Collective modes near a Pomeranchuk instability in two dimensions

Avraham Klein ,1 Dmitrii L. Maslov,2 Lev P. Pitaevskii,3,4 and Andrey V. Chubukov1 1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 2Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 32611-8440, USA 3INO-CNR BEC Center and Dipartimento di Fisica, Università degli Studi di Trento, 38123 Povo, Italy 4Kapitza Institute for Physical Problems, Russian Academy of Sciences, Moscow, 119334, Russia

The only peculiarity of the charge-and spin-current order parameter is that its time evolution occurs on longer scales than for other order parameters. We also analyze collective modes away from the critical point, and find that the modes evolve with F c(s) l on a multisheet Riemann surface. For certain intervals of F c(s) l , the modes either move to an unphysical Riemann sheet or stay on the physical sheet but away from the real frequency axis. In that case, the modes do not give rise to peaks in the imaginary parts of the corresponding susceptibilities.

I. INTRODUCTION
A Pomeranchuk transition is an instability of a Fermi liquid (FL) toward a spontaneous order which breaks rotational symmetry but leaves translational symmetry intact [1]. Examples include ferromagnetism [2][3][4] and various forms of nematic order in quantum Hall systems, Sr 3 Ru 2 O 7 , and cuprate and Fe-based superconductors [5,6]. For a rotationally invariant system in two dimensions (2D), deformations of the Fermi surface (FS) can be classified by the value of the angular momentum l. In general, a deformation with only one particular l develops at a Pomeranchuk transition. A Pomeranchuk order parameter c(s) l (q) = k f c(s) l (k) a † k+q/2,α t c(s) α,α a k−q/2,α is bilinear in fermions and has the spin structure t c α,α = δ α,α or t s α,α = σ z α,α in the charge (c) and spin (s) channels, correspondingly (σ z is the Pauli matrix). The order parameter is assumed to vary slowly, i.e., q min{a −1 0 , k F }, where a 0 is the lattice constant and k F is the Fermi momentum. Under rotations, the form factors f c(s) l (k) transform as basis functions of the angular momentum and, in general, also depend on the magnitude of |k| ≡ k. For example, f c(s) 1 (k) = cos θ f c(s) (k) or f c(s) 1 (k) = sin θ f c(s) (k), where θ is the azimuthal angle of k, in a coordinate system chosen such that the x axis points in the direction of q. According to the FL theory [7,8], a Pomeranchuk order with angular momentum l emerges when the corresponding Landau parameter F c(s) l approaches the critical value of −1 from above.
In this paper we focus on dynamical aspects of a Pomeranchuk instability. We consider primarily the 2D case because examples of Pomeranchuk transitions have been discussed mostly for 2D systems [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. We consider an isotropic FL but do not specifically assume Galilean invariance, i.e., the single-particle dispersion in our model is not necessarily quadratic in |k|. The main object of our study is the dynamical susceptibility χ l (q, ω), which corresponds to a particular order parameter c(s) l . In 2D, there are two types of susceptibilities for any given l: a longitudinal one, with the form factor proportional to cos lθ, and a transverse one, with the form factor proportional to sin lθ (for l = 0, there is only one type, with an isotropic form factor). The longitudinal susceptibility corresponds to FS fluctuations that are not volume conserving, analogous to axisymmetric fluctuations in three dimensions (3D), while the transverse susceptibility describes volume-preserving fluctuations, analogous to nonaxisymmetric fluctuations in 3D [7]. In the low-energy limit (q → 0 and ω → 0) the dynamical susceptibility is a function of the ratio s = ω/v * F q, where v * F is the renormalized Fermi velocity. As a function of complex variable, χ (s) has both poles and branch cuts in the complex plane. We focus on the retarded susceptibility, which is analytic in the upper halfplane Ims > 0, except for the case when the system is below the Pomeranchuk instability, and, in general, has poles and branch cuts in the lower half-plane Ims < 0. The poles of χ (s) correspond to zero-sound collective modes whose frequency and momentum are related by ω = sv * F q. If the Landau parameter is positive and nonzero for only one value of l, there is one longitudinal and at most one transverse zero-sound mode for any l = 0. These are conventional propagating modes with Res > 1 and infinitesimally small Ims in the clean limit, when the fermionic lifetime is infinite. The branch cuts are a consequence of the nonanalyticity of the free-fermion bubble (the Lindhard function). Their significance is that the zero-sound poles are defined on a multisheet Riemann surface. In 2D, this nonanalyticity is particularly simple, being just a square root (see Sec. II A). As a result, the Riemann surface is a genus 0, two-sheet surface. We shall refer to the first sheet, which includes the analytic half-plane, as the "physical sheet." In our work, we mostly discuss the properties of the physical sheet.
We obtain explicit results for the frequencies of collective modes in the whole range of −1 < F c(s) l < ∞. First, we consider a clean FL (Sec. II). We present explicit results for the zero-sound modes with l = 0, 1, 2 (Secs. II B, II C, and II D, correspondingly) and analyze the structure of the zero-sound modes for arbitrary l (Sec. II E). We show that for l = 0 and in the transverse channel with l = 0, all zero-sound modes acquire a finite decay rate already for an arbitrary small negative F c(s) l , i.e., s = ±a − ib, where a, b are real and b > 0. A positive b implies that a perturbation of the order parameter decays exponentially with time. The frequency of one of the modes vanishes at the Pomeranchuk transition. We call this mode the critical one. In the longitudinal channel, the decay rate of one of the modes remains infinitesimally small until the corresponding |F c(s) l | exceeds a threshold value. Immediately below the threshold, this mode is located at s = ±a − ib with a > 1 and b 1. Even though the mode frequency is almost real, the corresponding pole is located below the branch cut and thus cannot be reached from the real axis of the physical sheet. Accordingly, the imaginary part of the susceptibility does not have a peak above the particle-hole continuum for real s, i.e., the mode is "hidden." Below the transition, i.e., for F c(s) l < −1, the pole is located in the upper frequency half-plane, and a perturbation of the order parameter grows exponentially with time, which indicates that a FL becomes unstable with respect to a Pomeranchuk order. We obtain explicit results for the frequencies of collective modes in the whole range of −1 < F c(s) l < ∞. In Sec. III, we analyze how the dispersion of collective modes is modified in the presence of impurity scattering. In the dirty limit, the critical modes in both longitudinal and transverse channels become overdamped for all l. Impurity scattering also smears the threshold, described in the previous paragraph, i.e., the longitudinal collective modes have nonzero damping rates for any F c(s) l < 0. In Sec. IV, we analyze the susceptibility in the time domain which determines the time evolution of the order parameter following an initial perturbation. We obtain explicit forms of χ c(s) l (q, t ) for l = 0 and 1. Above the Pomeranchuk transition, the time dependence of χ c(s) l (q, t ) in the clean limit is a combination of an exponentially decaying part, which comes from the poles of χ c(s) l (q, ω), and of an oscillatory (and algebraically decaying) part, which comes from its branch cuts. At the transition, χ c(s) 0 (q, t ) reaches a time-independent limit at t → ∞, while χ c(s) 1 (q, t ) grows linearly with time in the clean case and saturates at a finite value in the presence of disorder. Below the transition, the poles of χ c(s) 0,1 (q, ω) are located in the upper half-plane of ω. Consequently, both χ c(s) 0 (q, t ) and χ c(s) 1 (q, t ) increase exponentially with time. This means that any small fluctuation of the corresponding order parameter is amplified, and thus the ground state with no Pomeranchuk order is unstable. In the case of finite disorder, the branch-cut contribution also begins to decay exponentially, on top of its algebraic and oscillatory behavior.
In Sec. V, we consider the special case of an order parameter that coincides with either the charge or spin current. Previous studies [26][27][28] found that the corresponding static susceptibility χ c(s) 1 (q, 0) does not diverge at the tentative Pomeranchuk instability at F c(s) 1 = −1 because of the Ward identities that follow from conservation of total charge and spin. We analyze the dynamical susceptibility for such an order parameter. We show, using both general reasoning and direct perturbation theory for the Hubbard model, that while the static susceptibility indeed remains finite at F c(s) 1 = −1, the dynamical one still has a pole, which moves to the upper frequency half-plane below the transition. The residue of this pole vanishes as (1 + F c(s) 1 ) 2 at F c(s) 1 = −1, but is finite both for F c(s) 1 > −1 and F c(s) 1 < −1. We argue that the presence of the pole in the upper frequency half-plane for F c(s) 1 < −1 indicates that the state with no Pomeranchuk order becomes unstable, like for any other type of the order parameter. We derive a Landau functional for the charge-and spin-current order parameter and show that it has a conventional form, except that the coupling between the order parameter and an external perturbation has an additional factor of 1 + F c(s) 1 . We argue that the charge-and spin-current order does develop at 1 + F c(s) 1 < 0, just as for a generic l = 1 order parameter, but it takes longer to reach equilibrium after an instantaneous perturbation. This result differs from earlier claims that there is no Pomeranchuk transition to a state with the charge-and spin-current order parameter [26][27][28].
