Collective Modes Near a Pomeranchuk Instability in Two Dimensions

PHYSICAL REVIEW RESEARCH 1, 033134 (2019)

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

(Received 15 August 2019; revised manuscript received 3 October 2019; published 27 November 2019)

We consider zero-sound collective excitations of a two-dimensional Fermi liquid. For each value of the angular momentum l, we study the evolution of longitudinal and transverse collective modes in the charge (c)andspin c(s) c(s) (s) channels with the Landau parameter Fl , starting from positive Fl and all the way to the Pomeranchuk c(s) =− transition at Fl 1. In each case, we identify a critical zero-sound mode, whose velocity vanishes at the c(s) < − Pomeranchuk instability. For Fl 1, this mode is located in the upper frequency half-plane, which signals an instability of the ground state. In a clean Fermi liquid, the critical mode may be either purely relaxational or almost propagating, depending on the parity of l and on whether the response function is longitudinal or transverse. These differences lead to qualitatively different types of time evolution of the order parameter following an initial perturbation. A special situation occurs for the l = 1 order parameter that coincides with the spin or charge current. In this case, the residue of the critical mode vanishes at the Pomeranchuk transition. However, the critical mode can be identiﬁed at any distance from the transition, and is still located in the upper c(s) < − frequency half-plane for F1 1. 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 c(s) away from the critical point, and ﬁnd that the modes evolve with Fl on a multisheet Riemann surface. For c(s) certain intervals of Fl , 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.

DOI: 10.1103/PhysRevResearch.1.033134 I. INTRODUCTION c(s) = θ c(s) θ or f1 (k) sin f (k), where is the azimuthal angle of A Pomeranchuk transition is an instability of a Fermi liquid k, in a coordinate system chosen such that the x axis points (FL) toward a spontaneous order which breaks rotational sym- in the direction of q. According to the FL theory [7,8], a Pomeranchuk order with angular momentum l emerges when metry but leaves translational symmetry intact [1]. Examples c(s) include ferromagnetism [2–4] and various forms of nematic the corresponding Landau parameter Fl approaches the critical value of −1 from above. order in quantum Hall systems, Sr3Ru2O7, and cuprate and Fe-based superconductors [5,6]. For a rotationally invariant In this paper we focus on dynamical aspects of a Pomer- system in two dimensions (2D), deformations of the Fermi anchuk instability. We consider primarily the 2D case because surface (FS) can be classified by the value of the angular mo- examples of Pomeranchuk transitions have been discussed mentum l. In general, a deformation with only one particular mostly for 2D systems [9–26]. We consider an isotropic FL but do not specifically assume Galilean invariance, i.e., l develops at a Pomeranchuk transition. A Pomeranchuk or- c(s) = c(s) † c(s) the single-particle dispersion in our model is not necessar- der parameter l (q) k fl (k) ak+q/2,αtα,α ak−q/2,α c ily quadratic in |k|. The main object of our study is the is bilinear in fermions and has the spin structure tα,α = δα,α s z dynamical susceptibility χ (q,ω), which corresponds to a or t = σ in the charge (c) and spin (s) channels, cor- l α,α α,α c(s) respondingly (σ z is the Pauli matrix). The order parameter is particular order parameter l . In 2D, there are two types −1 of susceptibilities for any given l: a longitudinal one, with assumed to vary slowly, i.e., q min{a , kF }, where a0 is 0 the form factor proportional to cos lθ, and a transverse one, the lattice constant and kF is the Fermi momentum. Under ro- c(s) with the form factor proportional to sin lθ (for l = 0, there is tations, the form factors fl (k) transform as basis functions of the angular momentum and, in general, also depend on the only one type, with an isotropic form factor). The longitudi- magnitude of |k|≡k. For example, f c(s)(k) = cos θ f c(s)(k) nal susceptibility corresponds to FS fluctuations that are not 1 volume conserving, analogous to axisymmetric fluctuations in three dimensions (3D), while the transverse susceptibility describes volume-preserving fluctuations, analogous to non- axisymmetric fluctuations in 3D [7]. In the low-energy limit Published by the American Physical Society under the terms of the (q → 0 and ω → 0) the dynamical susceptibility is a function = ω/ ∗ ∗ Creative Commons Attribution 4.0 International license. Further of the ratio s vF q, where vF is the renormalized Fermi distribution of this work must maintain attribution to the author(s) velocity. As a function of complex variable, χ(s) has both and the published article’s title, journal citation, and DOI. poles and branch cuts in the complex plane. We focus on

2643-1564/2019/1(3)/033134(28) 033134-1 Published by the American Physical Society KLEIN, MASLOV, PITAEVSKII, AND CHUBUKOV PHYSICAL REVIEW RESEARCH 1, 033134 (2019) the retarded susceptibility, which is analytic in the upper half- χ c(s) , = l (q t )forl 0 and 1. Above the Pomeranchuk transition, plane Ims > 0, except for the case when the system is below χ c(s) , the time dependence of l (q t ) in the clean limit is a the Pomeranchuk instability, and, in general, has poles and combination of an exponentially decaying part, which comes branch cuts in the lower half-plane Ims < 0. The poles of χ(s) χ c(s) ,ω from the poles of l (q ), and of an oscillatory (and alge- correspond to zero-sound collective modes whose frequency braically decaying) part, which comes from its branch cuts. At ω = ∗ and momentum are related by svF q. If the Landau pa- the transition, χ c(s)(q, t ) reaches a time-independent limit at rameter is positive and nonzero for only one value of l, there is 0 t →∞, while χ c(s)(q, t ) grows linearly with time in the clean one longitudinal and at most one transverse zero-sound mode 1 case and saturates at a finite value in the presence of disorder. for any l = 0. These are conventional propagating modes with Below the transition, the poles of χ c(s)(q,ω) are located in Res > 1 and infinitesimally small Ims in the clean limit, when 0,1 ω χ c(s) , the fermionic lifetime is infinite. The branch cuts are a conse- the upper half-plane of . Consequently, both 0 (q t ) and χ c(s) , quence of the nonanalyticity of the free-fermion bubble (the 1 (q t ) increase exponentially with time. This means that Lindhard function). Their significance is that the zero-sound any small fluctuation of the corresponding order parameter poles are defined on a multisheet Riemann surface. In 2D, this is amplified, and thus the ground state with no Pomeranchuk nonanalyticity is particularly simple, being just a square root order is unstable. In the case of finite disorder, the branch-cut (see Sec. II A). As a result, the Riemann surface is a genus contribution also begins to decay exponentially, on top of its 0, two-sheet surface. We shall refer to the first sheet, which algebraic and oscillatory behavior. includes the analytic half-plane, as the “physical sheet.” In our In Sec. V, we consider the special case of an order pa- work, we mostly discuss the properties of the physical sheet. rameter that coincides with either the charge or spin current. We obtain explicit results for the frequencies of collective Previous studies [26–28] found that the corresponding static − < c(s) < ∞ χ c(s) , modes in the whole range of 1 Fl .First,we susceptibility 1 (q 0) does not diverge at the tentative consider a clean FL (Sec. II). We present explicit results c(s) =− Pomeranchuk instability at F1 1 because of the Ward for the zero-sound modes with l = 0, 1, 2 (Secs. II B, II C, identities that follow from conservation of total charge and and II D, correspondingly) and analyze the structure of the spin. We analyze the dynamical susceptibility for such an zero-sound modes for arbitrary l (Sec. II E). We show that for order parameter. We show, using both general reasoning and l = 0 and in the transverse channel with l = 0, all zero-sound direct perturbation theory for the Hubbard model, that while modes acquire a finite decay rate already for an arbitrary small the static susceptibility indeed remains finite at F c(s) =−1, c(s) =± − > 1 negative Fl , i.e., s a ib, where a, b are real and b 0. the dynamical one still has a pole, which moves to the upper A positive b implies that a perturbation of the order parameter frequency half-plane below the transition. The residue of this decays exponentially with time. The frequency of one of the + c(s) 2 c(s) =− pole vanishes as (1 F1 ) at F1 1, but is finite both modes vanishes at the Pomeranchuk transition. We call this c(s) > − c(s) < − for F1 1 and F1 1. We argue that the presence mode the critical one. In the longitudinal channel, the decay c(s) < − of the pole in the upper frequency half-plane for F1 1 rate of one of the modes remains infinitesimally small until the indicates that the state with no Pomeranchuk order becomes corresponding |F c(s)| exceeds a threshold value. Immediately l unstable, like for any other type of the order parameter. We s =±a − ib below the threshold, this mode is located at with derive a Landau functional for the charge- and spin-current a > b 1 and 1. Even though the mode frequency is almost order parameter and show that it has a conventional form, real, the corresponding pole is located below the branch cut except that the coupling between the order parameter and an and thus cannot be reached from the real axis of the physical + c(s) external perturbation has an additional factor of 1 F1 .We sheet. Accordingly, the imaginary part of the susceptibility argue that the charge- and spin-current order does develop does not have a peak above the particle-hole continuum for at 1 + F c(s) < 0, just as for a generic l = 1 order parameter, s 1 real , i.e., the mode is “hidden.” Below the transition, i.e., but it takes longer to reach equilibrium after an instantaneous for F c(s) < −1, the pole is located in the upper frequency l perturbation. This result differs from earlier claims that there half-plane, and a perturbation of the order parameter grows is no Pomeranchuk transition to a state with the charge- and exponentially with time, which indicates that a FL becomes spin-current order parameter [26–28]. unstable with respect to a Pomeranchuk order. We obtain Before we move on, a comment is in order. It is well explicit results for the frequencies of collective modes in the known that the range of FL behavior shrinks as the system whole range of −1 < F c(s) < ∞. l approaches a Pomeranchuk instability and disappears at the In Sec. III, we analyze how the dispersion of collective transition point, where the system displays non-Fermi-liquid modes is modified in the presence of impurity scattering. In behavior down to the lowest energies. In our analysis, we will the dirty limit, the critical modes in both longitudinal and be studying the collective modes at finite s = ω/(v∗ q) and l F transverse channels become overdamped for all . Impurity assume that ω and q are both small enough so that at any given scattering also smears the threshold, described in the previ- distance to the critical point the system remains a Fermi liquid. ous paragraph, i.e., the longitudinal collective modes have c(s) < nonzero damping rates for any Fl 0. II. DYNAMICAL QUASIPARTICLE SUSCEPTIBILITY In Sec. IV, we analyze the susceptibility in the time domain NEAR A POMERANCHUK TRANSITION IN A CLEAN dω FERMI LIQUID χ c(s)(q, t ) = χ c(s)(q,ω)eiωt , (1) l 2π l A. Quasiparticle susceptibility which determines the time evolution of the order parameter According to the Kubo formula, the correlation function c(s) following an initial perturbation. We obtain explicit forms of of an order parameter l is related to the susceptibility

