Hidden and mirage collective modes in two dimensional Fermi liquids

We show that a two-dimensional (2D) isotropic Fermi liquid harbors two new types of collective modes, driven by quantum fluctuations, in addition to conventional zero sound:"hidden"and"mirage"modes. The hidden modes occur for relatively weak attractive interaction both in the charge and spin channels with any angular momentum $l$. Instead of being conventional damped resonances within the particle-hole continuum, the hidden modes propagate at velocities larger than the Fermi velocity and have infinitesimally small damping in the clean limit, but are invisible to spectroscopic probes. The mirage modes are also propagating modes outside the particle-hole continuum that occur for sufficiently strong repulsion interaction in channels with $l\geq 1$. They do give rise to peaks in spectroscopic probes, but are not true poles of the dynamical susceptibility. We argue that both hidden and mirage modes occur due to a non-trivial topological structure of the Riemann surface, defined by the dynamical susceptibility. The hidden modes reside below a branch cut that glues two sheets of the Riemann surface, while the mirage modes reside on an unphysical sheet of the Riemann surface. We show that both types of modes give rise to distinct features in time dynamics of a 2D Fermi liquid that can be measured in pump-probe experiments.

We show that a two-dimensional (2D) isotropic Fermi liquid harbors two new types of collective modes, driven by quantum fluctuations, in addition to conventional zero sound: "hidden" and "mirage" modes. The hidden modes occur for relatively weak attractive interaction both in the charge and spin channels with any angular momentum l. Instead of being conventional damped resonances within the particle-hole continuum, the hidden modes propagate at velocities larger than the Fermi velocity and have infinitesimally small damping in the clean limit, but are invisible to spectroscopic probes. The mirage modes are also propagating modes outside the particle-hole continuum that occur for sufficiently strong repulsion interaction in channels with l ≥ 1. They do give rise to peaks in spectroscopic probes, but are not true poles of the dynamical susceptibility. We argue that both hidden and mirage modes occur due to a non-trivial topological structure of the Riemann surface, defined by the dynamical susceptibility. The hidden modes reside below a branch cut that glues two sheets of the Riemann surface, while the mirage modes reside on an unphysical sheet of the Riemann surface. We show that both types of modes give rise to distinct features in time dynamics of a 2D Fermi liquid that can be measured in pump-probe experiments.  [16]).

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][2][3][4][6][7][8][9][10][11][12][13][14][15]. These qualitative notions are consistent with rigorous results for a 3D FL [1][2][3][4]6].
In this Letter, 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 sharp 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 c(s) l (q, ω) 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 c(s) l (q, ω) 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 c(s) l (q, ω) near a branch cut has only a benign logarithmic non-analyticity. In 2D, however, the non-analyticity is algebraic ( √ x). In this situation, the analytic structure of χ c(s) l (q, ω) is encoded in a two-sheet genus 0 algebraic Riemann surface (a sphere) [17][18][19]. It has a physical sheet, on which χ c(s) l (q, ω) is analytic in the upper half-plane by causality, and a nonphysical sheet. The ZS modes appear as poles of χ c(s) l (q, ω). Both the genus and the number of ZS poles are topological invariants of χ c(s) l ω axis of the physical sheet and does not give rise to a sharp peak in Imχ c(s) l (q, ω) 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 identified spectroscopically by probing Imχ c(s) l (q, ω), 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 c(s) l (q, t) 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 c(s) l (q, t) 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 c(s) l (q, t) 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 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 c(s) l (q, t) 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 c(s) l (q, t) changes from cos(v F qt + π/4)/t 1/2 at intermediate t to cos(v F qt − π/4)/t 3/2 at the longest t, and the location of the hidden pole can be extracted from the crossover scalet 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 The quasiparticle susceptibility at small ω and q but fixed ω/v F q = s is expressed solely in terms of Landau parameters F c(s) l in the charge or spin sectors [1-4, 6, 7, 13-15]. An explicit form of χ c(s) l (q, ω) is rather cumbersome but becomes much simpler if one of the Landau parameters, F c(s) 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 (k, ω) = (ω + iγ/2 − v F (|k| − k F )) −1 and form-factors f l (θ) at the vertices, where θ is the angle between k and q, f 0 = 1, and f l (θ) = √ 2 cos lθ ( √ 2 sin lθ) 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. [14,31]). This is true even in the clean limitγ → 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 (θ) = √ 2 cos θ, as an example of a mirage mode [32]. For l = 0, χ 0 (s) with vertex corrections due to impurity scattering included is given by [14,31] χ 0 (s) = 1 + is where γ =γ/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. 