Before we move on, a comment is in order. It is well known that the range of FL behavior shrinks as the system approaches a Pomeranchuk instability and disappears at the transition point, where the system displays non-Fermi-liquid behavior down to the lowest energies. In our analysis, we will be studying the collective modes at finite s = ω/(v * F q) and assume that ω and q are both small enough so that at any given distance to the critical point the system remains a Fermi liquid. χ c(s) l (q, ω) with respect to the conjugated "field." In its turn, χ c(s) l (q, ω) is given by a fully renormalized particle-hole bubble with external momentum q and external frequency ω. For free fermions, the particle-hole bubble is just a convolution of two fermionic Green's functions, whose momenta (frequencies) differ by q (ω). In the time-ordered representation, we define the normalized susceptibility as where N F is the density of states at the Fermi energy E F , Green's function, k is the single-particle dispersion, D is the spatial dimensionality, and a factor of 2 comes from summing over spins. We will be interested only in the case of small q and ω, i.e., q k F and ω E F . In this case, integration over the internal fermionic momentum and frequency is confined to the regions of small ε and k − E F , i.e., the susceptibility comes from the states near the FS or, for brevity, from "lowenergy fermions." For interacting fermions, the particle-hole bubble is modified in several ways [7,8,29]. First, the self-energy corrections transform a free-fermion Green's function near the FS into a quasiparticle Green's function, in which the bare velocity v F is replaced by the renormalized velocity v * F = v F (m/m * ), where m * is the renormalized mass, and the Green's function is multiplied by the quasiparticle residue Z < 1. Second, interactions between low-energy fermions generate multibubble contributions to the susceptibility. These renormalizations transform the free-fermion susceptibility χ c(s) free,l (q, ω) into the quasiparticle susceptibility χ c(s) qp,l (q, ω). (The effect of damping due to the residual interaction between quasiparticles is a subleading effect in the range of q and ω of interest to us, and will not be considered here.) Third, fermions far away from the FS ("high-energy fermions") also contribute to the full susceptibility χ c(s) l (q, ω). The general expression for the dynamic susceptibility is [29] Here, c(s) l is the side vertex, renormalized by high-energy fermions, and the standalone term χ c(s) inc,l represents the contribution solely from high-energy fermions. This last term does not have a singular dependence on q and ω and will not play any crucial role in our analysis. We emphasize that Eq. (3) is valid for any isotropic system, even if it is not Galilean invariant.
The quasiparticle contribution to the susceptibility depends on the fully renormalized (and antisymmetrized) interaction between low-energy fermions, usually denoted by αβ,γ δ (k − p). This interaction includes renormalizations by high-energy fermions but not by low-energy fermions. For a rotationally and SU(2)-invariant system, which we consider here, αβ,γ δ (k − p) can be expanded over harmonics characterized by orbital momenta l, and the properly normalized coefficients of this expansions are known as Landau parameters F c(s) where and θ k , θ p are the azimuthal angles of k, p. The static quasiparticle susceptibility χ c(s) qp,l (q, 0) is expressed in terms of just a single F c(s) l : The dynamical quasiparticle susceptibility cannot, in general, be expressed in terms of a single Landau parameter, unless all Landau parameters except for a single F c(s) l are small. In this special case, where q * = (m/m * )q and χ free,l (q * , ω) is normalized to χ free,l (q * , 0) = 1 (we recall that we consider small q * k F ). For all order parameters, except for the charge or spin current, the vertex c(s) l in Eq. (3) is expected to remain finite at the Pomeranchuk transition. The behavior of the full susceptibility is then determined entirely by the quasiparticle χ c(s) qp,l (q, ω). Although the calculations are straightforward and some of the results have appeared before [9,26,28,[30][31][32][33][34][35][36][37][38][39], we include below the details of the derivation of χ c(s) qp,l (q, ω) in 2D, as we will be interested in the pole structure of the susceptibility not only near a Pomeranchuk transition, but also away from it. In what follows we first consider separately the cases of l = 0, 1, 2 and trace out the evolution of the poles of χ c(s) qp,l (q, ω) in the complex plane of frequency. For a reader wishing to avoid going through the computational details, we depict the evolution graphically in Figs. 2 and 4, and summarize our main results for l = 0, 1, 2 in Table I. We then analyze the case of arbitrary l. In these calculations, up to Sec. II F, we assume that a single Landau parameter F c(s) l is much larger than the rest and compute χ c(s) qp,l (q, ω) using Eq. (7). In Sec. II F, we consider the case when F c(s)  In this case, the form factor f c(s) 0 (k F ) is just a constant. The form of the retarded free-fermion susceptibility along the real frequency axis is well known: where we used that v F q * = v * F q and defined s = ω/v * F q. Viewed as a function of complex s, χ free,0 (s) has branch cuts, which start at s = −iδ below the real axis and run along the segments (−∞, −1) and (1, ∞) along the real axis.
Traditionally, δ in Eq. (8) is interpreted as an infinitesimally small damping rate whose physical origin does need not to be specified and whose sole purpose is to shift the branch cut into the lower half-plane of complex s. We will see, however, that such approach is not sufficient for our purposes because it would not allow us to resolve the relative positions of the zero-sound poles and branch cuts of the susceptibility in the complex plane of s. For this reason, we will consider a specific damping mechanism, namely, scattering by shortrange impurities, and treat δ as a finite albeit small number.
The order parameter in the l = 0 channel (charge or spin) is conserved, i.e., the susceptibility must satisfy χ c(s) 0 (q = 0, ω) = 0 (see, e.g., Refs. [40,41]). Once δ is finite, Eq. (8) does not satisfy this condition because it was obtained either by adding iδ self-energy corrections to the Green's functions or, which is equivalent, by solving the kinetic equation in the relaxation time approximation. To ensure that charge and spin are conserved, one also has to include vertex corrections to the particle-hole bubble or go beyond the relaxation-time approximation. 1 The corresponding free-fermion susceptibility is given by [42] χ free,0 (s) = 1 + is where δ now stands for the dimensionless impurity scattering rate. The −δ term next to 1 − (s + iδ) 2 in (9) comes from vertex corrections. Until Sec. III, we will be assuming that impurity scattering is weak, i.e., δ min{Res, Ims}. For F c(s) 0 > 0 we expect to have well-defined collective modes with |s| > 1. In this case, one can safely neglect δ in Eq. (9) and replace 1 − (s + iδ) 2 by −i sgns √ s 2 − 1. Equation (9) is then reduced to 1 The collision integral in the relaxation-time approximation I coll = −δ × ( f − f 0 ), with ( f ) f 0 being (non)equilibrium distribution function, is not an appropriate form for impurity scattering, which is elastic and therefore must conserve a number of particles with given energy. Consequently, I coll must vanish upon averaging over the directions of the momentum, which is not the case for I coll . The correct form of the collision integral for impurity scattering is −δ × ( f −f ), wheref is the angular average of f . The kinetic equation with this collision integral reproduces Eq. (9). Substituting this form into Eq. (7), we obtain The locations of the poles are determined from the equation One can check that the solution s 1,2 = ±s p with s p > 1 indeed exists only for F c(s) 0 > 0: We now widen the scope of our analysis and search for solutions with complex s. To this end, we need to keep δ terms in χ free,0 (s). The quasiparticle susceptibility for s in the lower half-plane is obtained by substituting (9) into (7). This yields Using Eq. (14), we can study the poles of χ c(s) qp,0 (s) everywhere in the lower half-plane of complex s and in the whole range of F c(s) 0 .
The positions of the poles in the lower half-plane of s are determined by For F c(s) 0 > 0, the solutions of (15) are where Up to the −iδ 0 term, this result coincides with Eq. (13), as it should. We see thatδ 0 is positive but smaller than δ in (15). This implies that the poles are located above the branch cuts. The purpose of starting with Eq. (9) with small but finite δ was to resolve the difference between δ, which determines the locations of the branch cuts, andδ 0 , which determines the distance between the poles and the real axis. Near the poles, the susceptibility reduces to This expression is valid for complex s above the branch cut at Ims = −iδ. This includes the real axis. For real s and vanishingly smallδ 0 , Imχ c(s) 0 (s) has δ-functional peaks at s = ±s p,0 with s p,0 > 1, i.e., outside the particle-hole continuum [see Fig. 1(a)].
The pole at s = −is i,0 describes a purely relaxational zerosound mode. As long as 1 + F c(s) 0 > 0, the pole in χ c(s) qp,0 (q, ω) is in the lower half-plane of s, i.e., excitations decay exponentially with time. Once 1 + F c(s) 0 becomes negative, the pole moves into the upper half-plane. Then, excitations grow exponentially with time, i.e., the system becomes unstable (see Sec. IV for more detail). This is corroborated by the fact that the static susceptibility diverges as the system approaches a Pomeranchuk instability: As F c(s) 0 increases above −1, i.e., |F c(s) 0 | gets smaller, the frequency of the relaxational mode in (19) increases in magnitude. It reaches s i,0 = ∞ at F c(s) 0 = −1/2. At this value of F c(s) 0 , the mode bifurcates into two (s i,0 → ±s p,0 ), and each new mode moves from imaginary to almost real s along infinite quarter circles in the complex s plane.
For − 1 2 < F c(s) 0 < 0 the mode frequency is given by s = ±s p,0 − iδ 0 , wherē The real part varies froms p,0 = ∞ at F c(s)  17)]; however, nowδ 0 δ, i.e., the poles are located below the branch cut at Ims = −δ. At vanishingly small δ, which we consider here, the poles are glued to the lower edge of the branch cut immediately below the real axis. The evolution of the real and imaginary parts of the poles with F c(s) 0 is shown in Fig. 2.
The existence of the poles glued to the lower edge of the branch cut is a tricky phenomenon. At first glance, they describe undamped collective excitations with velocity larger than the Fermi velocity [note thats p,0 > 1 in Eq. (21)]. Indeed, the susceptibility near the poles is This form is very similar to that in Eq. (18) for positive F c(s) 0 . However, Eq. (22) is valid only for complex s below the lower edge of the branch cut at |s| > 1, and cannot be extended to real s. More precisely, Eq. (22) cannot be extended to the real axis on the physical sheet of the Riemann surface, which we recall is the sheet for which χ c(s) qp,0 (s) is analytic in the upper half-plane. Instead, it can be extended to the real axis of the unphysical sheet, the one for which 1 − (s + iδ) 2 = i (s + iδ) 2 − 1. This means that the pole below the branch cut has no effect on the behavior of Imχ c(s) qp,0 (s) on the real axis, and the imaginary part of the susceptibility for real s, with χ (s) ≡ Reχ c(s) free,0 (s) and χ (s) ≡ Imχ c(s) free,0 (s), has no peak above the continuum. Therefore, the modes for −1/2 < F c(s) 0 < 0 are "hidden," in a sense that they cannot be detected by a spectroscopic measurement which probes Imχ c(s) qp,0 (s).

