, i.e., in our notations this is a weakly damped conventional ZS pole (by x ≲ y we mean that x is smaller than y by an asymptotically small quantity). Here we have s zs % 2= ffiffi ffi 3 p ; γ zs % ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ðjF cðsÞ 1 j À 1=9Þ=2 q . Up to two leading orders in γ zs , the pole contribution is χ cðsÞ 1;pole ðt Ã Þ ¼ À e Àγ zs t Ã cos s zs t Ã γ zs þ 3 ffiffi ffi 3 p sin s zs t Ã þ Oðγ zs Þ : We verified that both terms in the pole contribution are cancelled out by the corresponding contributions from the branch cut. The branch cut contribution can again be represented as the sum of two terms, like in (50), (51), (52), but now s 3 is complex conjugate of s 1 : s 3 = s h + iγ h . The term that cancels (58) is obtained by expanding in ϵ = x − s h and evaluating integrals up to two leading orders in γ h . The cancellation implies that there are no oscillations in χ cðsÞ 1 ðt Ã Þ with frequency s zs , even when the system is slightly outside the range where the ZS pole is a hidden one. The remaining contribution from the branch cut has the same form as in other regimes: at largest t*, We now study the crossover from the behavior at F cðsÞ 1 t À1=9, where we just found that the pole contribution is cancelled by the contribution from the branch cut, to the behavior at F cðsÞ 1 \ À1, where we found earlier that there is no such cancellation. As F cðsÞ 1 decreases, the trajectory of s 1 evolves in the complex plane, mirrored by the other s 2..4 . During this evolution, γ zs is finite but numerically small. For this reason, below we restrict ourselves to the leading contribution in γ zs .
Within this approximation, the pole contribution is the first term in (58). For the branch cut contribution we find, not requiring s zs to be close to 2= ffiffi ffi 3 p , χ cðsÞ 1;branch ðt Ã Þ ¼ À 1Àszs dx e Àixt Ã x 2 þ γ 2 zs ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s zs À 1 þ x s zs À 1 r þ c:c: For s zs < 1, the lower limit of the integral is positive. This happens when where F vis 1 ¼ À0:162. In this range of F cðsÞ 1 , one can safely set γ zs to zerothe integral does not diverge. As a consequence, χ cðsÞ 1;branch;1 ðt Ã Þ does not contain the factor / γ À1 zs and cannot cancel χ cðsÞ 1;pole ðt Ã Þ / cosðs zs t Ã Þ=γ zs in (58). The leading contribution to the integral in (60) comes from x ≈ 1 − s zs , and the integration yields as in (59). We see that the behavior of χ cðsÞ 1 ðt Ã Þ is qualitatively the same as for F ≥ −1: the pole contribution yields oscillations with frequency s zs and remains the largest contribution to χ cðsÞ 1 ðt Ã Þ up to t*~t cross,2 . At t* > t cross,2 , the branch cut contribution becomes the largest one and χ cðsÞ 1 ðt Ã Þ oscillates at the (dimensionless) frequency equal to one.
However, when s zs > 1, which happens for F vis 1 < F cðsÞ 1 < À1=9, the lower limit of integration in Eq. (60) is negative, and the integral contains a singular contribution from x → 0. Using we find that this singular piece cancels out the contribution from the pole. Evaluating the other relevant contribution from x ≈ 1 − s zs , we find This result is valid for t*|s zs − 1| ≫ 1. The cosðt Ã À π=4Þ=ðt Ã Þ 3=2 is precisely the expected time dependence for the case when the contribution to χ cðsÞ 1 ðt Ã Þ comes solely from the end points of the branch cut. We see therefore that oscillations with frequency s zs exist as long as F cðsÞ 1 < F vis 1 . For F vis 1 < F cðsÞ 1 < À1=9 only oscillations, coming from the branch points, with frequency equal to one are present.