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 ≈ (γ zs − γ) −1 . Inset: a zoomed-in view showing the crossover at t ∼ t cross . χ c(s) 1 (t * ) is multiplied by e γt * to enhance visibility. The disorder strength is γ = 0.2. and at F c(s) 0 → −1 its frequency vanishes, signaling a Pomeranchuk instability [6,14]. The hidden ZS mode emerges at −1/2 < F c(s) l < 0. Here the two modes are again located near the real axis, at ω = v F q (±s h − iγ h ), where 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 c(s) 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 c(s) (q, ω) at ω = v F s h q. The evolution of the poles with F c(s) 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 −1/9 < F , 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 c(s) 1 (q, ω); 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 c(s) l (q, t). At first glance, it seems redundant to study χ c(s) l (q, t), which is just a Fourier transform of χ c(s) l (q, ω) for real ω, expressed via Imχ c(s) l (q, ω) as χ c(s) l (q, t > 0) = (2/π) ∞ 0 sin(ωt)Imχ c(s) l (q, ω) by causality. A hidden mode does not give rise to a peak in Imχ c(s) l (q, ω) 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 Our reasoning is based on the argument that χ c(s) l (q, t) 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 The pole contribution behaves as χ c(s) l (q, t) ∝ sin(s a t * )e −γat * , where a = zs, h, m. For a conventional ZS mode γ zs < γ, and the long-t behavior of χ c(s) l (q, t) is dominated by oscillations at the ZS frequency. For a mirage mode γ < γ m , 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, which depicts χ c(s) 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 c(s) 0 (q, t). The only oscillations in χ c(s) 0 (q, t) are due to the branch points, with a period T = 2π/v F q. However, a more careful study shows [16] 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, χ c(s) 0 (q, t) ∝ cos(t * + π/4)/(t * ) 1/2 , while at longer t, χ c(s) 0 (q, t) ∝ cos(t * − π/4)/(t * ) 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 * cross is determined by the position of a hidden pole in relation to the branch point. For small F c(s) 0 it is just t * cross = |s h − (1 − iγ)| −1 ; this relation is verified numerically in the SM [16]. Hence, a hidden pole can be extracted from time-dependent measurements even though it does not show up in spectroscopic probes.

(q, t) at intermediate and long times for
For completeness, we also briefly discuss the behavior of χ c(s) where the pole is Landau overdamped even in the absence of disorder, i.e., ω = −iv F qγ zs [14]. 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 t −1 cross = (γ zs − γ) −1 to logarithmic accuracy. We also present the results for χ   1 (q, t) displays weakly damped oscillations (Fig. 3b). When F c(s) 1 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 c(s) 0 and F c(s) 1 , have comparable magnitudes.
Summary and discussion. In this Letter, we argued that zero-sound collective excitations in a 2D Fermi liquid have two unexpected features. First, for any angular momentum l and in some range of a negative Landau parameter F c(s) l , a zero-sound 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 zero-sound pole of χ c(s) 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 c(s) l above some critical value, a zero-sound pole moves from the physical Riemann surface to the unphysical one and becomes a "mirage" one. In this situation, Imχ c(s) l (q, ω) still has a peak at the pole frequency in the clean limit. However, the long-time behavior of χ c(s) l (q, t) is now determined by the branch points rather than by the pole. Our work establishes that dynamics of a 2D Fermi liquid, 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 zero-sound 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 [33]. Second, to observe a zero-sound mode, one either needs to either employ finite-q versions of the pump-probe techniques, e.g. time resolved RIXS [34] and neutron scattering [35], 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 a 0 < −1/2. Previous measurements on a GaAs/AlGaAs quantum well [36,37] indicate that F a 0 for this system is exactly in the required range. In this Supplementary Material we present the details of our calculations of the charge/spin susceptibility in the time domain, χ c(s) l (q, t), and discuss the analytic structure of the Riemann surface of χ c(s) l (q, ω). In Sec. we discuss the framework to calculate χ c(s) l (q, t) for a generic l in the charge or spin channel. In Secs. and we give detailed derivations of χ c(s) l (q, t) in the l = 0 and the l = 1 longitudinal channels and briefly discuss how these calculations can be extended to arbitrary l. In Sec. 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 these supplementary notes, we assume an isotropic system, such that at low enough momenta and frequency the fermionic dispersion can be approximated as F m/m * and m * is the Fermi liquid (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.