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χ c(s) ,ω αβ,γδ(k − p) can be expanded over harmonics characterized l (q ) with respect to the conjugated “field.” In its turn, χ c(s)(q,ω) is given by a fully renormalized particle-hole bub- by orbital momenta l, and the properly normalized coefficients l F c(s) ble with external momentum q and external frequency ω.For of this expansions are known as Landau parameters l : free fermions, the particle-hole bubble is just a convolution of = c − δ δ + s − σ · σ , two fermionic Green’s functions, whose momenta (frequen- αβ,γδ (k p) αγ βδ (k p) αγ βδ cies) differ by q (ω). In the time-ordered representation, we ∞ c(s) − = c(s) + c(s) θ − θ , define the normalized susceptibility as (k p) 0 2 l cos l( k p) l=1 2i dDk dε χ c(s) ,ω =− c(s) 2 c(s) c(s) , (q ) f (k) F = ν , (4) free l N (2π )D 2π l l F l F 1 1 × G k + q,ε+ ω where free 2 2 ∗ 2 m 1 1 νF = 2NF Z (5) × G k − q,ε− ω , (2) m free 2 2 and θk,θp are the azimuthal angles of k, p. The static quasi- where NF is the density of states at the Fermi energy EF , c(s) particle susceptibility χ , (q, 0) is expressed in terms of just Gfree(k,ε) = 1/[ε − k + EF + iδ sgnε] is the (time-ordered) qp l c(s) Green’s function, k is the single-particle dispersion, D is the a single Fl : spatial dimensionality, and a factor of 2 comes from summing ν over spins. We will be interested only in the case of small q χ c(s)(q, 0) = F . (6) and ω, i.e., q k and ω E . In this case, integration over qp,l + c(s) F F 1 Fl the internal fermionic momentum and frequency is confined to the regions of small ε and k − EF , i.e., the susceptibility The dynamical quasiparticle susceptibility cannot, in general, comes from the states near the FS or, for brevity, from “low- be expressed in terms of a single Landau parameter, unless all c(s) energy fermions.” Landau parameters except for a single Fl are small. In this For interacting fermions, the particle-hole bubble is modi- special case, fied in several ways [7,8,29]. First, the self-energy corrections χ ∗,ω transform a free-fermion Green’s function near the FS into χ c(s) ,ω = ν free,l (q ) , qp,l (q ) F c(s) ∗ (7) a quasiparticle Green’s function, in which the bare velocity 1 + F χ , (q ,ω) ∗ = / ∗ l free l vF is replaced by the renormalized velocity vF vF (m m ), where m∗ is the renormalized mass, and the Green’s function ∗ ∗ ∗ where q = (m/m )q and χfree,l (q ,ω) is normalized to is multiplied by the quasiparticle residue Z < 1. Second, ∗ ∗ χfree,l (q , 0) = 1 (we recall that we consider small q kF ). interactions between low-energy fermions generate multibub- For all order parameters, except for the charge or spin current, ble contributions to the susceptibility. These renormalizations the vertex c(s) in Eq. (3) is expected to remain finite at the χ c(s) ,ω l transform the free-fermion susceptibility free,l (q ) into the Pomeranchuk transition. The behavior of the full susceptibil- χ c(s) ,ω χ c(s) ,ω quasiparticle susceptibility qp,l (q ). (The effect of damping ity is then determined entirely by the quasiparticle qp,l (q ). due to the residual interaction between quasiparticles is a Although the calculations are straightforward and some of subleading effect in the range of q and ω of interest to us, the results have appeared before [9,26,28,30–39], we include χ c(s) ,ω and will not be considered here.) Third, fermions far away below the details of the derivation of qp,l (q ) in 2D, as we from the FS (“high-energy fermions”) also contribute to the will be interested in the pole structure of the susceptibility not χ c(s) ,ω full susceptibility l (q ). only near a Pomeranchuk transition, but also away from it. The general expression for the dynamic susceptibility is In what follows we first consider separately the cases of l = , , χ c(s) ,ω [29] 0 1 2 and trace out the evolution of the poles of qp,l (q ) in the complex plane of frequency. For a reader wishing to χ c(s) ,ω = c(s) 2χ c(s) ,ω + χ c(s) . l (q ) l qp,l (q ) inc,l (3) avoid going through the computational details, we depict the evolution graphically in Figs. 2 and 4, and summarize our c(s) Here, l is the side vertex, renormalized by high-energy main results for l = 0, 1, 2 in Table I. We then analyze the χ c(s) fermions, and the standalone term inc,l represents the contri- case of arbitrary l. In these calculations, up to Sec. II F, bution solely from high-energy fermions. This last term does c(s) we assume that a single Landau parameter Fl is much not have a singular dependence on q and ω and will not play χ c(s) ,ω larger than the rest and compute qp,l (q )usingEq.(7). any crucial role in our analysis. We emphasize that Eq. (3) c(s) c(s) is valid for any isotropic system, even if it is not Galilean In Sec. II F, we consider the case when F0 and F1 are c(s) invariant. comparable, while all Fl>1 can be neglected. The quasiparticle contribution to the susceptibility depends For definiteness, in this and the next two sections we on the fully renormalized (and antisymmetrized) interaction approximate the√ form factors by their values on the FS, c(s) = θ between low-energy fermions, usually denoted by αβ,γδ(k − as fl (kF )√ 2 cos(l ) in the longitudinal channel and c(s) = θ p). This interaction includes renormalizations by high-energy fl (kF ) 2sin(l ) in the transverse channel, where fermions but not by low-energy fermions. For a rotation- kF = kF k/k and θ is the angle between the direction of kF ally and SU(2)-invariant system, which we consider here, and the x axis.

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B. l = 0 Substituting this form into Eq. (7), we obtain c(s) |s| In this case, the form factor f0 (kF ) is just a constant. The 1 − √ χ c(s) = ν s2−1 . form of the retarded free-fermion susceptibility along the real , (s) F | | (11) qp 0 1 + F c(s) 1 − √ s frequency axis is well known: 0 s2−1 ω ∗ i The locations of the poles are determined from the equation χfree,0(q ,ω) = 1 + ∗ 2 − ω + δ 2 (vF q ) ( i ) |s| 1 + F c(s) − F c(s) √ = 0. (12) is 0 0 2 − = 1 + , (8) s 1 − s + iδ 2 (1 ( ) One can check that the solution s1,2 =±sp with sp > 1 indeed ∗ = ∗ = ω/ ∗ c(s) > where we used that vF q vF q and defined s vF q. exists only for F0 0: χ Viewed as a function of complex s, free,0(s) has branch cuts, c(s) =−δ 1 + F which start at s i below the real axis and run along the s = 0 . (13) −∞, − , ∞ p segments ( 1) and (1 ) along the real axis. 1 + 2F c(s) Traditionally, δ in Eq. (8) is interpreted as an infinitesi- 0 mally small damping rate whose physical origin does need We now widen the scope of our analysis and search for not to be specified and whose sole purpose is to shift the solutions with complex s. To this end, we need to keep δ terms branch cut into the lower half-plane of complex s. We will see, in χfree,0(s). The quasiparticle susceptibility for s in the lower however, that such approach is not sufficient for our purposes half-plane is obtained by substituting (9)into(7). This yields because it would not allow us to resolve the relative positions 1 + i √ s of the zero-sound poles and branch cuts of the susceptibility c(s) 1−(s+iδ)2−δ χ , (s) = νF . (14) in the complex plane of s. For this reason, we will consider qp 0 1 + F c(s) 1 + i √ s a specific damping mechanism, namely, scattering by short- 0 1−(s+iδ)2−δ range impurities, and treat δ as a finite albeit small number. Using Eq. (14), we can study the poles of χ c(s) (s) everywhere The order parameter in the l = 0 channel (charge or spin) qp,0 c(s) in the lower half-plane of complex s and in the whole range of is conserved, i.e., the susceptibility must satisfy χ (q = 0 F c(s). 0,ω) = 0 (see, e.g., Refs. [40,41]). Once δ is finite, Eq. (8) 0 The positions of the poles in the lower half-plane of s are does not satisfy this condition because it was obtained either determined by by adding iδ self-energy corrections to the Green’s functions or, which is equivalent, by solving the kinetic equation in the + c(s) 1 F0 is relaxation time approximation. To ensure that charge and spin =− . (15) F c(s) − s + iδ 2 − δ are conserved, one also has to include vertex corrections to 0 1 ( ) the particle-hole bubble or go beyond the relaxation-time ap- For F c(s) > 0, the solutions of (15)are proximation.1 The corresponding free-fermion susceptibility 0 is given by [42] s1,2 =±sp,0 − iδ˜0, (16) is where χfree,0(s) = 1 + , (9) (1 − (s + iδ)2 − δ + c(s) + c(s) 1 F0 1 F0 , = δ˜ = δ . sp 0 and 0 c(s) (17) where δ now stands for the dimensionless impurity scattering c(s) + 1 + 2F 1 2F0 rate. The −δ term next to 1 − (s + iδ)2 in (9) comes from 0 vertex corrections. Until Sec. III, we will be assuming that Up to the −iδ˜0 term, this result coincides with Eq. (13), as it δ { , } impurity scattering is weak, i.e., min Res Ims . should. We see that δ˜0 is positive but smaller than δ in (15). c(s) > For F0 0 we expect to have well-defined collective This implies that the poles are located above the branch cuts. modes with |s| > 1. In this case, one can safely neglect√ δ The purpose of starting with Eq. (9) with small but finite δ in Eq. (9) and replace 1 − (s + iδ)2 by −i sgns s2 − 1. was to resolve the difference between δ, which determines Equation (9) is then reduced to the locations of the branch cuts, and δ˜0, which determines the |s| distance between the poles and the real axis. Near the poles, χfree,0(s) = 1 − √ . (10) the susceptibility reduces to s2 − 1 F c(s) χ c(s) ∝ 0 qp,0(s) / 1 + 2F c(s) 3 2 1 = 0 The collision integral in the relaxation-time approximation Icoll −δ × − ( f f0 ), with ( f ) f0 being (non)equilibrium distribution func- × 1 − 1 . tion, is not an appropriate form for impurity scattering, which is (18) s + sp,0 + iδ˜0 s − sp,0 + iδ˜0 elastic and therefore must conserve a number of particles with given energy. Consequently, Icoll must vanish upon averaging over This expression is valid for complex s above the branch cut =−δ the directions of the momentum, which is not the case for Icoll. at Ims i . This includes the real axis. For real s and δ˜ χ c(s) δ = The correct form of the collision integral for impurity scattering vanishingly small 0,Im 0 (s) has -functional peaks at s is −δ × ( f − f¯), where f¯ is the angular average of f . The kinetic ±sp,0 with sp,0 > 1, i.e., outside the particle-hole continuum equation with this collision integral reproduces Eq. (9). [see Fig. 1(a)].

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= χ c(s) = ω/ ∗ χ c(s) c(s) > FIG. 1. The imaginary part of the susceptibility in the l 0 channel, qp,0(s), where s vF q.(a)Im qp,0(s) for different F0 0in δ = −3 χ c(s) c(s) < a clean system (a small impurity scattering rate 10 was added to make the poles visible). (b) Im qp,0(s) for different F0 0ina χ c(s) | | < clean system. Note that Im qp,0(s) is nonzero only for s 1 in this case. In this and all other figures, we omit the “qp” subscript and “c(s)” superscript of the susceptibilities, and set νF = 1. c(s) The existence of the poles glued to the lower edge of For negative F0 we search for complex solutions of Eq. (12) in the form s =±a − ib.ForF c(s) < − 1 , there exists the branch cut is a tricky phenomenon. At first glance, they 0 2 describe undamped collective excitations with velocity larger a purely imaginary solution s =−is , , where i 0 > than the Fermi velocity [note thats ¯p,0 1inEq.(21)]. Indeed, − c(s) the susceptibility near the poles is 1 F0 , = . si 0 (19) c(s) c(s) 2 F − 1 c(s) F0 0 χ , (s) ∝ qp 0 − | c(s)| 3/2 1 2 F0 The pole at s =−isi,0 describes a purely relaxational zero- + c(s) > χ c(s) ,ω 1 1 sound mode. As long as 1 F0 0, the pole in qp,0(q ) × − . (22) is in the lower half-plane of s, i.e., excitations decay expo- s + s¯p,0 + iδ¯0 s − s¯p,0 + iδ¯0 nentially with time. Once 1 + F c(s) becomes negative, the 0 This form is very similar to that in Eq. (18) for positive F c(s). pole moves into the upper half-plane. Then, excitations grow 0 However, Eq. (22) is valid only for complex s below the lower exponentially with time, i.e., the system becomes unstable edge of the branch cut at |s| > 1, and cannot be extended (see Sec. IV for more detail). This is corroborated by the fact to real s. More precisely, Eq. (22) cannot be extended to the that the static susceptibility diverges as the system approaches real axis on the physical sheet of the Riemann surface, which a Pomeranchuk instability: χ c(s) we recall is the sheet for which qp,0(s) is analytic in the νF upper half-plane. Instead, it can be extended to the real axis χ c(s) (q, 0) = . (20) qp,0 + c(s) − s + iδ 2 = 1 F0 of the unphysical sheet, the one for which 1 ( ) i (s + iδ)2 − 1. This means that the pole below the branch As F c(s) increases above −1, i.e., |F c(s)| gets smaller, the χ c(s) 0 0 cut has no effect on the behavior of Im qp,0(s) on the real frequency of the relaxational mode in (19) increases in mag- axis, and the imaginary part of the susceptibility for real s, =∞ c(s) =− / nitude. It reaches si,0 at F0 1 2. At this value c(s) νF χ , (s) of F , the mode bifurcates into two (s , →±s¯ , ), and c(s) free 0 0 i 0 p 0 Imχ , (s) = each new mode moves from imaginary to almost real s along qp 0 + c(s)χ 2 + c(s)χ 2 1 F0 free,0(s) F0 free,0(s) infinite quarter circles in the complex s plane. − 1 < c(s) < = (23) For 2 F0 0 the mode frequency is given by s ± − δ¯ χ ≡ χ c(s) χ ≡ χ c(s) s¯p,0 i 0, where with (s) Re free,0(s) and (s) Im free,0(s), has no peak above the continuum. Therefore, the modes for −1/2 < − F c(s) − F c(s) c(s) < 1 0 1 0 F0 0 are “hidden,” in a sense that they cannot be detected s¯ , = , δ¯ = δ . (21) p 0 0 c(s) χ c(s) − c(s) 1 − 2 F by a spectroscopic measurement which probes Im qp,0(s). 1 2 F0 0

=∞ c(s) =− / + C. l = 1 The real part varies froms ¯p,0 at F0 1 2 0to c(s) s¯p,0 = 1atF = 0. The pole positions are similar to those For l 1 we have to distinguish√ between the longitudinal 0 θ for positive F c(s) [see Eq. (17)]; however, now δ¯ δ, i.e., susceptibility with the form factor 2√ cos and the transverse 0 0 θ the poles are located below the branch cut at Ims =−δ.At susceptibility with the form factor 2sin . We consider vanishingly small δ, which we consider here, the poles are the two cases separately. Here and in what follows, we will glued to the lower edge of the branch cut immediately below suppress the c(s) superscript in the longitudinal and transverse χ c(s),long → the real axis. The evolution of the real and imaginary parts of susceptibilities for brevity, i.e., we will relabel l c(s) χ long χ c(s),tr → χ tr the poles with F0 is shown in Fig. 2. l and l l .