In the analysis above we expanded in γ zs , i.e., we assumed that the damping remains small in the crossover regime around F vis 1 . The approximation of small γ zs would be rigorously valid if the pole trajectory in the complex plane would remain close to the real axis for all À1 < F cðsÞ 1 < À1=9. In that case we would expect oscillations to persist for a long time, both at F oscillations would occur with frequency equal to one at all t* ≫ 1. We see therefore that the branch contribution "eats up" the pole contribution once the coordinate of the pole in the complex plane moves to below the branch cut. In reality, γ zs is small (or order γ) near F is varied around F vis 1 , see Fig. 9 b and its caption.
Calculations using the contour of Fig. 5. We now obtain the same results by using the integration contour of Fig. 5. Again, the use of this contour will allow us to avoid canceling out pole and branch contributions. It also allows one to see more transparently how the poles on the unphysical sheet contribute to the dynamics. We study both the regime of hidden poles and the crossover regime between F cðsÞ 1 ¼ À1 and F cðsÞ 1 À 1=9. For consistency we define s 1 = s zs − iγ zs and σ zs = s 1 − (1 − iγ). With the contour of Fig. 5, the pole contribution is zero for the same reason as for the l = 0 case (cf. Section 4), and the dynamics is determined entirely by the branch-cut contribution, which is given by where we used Eq. (12) and shifted the integration variable via y = 1 − x.
To proceed further, we infer from Eq. (41) that the y integral is dominated by the region y ≪ |σ i |, i.e., by whichever pole is nearest to the branch point, see Eq. (19). In our notations, it is σ 1 ≡ σ zs . For |σ zs | ≪ 1 we may expand the integral in small y and extend the integration limits to infinity. First, we consider the situation when F cðsÞ 1 < 0 and jF cðsÞ 1 j ( 1=9, i.e., when s 1,2 reside below the branch cut (see Fig. 9) and are close to the branch point. In this situation |s 3,4 | ≫ 1 and the y dependence in the (y + σ 3 )(y + σ 4 ) factor in Eq. (66) can be neglected. Then Eq. (66) is identical to Eq. (36), up to unimportant constant factors, i.e., the hidden pole behavior for l = 1 is the same as for l = 0. Next, we consider the situation when F cðsÞ 1 decreases and becomes smaller than −1/9. We evaluate the integral in Eq.
where A j ¼ P i≠j ðσ i À σ j Þ À1 are the partial fraction decompositions of ∏ j (x + σ j ), and Àσ p e Àiσt þ e iπ=4 ZðσtÞ; (68) where Z(a) was defined in Eq. (32) and Θ(a) is the Heaviside function. (Note that since s 2,3 are not near the branch point at 1 − iγ, they have σ j ≈ −2 while the integral is dominated by the region y~|σ 1 |, |σ 4 |. However, their contribution is included in the complex conjugate term in χ 1,branch .) Equations (67) and (68) are applicable in both the hidden pole regime and the crossover regime, as long as |σ 1 | ≪ 1. Let us examine them in the crossover regime. Although the sum in Eq. (67) is over all four poles, the Heaviside functions in Eq. (68) are nonzero only for s 1 . It can be verified that the sudden appearance of the pole contribution for s 1 is mirrored by a jump in ∑ j A j Z(σ j t), so that the crossover is actually smooth-the pole progressively "emerges" from behind the branch cut. This behavior is the analog of the progressive "eating up" of the poles that we obtained via integration over the contour of Fig. 4, see Eq. (60).
Mirage poles. Finally, we discuss the mirage poles. For 0 < F cðsÞ 1 < 3=5, the conventional ZS pole s 1 is located outside particle-hole continuum, and its position in the lower half-plane of frequency is between the real frequency axis and the branch cut, i.e., Res 1 > 1 and −γ < Ims 1 < 0. At F cðsÞ 1 ¼ 3=5, Ims 1 becomes equal to γ, and for larger F cðsÞ 1 , the pole moves to the unphysical Riemann sheet, i.e. in our notations it becomes a mirage pole (see ref. 15 ).
As before, we first compute χ cðsÞ 1 ðt Ã Þ using the integration contour in Fig.  4 where ffiffiffiffiffiffiffiffiffi ffi Equation (72) is valid only for s above the branch cut, i.e., for |Ims| < γ. This is satisfied on the upper branch of the cut, but not on the lower branch.