In this section we provide details of our calculations of the response functions in the frequency and time domains, χ c(s) l (q, ω) and χ c(s) l (q, t). 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 Fermi surface (FS). The quasiparticle contribution to the dynamical susceptibility was obtained by Leggett back in 1965 (Ref. 38). 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 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 c(s) l (q, ω) 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 [38]. These additional terms are relevant for the full form of the susceptibility near Pomeranchuk instabilities towards states with special order parameter [12,14,39,40] 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 + 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. 13, 14, and 31. Here we just state the final results for χ c(s) qp,l (s) and focus on calculating its time-domain form. To shorten the notations, henceforth we skip the subindex "qp" in χ c(s) qp,l (q, ω), as we did in the main text. The retarded time-dependent susceptibility is a Fourier transform of χ c(s) l (q, ω): where t * = v F qt. In physical terms, χ c(s) l (q, t) 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 c(s) l (s) 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" zero-sound 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 c(s) l (s) has branch points at s = ±1 − iγ, where γ =γ/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 c(s) l,pole (t * ) is a contribution from the residues of the poles of χ c(s) l (s) on the physical sheet: 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 where ∆ c(s) χ l (x) is the discontinuity of χ c(s) l (s) at the branch cut: FIG. 5: 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).
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 c(s) l (s) 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: In addition, the second contribution in Eq. (7) now comes from the difference between the values of χ c(s) l (s) 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. 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 c(s) 0 (s) 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 ∆χ c(s) 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.
An overdamped ZS resonance occurs for −1 < F c(s) 0 < −1/2. 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 Now we turn to χ c(s) 0,branch (t * ), 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 where are the pole coordinates measured from the branch point at s = 1 − iγ Keeping γ only in the exponential, we re-write Eq. (18) as Comparing χ c(s) 0,pole and χ c(s) 0,branch , we see that at F c(s) 0 −1, where γ zs 1 (but still γ zs > γ), the pole contribution dominates up to t ∼ t cross , where For t t cross , the branch-cut contribution becomes the dominant one. At F c(s) 0 not close to −1, t cross ∼ 1. In this situation, the branch-cut contribution dominates over the pole one for all t * 1.
is analyzed along the same lines as for the overdamped case. The main difference is that for a propagating mode γ zs < γ, and, hence, the pole contribution remains the dominant one at all times, i.e. there is no crossover to oscillations from the branch point (this incidentally is indicated by the divergence of t cross in Eq. (21) as γ zs crosses γ). The pole contribution is now obtained by summing up the residues of the two poles at s 1,2 = ±s zs −iγ zs , where s zs = (1+F Keeping γ only in the exponential, we find 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 c(s) 0,branch;a , we expand near x = s h as x = s h + and keep the leading terms in . We obtain χ c(s) 0,branch;a (t * ) = e −γt * 2π whereγ = γ h − γ > 0. The integral in (26) yields, by Cauchy theorem Substituting into (26) we obtain χ c(s) 0,branch;a (t * ) = −2 Observe that the exponential factor in (25) is e −γ h t * , despite that the overall factor in (24) is e −γt * . The extra factor e −(γ h −γ)t * appears after the integration in (27).
Comparing (23) and (28), we see that χ c(s) 0,branch;a (t * ) cancels out the pole contribution: Because of the cancellation between χ However, the full form of χ c(s) 0,branch;b (t * ) is more involved, and the 1/(t * ) 3/2 behavior sets in only after some characteristic time t cross,1 , which becomes progressively larger as |F c(s) 0 | 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 , a = σ h t * can remain small even when t * is large. Accordingly, we treat a as a variable which can have any value. In the two limits a 1 and a 1 we have Accordingly, in the two limits χ c(s) 0,branch;b (t * ) 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  Fig. 8 we depict φ(t * ) extracted from numerical evaluation of χ c(s) [41] 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-too-small F c(s) 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 c(s) 0 (t * ) in the case of a hidden pole, i.e. at −1/2 < F c(s) 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 brunch-cut 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 c(s) . In the limit γ → 0 the dependence is given by: At small but finite γ, the slope at t * → 0 changes to χ In this section we provide a detailed derivation of χ c(s) 1 (t * ) in the longitudinal channel. The free-fermion susceptibility is In the limit γ → 0, the pole coordinates are the solutions of This gives 4 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 = −s * 1 , s 4 = −s * 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 ∆χ c(s) Before proceeding to a calculation of χ c(s) , the poles s 3 , s 4 are on the imaginary axis of the lower half plane of the physical sheet, and on the imaginary axis of the upper half-plane of the unphysical sheet. (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 c(s) 1 (t * ). 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 c(s) 1 we need to take into account contributions from all four poles. |)/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 is described in the text around Eq. 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 c(s) 1 (q, t) oscillates at different frequencies that correspond to poles for different F c(s) 1 , until oscillations from the branch points take over at long times. respectively, where σ j = s j − (1 − iγ), similar to Eq. (19). For γ → 0, the pole contribution is The branch cut contribution has the same form as in the l = 0 case, cf. Eq. (30): For F c(s) 1 ≈ −1, 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/(t * ) 3/2 at large t * . Still, at any |F c(s) 1 | < 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).