C. l = 1
For l 1 we have to distinguish between the longitudinal susceptibility with the form factor √ 2 cos θ and the transverse susceptibility with the form factor √ 2 sin θ . We consider the two cases separately. Here and in what follows, we will suppress the c(s) superscript in the longitudinal and transverse susceptibilities for brevity, i.e., we will relabel χ c(s),long < −1 the pole is purely imaginary and above the real axis, which indicates that the FL state is unstable. With increasing F c(s) 0 , the pole moves down along the imaginary axis, which corresponds to an overdamped zero-sound mode, and reaches −i∞ at F c(s) 0 = − 1 2 . It then "jumps" to the lower edge of the branch cut. A pole located at the lower edge of the branch cut corresponds to a "hidden" zero-sound mode, which cannot be detected in measurements of χ c(s) 0 (s) for real s (i.e., real frequencies). At F c(s) 0 = 0 the pole moves to the upper edge of the branch cut, where it becomes a well-defined zero-sound mode, detectable by spectroscopic methods.

l = 1, longitudinal channel
The computation of the free-fermion susceptibility with √ 2 cos θ form factors at the vertices is quite straightforward. In notations of the previous section, the retarded susceptibility is given by For real |s| > 1, Eq. (24) reduces to χ long free,1 (s) = 1 + 2s 2 Substituting this form into Eq. (7), we obtain an equation for the poles: A solution of Eq. (26) in the form s 1,2 = ±s p,1 with s p,1 > 1, i.e., outside the continuum, exists only for F c(s) increases, the magnitude of s p,1 also increases, and at large F c(s) 1 becomes s p,1 ≈ (3F c(s) 1 /4) 1/2 . Correspondingly, Imχ long qp,1 (s) has peaks on the real axis at s = ±s p,1 .
To find the actual position of the poles in the complex plane, we will again need to treat δ as a finite, albeit small, quantity. As for the l = 0 case, we associate δ with weak impurity scattering. Because a generic l = 1 order parameter is not a conserved quantity, vertex corrections are not crucial. 2 Nevertheless, they are necessary to correctly determine the location of the poles.
The expression for χ long free,1 (s) in the presence of impurity scattering will be derived in Sec. III. Here, we just borrow the result χ long free,1 (s) = 1 + 2s 2 The equation for the poles becomes If we assume that s is in the lower half-plane above the branch cut, i.e., −δ Ims < 0 and 1 − (s + iδ) 2 = −i sgns (s + iδ) 2 − 1, we find that the solution actually exists only for 0 F c(s) For 0 < F c(s) 1 < 3 5 , s p,1 varies between 1 and 2/ √ 3, andQ 1 < 1, as we assumed. For F c(s) 1 = 3 5 , we have s p,1 = 2/ √ 3 and Q 1 = 1, i.e., the pole merges with the branch cut. For larger F c(s) 1 , we haveQ 1 > 1, violating our assumption that the pole is above the branch cut, so that Eq. (28) has no solution. A more careful analysis shows that the pole has moved to the unphysical Riemann sheet on which 1 − (a + ib) 2 near the because the self-energy and vertex corrections due to impurity scattering do not cancel each other. branch cut is defined as 1 − (a + ib) 2 which we used to search for the poles on the physical Riemann sheet.
The absence of the zero-sound pole for F c(s) 1 3 5 is a surprising result, but it has little effect on the form of χ long qp,1 (s) for real s. The latter has a conventional form for all positive F c(s) 1 : and Imχ long qp,1 (s) has peaks at s = ±s p,1 , as shown in Fig. 3. There also exists another solution for F c(s) 1 > 0, which is purely imaginary: s = −is i,1 . Assuming that s i, 1 δ, we obtain an equation for s i,1 from Eq. (28): For small positive F c(s) increases, s i,1 decreases and eventually saturates at s i,1 = 1/ √ 3. This additional solution will be relevant for the case of finite damping, analyzed in Sec. III. Note that Eq. (31) has a solution only for positive s i,1 , i.e., the pole is in the lower half-plane, as it should be.
For negative F c(s) 1 , we again search for complex solutions in the form Right above the Pomeranchuk instability, i.e, at F c(s) In contrast to the l = 0 case, the collective modes are almost propagating because a b. Below the Pomerachuk transition, i.e., for F c(s) 1 < −1, both poles become purely imaginary and split away from each other along the imaginary s axis: One of these poles is now in the upper frequency half-plane, i.e., a perturbation with the structure of the longitudinal l = 1 order parameter grows exponentially (see Sec. III). This indicates a Pomeranchuk instability.
As |F c(s) 1 | decreases, a 1 monotonically increases, while b 1 first increases and then changes trend and starts decreasing (see Fig. 4). The poles reach the lower edges of the branch cuts at F c(s) 1,cr . At this critical value of F c(s) 1 , a cr,1 = 2/ √ 3 > 1 and b = 0 (up to a term of order δ). For |F c(s) 1 | slightly below F c(s) 1,cr , a 1 and b 1 are approximately given by When F c(s) 1 approaches F c(s) 1,cr , the poles approach the real axis along the paths that are almost normal to it.
The existence of the solution with a 1 > 1 but finite b 1 for 1,cr is at first glance questionable because conventional wisdom suggests that a mode with Res > 1 is located outside the particle-hole continuum and thus should be purely propagating. However, as for the l = 0 case, these poles are located below the branch cuts, cannot be accessed from the real axis and do not lead to a peak in Imχ long < 0, the poles are located at s 1,2 = ±s p,1 − iδ 1 , wheres p,1 is determined from The magnitude ofs p,1 varies betweens p, the poles approach the end points of the branch cuts. As follows from Eq. (38),Q 1 1 fors p,1 in this interval, henceδ 1 δ, i.e., the poles are located below the lower edges of the cuts, as expected. This is very similar to what we found in the l = 0 case for − 1 2 < F c(s) at s = ±s p,1 − iδ 1 : However, Eq. (39) is again only valid for complex s in the lower half-plane below the branch cut, and cannot be extended to real s. These poles correspond to hidden modes, and the susceptibility does not have peaks above the particle-hole continuum. We plot Imχ long qp,1 (s) for real s in Fig. 3. The evolution of the real and imaginary parts of the poles with F c(s) 1 is shown in Fig. 4.