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FIG. 2. Evolution of the poles of the dynamical susceptibility in the l = 0 channel. (a) The real [blue (dark)] and imaginary [yellow (light)] χ c(s) c(s) parts of the pole of the quasiparticle susceptibility qp,0(s) as a function of the Landau parameter F0 . For clarity, we show only one pole (for c(s) > − 1 c(s) F0 2 there are two poles with real parts of opposite signs). (b) The path followed by the pole in the complex plane with increasing F0 . c(s) < − c(s) For F0 1 the pole is purely imaginary and above the real axis, which indicates that the FL state is unstable. With increasing F0 ,the − ∞ c(s) =−1 pole moves down along the imaginary axis, which corresponds to an overdamped zero-sound mode, and reaches i at F0 2 .Itthen “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, χ c(s) c(s) = which cannot be detected in measurements of 0 (s) for real s (i.e., real frequencies). At F0 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.

1. l = 1, longitudinal channel Nevertheless, they are necessary to correctly determine the √ The computation of the free-fermion susceptibility with location of the poles. χ long 2 cos θ form factors at the vertices is quite straightforward. The expression for free,1(s) in the presence of impurity In notations of the previous section, the retarded susceptibility scattering will be derived in Sec. III. Here, we just borrow the is given by result + δ 1 + i √ (s i ) is 1−(s+iδ)2 χ long = + 2 + . χ long = + 2 . , (s) 1 2s 1 (24) free,1(s) 1 2s δ (27) free 1 1 − (s + iδ)2 1 − √ 1−(s+iδ)2 | | > For real s 1, Eq. (24) reduces to The equation for the poles becomes | | long 2 s + c(s) − + δ 2 + + δ χ , (s) = 1 + 2s 1 − √ . (25) 1 F1 2 1 (s i ) i(s i ) free 1 2 − − = 2s . (28) s 1 c(s) − + δ 2 − δ F1 1 (s i ) Substituting this form into Eq. (7), we obtain an equation for the poles: 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 = c(s) 1 + F |s| − + δ 2 − 1 =−2s2 + 2s2 √ . (26) i sgns (s i ) 1, we find that the solution actually ex- c(s) 2 − c(s) 3 c(s) F1 s 1 ists only for 0 F1 5 . For these F1 , the poles are located at s1,2 =±sp,1 − iδ˜1, where sp,1 is the solution of A solution of Eq. (26) in the form s , =±s , with s , > 1, 1 2 p 1 p 1 c(s) > δ˜ = ˜ δ c(s) > Eq. (26) (which exists for all F1 0), and 1 Q1 , where i.e., outside the continuum, exists only for F1 0. For small c(s) c(s) 2 c(s) F , s , = 1 + 2(F ) .AsF increases, the magnitude s2 1 p 1 1 1 = p,1 . c(s) ≈ Q˜ 1 (29) of sp,1 also increases, and at large F1 becomes sp,1 2 − s2 + s , s2 − 1 (3F c(s)/4)1/2. Correspondingly, Imχ long(s) has peaks on the p,1 p 1 p,1 1 qp,1 √ real axis at s =±sp,1. c(s) 3 < < , / ˜ < For 0 F1 5 , sp 1 varies between 1 and 2 3, and√Q1 To find the actual position of the poles in the complex c(s) 3 F = s , = / plane, we will again need to treat δ as a finite, albeit small, 1, as we assumed. For 1 5 ,wehave p 1 2 3 and ˜ = quantity. As for the l = 0 case, we associate δ with weak Q1 1, i.e., the pole merges with the branch cut. For larger c(s) ˜ > impurity scattering. Because a generic l = 1 order parameter F1 ,wehaveQ1 1, violating our assumption that the pole is not a conserved quantity, vertex corrections are not crucial.2 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

2 In the clean case, χl (q = 0,ω) for an order parameter with l = 0is ﬁnite because the self-energy and vertex corrections due to electron- electron interaction do not cancel each other [24]. For noninteracting because the self-energy and vertex corrections due to impurity scat- fermions, but in the presence of disorder, χl =0(q = 0,ω) is ﬁnite tering do not cancel each other.

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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. − c(s) c(s) c(s) ≡−1 In the interval 1 F1 F1,cr , where F1,cr 9 ,we find s1,2 =±a1 − ib1, where 1 + F c(s) = 1 − c(s) + c(s) 1/2, a1 1 F1 1 3 F1 c(s) 4 F1 1 − F c(s) = 1 + c(s) c(s) − 1/2. b1 1 F1 3 F1 1 (35) FIG. 3. Imχ long(s) for different F . (A small impurity scattering c(s) qp,1 1 4 F −3 1 δ = 10 was added to make the pole at positive F1 visible.) | c(s)| As F1 decreases, a1 monotonically increases, while b1 first increases and then changes trend and starts decreasing (see branch cut is defined as 1 − (a + ib)2 = i (a + ib)2 − 1 2 2 Fig. 4). The poles reach the lower edges of the√ branch cuts instead of 1 − (s + iδ) =−i sgns (s + iδ) − 1, which c(s) c(s) at F . At this critical value of F , a , = 2/ 3 > 1 and we used to search for the poles on the physical Riemann sheet. 1,cr 1 cr 1 c(s) c(s) c(s) 3 b = 0 (up to a term of order δ). For |F | slightly below F , , The absence of the zero-sound pole for F1 5 is a 1 1 cr χ long a1 and b1 are approximately given by surprising result, but it has little effect on the form of qp,1 (s) for real s. The latter has a conventional form for all positive / c(s) 3 c(s) c(s) 1 2 F : b1 = F − F , , 1 2 1 1 cr √ long 1 1 χ (s) ∝ − , (30) = 2 − 243 3 c(s) − c(s) qp, 1 + + δ˜ − + δ˜ a1 √ F F , s sp,1 i 1 s sp,1 i 1 3 2 1 1 cr long and Imχ (s) has peaks at s =±s , , as shown in Fig. 3. 2 qp,1 p 1 = √ − O b2 . (36) c(s) > 3 1 There also exists another solution for F1 0, which is purely imaginary: s =−isi,1. Assuming that si,1 δ,we c(s) c(s) When F1 approaches F1,cr , the poles approach the real axis obtain an equation for si,1 from Eq. (28): ⎛ ⎞ along the paths that are almost normal to it. > + c(s) The existence of the solution with a1 1 but finite b1 for 1 F1 2 si,1 c(s) c(s) = ⎝ + ⎠. F F , is at first glance questionable because conven- c(s) si,1 1 (31) 1 1 cr 2F + 2 tional wisdom suggests that a mode with Res > 1 is located 1 1 si,1 √ outside the particle-hole continuum and thus should be purely c(s) ≈ / c(s) c(s) propagating. However, as for the l = 0 case, these poles are For small positive F1 , si,1 1 2 F1 1. As F1 √in- located below the branch cuts, cannot be accessed from the creases, s , decreases and eventually saturates at s , = 1/ 3. i 1 i 1 χ long This additional solution will be relevant for the case of finite real axis and do not lead to a peak in Im 1 (s) for real s. c(s) < c(s) < = damping, analyzed in Sec. III. Note that Eq. (31) has a solution For F1,cr F1 0, the poles are located at s1,2 ± − δ¯ only for positive si,1, i.e., the pole is in the lower half-plane, s¯p,1 i 1, wheres ¯p,1 is determined from as it should be. ⎛ ⎞ c(s) c(s) For negative F , we again search for complex solutions s¯ , 1 − F 1 s2 ⎝ + p 1 ⎠ = 1 in the form ¯p,1 1 c(s) (37) 2 − F s¯p,1 1 1 s1,2 =±a − ib, b > 0. (32) δ¯ = ¯ δ c(s) ≈− and 1 Q1 with Right above the Pomeranchuk instability, i.e, at F1 1 c(s) 2 > − s¯ , but F1 1, we find ¯ = p 1 . Q1 (38) 1/2 2 − s¯2 − s¯ , s¯2 − 1 1 + F c(s) 1 + F c(s) p,1 p 1 p,1 a = 1 , b = 1 . (33) √ 2 4 = / c(s) = The magnitude ofs ¯p,1 varies betweens ¯p,1 2 3atF1 c(s) = + c(s) 2 − c(s) In contrast to the l = 0 case, the collective modes are almost F1,cr ands ¯p,1 1 2(F1 ) for F1 1, i.e., at vanish- c(s) propagating because a b. Below the Pomerachuk transi- ing F1 the poles approach the end points of the branch cuts. c(s) < − ¯ tion, i.e., for F1 1, both poles become purely imaginary As follows from Eq. (38), Q1 1for¯sp,1 in this interval, and split away from each other along the imaginary s axis: hence δ˜1 δ, i.e., the poles are located below the lower 1/2 edges of the cuts, as expected. This is very similar to what + c(s) 1 c(s) 1 F1 we found in the l = 0 case for − < F < 0. Like in that s , ≈±i . (34) 2 0 1 2 2 = c(s) < c(s) < case, the l 1 susceptibility for F1,cr F1 0 has poles

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χ c(s) FIG. 4. The poles of qp,1(s) in the longitudinal (a), (b) and transverse (c) channels. The use of colors and notations is the same as in Fig. 2 [in the lower panels: blue (dark), real part of pole; yellow (light), imaginary part of pole]. The circle in the upper panel of (a) denotes a point where a pole of the l = 1 longitudinal mode moves to the unphysical Riemann sheet. See Sec. II C for a detailed discussion. at s =±s¯p,1 − iδ¯1: To obtain the solutions in the complex plane s,weintro- duce impurity scattering in the same way as in the previous χ long ∝ 1 − 1 . cases. Equation (40) is then replaced by qp,1 (s) (39) s + s¯p,1 + iδ¯1 s − s¯p,1 + iδ¯1 χ tr (s) = 1 − 2s(s + iδ − i 1 − (s + iδ)2 ). (43) However, Eq. (39) is again only valid for complex s in the free,1 lower half-plane below the branch cut, and cannot be extended There are no additional terms due to vertex corrections be- to real s. These poles correspond to hidden modes, and the cause the form factor is an odd function of the angle θ and susceptibility does not have peaks above the particle-hole long thus vertex corrections vanish upon angular integration. χ c(s) continuum. We plot Im qp,1 (s) for real s in Fig. 3.The Substituting Eq. (43)into(7), we find that for F > 1, evolution of the real and imaginary parts of the poles with 1 c(s) where Eq. (42) has a solution for real s, there is actually no F1 is shown in Fig. 4. χ tr solution for the pole of qp,1(s) in the complex plane of s, above the branch cut. Still, for real s,Imχ tr (s) displays sharp = qp,1 2. l 1, transverse channel c(s) > peaks even for F1 1. We next consider the transverse quasiparticle susceptibil- < c(s) < For 0 F1 1 we assume that s is below the branch cut ity in the l = 1 channel.√ The retarded susceptibility of free − s + iδ 2 = fermions with the 2sinθ form factors at the vertices is and rewrite the square root in Eq. (43)as 1 ( ) i sgns (s + iδ)2 − 1. If we just neglect δ after that, we find tr 2 2 s =± χ , (s) = 1 − 2s + 2i(1 − s ) . another propagating mode, located at s1,2 s¯p,1, where free 1 − + δ 2 1 (s i ) s¯p,1 1 is the solution of (40) 1 + F c(s) 1 = + 2 − . For real |s| > 1, Eq. (40) is reduced to 2¯sp,1(¯sp,1 (¯sp,1) 1) (44) F c(s) 1 χ tr (s) = 1 − 2s2 + 2|s| s2 − 1. (41) free,1 However, if δ is treated as a small but finite quantity, we find χ tr −1 = | | >δ Substituting this form into Eq. (7), we find that the positions that there is no solution of ( qp,1(s)) 0 with Im¯sp,1 , of the poles on the real frequency axis and outside the particle- i.e., there is no pole below the branch cut. Combining this c(s) > hole continuum are determined by with the absence of the pole for F1 1, we conclude that the l = 1 transverse susceptibility does not have a pole on the 1 + F c(s) physical sheet for F c(s) > 0. However, as was the case for the 1 = 2s2 − 2|s| s2 − 1. (42) 1 F c(s) longitudinal mode, the poles do exist on the unphysical sheet. 1 c(s) χ tr For negative F1 the pole of qp,1(s) is on the imaginary In contrast to the longitudinal case, the solutions of this axis: s =−isi,1.Thevalueofsi,1 is determined by c(s) equation s , =±s , exist not for any positive F but only 1 2 p 1 1 c(s) = − c(s) for F 1(Ref.[35]). Slightly above the threshold, sp,1 1 F1 2 1 √ = 2(s , ) + 2s , 1 + (s , )2. (45) c(s) 2 c(s) c(s) c(s) i 1 i 1 i 1 + − / , ≈ / 1 (F1 1) 8. For large positive F1 , sp 1 F1 2. F1