The function Q 2 (x m ) satisfies Q 2 ð2= ffiffi ffi 3 p Þ ¼ 1 and increases with x m for larger x m , which correspond to F Combining (70) and (74), we see that in the range where a ZS pole is a mirage one, χ cðsÞ 1 ðt Ã Þ ¼ Àðχ cðsÞ 1;branch;am ðt Ã Þ þ χ cðsÞ 1;branch;bm ðt Ã ÞÞ has a contribution oscillating with (dimensionless) frequency x m and the contribution oscillating with (dimensionless) frequency equal to one. When F cðsÞ 1 ¼ Oð1Þ, the second contribution is the dominant one in some range of t* > 1, because the first contribution contains 1=ðt Ã Þ 3=2 . However, above a certain t* the contribution from the branch point becomes the dominant one as it contains the smaller factor in the exponent. This crossover from oscillations with frequency x m to oscillations with frequency 1 provides a way to detect a mirage pole experimentally.
For 0 < F cðsÞ 1 < 3=5, the ZS pole is located in the lower half-plane of frequency on the physical Rieman sheet. In this situation, χ where now 0 < x m < 2= ffiffi ffi 3 p and Q 2 (x m ) < 1. The contribution from the branch points is still given by (70). There is no crossover in this case because the exponential factor in the pole contribution is smaller than in the branch cut contribution. We note in passing that there is also a sign change between χ cðsÞ 1 ðt Ã Þ and Àχ cðsÞ 1;branch;bm ðt Ã Þ in (74), (i.e., the phase of sinðx m Þt Ã ) oscillations changes by π between the regions where a ZS pole is a conventional one and where it is a mirage one.
Calculations using the contour of Fig. 5. The same results can be obtained using the contour in Fig. 5. For the contour of Fig. 5, the pole contribution is non-zero and is given by where s 1 = s m −iγ m is the mirage pole according to our conventions. This is just −1 times the result for a conventional ZS mode residing above the branch cut on the physical sheet, Eq. (42). The phase shift is due to the pole The crossover time is i.e., it is analogous to the crossover time for a conventional pole with γ z s < γ, see Eq. (46).

Arbitrary l
Our results for l = 0 and l = 1 can be readily generalized to any channel. Using the contour of Fig. 5, we see that for a given channel with 2n poles on the Riemann surface, the solution is given by the contributions of mirage and conventional poles with γ zs < γ, along with the branch points contribution χ branch ðt Ã Þ ¼ Q 0 X j¼1::2n A j e iπ=4 Zðσ j t Ã Þ; where Z(a) is given by Eq. (32), A j ¼ P i≠j ðσ i À σ j Þ À1 and Q 0 is a constant, calculated directly from Δχ cðsÞ l ðxÞ and given by To study a crossover regime where a pole s 1 emerges from behind a branch cut, simply replace e iπ/4 Z(σ j t*) in (80) by Z, given in Eq. (67). dominate over all others, as can be expected for a generic interaction which decreases monotonically with momentum transfer. Our results can be readily generalized for the case of more nonzero F where χ 0 and χ 1 are given by Eqs. (13) and (39), respectively, while χ 01 (s) is the fermion bubble with l = 0 and l = 1 form-factors at the vertices The equations for the poles in the l = 0 and (longitudinal) l = 1 channels are the same because Eqs. does not change qualitatively. A new element, however, is that the mirage mode occurs both in the l = 0 and l = 1 channels (again, because they have a common pole). Also, the conditions for the existence of the mirage mode become less stringent compared to the F For a charged FL, the situation is somewhat different. The new diagrammatic element are the chains of bubbles connected by the unscreened Coulomb interaction, U q = 2πe 2 /q. Such chains are present in the l = 0 charge channel and in the l ≥ 1 longitudinal charge channel, but not in the transverse charge channel and the spin channel. Each bubble in the chain is renormalized by a FL interaction, parameterized by the Landau function. The Landau function comprises infinite series of diagrams containing the screened Coulomb interaction. Resumming the diagrammatic series, one obtains the full charge susceptibilities in the form