l = 1, transverse channel
We next consider the transverse quasiparticle susceptibility in the l = 1 channel. The retarded susceptibility of free fermions with the √ 2 sin θ form factors at the vertices is For real |s| > 1, Eq. (40) is reduced to Substituting this form into Eq. (7), we find that the positions of the poles on the real frequency axis and outside the particlehole continuum are determined by In contrast to the longitudinal case, the solutions of this equation s 1,2 = ±s p,1 exist not for any positive F c(s) 1 but only for F c(s) 1 1 (Ref. [35]). Slightly above the threshold, s p, To obtain the solutions in the complex plane s, we introduce impurity scattering in the same way as in the previous cases. Equation (40) is then replaced by There are no additional terms due to vertex corrections because the form factor is an odd function of the angle θ and thus vertex corrections vanish upon angular integration. Substituting Eq. (43) into (7), we find that for F c(s) 1 > 1, where Eq. (42) has a solution for real s, there is actually no solution for the pole of χ tr qp,1 (s) in the complex plane of s, above the branch cut. Still, for real s, Imχ tr qp,1 (s) displays sharp peaks even for F c(s) < 1 we assume that s is below the branch cut and rewrite the square root in Eq. (43) as 1 − (s + iδ) 2 = i sgns (s + iδ) 2 − 1. If we just neglect δ after that, we find another propagating mode, located at s 1,2 = ±s p,1 , wherē s p,1 1 is the solution of However, if δ is treated as a small but finite quantity, we find that there is no solution of (χ tr qp,1 (s)) −1 = 0 with |Ims p,1 | > δ, i.e., there is no pole below the branch cut. Combining this with the absence of the pole for F c(s) 1 > 1, we conclude that the l = 1 transverse susceptibility does not have a pole on the physical sheet for F c(s) 1 > 0. However, as was the case for the longitudinal mode, the poles do exist on the unphysical sheet.
For negative F c(s) 1 the pole of χ tr qp,1 (s) is on the imaginary axis: s = −is i,1 . The value of s i,1 is determined by 033134-8 The solution exists for all negative F c(s) )/2. As before, when 1 + F c(s) 1 changes sign and becomes negative, the pole moves from the lower to the upper frequency half-plane, i.e. an l = 1 perturbation in the shape of the FS grows with time exponentially. This behavior is similar to the one for l = 0. Yet, a purely relaxational collective mode in the l = 1 transverse channel exists for all −1 < F c(s) 1 < 0, i.e., it appears without a threshold.
The retarded free-fermion susceptibility with the √ 2 cos 2θ form factors at the vertices is The equation for the poles of χ long qp,2 (s) outside the continuum, i.e., for s real and |s| > 1, now reads as Similarly to the cases of l = 0 and of the longitudinal channel for l = 1, the propagating solutions To obtain the solutions in the complex plane, we introduce impurity scattering in the same way as before. Combining the self-energy and vertex corrections, we obtain after some algebra The equation for the pole becomes Solving for the pole at small but finite δ, we find s 1,2 = ±s p,2 − iδ 2 , whereδ 2 =Q 2 δ. EvaluatingQ 2 , we find that it is smaller than 1 for F c(s) 2 < 0.420, when s p,2 < 1.072. For these F c(s) 2 , the pole is located above the branch cut, as it should be. For larger F c(s) 2 there are no poles near the real axis. This is similar to the behavior in the longitudinal channel for l = 1. We reiterate that the absence of a true pole in the complex plane does not affect the behavior of χ long 2 (s) for real s; in particular, Imχ long 2 (s) still displays sharp peaks at s = ±s p,2 . In mathematical terms, the pole moves to a different Riemann sheet at F c(s) For negative F c(s) 2 , Eq. (49) has two solutions. One of them is purely imaginary:  (49) is satisfied not only by s = 0, but also by s 1,2 = ±1/ √ 2. The latter solutions are on the real axis, but away from the branch cut. At small deviation from the critical value F c(s) Observe that b 2 remains positive even when 1 + F c(s) gets larger, the solutions first move away from the real axis but then reverse the trend and, at the threshold value F c(s) 2,cr = −0.0632, reach the lower edge of the branch cut at 2,cr to 0, the solutions "slide" along the lower edge of the branch cut toward s = ±1. This is very similar to what we found in the l = 0 case, < 0. At a small but finite δ, the two sliding solutions are s 1,2 = ±s p,2 − iδ 2 withδ 2 δ, i.e., the pole does exist but is located below the branch cut. The evolution of the poles with F c(s) 2 is shown in Fig. 5.

l = 2, transverse channel
The retarded free-fermion susceptibility with the √ 2 sin 2θ form factors at the vertices is given by The equation for the poles outside the continuum, i.e., for s real and |s| > 1, reads as For positive F c(s) 2 , the solutions s 1,2 = ±s p,2 exist for F c(s) To obtain the solutions in the complex plane s, we introduce impurity scattering in the same way as before. There are no additional terms due to vertex corrections because the form factor is an odd function of the angle θ . Equation (51) is then replaced by For F c(s) 1 > 1 3 , we assume that s is above the branch cut and use 1 − (s + iδ) 2 = −i sgns (s + iδ) 2 − 1. If we neglect δ after that, we obtain the same solution s p,2 as in (52). For 0 < F c(s) 2 < 1 3 , the same procedure but with an assumption that the 033134-9 FIG. 5. The poles of χ c(s) qp,2 (s) in the complex plane. The use of color and notation is the same as in Fig. 2 [blue (dark), real part of pole; yellow (light), imaginary part of pole]. See Sec. II D for a detailed discussion.
pole is below the branch cut yields another propagating mode, which slides along the lower edge of the branch cut, much like it happens for the pole of the transverse susceptibility for l = 1. Once we take into account that δ is small but finite, we find that the solution along the real axis does not survive in either of the cases, i.e., there is no pole close to the real axis on the physical Riemann sheet. This is similar to the situation for the l = 1 transverse channel.
As for the l = 1 case, there also exists another mode for F c(s) For small F c(s) there is no solution on either real or imaginary frequency axes, and we search for the solutions in the form s = ±a 2 − ib 2 , where both a 2 and b 2 are finite. In this situation, one can safely neglect δ and write the equation for the poles as An analysis of this equation shows that the solution exists for all F c(s) is shown in Fig. 5. Table I contains the summary of the results for the pole positions for l = 0, 1, 2. The table reveals several trends, which we now study in more detail by extending our analysis to arbitrary l.