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c(s) c(s) c(s) The solution exists for all negative F1 . When F1 ap- For negative F2 ,Eq.(49) has two solutions. One of ≈ / | c(s)|1/2 + =− c(s) ≈− = proaches zero from below, si,1 1 2 F1 . Near 1 them is purely imaginary: s isi,2.ForF2 1, si,2 c(s) = ≈ + c(s) / =− + −| c(s)| / c(s) ≈ / | c(s)|1/4 F1 0, we have si,1 (1 F1 ) 2, i.e., s i(1 (1 F2 ) 2; for small negative F2 , si,2 1 (2 F2 ). F c(s))/2. As before, when 1 + F c(s) changes sign and becomes Another solution does not become critical at the Pomeranchuk 1 1 c(s) =− negative, the pole moves from the lower to the upper fre- transition. To detect this mode, we notice that, for F2 1, = = quency half-plane, i.e. an l = 1 perturbation in the shape of Eq.√ (49) is satisﬁed not only by s 0, but also by s1,2 the FS grows with time exponentially. This behavior is similar ±1/ 2. The latter solutions are on the real axis, but away to the one for l = 0. Yet, a purely relaxational collective mode from the branch cut. At small deviation from the critical c(s) c(s) = − < < =− , =± − in the l 1 transverse channel exists for all 1 F1 0, value F2 1, these solutions evolve into s1 2 a2 i.e., it appears without a threshold. ib , where 2 + c(s) 1 1 F2 D. l = 2 a2 = √ 1 + , 2 4 1. l = 2, longitudinal channel + c(s) 2 √ The retarded free-fermion susceptibility with the 1 F2 b2 = √ . (50) 2 cos 2θ form factors at the vertices is 8 2

long 2 4 2 2 s + c(s) < χ , (s) = 1 − 4s + 8s + 2i(2s − 1) . Observe that b2 remains positive even when 1 F2 0. free 2 1 − (s + iδ)2 c(s) As F2 gets larger, the solutions first move away from the (46) real axis but then reverse the trend and, at the threshold value c(s) F , =−0.0632, reach the lower edge of the branch cut at long 2 cr χ c(s) c(s) The equation for the poles of qp,2 (s) outside the continuum, a , ≈±1.046. As F is varied from F to 0, the solutions | | > 2 cr 2 2,cr i.e., for s real and s 1, now reads as “slide” along the lower edge of the branch cut toward s =±1. c(s) l = 1 + F |s| This is very similar to what we found in the 0 case, − 2 = 8s4 − 4s2 − 2(2s2 − 1)2 √ = 0. (47) for − 1 < F c(s) < 0, and in the l = 1 longitudinal channel, c(s) 2 − 2 0 F2 s 1 c(s) < c(s) < δ for F1,cr F1 0. At a small but finite , the two sliding Similarly to the cases of l = 0 and of the longitudinal channel solutions are s1,2 =±s¯p,2 − iδ¯2 with δ¯2 δ, i.e., the pole does for l = 1, the propagating solutions s1,2 =±sp,2 exist for all exist but is located below the branch cut. The evolution of the c(s) c(s) c(s) ≈ + c(s) 2 poles with F is shown in Fig. 5. positive F2 . For small F2 , sp,2 (1 2(F2 ) ); for large 2 c(s) c(s) 1/2 F , s , ≈ (F /2) . 2 p 2 2 = To obtain the solutions in the complex plane, we introduce 2. l 2, transverse channel impurity scattering in the same way as before. Combining √ The retarded free-fermion susceptibility with the the self-energy and vertex corrections, we obtain after some 2sin2θ form factors at the vertices is given by algebra 3 2 − tr 2 4 8is (s 1) χ , (s) = 1 + 4s − 8s − . (51) long s free 2 − + δ 2 χ , (s) = 1 − 2i 1 (s i ) free 2 1 − (s + iδ)2 − δ The equation for the poles outside the continuum, i.e., for × (s + iδ − i 1 − (s + iδ)2 )2[1 − 2s(s + iδ)]. s real and |s| > 1, reads as (48) 1 + F c(s) 2 = 4 − 2 − 2| | 2 − . c(s) 8s 4s 8s s s 1 (52) The equation for the pole becomes F2 c(s) c(s) =± c(s) > 1 + F is For positive F2 , the solutions s1,2 sp,2 exist for F2 2 = + δ − − + δ 2 2 1 c(s) 1 c(s) c(s) (s i i 1 (s i ) ) .ForF just slightly above , s , ≈ 1 + (81/128)(F − 2F 1 − (s + iδ)2 − δ 3 2 3 p 2 2 2 2 c(s) c(s) 1/2 1/3) .ForlargeF , s , ≈ (F /2) . × − + δ . 2 p 2 2 [1 2s(s i )] (49) To obtain the solutions in the complex plane s,weintro- duce impurity scattering in the same way as before. There are Solving for the pole at small but finite δ, we find s1,2 = ±s , − iδ˜ , where δ˜ = Q˜ δ. Evaluating Q˜ , we find that it is no additional terms due to vertex corrections because the form p 2 2 2 2 2 θ c(s) < . < . factor is an odd function of the angle . Equation (51) is then smaller than 1 for F2 0 420, when sp,2 1 072. For these c(s) replaced by F2 , the pole is located above the branch cut, as it should be. c(s) χ tr = − + δ + δ − − + δ 2 2. For larger F2 there are no poles near the real axis. This is free,2(s) 1 4s(s i )(s i i 1 (s i ) ) = similar to the behavior in the longitudinal channel for l 1. (53) We reiterate that the absence of a true pole in the complex χ long For F c(s) > 1 , we assume that s is above the branch cut and plane does not affect the behavior of 2 (s) for real s;in 1 3 χ long =± use 1 − (s + iδ)2 =−i sgns (s + iδ)2 − 1. If we neglect δ particular, Im 2 (s) still displays sharp peaks at s sp,2. In mathematical terms, the pole moves to a different Riemann after that, we obtain the same solution sp,2 as in (52). For 0 < c(s) > . c(s) < 1 sheet at F2 0 420. F2 3 , the same procedure but with an assumption that the

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χ c(s) FIG. 5. The poles of 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, l = 0, 2 and almost real for odd l = 1. For transverse channels which slides along the lower edge of the branch cut, much the situation is the opposite: the mode near a Pomeranchuk like it happens for the pole of the transverse susceptibility for instability is purely imaginary for l = 1 and almost real for l = 1. Once we take into account that δ is small but finite, we l = 2. In this section we analyze whether this trend persists find that the solution along the real axis does not survive in for other values of l. either of the cases, i.e., there is no pole close to the real axis The retarded longitudinal and transverse susceptibilities of on the physical Riemann sheet. This is similar to the situation free fermions can be obtained analytically for any l.Wehave for the l = 1 transverse channel. χ long,tr = ± , As for the l = 1 case, there also exists another mode for free,l>0(s) K0 K2l (57) c(s) F > 0, at s =−is , on the imaginary axis. The value of 2 i 2 where si,2 is determined from θ =− θ cos K2l cos 2l (58) 1 + F c(s) s + iδ − cos θ 2 = 2 + + 2 2. (si,2 ) (si,2 1 (si,2 ) ) (54) s c(s) 2 2l 4F2 = δl,0 + i (s − i 1 − (s + iδ) ) . (59) 1 − (s + iδ)2 c(s) c(s) 1/4 c(s) For small F > 0, s , ≈ 1/(2F ) ;forlargeF , s , ≈ √ 2 i 2 2 2 i 2 The equation for the pole on real frequency axis outside the 1/2 2. | | > c(s) continuum, i.e., for s 1, is For negative F2 there is no solution on either real or + c(s) | | imaginary frequency axes, and we search for the solutions in 1 Fl s 2l =± − = (1 ± (|s|− s2 − 1) ). (60) the form s a2 ib2, where both a2 and b2 are finite. In F c(s) (s + iδ)2 − 1 this situation, one can safely neglect δ and write the equation l for the poles as The upper and lower signs correspond to the longitudinal and transverse channels, respectively. One can easily verify that, − c(s) | | > 1 F2 2 2 for any l, a solution with real s 1 exists only for positive = s − s + is − s2 . c(s) c(s) (1 2 2 1 ) (55) 4 F Fl . 2 c(s) For negative Fl , we search for complex solutions. In this An analysis of this equation shows that the solution exists for case, we rewrite (60)as all F c(s) < 0. Near F c(s) =−1, 2 2 c(s) 1 − F is 1/2 l = √ ± s − i − s2 2l . − c(s) − c(s) c(s) (1 ( 1 ) ) (61) 1 F2 1 F2 F 1 − s2 a ≈ , b ≈ . (56) l 2 4 2 4 In what follows, we consider the longitudinal and trans- √ c(s) ≈ ≈ / | c(s)|1/4 verse channels separately, first for even l and then for odd l. For small negative F2 , a2 b2 1 (2 2 F2 ). The c(s) ≈− | c(s)| c(s) We consider separately the limits of Fl 1 and Fl evolution of this pole with F2 is shown in Fig. 5. 1, and then interpolate between the two limits. We show that Table I contains the summary of the results for the pole there are multiple solutions with complex s in each channel. = , , positions for l 0 1 2. The table reveals several trends, The structure of the solutions in the longitudinal channel for which we now study in more detail by extending our analysis even l are very similar to those in the transverse channel for to arbitrary l. odd l. We do not discuss here the solutions in the transverse c(s) channel for positive Fl , but below the threshold on the χc(s) > E. Poles of qp,l for arbitrary l solution with real s and s 1. 1. Equations for the poles 2. Even l, longitudinal channel We now focus in more detail on negative F c(s) and, l c(s) ≈− in particular, on the behavior of collective modes near a For Fl 1, we first search for a solution with small | | Pomeranchuk instability. Comparing the results for for the s . Expanding Eq. (61)ins, we find a pole on the imaginary l = 0, 1, 2 modes, we see a difference between even and axis odd l. Namely, near a Pomeranchuk instability the critical 1 + F c(s) s = s ≈−i l . (62) mode in the longitudinal channel is purely imaginary for even i 2

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= , , c(s) TABLE I. The positions of the poles for l 0 1 2. The table describes for what values of Fl a pole sp 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 c(s) two poles they are specified by (a), (b) markings. For ranges of Fl that do not appear in the table, the pole is on the unphysical sheet. For = < c(s) < 3 example, in the l 1 longitudinal channel, one of the two modes is always on the unphysical sheet, except for when 0 F1 5 .Table − < c(s) entries marked by “–” denote that a given type of pole does not occur for any 1 Fl . The table includes entries only for poles having Resp 0. Please note that every pole with Resp > 0 has a counterpart with a negative real part −Resp 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 = − < c(s) < − / − / < c(s) < < c(s) l 0(1) 1 F0 1 2(d) 1 2 F0 00F0 = − < c(s) < − / − / < c(s) < < c(s) < / l 1, longitudinal (2) 1 F1 1 9(a)[p] 1 9 F1 0(a) 0 F1 3 5(a) < c(s) 0 F1 (b) [d] = − < c(s) < l 1, transverse (1) 1 F1 0[d] – – = − < c(s) < − . < c(s) < < c(s) < . l 2, longitudinal (2) 1 F2 0(a)[d] 0 06 F2 0(b) 0 F2 0 42 (b) − < c(s) < − . 1 F2 0 06 (b) [p] = − < c(s) < l 2, transverse (2) 1 F2 0(a)[p] – – < c(s) 0 F2 (b) [d]