In the hidden pole regime, which occurs for −1/9 < F c(s) 1 < 0, the pole contribution is still given by Eq. (42). To leading order in γ, it is where The pole frequency is In the two limits, → 0. To leading order in γ, the branch-cut contribution can be expressed as the sum of the two terms: The first term contains the frequency of the pole s 1 on the physical Riemann sheet: where we recall that s 1 = s h − iγ h and γ h ≥ γ. The second term contains the frequency of the pole s 3 on the unphysical Riemann sheet: where s 3 = s 3 − iγ 3 with γ 3 < 0 and As for l = 0, the two largest contributions to χ c(s) 1,branch;1 (t * ) in (51) at t * 1 come from x ≈ s h and from x ≈ 1.
Accordingly, we further split χ Because γ h > γ, the two terms in the last bracket in (54) are of the same sign and add up to a factor of 2. Then This term exactly cancels out χ c(s) 1,pole (t * ) from (47). The second contribution, χ c(s) 1,branch;b , yields oscillations with frequency equal to one. It evinces a crossover from χ c(s) 1,branch;b ∝ cos(t + π/4)/(t * ) 1/2 behavior at t * < t cross,3 to χ c(s) 1,branch;b ∝ cos(t − π/4)/((s zs − 1)(t * ) 3/2 ) behavior at t * > t cross,3 , where again This t cross,3 is the analog of t cross,1 for l = 0, Eq. (35). The term χ c(s) 1,branch;2 (t * ) can also be split into two contributions, one from x ≈ s 3 and another one from x ≈ 1. Evaluating the first contribution, we find that, up to an overall factor, χ c(s) 1,branch;2a (t * ) ∝ sin(s 3 t * ) 1 + Because γ 3 < 0, the second term in the round brackets equals −1 and cancel the first one. As a result, there is no sin(s 3 t * ) term in χ c(s) 1 (t * ). The second contribution, χ c(s) 1,branch;2b (t * ), has the same structure as χ c(s) 1,branch;1b (t * ) and just adds up to the prefactor of an oscillation with frequency equal to one. In this section we consider the range of −1 < F c(s) 1 < −1/9, excluding the immediate vicinity of −1, which has been already considered in Sec. . For F c(s) 1 −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/ √ 3, γ zs ≈ 3(|F c(s) 1 | − 1/9)/2. Up to two leading orders in γ zs , the pole contribution is 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 = x − s h and evaluating integrals up to two leading orders in γ h . The cancellation implies that there are no oscillations in χ c(s) 1 (t * ) 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 c(s) 1 −1/9, where we just found that the pole contribution is cancelled by the contribution from the branch cut, to the behavior at F c(s) 1 −1, where we found earlier that there is no such cancellation. As F c(s) 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/ √ 3, For s zs < 1, the lower limit of the integral is positive. This happens when where F vis 1 = −0.162. In this range of F c(s) 1 , one can safely set γ zs to zero -the integral does not diverge. As a consequence, χ c(s) 1,branch;1 (t * ) does not contain the factor ∝ γ −1 zs and cannot cancel χ c(s) 1,pole (t * ) ∝ cos(s zs t * )/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 c(s) 1 (t * ) is qualitatively the same as for F ≥ −1: the pole contribution yields oscillations with frequency s zs and remains the largest contribution to χ c(s) 1 (t * ) up to t * ∼ t cross,2 . At t * > t cross,2 , the branch cut contribution becomes the largest one and χ c(s) 1 (t * ) oscillates at the (dimensionless) frequency equal to one.