E. Poles of χ c(s) qp,l for arbitrary l 1. Equations for the poles
We now focus in more detail on negative F c(s) l and, in particular, on the behavior of collective modes near a Pomeranchuk instability. Comparing the results for for the l = 0, 1, 2 modes, we see a difference between even and odd l. Namely, near a Pomeranchuk instability the critical mode in the longitudinal channel is purely imaginary for even l = 0, 2 and almost real for odd l = 1. For transverse channels the situation is the opposite: the mode near a Pomeranchuk instability is purely imaginary for l = 1 and almost real for l = 2. In this section we analyze whether this trend persists for other values of l.
The retarded longitudinal and transverse susceptibilities of free fermions can be obtained analytically for any l. We have where The equation for the pole on real frequency axis outside the continuum, i.e., for |s| > 1, is The upper and lower signs correspond to the longitudinal and transverse channels, respectively. One can easily verify that, for any l, a solution with real |s| > 1 exists only for positive F c(s) l . For negative F c(s) l , we search for complex solutions. In this case, we rewrite (60) as In what follows, we consider the longitudinal and transverse channels separately, first for even l and then for odd l. We consider separately the limits of F c(s) l ≈ −1 and |F c(s) l | 1, and then interpolate between the two limits. We show that there are multiple solutions with complex s in each channel. The structure of the solutions in the longitudinal channel for even l are very similar to those in the transverse channel for odd l. We do not discuss here the solutions in the transverse channel for positive F c(s) l , but below the threshold on the solution with real s and s > 1.

Even l, longitudinal channel
For F c(s) l ≈ −1, we first search for a solution with small |s|. Expanding Eq. (61) in s, we find a pole on the imaginary axis 033134-10 a pole s p is in the lower half-plane of the physical sheet. For each pole, the table notes if the pole is in the complex plane (i.e., has a finite imaginary part), is glued to the lower edge of the branch cut (i.e., has infinitesimal imaginary part but is hidden below the branch cut), or is glued to the real axis (i.e., has infinitesimal imaginary part and is above the branch cut). When the pole is in the complex plane, it either has a finite real part (denoted by "p" for propagating) or is pure imaginary (denoted by "d" for purely damped). The first column denotes the channel and the number of poles it has. In the case of two poles they are specified by (a), (b) markings. For ranges of F c(s) l that do not appear in the table, the pole is on the unphysical sheet. For example, in the l = 1 longitudinal channel, one of the two modes is always on the unphysical sheet, except for when 0 < F c(s) 1 < 3 5 . Table entries marked by "-" denote that a given type of pole does not occur for any −1 < F c(s) l . The table includes entries only for poles having Res p 0. Please note that every pole with Res p > 0 has a counterpart with a negative real part −Res p and the same imaginary part.

Channel (no. of poles)
Pole is in the lower half-plane Pole is glued to the lower branch cut edge Pole is glued to the real axis There exist additional noncritical solutions for which s remains finite at For |F c(s) l | smaller than the threshold, the poles remain below the branch cut at s = ±a − iδ, a 1. For vanishingly small δ, which we consider in this section, the poles are glued to the lower edge of the branch cut and slide along the branch cut toward its lower end at |s| = 1 as |F c(s) l | decreases. We see that for any even l there exists exactly one such threshold solution, while other solutions appear already for infinitesimally small negative F c(s) l .

Even l, transverse channel
We start again with F c(s) l ≈ −1 and consider the solution with vanishingly small s. For the transverse channel [for which we have to choose the minus sign in Eq. (61)], the leading, linear-in-s term on the right-hand side of Eq. (61) is absent, and one needs to include the subleading terms. A straightforward analysis then shows that the poles of the transverse susceptibility are located near the real axis, at When |F c(s) l | becomes larger than 1, this solution moves into the upper half-plane, signaling an instability toward the development of a Pomeranchuk order.
There also exist other solutions that remain finite at

Odd l, longitudinal channel
The analysis for odd l proceeds along the same lines. We do not present the details of calculations and just state the results. For

F. Case of two comparable Landau parameters
As a more realistic example, we consider the case when two Landau parameters, e.g., F c(s) 0 and F c(s) 1 , are comparable in magnitude, while the rest of the Landau parameters are negligibly small. In this situation, the relation between the quasiparticle and free susceptibilities is more complicated than in Eq. (7) where K n are given by Eq. (58). Explicitly, The denominators of χ c(s) qp,0 and χ long qp,1 vanish when Suppose that F c(s) 1 is negative and close to −1 while 1 + F c(s) can be of either sign). In the previous sections, we saw that a critical zero-sound mode corresponds to small s. Substituting the forms of K n into Eq. (67) and assuming that s is small, we obtain For large F c(s) 1 , s ≈ F c(s) 1 /2.

G. 3D systems
For comparison, we also briefly discuss the behavior of zero-sound excitations near a Pomeranchuk instability in a 3D system. We present the results for l = 0 and 1 and, in each case, consider only one nonzero Landau parameter F c(s) l < 0.

l = 0
Zero-sound modes in the l = 0 channel were analyzed in Refs. [30,43]. The free-fermion susceptibility with the form factor f c(s) 0 (k F ) = 1 is The equation for the pole reads as The pole is completely imaginary:

for a Pomeranchuk instability is F c(s)
In the longitudinal channel, the form factor is f c(s) 1 (k F ) = Y 0 1 (θ ). The free-fermion susceptibility is The equation for the zero-sound pole is When F c(s) 1 ≈ −1, the solution is The form factor in the transverse channel is f c(s) The free-fermion susceptibility is The equation for the zero-sound pole is One can easily verify that the pole is located on the imaginary axis, at s = −ia, where a is the solution of Near F c(s) . This is again similar to 2D, except for the solution s = −ia in 3D exists for all negative F c(s) 1 , i.e., there is no threshold.