There exist additional noncritical solutions for which s re- | c(s)| when Fl exceeds a threshold value. At the threshold, mains finite at F c(s) =−1. To obtain these solutions, we l s1,2 =±a − iδ˜ with a > 1 and δ˜ δ. choose the plus sign in Eq. (61), set F c(s) =−1, and solve | c(s)| √ l For Fl smaller than the threshold, the poles remain the resultant equation 1 + (s − i 1 − s2 )2l = 0. There are l below the branch cut at s =±a − iδ, a 1. For vanishingly solutions sm = arccos [π(2m + 1)/2l], where 0 m < l is an small δ, which we consider in this section, the poles are integer. They form l/2 pairs of solutions with s1,2;p =±ap, glued to the lower edge of the branch cut and slide along the c(s) < < / = , = | |= | | ap √1, 0 p l 2. For l 2, we have a single pair s1 2;0 branch cut toward its lower end at s 1as Fl decreases. ±1/ 2, consistent with what we found earlier. At small We see that for any even l there exists exactly one such deviations from F c(s) =−1, in any direction, these solutions threshold solution, while other solutions appear already for l c(s) =± − ∝ + c(s) 2 infinitesimally small negative F . become complex s1,2;p ap ibp, bp (1 Fl ) .The l imaginary part of these solutions remains negative even for c(s) < − Fs 1. 3. Even l, transverse channel Next, consider the interval 0 < −F c(s) 1. In this limit, l F c(s) ≈− the magnitude of s must be large for the right-hand side of We start again with l 1 and consider the solution Eq. (61) to match 1/|F c(s)|1 on the left-hand side of the with vanishingly small s. For the transverse channel [for √l which we have to choose the minus sign in Eq. (61)], the − s2 ≈ is s same equation. Using 1 for in the lower half- leading, linear-in-s term on the right-hand side of Eq. (61) plane, we reduce (61)atsmall|F c(s)| to l is absent, and one needs to include the subleading terms. A straightforward analysis then shows that the poles of the 1 = s 2l . transverse susceptibility are located near the real axis, at c(s) (2 ) (63) Fl / + c(s) 1 2 + c(s) − = 1 Fl 1 Fl This equation has l 1 solutions with sm s1,2 ≈± − i . (64) −iπm/l / | c(s)|1/2l < < 2l 4 e 2 Fl , where 0 m l is an integer. The solution with m = l/2 is purely imaginary, and the other l − 2 = − / , =± − solutions form p (l 2) 2 pairs of s1 2; p ap ibp. When |F c(s)| becomes larger than 1, this solution moves = l The purely imaginary solution sl/2 evolves toward sl/2 0 into the upper half-plane, signaling an instability toward the c(s) − as Fl approaches 1 and moves into the upper half-plane development of a Pomeranchuk order. | c(s)| > when Fl 1, signaling a Pomeranchuk instability. The There also exist other solutions that remain finite c(s) =− at F c(s) →−1. These solutions are obtained by setting other solutions s1,2;p evolve toward finite values at Fl 1. l =± − c(s) =− Comparing the number of solutions with s1,2;p ap ibp Fl 1inEq.(√ 61) and solving the resultant equation c(s) =− < − c(s) − − − 2 2l = − = at Fl 1 and 0 Fl 1, we see that they differ 1 (s i 1 s ) 0. There are l 2 solutions sn by one pair, which exists for the former case but not for arccos(πn/l), where 0 < n < l and n = l/2. Such solutions the latter. From the analysis of the l = 2 case, we know do not exist for l = 2, i.e., there is only a solution that vanishes =± − c(s) →− that the solution s1,2 a ib with a nonzero b emerges at Fl 1.

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< − c(s) /| c(s)|=− 2l negligibly small. In this situation, the relation between the For 0 Fl 1, we need to solve 1 Fl (2s) . The solutions are s = e−iπ(1+2n)/(2l)/2|F e|1/2l with 0 n < quasiparticle and free susceptibilities is more complicated n l = l. The number of solutions is l, and they form l/2 pairs with than in Eq. (7) because the l 0 and 1 channels are coupled at finite s via the F c(s)F c(s) term. Resumming the coupled s1,2;p =±ap − ibp,0 p < l/2. One pair evolves toward 0 1 c(s) long = c(s) − − random phase approximation (RPA) series for χ , and χ , s1,2;p 0, as Fl approaches 1, while the other l 2 qp 0 qp 1 ± c(s) =− or, equivalently, solving the FL kinetic equation, one arrives solutions tend to finite values ap at Fl 1. We see that the number of noncritical solutions is the same, i.e., l − 2, at [26,34] c(s) c(s) c(s) < − =− 2F K2 both for 0 Fl 1 and at Fl 1. − 1 1 K0 + c(s) + c(s) 1 F1 (K0 K2 ) χ , (s) = νF , qp 0 2F c(s)F c(s)K2 4. Odd l, longitudinal channel + c(s) − 0 1 1 1 F0 K0 + c(s) + 1 F1 (K0 K2 ) The analysis for odd l proceeds along the same lines. We do 2F c(s)K2 K + K − 0 1 not present the details of calculations and just state the results. 0 2 1+F c(s)K c(s) χ long = ν 0 0 , F ≈− l + l + , (s) F c(s) c(s) (65) For l 1, there are 1 solutions, which form ( qp 1 2F F K2 + c(s) + − 0 1 1 1)/2 pairs s1,2;p =±ap − ibp,0 p < (l + 1)/2. One pair is 1 F1 (K0 K2 ) + c(s) 1 F0 K0 thesameasinEq.(64), the other solutions tend to finite sm = arccos[π(1 + 2m)/(2l)], with 0 m < l, m = (l − 1)/2, at where Kn are given by Eq. (58). Explicitly, c(s) =− < − c(s) − s Fl 1. For 0 Fl 1, there are l 1 solutions = + , − π / / K0 1 i = i m l / | e|1 2l < < − / 2 sm e 2 Fl , with 0 m l. They form (l 1) 2 1 − (s + iδ) =± − < − / pairs s1,2;p ap ibp,0 p (l 1) 2. Comparing the 3 c(s) ≈− < − c(s) 2 s number of solutions at F 1 and for 0 F 1, K0 + K2 = 1 + 2s + 2i , l l − + δ 2 we see that there exists one pair of solutions s1,2 =±a − ib 1 (s i ) with b > 0, which emerges once |F c(s)| exceeds a threshold 2 l = + s . |F c(s)| and K1 s i (66) value. For l smaller than the threshold, this pair of 1 − (s + iδ)2 solutions remains glued to the lower edge of the branch cut χ c(s) χ long immediately below the real axis. The denominators of qp,0 and qp,1 vanish when + c(s) + c(s) + = c(s) c(s) 2. 5. Odd l, transverse channel 1 F0 K0 1 F1 (K0 K2 ) 2F0 F1 K1 (67) c(s) ≈− c(s) − + For Fl 1, there is one purely imaginary solution Suppose that F1 is negative and close to 1 while 1 − = c(s) c(s) with vanishing s,asinEq.(62), and l 1 solutions sn F > 0(F can be of either sign). In the previous sections, π / < < < − c(s) 0 0 arccos( n l), 0 n l.For0 Fl 1, there are l so- we saw that a critical zero-sound mode corresponds to small = −iπ(1+2m)/(2l)/ | c(s)|1/2l < lutions sm e 2 Fl , with 0 m l.One s. Substituting the forms of Kn into Eq. (67) and assuming that solution, with m = (l − 1)/2, is purely imaginary, while the s is small, we obtain other solutions form (l − 1)/2 pairs s , =±a − ib ,0 1 2;p p p + c(s) + c(s) = 2 + 3 + c(s) + c(s) . < − / c(s) ≈ 1 F0 1 F1 2s 2is iF0 1 F1 s p (l 1) 2. Comparing the number of solutions at Fl − < − c(s) (68) 1 and for 0 Fl 1, we see that the number is the same, i.e., all solutions develop already at infinitesimally + c(s) c(s) Iterating this equation in 1 F1 1, we obtain its approx- small Fl . The purely imaginary solution moves into the imate solution as c(s) upper frequency half-plane when F + 1 becomes negative, / l c(s) 1 2 1 + F / signaling a Pomeranchuk instability, while other solutions =± 1 + c(s) 1 2 s 1 F0 s1,2;p =±ap − ibp remain in the lower frequency half-plane 2 c(s) < − even for Fl 1 − i + c(s) . Comparing the solutions for even and odd l, we see that 1 F1 (69) c(s) ≈− 4 at Fl 1, the solutions in the longitudinal channel for even l are quite similar to those in the transverse channel at This form does not differ qualitatively from Eq. (33)forthe c(s) case F c(s) = 0, i.e., both the real and imaginary parts of the odd l and vice versa. For smaller negative Fl there is a 0 c(s) − difference between the longitudinal and transverse channels zero-sound velocity vanish when F1 approaches 1, and at any l. Namely, there exists one solution in the longitudinal the imaginary part vanishes faster. The only effect of nonzero c(s) channel which remains glued to the lower edge of the branch F0 is to renormalize the prefactor of Res. cut at |s| > 1, until |F c(s)| exceeds a threshold value, while c(s) − c(s) l In the opposite case, when F0 is close to 1 while F1 in the transverse channel all solutions with Ims < 0emerge is not close to −1 but otherwise arbitrary, we find from (67) c(s) already at infinitesimally small Fl . 1 − F c(s) s ≈−i 1 + F c(s) + 1 + F c(s) 2 1 . (70) 0 0 + c(s) F. Case of two comparable Landau parameters 1 F1 As a more realistic example, we consider the case when We see that the pole remains on the imaginary axis and c(s) c(s) two Landau parameters, e.g., F0 and F1 , are comparable moves from the lower to upper frequency half-plane when + c(s) c(s) in magnitude, while the rest of the Landau parameters are 1 F0 changes sign. The Landau parameter F1 affects

033134-12 COLLECTIVE MODES NEAR A POMERANCHUK … PHYSICAL REVIEW RESEARCH 1, 033134 (2019) only the subleading term. We expect this behavior to hold The equation for the zero-sound pole is when F c(s) with l > 1 are also present, as long as F c(s) are not l l>1 c(s) − 1 − F (s + iδ)2 s + iδ + 1 close to 1. 1 = s s + iδ − −iπ + ln . The simultaneous presence of F c(s) and F c(s), however, c(s) 2 1 − s − iδ 0 1 3 F1 changes the threshold for the existence of a propagating zero- sound mode. (For 3D systems, this effect was noticed in (75) c(s) c(s) Ref. [43].) For example, if only F0 is nonzero, a propagating When F ≈−1, the solution is c(s) c(s) 1 mode exists only for positive F0 .IfF1 is also nonzero and c(s) π − c(s) positive, a propagating mode exists also for negative F . / 1 F1 0 s =± 1 2 − i , = . (76) c(s) > 4 3 Moreover, for large enough F1 0, a propagating mode = exists even at the l 0 Pomeranchuk instability, i.e., when This is very similar to the 2D case [cf. Eq. (33)]. In contrast c(s) =− c(s) =− c(s) > F0 1. Namely, setting F0 1 and varying F1 to the 2D case, however, a complex solution in 3D exists for 0, we find the solution of Eq. (67) in the form of a propagating all negative F c(s), i.e., there is no threshold. When |F c(s)| is c(s) > 1 1 zero-sound mode for F1 1. The mode frequency is small, / 1 + F c(s) 1 3 =± 1 , | | > . 1 − π/ s s 1 (71) s ≈ e i 6. (77) c(s) π c(s) 2 F1 3 F1 The form factor in the transverse channel is f c(s)(k ) = c(s) ≈ c(s)/ 1 F For large F1 , s F1 2. ±1 θ,φ Y1 ( ). The free-fermion susceptibility is − 2 + G. 3D systems 3 2 3s(1 s ) s 1 χ , s = − s − . free 0( ) 1 ln − (78) For comparison, we also briefly discuss the behavior of 2 4 s 1 zero-sound excitations near a Pomeranchuk instability in a 3D The equation for the zero-sound pole is system. We present the results for l = 0 and 1 and, in each c(s) case, consider only one nonzero Landau parameter F c(s) < 0. 4 1 + F s + 1 l 1 =−2s2 − s(1 − s2 ) −iπ + ln . (79) c(s) 1 − s 3 F1 1. l = 0 One can easily verify that the pole is located on the imaginary = Zero-sound modes in the l 0 channel were analyzed in axis, at s =−ia, where a is the solution of Refs. [30,43]. The free-fermion susceptibility with the form c(s) c(s) factor f (kF ) = 1is 4 1 − F 1 + ia 0 1 = πa(1 + a2 ) + 2a2 + ia ln . (80) c(s) 1 − ia s s + iδ + 1 3 F1 χfree,0(s) = 1 − ln . (72) 2 s + iδ − 1 c(s) =− ≈ / π −| c(s)| Near F1 1, a (4 3 )(1 F1 ). For small negative c(s) ≈ / π| c(s)| 1/3 The equation for the pole reads as F1 , a [4 (3 F1 )] . This is again similar to 2D, except for the solution s =−ia in 3D exists for all negative 1 s 1 + s + iδ c(s) − 1 =− −iπ + ln . (73) F1 , i.e., there is no threshold. c(s) 2 1 − s − iδ F0 The pole is completely imaginary: s =−ib [hence, iδ in III. FINITE DISORDER (73) is irrelevant]. In contrast to the 2D case, such solu- A. General formalism F c(s) tion exists for all negative 0 , i.e., there is no threshold. In this section we analyze how the results of the previous c(s) ≈− ≈ /π −| c(s)| < − c(s) For F0 1, b (2 )(1 F0 ). For 0 F0 1, sections change in the presence of finite disorder. As in the ≈ / π| c(s)| = b 1 ( F0 ). previous section, we consider separately the cases of l 0, 1, 2. Other cases can be analyzed in the same manner as 2. l = 1 these two. For a reader who prefers to avoid going through the exhaustive derivations, we present in Figs. 7 and 9 a The eigenfunctions of angular momentum l = 1 are spher- graphical summary of the evolution of the poles for the l = 0 ical harmonics Y m(θ,φ). We normalize Y m as Y 0(θ ) = √ 1 √ 1 1 and the l = 1 longitudinal channels. These figures depict the θ ±1 θ,φ =∓ / θ ±iφ 3 cos and Y1 ( ) 3 2sin e . Then, the criti- qualitative changes that occur for finite disorder. c(s) c(s) =− cal value of F1 for a Pomeranchuk instability is F1 1. The free-fermion susceptibility in the presence of scatter- c(s) = In the longitudinal channel, the form factor is f1 (kF ) ing by short-range impurities consists of two parts: the bubble 0 θ Y1 ( ). The free-fermion susceptibility is part and the vertex part: 2 χ ,ω = χ B ,ω + χ V ,ω (s + iδ) s + iδ + 1 free,l (q m ) free,l (q m ) free,l (q m ) (81) χ , s = + s s + iδ − . free 1( ) 1 3 ln + δ − 2 s i 1 (cf. Fig. 6). The bubble part is formed from the (Matsubara) −1 (74) Green’s functions G(k,νm ) = (iνm − k + i sgnνmγ/˜ 2) ,