However, when s zs > 1, which happens for F vis 1 < F c(s) 1 < −1/9, 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 χ c(s) This result is valid for t * |s zs − 1| 1. The cos(t * − π/4)/(t * ) 3/2 is precisely the expected time dependence for the case when the contribution to χ c(s) 1 (t * ) comes solely from the end points of the branch cut. We see therefore that oscillations with frequency s zs exist as long as F c(s) 1 < −1/9 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 −1 < F c(s) 1 < −1/9. In that case we would expect oscillations to persist for a long time, both at F c(s) 1 < F vis 1 and at −1/9 < F c(s) 1 oscillations would occur with frequency s zs at intermediate t * (but still t * 1) and with frequency equal to one at even larger t * . For F vis 1 < F c(s) 1 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 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 c(s) 1 = −1 and F c(s) 1 − 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. Sec. ), 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. This yields First, we consider the situation when F c(s) 1 < 0 and |F c(s) 1 | 1/9, i.e., when s 1,2 reside below the branch cut (see Fig. 9b) 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 c(s) 1 decreases and becomes smaller than −1/9. We evaluate the integral in Eq. (66) exactly by contour integration in the first quadrant of complex y and obtain where A j = i =j (σ i − σ j ) −1 are the partial fraction decompositions of j (x + σ j ), and where Z(a) was defined in Eq. (32) and Θ(a) is the Heaviside function. [42] 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).
To obtain a qualitative understanding of how the poles emerge, we expand Eqs. (67) and (68) in small γ zs − γ. This approximation is analogous to the one we made above when studying the crossover using the contour of Fig. 4, i.e. of keeping only the leading contribution in γ zs . Using our results for the contour of Fig. 5, the only necessary step is to take the limit Imσ j → 0 in Eqs. (67) and (68), which yields, i.e. oscillations at a frequency s zs = 1 begin to emerge precisely when s zs < 1. Eq. (69) is valid when |(1 − s zs )t * | 1.

Mirage poles
Finally, we discuss the mirage poles. For 0 < F c(s) 1 < 3/5, the conventional ZS pole s 1 is located outside particlehole 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 c(s) 1 = 3/5, Ims 1 becomes equal to γ, and for larger F c(s) 1 , the pole moves to the unphysical Riemann sheet, i.e. in our notations it becomes a mirage pole (see Ref. [14]).
As before, we first compute χ c(s) 1 (t * ) using the integration contour in Fig. 4. Because there are no poles on the physical Riemann sheet for F c(s) 1 > 3/5, the whole contribution comes from the branch cut: χ c(s) The integral over the branch cut has two relevant contributions. The first one, χ At F c(s) 1 where Eq. (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 (2/ √ 3) = 1 and increases with x m for larger x m , which correspond to F c(s) 1 The condition Q 2 (x m ) > 1 implies that there is no pole in (72) above the branch cut, where this expression is valid. Evaluating the branch cut contribution along s = x − i(γ − 0+), we find that the largest piece comes from x ≈ x m and yields χ c(s) 1,branch;bm (t * ) = Combining (70) and (74), we see that in the range where a ZS pole is a mirage one, χ c(s) 1,branch;am (t * ) + χ c(s) 1,branch;bm (t * )) has a contribution oscillating with (dimensionless) frequency x m and the contribution oscillating with (dimensionless) frequency equal to one. When F c(s) 1 = O(1), the second contribution is the dominant one in some range of t * > 1, because the first contribution contains 1/(t * ) 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 c(s) 1 < 3/5, the ZS pole is located in the lower half-plane of frequency on the physical Rieman sheet. In this situation, χ c(s) 1 (t * ) contains contributions both from the pole and from the branch cut. The combined contribution from the pole and the upper edge of the branch cut is χ c(s) where now 0 < x m < 2/ √ 3 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 c(s) 1 (t * ) and −χ c(s) 1,branch;bm (t * ) in (74), i.e., the phase of sin(x m )t * ) oscillations changes by π between the regions where a ZS pole is a conventional one and where it is a mirage one.
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 being on the unphysical sheet. The contribution of χ c(s) 1,branch (t * ) is dominated by the branch points and is given by 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 (t * ) = Q 0 j=1..2n A j e iπ/4 Z(σ j t * ), where Z(a) is given by Eq. (32), A j = i =j (σ i − σ j ) −1 and Q 0 is a constant, calculated directly from ∆χ c(s) l (x) 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).
The case of comparable F c(s) 0 and F c(s) 1 In the main text and in the previous sections, we assumed that one Landau parameter dominates over all others. In this section, we discuss what happens when two Landau parameters are comparable. We focus on the most physically relevant case when F c(s) 0 and F c(s) 1 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 (1 + F c(s) 0 where χ 0 and χ 1 are given by Eqs. (13) and (39) 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 χ c l (q, ω) = χ c l (q, ω) + (χ c l0 (q, ω)) 2 1 − U q χ c 0 (q, ω) , l ≥ 1, where χ c l (q, ω) is the quasiparticle susceptibility renormalized by the FL interaction and χ c l0 (q, ω) is the "mixed" quasiparticle susceptibility with vertices at the opposite corners given by √ 2 cos lθ and 1, correspondingly. The pole