A. General formalism
In this section we analyze how the results of the previous sections change in the presence of finite disorder. As in the previous section, we consider separately the cases of l = 0, 1, 2. Other cases can be analyzed in the same manner as these two. For a reader who prefers to avoid going through the exhaustive derivations, we present in Figs. 7 and 9 a graphical summary of the evolution of the poles for the l = 0 and the l = 1 longitudinal channels. These figures depict the qualitative changes that occur for finite disorder.
The free-fermion susceptibility in the presence of scattering by short-range impurities consists of two parts: the bubble part and the vertex part: (cf. Fig. 6). The bubble part is formed from the (Matsubara) whereγ is the impurity scattering rate: The vertex part is where D(q, ω m ; ν m ) is the diffusion propagator [42] Diagrammatically, D(q, ω m ; ν m ) is represented by the sum of ladder diagrams in the particle-hole channel (the sequence of diagrams in the square brackets in Fig. 6). The retarded forms of the susceptibilities are obtained by choosing ω m > 0 and replacing iω m → ω in the final results. The vertex part is especially important for the l = 0 case because the corresponding order parameters (charge or spin) are conserved quantities, and hence the bubble and vertex parts of the susceptibility must cancel each other at q = 0. For l > 0, the corresponding order parameters are not conserved, but the vertex parts must be also included in order to obtain the correct positions of the zero-sound poles in the complex plane. The bubble and vertex parts for the l = 0 case are given by where γ =γ /v * F q. Adding these up, we obtain which is the result quoted in Eq. (9), except that we have changed the notations δ → γ to emphasize that γ does not have to be small. This result, as well as a corresponding result for the l = 1 case, holds forγ E F while the ratio γ /s can be arbitrary. At q → 0, i.e., at s → ∞, the susceptibility vanishes, which guarantees that the charge and spin are conserved.
For |s| γ , Eq. (86) reduces to the well-known diffusive form [44,45] where D = (v * F ) 2 /2γ is the diffusion coefficient in 2D. Substituting Eq. (86) into Eq. (7) and solving for the poles, we find that for F c(s) 0 < − 1 2 the pole is on the imaginary axis, at For γ → 0, this reduces to Eq. (19). For large γ , s 1 = −i(1 − |F c(s) 0 |)/2γ . In this limit, we have a diffusion pole at = −iD * q 2 , where D * = D(1 − |F c(s) 0 |) is the renormalized diffusion coefficient [46]. In the ballistic regime at small γ , the damping term accounts for a small correction to the result for a clean Fermi liquid [cf. Eq.
appears in the lower half-plane, initially at s 2 = −i∞. As small damping, this solution and the s 1,2 solution are not connected. We now show that the behavior of χ long 1 changes qualitatively in the presence of disorder. The bubble and vertex parts of the free-fermion susceptibility are now given by Adding these up, we obtain χ long free,1 (s) = 1 + 2s 2 which is the result quoted in Eq. (27), up to a replacement γ → δ. Note that the vertex part vanishes at q → 0, i.e., at s → ∞, while the bubble part is reduced to a form which is identical to the Drude conductivity at finite frequency ω. This indicates that the charge and spin currents are not conserved in the presence of disorder. The analysis of the evolution of the poles with F c(s) 1 for different γ is straightforward but somewhat involved. We omit the details of the calculations and present only the results. These results are summarized graphically in the panels of Fig. 9.
The beginning stage of the evolution is the same for all γ : for 1 + F c(s) For γ < 1 2 the evolution of the poles is similar to that for vanishingly small γ (see Fig. 3 where The values of F c(s) 1,R and of a 1,R decrease as γ increases, but F c(s) 1,R remains positive, and a 1,R remains larger than 1 as long as γ < 1. For large F c(s) 1 the pole on the unphysical sheet is at )/2, i.e., Ims 1 increases with F c(s) 1 . Figure 10 depicts the imaginary part of χ long qp,1 (s) for a finite disorder.
The purely imaginary pole s = −is i,1 , which exists only for F c(s) For γ < 1 2 , this limiting value s i,1 > γ . For 0.923 < γ < 1, the poles, which initially move away from the imaginary axis, return to this axis at some positive value of F c(s) 1 , at which the purely imaginary is still located at a higher point on the imaginary axis (see Fig. 9, fourth panel). The subsequent evolution with increasing F c(s) and γ 1, it approaches the point s ≈ −i/2γ . We note that there are certain similarities between the evolution of the poles with F c(s) 1 and the behavior of the plasmon modes in a 2D electron gas with conductivity exceeding the speed of light σ > c/2π , where σ is in CGS units. This problem was studied some time ago [47] and has recently been revisited in Ref. [48].

l = 1, transverse channel
For vanishingly weak damping (γ → 0), the pole moves along the imaginary axis (s = −is i ) for −1 < F c(s) For finite γ , the evolution remains essentially the same. There are still no solutions for F c(s) < 0 the pole is on the imaginary axis, at s = −is i,1 , where At F c(s) Note that there is no diffusive behavior for large γ . In this limit, For finite γ , the poles are determined from the equation As for the l = 1 case, the behavior of the poles is quite involved, particularly for γ > 1, and we refrain from presenting all the details. We note only that at F c(s) 2 ≈ −1 the purely imaginary pole is located at s ≈ −i(1 − |F c(s) This pole is not a diffusive one, which to be is expected because the l = 2 order parameter is not a conserved quantity [49].

l = 2, transverse channel
For vanishingly weak damping (γ → 0), the poles s = ±a 2 − ib 2 are in the complex plane of s for negative F c(s) 2 . As F c(s) 2 increases from −1 toward 0, the poles move from the vicinity of the real axis at F c(s)

For positive F c(s)
2 , there is only a single pole on the imaginary axis.
For finite γ , the equation for the pole is It has two solutions. At small 1 + F c(s) 2 both are on the imaginary axis: one is a 2 = 0, b 2 ≈ (1 − |F c(s) 2 |)( 1 + γ 2 + γ ) 2 /4γ and another one is a 2 = 0, b 2 ≈ −γ . Note that neither mode is diffusive for large γ . As 1 + F c(s) 2 increases, the two solutions move toward each other and merge at some critical value F c(s) 2 = F c(s) 2,cr . For small γ , F c(s) 2,cr ≈ −1 + γ 2 and the solutions merge at b 2 ≈ γ /2. At F c(s) 2 = F c(s) 2,cr + 0, the poles split and move away from the imaginary axis, i.e., a 2 becomes finite. The subsequent evolution is essentially the same as for vanishingly small γ . For large positive F c(s) 2 , the pole is located on the imaginary axis at b 2 = 1/2/ √ 2 for small γ and at b 2 = γ + 1/(4γ ) for large γ . 1 vanishes at a Pomeranchuk transition. To simplify the expressions, below we write χ c(s) qp,l (t * ) simply as χ c(s) l (t * ).

B. l = 0
We recall that near the Pomeranchuk transition the only pole of χ c(s) 0 (s) in the lower half-plane is located at s = s i ≈ −i(1 − |F c(s) 0 |) [see Eq. (19)]. Near this pole, Evaluating the residue, we obtain To obtain the branch-cut contribution, we recall that Hence, For large x, the right-hand side of Eq. (107) approaches a constant value (= −2), and the integral over x in (103) formally diverges. This divergence is artificial and can be eliminated by introducing a factor of exp(−αx) with α > 0 and taking the limit of α → 0 at the end of the calculation. For F c(s) 0 ≈ −1, the leading contribution to the integral in Eq. (103) comes from nonanalyticity of the integrand at x = 1. For t * 1, we use dy √ y cos yt * = − √ π/(2t * ) 3/2 , dy √ y sin yt * = √ π/(2t * ) 3/2 , and obtain Comparing Eqs. (105) and (108), we see that the pole contribution is the dominant one for 1 t * (3/2)| ln(1 − |F c(s) l |)|/(1 − |F c(s) l |), while at longer times the time dependence of the response function comes from the end point of the branch cut.
In Fig. 12 we show χ c(s) 0 (t * ) computed numerically using Eq. (101). As is obvious from this equation, χ c(s) 0 (t * ) increases linearly with t * at short times t * 1 (the pole and branch contributions cancel each other at t * = 0). At intermediate , χ c(s) 0 (t * ) exhibits an exponentially decay augmented by weak oscillations, in agreement with Eqs. (105) and (108). This behavior is shown in the left panel of Fig. 12. At long times, χ c(s) 0 (t * ) oscillates and decreases algebraically with time, in agreement with Eq. (108). This behavior is shown in the right panel of Fig. 12.
As F c(s) 0 becomes closer to −1, the exponential decay of χ c(s) 0 (t * ) with t * becomes slower and the crossover to a powerlaw behavior shifts to larger t * . Right at the Pomeranchuk instability, when F c(s) 0 = −1, the form χ c(s) 0 (t * ) can be found directly from Eq. (101). In this case, Imχ c(s) 0 (s) = ν F θ (1 − |s|) √ 1 − s 2 /s. Substituting into (101), we find that χ c(s) 0 (t * ) starts off linearly for t * 1, exhibits an oscillatory behavior for t * ∼ 1, and approaches the limiting value of χ c(s) For F c(s) 0 < −1, a long-range order develops. Within our approach, we can analyze the initial growth rate of the order parameter c(s) < −1 requires some care because integrating Eq. (100) over the same contour as in Fig. 11 we would find that χ c(s) 0 (t * < 0) becomes finite, i.e., that causality is lost. This issue was analyzed in Ref. [30] (see also, e.g., Ref. [51]), where it was shown that, to preserve causality, one has to modify the integration contour such that it goes above all poles, as shown in Fig. 13(c). Integrating along the modified contour, we find that χ c(s) 0 (t * < 0) = 0, as required by causality. For t * > 0 we now have c(s) i.e., a perturbation grows exponentially with time. This obviously indicates that the FL state without Pomeranchuk order becomes unstable. To see how the system eventually relaxes to the final equilibrium state with c(s) 0 (t * ) = 0 , we would need to recalculate χ c(s) 0 (t ) in the broken-symmetry state.