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The bubble and vertex parts for the l = 0 case are given by is χ B = + , free,0 1 1 − (s + iγ )2 FIG. 6. The susceptibility of noninteracting fermions in the presence of impurity scattering. The diagram on the left corresponds γ/ − + γ 2 V is 1 (s i ) to χ B in Eq. (82), while the diagrams in the square brackets χ , = , (85) free,l free 0 2 χV 1 − (s + iγ ) − γ correspond to free,l in Eq. (83). The solid lines represent disorder- averaged fermionic propagators, the dashed lines represent the cor- where γ = γ/˜ v∗ q. Adding these up, we obtain relation functions of the impurity potential. F s χfree,0(s) = 1 + i , (86) 1 − (s + iγ )2 − γ whereγ ˜ is the impurity scattering rate: which is the result quoted in Eq. (9), except that we have 2 2 d k dνm 2 δ → γ γ χ B (q,ω ) =− f c(s)(k) changed the notations to emphasize that does not free,l m 2 l NF (2π ) 2π have to be small. This result, as well as a corresponding result = γ γ/ × G(k + q/2,νm + ωm/2, ) for the l 1 case, holds for ˜ EF while the ratio s can be arbitrary. At q → 0, i.e., at s →∞, the susceptibil- × − / ,ν − ω / . G(k q 2 m m 2) (82) ity vanishes, which guarantees that the charge and spin are conserved. The vertex part is For |s|γ ,Eq.(86) reduces to the well-known diffusive 2 2 ν form [44,45] V 2 d k d k d m c(s) χ , (q,ωm ) =− f (k) free l 2 2 l 2 NF (2π ) (2π ) 2π 1 Dq χ , (s) = = , (87) × c(s) ∗ + / ,ν + ω / free 0 1 − 2iγ s Dq2 − iω fl (k ) G(k q 2 m m 2) × − / ,ν − ω / = ∗ 2/ γ G(k q 2 m m 2) where D (vF ) 2˜ is the diffusion coefﬁcient in 2D. Substituting Eq. (86) into Eq. (7) and solving for the poles, ×G(k + q/2,νm + ωm/2) we ﬁnd that for F c(s) < − 1 the pole is on the imaginary axis, × − / ,ν − ω / D ω , ν , 0 2 G(k q 2 m m 2) ( m q; m ) at ⎛ ⎞ (83) − c(s) c(s) − 1 F0 ⎝ 2 F0 1 ⎠ s = s1 =−iγ 1 + − 1 . (88) where D(q,ωm; νm ) is the diffusion propagator [42] c(s) − γ 2 2 F0 1

γ˜ θ(ωm +|νm|/2)θ(|νm|/2 − ωm ) D(q,ωm; νm ) = . For γ → 0, this reduces to Eq. (19). For large γ , s1 = π ∗ 2 2 2 NF (v q) + (|ωm|+γ˜ ) − γ˜ − −| c(s)| / γ F i(1 F0 ) 2 . In this limit, we have a diffusion pole at =− ∗ 2 ∗ = −| c(s)| (84) iD q , where D D(1 F0 ) is the renormalized diffusion coefficient [46]. In the ballistic regime at small γ , D ,ω ν Diagrammatically, (q m; m ) is represented by the sum of the damping term accounts for a small correction to the result ladder diagrams in the particle-hole channel (the sequence of for a clean Fermi liquid [cf. Eq. (19)]. diagrams in the square brackets in Fig. 6). + c(s) When 1 F0 becomes negative, s1 moves into the upper The retarded forms of the susceptibilities are obtained by half-plane of s, which signals a Pomeranchuk instability. ω > ω → ω c(s) choosing m 0 and replacing i m in the final results. For positive 1 + F , the pole s moves down along the = 0 1 The vertex part is especially important for the l 0 case imaginary axis as 1 + F c(s) increases, but remains finite at because the corresponding order parameters (charge or spin) 0 F c(s) =−1 , in contrast to the behavior in the clean limit. are conserved quantities, and hence the bubble and vertex 0 2 | c(s)|− parts of the susceptibility must cancel each other at q = 0. For Expanding the square root in Eq. (88)in2F0 1, we > =−/ γ c(s) =−1 c(s) l 0, the corresponding order parameters are not conserved, find s1 i (4 )atF0 2 . At larger F1 , another but the vertex parts must be also included in order to obtain the solution ⎛ ⎞ correct positions of the zero-sound poles in the complex plane. 1 − F c(s) 1 − 2F c(s) s =−iγ 0 ⎝ 1 − 0 + 1⎠ (89) 2 − c(s) γ 2 B. l = 0 1 2 F0 We recall that in a clean Fermi liquid with positive F c(s) 0 appears in the lower half-plane, initially at s2 =−i∞.As the pole of χ c(s)(s) is on the real axis at |s| > 1. For nega- c(s) 0 F0 keeps increasing, s1 and s2 move toward each other. For c(s) =− − γ< = = tive F0 , the pole is on the imaginary axis, at s i(1 1, the two solutions merge into a single pole s1 s2 | c(s)| / | c(s)|− 1/2 − < c(s) < − 1 c(s) = −i(1 + γ 2 )/(2γ )atF c(s) = (γ 2 − 1)/2 < 0[seeFig.7(a)]. F0 ) (2 F0 1) , when 1 F0 2 .AtF0 0 1 c(s) − , the pole at s =−∞splits into two, and the new poles For F0 slightly larger than this value, the double pole bifur- 2 =±∞ c(s) instantly move to the s points on the real axis. At cates into two poles with finite real parts. For even larger F0 , c(s) =± =± − larger, but still negative, F0 , the poles move toward s 1 the two poles move along arclike trajectories s1,2 a ib along the lower edge of the branch cut. in the complex plane. In contrast to the case of γ = 0, the

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χ c(s) c(s) FIG. 7. Evolution of the poles of qp,0(s) with Landau parameter F0 for a finite disorder, parametrized by the dimensionless scattering rate γ . We use blue (dark) and yellow (light) colors to show how poles merge and then bifurcate. The solid and dashed lines denote the χ c(s) c(s) behavior of the poles of qp,0(s) for negative and positive F0 , respectively. The black horizontal dotted lines denote the branch cuts of χ c(s) =−γ | | > c(s) − ∞ qp,0(s)atIms and Res 1. The arrows identify the direction of the poles’ motion with F0 increasing from 1to . poles remain at a finite distance from the lower edges of the C. l = 1 c(s) branch cuts, as long as F0 remains negative. 1. l = 1, longitudinal c(s) = = At F0 0, the poles reach the points s1,2 c(s) We start with recalling the situation at vanishingly small ± 1 − γ 2 − iγ .AtpositiveF , a increases and b becomes c(s) long 0 damping. For −1 < F < 0, the poles of χ (s)areat smaller than γ . This implies that the poles are now located 1 qp,1 s , =±a − ib , where a and b are given by Eq. (35). in-between the branch cuts and real axis. At F c(s) →∞, 1 2 1 1 1 1 0 For F c(s) just above −1, a ≈ [(1 −|F c(s)|)/2]1/2 and b ≈ a → (F c(s)/2)1/2 and b → γ/2. 1 1 1 1 0 (1 −|F c(s)|)/4, i.e., the the poles are almost on the real γ> s F c(s) =−1 1 For 1, the second pole 2 still emerges at 0 2 , c(s) | c(s)| axis. For larger F1 (smaller F1 ), the two poles evolve but the poles at s = s1 and s2 on the imaginary axis remain at c(s) such that a1 increases monotonically, while b1 first infinite distance from each other for all negative F0 and merge c(s) 2 creases and then decreases. The poles approach the lower F = γ − / > c(s) 1 only at 0 ( 1) 2 0[seeFig.7(b)]. At larger edges of the branch cuts along the real axis at F =− , c(s) =± − √ cr,1 9 F0 , the poles again follow the trajectories s1,2 a ib = / > c(s) c(s) when a1 2 3 1. For larger F1 , the poles remain with a increasing and b decreasing with increasing F0 .For slightly below the lower edge of the branch cut and move F c(s) 1, the poles reach the same values as for γ<1: =± < c(s) < 3 0 toward a1 1. For 0 F1 5 the poles are located a ≈ (F c(s)/2)1/2 and b ≈ γ/2. In Fig. 8 we plot Imχ c(s)(s)for c(s) > 3 0 0 slightly above the branch cuts. For F1 5 they move off c(s) γ< γ> c(s) real s for a range of F0 , both for 1 and 1 [Figs. 8(a) from the physical Riemann sheet. For all F > 0, there and 8(b), respectively]. 1 exists another, purely imaginary, solution s =−isi,1.For

χ c(s) γ FIG. 8. The imaginary part of qp,0(s) for ﬁnite disorder, characterized by the dimensionless coupling constant . (a) Weak disorder γ = . − < c(s) < − 1 χ c(s) − 1 ( 0 2). For 1 F0 2 ,theshapeofIm qp,0(s) has a characteristic overdamped form. As F0 increases from 2 to 0, the shape c(s) > χ c(s) changes its form due to the appearance of “hidden” poles below the branch cut. For F0 0, Im qp,0(s) has a conventional form of a damped γ = . χ c(s) c(s) zero-sound mode. (b) Strong disorder ( 1 5). In this case Im 0 (s) has an overdamped shape for all F0 , negative and positive.

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χ long c(s) FIG. 9. Evolution of the poles of qp,1 (s) with the Landau parameter F1 for finite disorder, parametrized by the dimensionless coupling constant γ , as specified in the legend. Like in Fig. 7, we use different colors to show how the poles merge and bifurcate. The × denotes the c(s) →∞ γ = limiting position of the pole for F1 . The inset in the first√ panel depicts how, for small , the pole bypasses the s 1 branching point > / c(s) > 3 before finally moving to an unphysical Riemann sheet for s 2 3(F1 5 ).

small damping, this solution and the s1,2 solution are not which is the result quoted in Eq. (27), up to a replacement connected. γ → δ. Note that the vertex part vanishes at q → 0, i.e., at χ long →∞ We now show that the behavior of 1 changes qualita- s , while the bubble part is reduced to a form which is tively in the presence of disorder. The bubble and vertex parts identical to the Drude conductivity at ﬁnite frequency ω.This of the free-fermion susceptibility are now given by indicates that the charge and spin currents are not conserved in the presence of disorder. + γ The analysis of the evolution of the poles with F c(s) for long,B s i 1 χ , (s) = 1 + 2s(s + iγ ) 1 + i , different γ is straightforward but somewhat involved. We omit free 1 1 − (s + iγ )2 the details of the calculations and present only the results. − + γ 2 + + γ 2 These results are summarized graphically in the panels of long,V [ 1 (s i ) i(s i )] χ , (s) =−2iγ s Fig. 9. free 1 − + γ 2 1 (s i ) The beginning stage of the evolution is the same for all γ + c(s) =± − 1 :for1 F1 1, the poles are located at s1,2 a1 × . (90) c(s) / 2 ≈ + / 1 2 γ 1 − (s + iγ ) − γ ib1, where a1 [(1 F1 ) 2] is independent of , and ≈ + c(s) / + γ 2 + γ b1 [(1 F1 ) 4]( 1 ). However, the behavior Adding these up, we obtain c(s) γ at larger F1 depends strongly on . We ﬁnd that there are three values of γ , at which the evolution of the poles changes − + γ 2 + + γ γ = 1 γ = . γ = long 2 1 (s i ) i(s i ) qualitatively: 2 , 0 923, and 1. χ , (s) = 1 + 2s , (91) free 1 1 − (s + iγ )2 − γ

033134-16 COLLECTIVE MODES NEAR A POMERANCHUK … PHYSICAL REVIEW RESEARCH 1, 033134 (2019) c(s) = − − γ 2 again at some positive F1 , when a1 1 (1 ) .The c(s) limiting value of the purely imaginary pole for F1 1is γ c(s) now smaller than . This implies that, for large F1 , this pole χ long gives the main contribution to 1 (t ) in the time domain. At γ = 0.923, the s1,2 poles touch the imaginary axis of s c(s) ≈ . = . at F1 0 031. The corresponding value of a1 1 391. At c(s) this F1 , the purely imaginary pole is located at the same point on the imaginary axis, i.e., there are three degenerate solutions. 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), at which the purely imaginary is still located at FIG. 10. Imχ long(s) for finite disorder γ = 1 and various inter- 1 qp,1 10 a higher point on the imaginary axis (see Fig. 9, fourth panel). action strengths. c(s) The subsequent evolution with increasing F1 involves two bifurcations. After the second bifurcation, the two solutions For γ<1 the evolution of the poles is similar to that for approach the upper edge of the branch cut and move to a 2 γ c(s) = c(s) vanishingly small (see Fig. 3), although the interval, where different Riemann sheet at F1 F1,R given by Eq. (93). the imaginary part of the pole frequency varies nonmonoton- At γ = 1, the first bifurcation occurs at F c(s) = 0, at the c(s) γ 1 ically with F1 , shrinks rapidly with increasing . The pole point s =−iγ . After the bifurcation, one solution moves up =± − =−γ positions s1,2 a1 ib1 cross the line s i first at some along the imaginary axis, while another moves down. The c(s) = − − γ 2 negative F1 , when a1 1 (1 ) , and then again at one that moves up eventually reaches the point s =−0.39i c(s) = = − γ 2 c(s) c(s) = . F1 0, when a1 1 . The pole moves to the other, at F1 1. The second bifurcation occurs at F1 0 022 c s ( ) at the point s =−2iγ . After that, the two solutions s1,2 = unphysical Riemann sheet at F1,R given by ±a1 − ib1 move toward the end point of the branch cut and c(s) 1 c(s) 2 − γ 2 − 2 − reach a1 = 1, b1 = γ at F = . 1 + F a1.R a1,R a1,R 1 1 3 1,R = , γ> c(s) < c(s) (92) For 1, the first bifurcation happens at F1 0 and, 2F , 2 − 1 R a1,R 1 after bifurcation, the first (second) solution moves up (down) c(s) = = the imaginary axis. At F1 0, these two solutions are at s where c(s) −i(γ ± γ 2 − 1). For F > 0, the third solution emerges 1/2 1 γ 4 − γ 2 + + − γ 2 on the imaginary axis and, eventually, it merges with the = 2 1 2 . a1,R (93) solution that moves down (see Fig. 9, fifth panel), that, the 3 two solutions bifurcate and move toward the branch cut. In c(s) distinction to the case γ<1, now they merge with the lower The values of F and of a , decrease as γ increases, but 1,R 1 R c(s) = ¯ c(s) c(s) edge of the branch cut at F1 F1,R , where F remains positive, and a , remains larger than 1 as long 1,R 1 R γ< c(s) as 1.√ For large F1 the pole on the unphysical sheet is c(s) 2 − γ 2 + 2 − 1 + F¯ a¯1,R a¯1,R a¯1 1 at s ≈ ( 3F c(s) − iγ F c(s))/2, i.e., Ims increases with F c(s). 1,R = 1 1 1 1 1 c(s) (95) ¯ 2 χ long 2F1,R a¯ − 1 Figure 10 depicts the imaginary part of qp,1 (s) for a finite 1,R disorder. √ =− The purely imaginary pole s isi,1, which exists only anda ¯1,R is given by Eq. (93). For large γ ,¯a1,R ≈ γ/ 3 and c(s) c(s) c(s) > , ≈ ¯ ≈ / γ 2 > ¯ for F1 0, moves up the imaginary axis from si 1 F1,R 3 8 1. For F1 F1,R the poles again move to / c(s) 1/2 c(s) > a different Riemann sheet. The solution that moves up the 1 2(F1 ) 1 for small F1 0 toward smaller values c(s) c(s) c(s) > c(s) for larger F .ForF 1, s , is determined from the imaginary axis survives for all F1 0 and, for F1 1 1 1 i 1 γ ≈−/ γ equation and 1, it approaches the point s i 2 . We note that there are certain similarities between the evo- c(s) 2 si,1 lution of the poles with F1 and the behavior of the plasmon s , 1 + = 1/2. (94) i 1 2 modes in a 2D electron gas with conductivity exceeding the 1 + (si,1 − γ ) − γ speed of light σ>c/2π, where σ is in CGS units. This 1 γ< , >γ problem was studied some time ago [47] and has recently been For 2 , this limiting value si 1 . 1 γ = , revisited in Ref. [48]. At 2 , the points at which the s1 2 poles cross the =−γ c(s) = s i line merge at F1 0, and the region of the non- c(s) 2. l = 1, transverse channel monotonic evolution of s1,2 for negative F1 disappears. At this γ , the limiting value of the purely imaginary pole at For vanishingly weak damping (γ → 0), the pole moves c(s) 1 c(s) , = γ = =− − < < F1 1 becomes si 1 2 . along the imaginary axis (s isi)for 1 F1 0, to- 1 <γ < . | c(s)| c(s) = + For 2 0 923, the poles evolve in the complex plane ward larger si,as F1 decreases. At F1 0 , si tends as shown in Fig. 9, third panel. The s , poles cross the line c(s) 1 2 to infinity. For positive F1 , there is no pole on the physical =−γ c(s) = = − γ 2 s i first at F1 0, when a1 1 , and then Riemann sheet.

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For finite γ , the evolution remains essentially the same. two solutions move toward each other and merge at some c(s) > − < c(s) = c(s) γ c(s) ≈− + γ 2 There are still no solutions for F1 0, while for 1 critical value F2 F2,cr . For small , F2,cr 1 and c(s) c(s) c(s) < =− , ≈ γ/ = + F1 0 the pole is on the imaginary axis, at s isi 1, the solutions merge at b2 2. At F2 F2,cr 0, the where poles split and move away from the imaginary axis, i.e., a 2 c(s) S 1 − F becomes finite. The subsequent evolution is essentially the = γ 2 + + 2 + γ , = 1 . γ c(s) si,1 ( (1 2S) ) S same as for vanishingly small . For large positive F2√ ,the 1 + 2S F c(s) 1 pole is located on the imaginary axis at b2 = 1/2/ 2for (96) small γ and at b2 = γ + 1/(4γ )forlargeγ . c(s) ≈− ≈ −| c(s)| + γ 2 + γ / At F1 1, si,1 (1 F1 )( 1 ) 2. Note that there is no diffusive behavior for large γ . In this limit, IV. SUSCEPTIBILITY IN THE TIME DOMAIN ≈ γ −| c(s)| ω ≈−γ −| c(s)| γ si,1 (1 F1 ), i.e., i ˜ (1 F1 ), where ˜ is A. General results c(s) → ≈ the dimensionful impurity scattering rate. At F0 0, si,1 In this section we study the real-time response of an order / | c(s)|1/2 γ 1 2 F1 for all . parameter on both sides of the Pomeranchuk transition by χ c(s) , analyzing the susceptibility in the time domain l (q t ). For D. l = 2 definiteness, we consider l = 0 and the longitudinal channel for l = 1. In both cases, 1. l = 2, longitudinal channel ∞ c(s) dω For vanishingly weak damping (γ → 0) and F < 0, one χ c(s) , = χ c(s) ,ω −iωt , 2 l (q t ) l (q )e (99) of the poles is on the imaginary axis while the other one is in −∞ 2π c(s) the complex plane. For small negative F2 , the latter pole is at χ c(s) ,ω c(s) where (q ) is the retarded susceptibility. Introducing ∗ ∗ l the lower edge of the branch cut. When F2 crosses zero, the = ω t vF qt and going over from integration over to integra- pole bypasses the end point of the branch cut, moves slightly tion over s = ω/v∗ q, we obtain above it, and continues to stay there as F c(s) increases from F 2 ∞ c(s) c(s) ∗ ≈ . c(s) ∗ 1 c(s) ds c(s) −ist 0uptoF2 0 4. For larger F2 , the pole is located on an χ (t ) ≡ χ (q, t ) = χ (s)e . (100) l ∗ l π l unphysical Riemann sheet. vF q −∞ 2 For finite γ , the poles are determined from the equation χ c(s) ∗ The time dependent l (t ) can be measured in pump- 1 + F c(s) is probe experiment, by applying an instantaneous perturbation 2 = (s + iγ − i 1 − (s + iγ )2 )2 ∗ ∗ c(s) c(s) h (t ) = hδ(t ) with the symmetry of the Pomeranchuk 2F 1 − (s + iγ )2 − γ l l 2 order parameter, to momentarily move the system away from × [1 − 2s(s + iγ )]. (97) the FL state without Pomeranchuk order [here δ(...)isthe δ c(s) ∗ As for the l = 1 case, the behavior of the poles is quite in- function]. The order parameter l (t ) will then relax to c(s) ∗ ∝ χ c(s) ∗ + c(s) > volved, particularly for γ>1, and we refrain from presenting zero as l (t ) h l (t ), if 1 Fl 0, and will grow c(s) ≈− + c(s) < all the details. We note only that at F2 1 the purely with time, if 1 Fl 0. ≈− −| c(s)| + γ 2 + χ c(s) ∗ < = imaginary pole is located at s i(1 F2 )( 1 Causality requires that l (t 0) 0. The vanishing γ / γ ≈− −| c(s)| γ ω ≈− − χ c(s) ∗ ∗ < ) 2. For large , s i(1 F2 ) , i.e., i(1 of l (t )fort 0 is guaranteed because the poles and |F c(s)|)˜γ . This pole is not a diffusive one, which to be is branch cuts of the retarded susceptibility χ c(s)(s) are located 2 l ∗ expected because the l = 2 order parameter is not a conserved in the lower frequency half-plane. For t ∗ < 0, e−ist vanishes quantity [49]. at s → i∞, and the integration contour can be closed in the χ c(s) upper half-plane of complex s, where l (s) is analytic. The 2. l = 2, transverse channel integral over s in Eq. (100) then vanishes. For t ∗ > 0, the For vanishingly weak damping (γ → 0), the poles s = integration contour should be closed in the lower half-plane of ± − c(s) s, where χ c(s) has both poles and branch cuts. In this situation, a2 ib2 are in the complex plane of s for negative F2 . l c(s) − χ c(s)(t ∗) is finite. As F2 increases from 1 toward 0, the poles move l c(s) ≈− ∼ − The susceptibility in the time domain can be obtained from the vicinity of the real axis at F1 √ 1[a2 (1 c(s) 1/2 c(s) 1/4 either by contour integration, or directly, by using the form |F |) b ]towarda ≈ b ≈ 1/2 2|F | at 0 < c(s) 2 2 2 2 2 of χ (s) above the branch cut and integrating over real s.In − c(s) c(s) l F2 1. For positive F2 , there is only a single pole on a clean system Eq. (100) can be rewritten as the imaginary axis. c(s) ∗ For finite γ , the equation for the pole is χ (t > 0) l ∞ + c(s) 1 c(s) ∗ c(s) ∗ 2 1 F2 = ds χ s st + χ s st s(s + iγ )(s + iγ − i 1 − (s + iγ )2 ) =− . (98) Re l ( ) cos Im l ( )sin c(s) π 0 F2 2 1 c(s) = χ c(s) ∗. It has two solutions. At small 1 + F both are on the dsIm l (s)sinst (101) 2 π 0 = , ≈ −| c(s)| + γ 2 + imaginary axis: one is a2 0 b2 (1 F2 )( 1 2 χ c(s) ∗ < = γ ) /4γ and another one is a2 = 0, b2 ≈−γ . Note that nei- Inthelastlineweusedthat 1 (t 0) 0 and that γ + c(s) χ c(s) | | < ther mode is diffusive for large .As1 F2 increases, the Im l (s) is nonzero only for s 1. Equation (101) is con-