l = 1, longitudinal channel
For small positive 1 + F c(s) 1 , the poles of χ long 1 (s) are given by Eq. (33). Near the poles, where · · · stands for nonsingular terms. Evaluating the residues, we obtain the pole contribution to χ long 1 (t * ) as (111) The branch-cut contribution has the same structure as for l = 0, i.e., χ long bcut,1 (t * ) ∝ cos(t * − π/4)/(t * ) 3/2 for t * 1. We see that the pole contribution remains dominant up to t * ∼ (t * ), we find that it grows linearly with t * for t * t * a , i.e., the system initially tends to move further away from equilibrium. For t * a t * t * b , the order parameter oscillates between the quasiequilibrium states with long . Finally, for t * t * b , long 1 (t * ) decays exponentially toward zero. At F c(s) 1 = −1, both t a and t b diverge and, following an instant perturbation at t = 0, the order parameter long 1 (t * ) ∝ h increases linearly with t * until the perturbation theory in h breaks down.
The difference with the l = 0 case, when c(s) 0 (t * → ∞) at F c(s) 0 = −1 is finite, can be understood by noticing that the behavior of χ long free,1 (t * ) for large t * is determined by that of Imχ  ), and, at vanishingly small δ, Imχ long (s) goes over to (π/2)δ(s)/s. Substituting this into Eq. (101), we find that χ long For F c(s) 1 < −1, both poles of χ long, 1 (s) are located on the imaginary axis, at s = ±(|1 + F c(s) 1 |/2) 1/2 . One of the poles is now in the upper half-plane of complex s. Modifying the integration contour the same way as for l = 0 to preserve causality, we obtain for t * > 0 χ long For t * t * a = (2/|1 + F c(s) 1 |) 1/2 , both χ long 1 (t * ) and long 1 (t * ) ∝ hχ long 1 (t * ) increase linearly with t * . For t * t * a , the perturbation grows exponentially, indicating that the FL state becomes unstable.
We note in passing that the need to bend the integration contour around the pole for F c(s) 1 < −1 can be also understood by considering the behavior of χ long 1 (t * ) at F c(s) 1 approaching −1 from above. In the limit F c(s) 1 → −1 + 0 + , the two poles of χ long 1 (s) coalesce into a single double pole at the origin, as shown in Fig. 13(b). Had we tried to compute χ long 1 (t * ) by integrating along the real axis of s, we would have intersected a divergence. To eliminate the divergence, one needs to bend the integration contour and bypass the double pole along a semicircle above it. The extension of this procedure for F c(s) 1 < −1 yields the contour shown in Fig. 13(c).

l = 1, transverse channel
For small 1 + F c(s) 1 , the pole in the transverse susceptibility for l = 1 is on the imaginary axis. The behavior of χ tr 1 (t * ) is then the same as for the l = 0 case.
After this change, the results for χ long 1 (t * ) remain the same as in the absence of disorder. The time dependence of χ long 1 in the presence of impurity scattering is shown in Fig. 14.
In all cases, finite γ modifies the branch-cut contribution, so that in addition to the algebraic decay, there is also an exponential decay. For l = 0 we find Similar expressions hold for l > 0. The presence of the exponentially decaying terms due to damping is particularly relevant for l > 0 and F c(s) l > 0, as it allows one to distinguish between the cases of smaller F c(s) l .
For the former, the zero-sound pole is present and located above the branch cut, at s = ±a − ibγ , where a > 1 and b < 1. For the latter, the zero-sound pole is located on the unphysical Riemann sheet. In both cases, Imχ c(s) l (s) at real s has a peak at s = ±a, but for larger F c(s) l its width isbγ withb > 1, i.e., it is larger than γ . Accordingly, for smaller F c(s) l , the dominant contribution to χ l (t * ) at large t * comes from the pole, and χ l (t * ) ∝ e −bγ t * cos(at * ). For larger F c(s) l , χ l (t * ) at large t * comes from the branch cut, and χ l (t * ) ∝ e −γ t * cos(t * − π/4)/(t * ) 3/2 . Because this property holds only for l > 0 and in the presence of disorder, it was not discussed in previous works [9,[31][32][33][34][35][36][37], which studied collective modes 033134-21 of a 2D Fermi liquid either in the l = 0 channel or in the absence of disorder.

V. SPECIAL CASES OF CHARGE-AND SPIN-CURRENT ORDER PARAMETERS
A. Ward identities and static susceptibility in the l = 1 channel In previous sections, we assumed that the behavior of the full susceptibility is at least qualitatively the same as that of the quasiparticle susceptibility, i.e., the collective modes, qp,l (q, ω). We now consider the special case of order parameters with l = 1, for which f c(s) 1 (k) = ( cos θ sin θ )∂ k /∂k, up to an overall factor. These order parameters correspond to charge or spin currents. The special behavior of a FL under perturbations of this form has been discussed in recent studies of the static susceptibility in the l = 1 channel [26][27][28]. Namely, for the spin-or charge-current order parameter, the vertices c(s) 1 satisfy the Ward identities which follow from conservation of the total number of fermions (the total "charge") and total spin. In the static limit, the Ward identities read as [28,52] m * m Z c(s) Under certain assumptions, these identities allow one to decide which of three factors on the left vanishes at the instability. First, we assume that the Z factor, being a highenergy property of the system, remains finite at the instability. . What was said above does not apply to the special case of a Galilean-invariant system. In this case, the charge current is equivalent to the momentum and thus is conserved. The Ward identity for the momentum implies that Z c 1 = 1, i.e., c 1 remains finite at 1 + F c 1 = 0. Equation (116) then implies that m * /m = 1 + F c 1 , which is the standard result for a Galileaninvariant FL. The static susceptibility still remains finite at 1 + F c 1 → 0, this time because the factor of m * /m in the numerator of χ c qp,1 cancels out with 1 + F c 1 in its denominator.
Furthermore, in the Galilean-invariant case the static l = 1 charge and spin susceptibilities are not renormalized at all by the electron-electron interaction [29]. On the other hand, m * /m = 1 + F c(s) 1 does vanish at the transition in the l = 1 charge channel. We believe that the vanishing mass indicates a global instability of a non-Pomeranchuk type, which is not associated with the l = 1 deformation of the FS.
where we recall that the last term represents the contribution from high-energy fermions. The static limit of the first term in the equation above, i.e., N F (1 + F c(s) 1 )(m/m * ), is indeed nonsingular at the transition, in agreement with the conclusions of the previous section. Nevertheless, one of the poles of χ long 1 (q, ω) moves into the upper frequency half-plane when 1 + F c(s) 1 becomes negative, i.e., a dynamical perturbation with the structure of spin or charge current grows exponentially with time, which is an indication of a Pomeranchuk instability. The peculiarity of the l = 1 case in that the residue of the pole vanishes right at the transition, but it is finite both above and below the transition.

C. Case of more than one nonzero Landau parameter
It is instructive to derive an analog of Eq. (118) for a more general case of several nonzero Landau parameters. We remind the reader that in this situation the pole structure of χ long 1 (q, ω) is more complex than when only F c(s) ) 2 as the overall factor. We argue that it does.
To demonstrate this, we need to express the full susceptibility via vertices¯ c(s) (q, ω), which include both highand low-energy renormalizations. Vertices¯ c(s) (q, ω) can be expanded into a series of partial harmonics:¯ c(s) (q, ω) = The vertex¯ c(s) l (q, ω) is given by a series of diagrams which contain momentum and frequency integrals of the product G p+ q where c(s) (θ, θ ) is the four-fermion (four-leg) vertex with external fermions right on the FS. By construction, c(s) (θ, θ ) contains only renormalizations from high-energy fermions (in the FL theory, such a vertex is called ω , see Ref. [7]).
where K 0,1,2 are defined by Eq. (66). Substituting the last two equations into Eq. (119), we obtain Comparing the last result with χ long qp,1 in Eq. (65), we see that χ long 1 (s) = ( c(s) 1 ) 2 χ long qp,1 (s), exactly as in the static case. This result implies that the residue of the pole is proportional to ( c(s) 1 ) 2 ∝ (1 + F c(s) 1 ) 2 and thus vanishes at the Pomeranchuk instability also for the case of two nonzero Landau parameters, when the pole structure of the susceptibility becomes more involved. Still, like in the case when only F c(s) , and it does not cancel the poles in χ long qp,1 . At 1 + F c(s) 1 < 0, one pole moves into the upper frequency half-plane, signaling a Pomeranchuk instability.

D. Perturbation theory for the vertex in the dynamical case
We now return to the case of a single Landau parameter F c(s) 1 and verify by a perturbative calculation that ( c(s) 1 ) 2 does not cancel the dynamical poles in the full l = 1 susceptibility. We perform the calculation to second order in the Hubbard (pointlike) interaction U . For simplicity we limit our attention to the spin channel and also consider a Galilean-invariant system. We will show that the dynamical vertex¯ s 1 (s) has the pole structure of Eq. (121) with s independent c(s) 1 . To demonstrate this, it suffices to show that the vertex s 1 , which acts as a source for the dynamical vertex¯ s 1 (s) in Eq. (120), does not vanish at s which corresponds to the pole of the dynamical vertex. Instead of calculating s 1 directly, we compute the product s 1 Z for reasons that will become clear later in the section. Using the Ward identity associated with the Galilean invariance, we express quasiparticle Z as [7] where the (2 + 1)-momentum p = (p, ω p ) is not necessarily close to the FS, and k = (k Fk , 0) + is infinitesimally close to the FS, i.e., = (q, ω) with both |q| and ω being infinitesimally small. The direction ofk in Eq. (124) is arbitrary. The ω αβ,αβ is the dressed four-fermion vertex and (G 2 p ) ω (k · p) is the regular part of the product G p+ /2 G p− /2 of two exact Green's functions, whose arguments differ by . (Note that the q and ω in the definition of the Z factor do not need to coincide with the corresponding variables describing the collective mode, but we choose them to be the same for simplicity.) The product G p+ /2 G p− /2 can be written as the sum of a regular part and a singular contribution from the FS [7]: wherep = p/|p| and, as before, s = ω/v * F |q|.

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Substituting Eq. (125) into (133) for Q 2 and choosing the direction of k to be along q, we obtain One can now verify that the first term in the right-hand side of Eq. (136) is zero, i.e., This can be done by substituting the explicit expressions for c(s) from Eq. (126) and changing integration variables [53]. We are then left with a contribution coming solely from the FS, where θ and θ are the azimuthal angles ofk andp, respectively. This term is independent of s and only contributes to the static vertex [26]. Going back to Eq. (131), we obtain where we used the relation m * /m = 1 + F c 1 valid for a Galilean-invariant system. This result agrees with the analysis in the previous section.

E. Charge-and spin-current order parameter: Ginzburg-Landau functional and time evolution
We now analyze the structure of the Landau functional that describes the l = 1 Pomeranchuk transition. Our purpose is to reconcile the apparent contradiction that on one hand the FL ground state becomes unstable for F c(s) 1 < −1, while on the other hand the static l = 1 susceptibility remains finite at F c(s) 1 = −1. The Ginzburg-Landau functional can be derived from the Hamiltonian of interacting fermions, coupled to an infinitesimal external perturbation h c(s) 1 , via a Hubbard-Stratonovich (HS) transformation with an auxiliary field c(s) 1 . For a generic l = 1 order parameter, the vertex remains finite at the Pomeranchuk transition. In this case, it is sufficient to consider only the quasiparticle part of the Hamiltonian and neglect the contributions from high-energy fermions. Then, the coupling to the external field is given by a bilinear term h c(s) 1 c(s) 1 , and the total susceptibility is identical to the quasiparticle susceptibility. For the charge-and spin-current order, the coupling is still proportional to h c(s) 1 c(s) 1 term, but the contributions from high-energy fermions to the proportionality coefficient cannot be neglected, as with these contributions the fully dressed coupling vanishes at the transition. To see this, we explicitly separate the four-fermion interaction into the components coming from the states near and away from the FS. Such an approach has been used in statistical FL theory and in renormalization group studies of FLs [54][55][56][57][58][59][60].
The generic form of the l = 1 component of αβ;γ δ (k, k ; q) is l=1 αβ;γ δ (k, k ; q) = −k ·k U c 1 f c (|k|, |k |)δ αγ δ βδ + U s 1 f s (|k|, |k |)σ αγ · σ βδ , (141) wherek = k/k F andk = k /k F , and k F should be treated here as just a normalization constant, which we choose for convenience to match the Fermi wave number of the quasiparticles. For charge-and spin-current orders, we replace the form factors f c,s (|k|, |k |) by constants and incorporate them into U c(s) 1 . An instability in the l = 1 channel can occur if U c(s) 1 > 0. Below, we approximate the full vertex function αβ;γ δ by its l = 1 component. Other components are not necessarily small, but we assume they are irrelevant for the lowenergy theory near the l = 1 Pomeranchuk instability. In this approximation, the effective interaction is separable into two parts that depend onk andk , and can be written as the sum of the charge and spin and, as before, t c μν = δ μν and t s μν = σ z μν . We next rewrite the sums over the fermionic momenta as 2 ,α t c(s) αγ a k− q 2 ,γ (1 − δ |k+ q 2 |,k F δ |k− q 2 |,k F ) (143) (and the same for the sum over k ). Here, δ a,b is nonzero only for |a − b| < , and is small compared to k F and will be taken to zero at the end of the calculation. The purpose of the projectors δ a,b is to split the fermions into those near 033134-25 the FS, which form the FL of quasiparticles, and those away from the FS, whose role is to renormalize the interaction between quasiparticles and their coupling to an external perturbation. Below, we denote fermions near the FS as ψ † (ψ ), and fermions away from the FS asψ † (ψ ). Using (143), we rewrite (142) as where the summation over k is confined to the vicinity of the FS. We see that the factor c(s) 1 only changes the response function to an external perturbation, but does not affect the thermodynamic stability of the FL state. If we compute the response function by differentiating the partition function twice with respect to h c(s) 1 (q), we find the same expression as in Eq.
The functional form of c(s) 1 (t * ) is the same as for a generic l = 1 order parameter, when high-energy renormalizations can be neglected, just the amplitude is smaller. For 1 + F c(s) 1 < 0, a deviation from the normal state grows as c(s) The functional form is again the same as for a generic l = 1 order parameter. The presence of the overall small factor (1 + F c(s) 1 ) 2 just implies that it takes a longer time for a deviation to develop. In particular, the ratio of c(s) 1 (t * ) in (151) and the initial perturbation h c(s) 1 becomes O(1) only after c(s) 1 (t * ) begins to grow exponentially.

VI. CONCLUSIONS
In this paper, we analyzed zero-sound collective bosonic excitations in different angular momentum channels in a metal with an isotropic, but not necessarily parabolic, dispersion k . We explicitly computed the longitudinal and transverse dynamical susceptibility χ c(s)