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B. l = 0 We recall that near the Pomeranchuk transition the only χ c(s) = ≈ pole of 0 (s) in the lower half-plane is located at s si − −| c(s)| i(1 F0 )[seeEq.(19)]. Near this pole, i χ c(s)(s) ≈ ν . (104) 0 F + − c(s) s i 1 F0 Evaluating the residue, we obtain

c(s) ∗ − ∗ −| c(s)| χ = ν t (1 Fl ). pole,0(t ) F e (105) To obtain the branch-cut contribution, we recall that x c(s) ∗ ± √ χ 1 2− FIG. 11. Integration contour for evaluation of l (t ), defined c(s) x 1 χ (x − iδ ∓ i ) = νF . (106) in Eq. (100). The contour is shown for the case of finite disorder, and l − c(s) ± √ x 1 Fl 1 2− the branch cuts are at s =−iγ + x, |x| > 1. The poles are located at x 1 finite distance below the lower edges of the branch cuts. Hence, χ c(s)(x − iδ − i ) − χ c(s)(x − iδ + i ) 0 0 √ venient for numerical calculations. To analyze the behavior of x x2 − 1 c(s) ∗ =−2ν . (107) χ F c(s) 2 c(s) 1 (t ) analytically, it is more convenient to integrate over 1 − F + x2 2 F − 1 the contour shown in Fig. 11, and evaluate the contributions 0 0 from the poles and branch cuts. This way, we get For large x, the right-hand side of Eq. (107) approaches a constant value (=−2), and the integral over x in (103) c(s) ∗ ∗ ∗ formally diverges. This divergence is artificial and can be χ (t ) = χ , (t ) − χ , (t ), (102) l pole l bcut l eliminated by introducing a factor of exp(−αx) with α>0 and taking the limit of α → 0 at the end of the calculation. ∗ c(s) where χpole,l (t ) is the sum of the residues of the poles, For F ≈−1, the leading contribution to the integral − 0 multiplied by i, and in Eq. (103) comes from nonanalyticity √ of the√ integrand at = ∗ ∗ =− π/ ∗ 3/2 x 1.√ For t 1,√ we use dy y cos yt (2t ) , ∞ ∗ = π/ ∗ 3/2 χ ∗ = dx ∗ dy y sin yt (2t ) , and obtain bcut,l (t ) cos xt 1 π ∗ ∗ 2 cos(t − π/4) c(s) c(s) χ , =−ν . × χ − δ − − χ − δ + , bcut 0(t ) F 2 (108) l (x i i ) l (x i i ) π c(s) ∗ 3/2 F0 (t ) (103) Comparing Eqs. (105) and (108), we see that the pole contribution is the dominant one for 1 t ∗ (3/2)| ln(1 − + where = 0 is the combined contribution from the two |F c(s)|)|/(1 −|F c(s)|), while at longer times the time depen- | | > χ c(s) − δ − l l edges of the branch cut along x 1. [By l (x i i ), dence of the response function comes from the end point of χ c(s) = we mean the retarded susceptibility l (s) computed at s the branch cut. − δ − < δ χ c(s) ∗ x i i , where x is a real variable and 0 .] In Fig. 12 we show 0 (t ) computed numerically using In what follows, we focus on the on the behavior of Eq. (101). As is obvious from this equation, χ c(s)(t ∗) increases χ c(s) ∗ = ∗ ∗ 0 l (t ) near the Pomeranchuk instability for l 0 and 1, linearly with t at short times t 1 (the pole and branch c(s) ≈− χ c(s) ∗ ∗ = when the corresponding Fl 1. The analysis of l (t ) contributions cancel each other at t 0). At intermediate c(s) times 1 t ∗ (3/2)ln |1 −|F c(s)||/(1 −|F c(s)|), χ c(s)(t ∗) for larger Fl requires a separate discussion, particularly l l 0 when the poles in χ c(s)(s) are near the lower edges of the exhibits an exponentially decay augmented by weak oscilla- l tions, in agreement with Eqs. (105) and (108). This behavior branch cuts, and will be presented elsewhere [50]. We will c(s) ∗ ∗ ∗ ∗ χ analyze the behavior of χ c(s)(t ) at large t 1. For such t , is shown in the left panel of Fig. 12. At long times, 0 (t ) l oscillates and decreases algebraically with time, in agreement the dominant contribution to χ c(s)(t ∗) comes from the quasi- l with Eq. (108). This behavior is shown in the right panel of particle part of the susceptibility [the first term in Eq. (3)], Fig. 12. the contribution from the incoherent part of the susceptibility c(s) As F becomes closer to −1, the exponential decay of is much smaller. Accordingly, χ c(s)(t ∗) = ( c(s))2χ c(s)(t ∗), 0 l l qp,l χ c(s) ∗ ∗ ∗ (t ) with t becomes slower and the crossover to a power- χ c(s) χ c(s) 0 ∗ where qp,l (t ) is the Fourier transform of qp,l (s). In this law behavior shifts to larger t . Right at the Pomeranchuk c(s) c(s) =− χ c(s) ∗ section we consider a generic case when l is finite near instability, when F0 1, the form 0 (t ) can be found χ c(s) ∗ χ c(s) = ν θ − a Pomeranchuk transition, and focus on qp,l (t ). In the next directly√ from Eq. (101). In this case, Im 0 (s) F (1 = | | − 2/ χ c(s) ∗ section we consider the special case of l 1 charge- and s ) 1 s s. Substituting into (101), we find that 0 (t ) c(s) ∗ spin-current order parameter, for which 1 vanishes at a starts off linearly for t 1, exhibits an oscillatory behavior ∗ ∼ χ c(s) ∗ = ν Pomeranchuk transition. To simplify the expressions, below for t 1, and approaches the limiting value of 0 (t ) F χ c(s) ∗ χ c(s) ∗ ∗ →∞ we write qp,l (t )simplyas l (t ). at t .

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χ c(s) ∗ ν = ∗ c(s) − FIG. 12. qp,0(t ) (with F 1) as a function of the dimensionless time t ,asdeﬁnedinEq.(100), for F0 near 1. The solid line χ c(s) ∗ is the numerically computed response and the dashed lines are the analytic expressions in Eqs. (104)and(108). Left: qp,0(t ) at short c(s) and intermediate times. At intermediate time, the time dependence is dominated by the exponentially decaying pole contribution. As F0 approaches −1, the decay time goes to inﬁnity. Right: long-time behavior, dominated by the oscillatory and power-law decaying contribution from the branch cut.

c(s) < − χ c(s) For F0 1, a long-range order develops. Within our would need to recalculate 0 (t ) in the broken-symmetry approach, we can analyze the initial growth rate of the order state. c(s) ∗ ∗ ∝ parameter 0 (t ) induced by an instant perturbation h(t ) δ ∗ c(s) ∗ ∝ χ c(s) ∗ 1. l = 1, longitudinal channel h (t ) such that 0 (t ) h 0 (t ). The computation of χ c(s)(t ∗)forF c(s) < −1 requires some care because integrat- + c(s) χ long 0 0 For small positive 1 F1 , the poles of 1 (s)aregiven ing Eq. (100) over the same contour as in Fig. 11 we would by Eq. (33). Near the poles, find that χ c(s)(t ∗ < 0) becomes finite, i.e., that causality is 0 1 lost. This issue was analyzed in Ref. [30] (see also, e.g., χ long(s) ∝ +··· , (110) 1 − − Ref. [51]), where it was shown that, to preserve causality, (s s1)(s s2 ) one has to modify the integration contour such that it goes where ··· stands for nonsingular terms. Evaluating the above all poles, as shown in Fig. 13(c). Integrating along the χ long ∗ c(s) ∗ residues, we obtain the pole contribution to 1 (t )as modified contour, we find that χ (t < 0) = 0, as required ∗ 0 by causality. For t > 0wenowhave 1−|F c(s)| ∗ − c(s) 1 long ∗ ∗ 1 F1 ∗ sin 2 t ∗ χ ∝ − . c(s) ∗ t (|F c(s)|−1) , (t ) t exp t t ∝ he 0 , pole 1 −| c(s)| 0 ( ) (109) 4 1 F1 ∗ 2 t i.e., a perturbation grows exponentially with time. This ob- (111) viously indicates that the FL state without Pomeranchuk The branch-cut contribution has the same structure as for = χ long ∗ ∝ ∗ − π/ / ∗ 3/2 ∗ order becomes unstable. To see how the system eventually l 0, i.e., bcut,1(t ) cos(t 4) (t ) for t 1. We c(s) ∗ = ∗ ∼ relaxes to the final equilibrium state with 0 (t ) 0,we see that the pole contribution remains dominant up to t

χ long + c(s) > FIG. 13. Positions of the poles and analyticity regions of 1 (s) in the Fermi-liquid phase without Pomeranchuk order [1 F1 0 + c(s) = + c(s) < (a)], at the transition point [1 F1 0 (b)], and in the ordered phase [1 F1 0 (c)]. In the ordered phase, the susceptibility in the time χ long ∗ domain 1 (t ) is an increasing function of time, and so its Fourier transform is only analytic at ﬁnite distance above the real axis.

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| −| c(s) |/ −| c(s)| ln(1 F1 ) (1 F1 ), which becomes progressively | c(s)| larger as F1 approaches one. We also see that the pole contribution contains two relevant scales ∗ = / − c(s) 1/2 ∗ = / − c(s) . ta 2 1 F1 and tb 4 1 F1 (112) ∗ ∗ Near the Pomeranchuk transition, tb ta 1. Applying an instant perturbation in the l = 1 channel h(t ∗) ∼ hδ(t ∗) long ∗ ∝ χ long ∗ and analyzing the behavior of 1 (t ) h 1 (t ), we ∗ ∗ ∗ find that it grows linearly with t for t ta , i.e., the system initially tends to move further away from equilibrium. ∗ ∗ ∗ For ta t tb , the order parameter oscillates between long ∗ =± χ long ∗ the quasiequilibrium states with 1 (t ) q-eq, where FIG. 14. qp,1 (t ) with and without impurity scattering. Solid: / ∝ / + c(s) 1 2 ∗ ∗ long ∗ numerical calculation for γ = 0and 1 , both for F c(s) =−0.95. q-eq h[2 (1 F1 )] . Finally, for t tb , 1 (t ) de- 2 1 cays exponentially toward zero. At F c(s) =−1, both t and Dashed: asymptotic expressions describing contributions that decay 1 a exponentially with characteristic times t ∗ [Eq. (112)forγ = 0and t diverge and, following an instant perturbation at t = 0, the b b Eq. (114)forγ = 1 ]. long ∗ ∝ ∗ 2 order parameter 1 (t ) h increases linearly with t until the perturbation theory in h breaks down. = c(s) ∗ →∞ The difference with the l 0 case, when 0 (t ) C. Response in the time domain in the presence of disorder c(s) =− at F0 1 is finite, can be understood by noticing that Near Pomeranchuk instabilities in the l = 0 and transverse χ long ∗ ∗ the behavior of free,1(t )forlarget is determined by l = 1 channels, the poles are on the imaginary axis, and χ long adding weak impurity scattering will not change the results that of Im free,1(s) for small s. Equation (91) shows that χ long = + + δ | | c(s) = obtained in Sec. IV B. Namely, the pole’s contribution to free,1(s) 1 2s(s i )for s 1. At F1 1, there- χ c(s) ∗ ∗ c(s) > − long 0 (t ) still decays exponentially with t for F0 1, fore, we have χ (s) ≈−1/2s(s + iδ), and, at vanishingly ∗ c(s) 1 becomes independent of t at F =−1, and increases ex- small δ,Imχ long(s) goes over to (π/2)δ(s)/s. Substituting this 0 ponentially with t ∗ for F c(s) < −1. into Eq. (101), we find that χ long(t ∗) ∝ t ∗. 0 1 In the longitudinal l = 1, the poles remain near the real c(s) < − χ long, For F1 1, both poles of 1 (s) are located on the axis also in the presence of γ . Finite damping changes the =±| + c(s)|/ 1/2 ∗ imaginary axis, at s ( 1 F1 2) . One of the poles timescale tb from Eq. (112)to is now in the upper half-plane of complex s. Modifying the integration contour the same way as for l = 0 to preserve ∗ = + γ 2 − γ / − c(s) . tb 4( 1 ) 1 F1 (114) causality, we obtain for t ∗ > 0 χ long ∗ c(s) After this change, the results for 1 (t ) remain the same as |F |−1 ∗ sinh 1 t in the absence of disorder. The time dependence of χ long in χ long(t ∗) ∝ t ∗ 2 . (113) 1 1 | c(s)|− the presence of impurity scattering is shown in Fig. 14. F1 1 ∗ 2 t In all cases, finite γ modifies the branch-cut contribution, so that in addition to the algebraic decay, there is also an ∗ ∗ = /| + c(s)| 1/2 χ long ∗ For t ta (2 1 F1 ) , both 1 (t ) and exponential decay. For l = 0 we find long(t ∗) ∝ hχ long(t ∗) increase linearly with t ∗.Fort ∗ t ∗, 1 1 a ∗ − π/ the perturbation grows exponentially, indicating that the FL ∗ −γ t ∗ cos(t 4) χ , (t ) ∝ e . (115) state becomes unstable. bcut 0 (t ∗)3/2 We note in passing that the need to bend the integration c(s) < − Similar expressions hold for l > 0. contour around the pole for F1 1 can be also understood χ long ∗ c(s) The presence of the exponentially decaying terms due to by considering the behavior of 1 (t )atF1 approaching > c(s) > c(s) + damping is particularly relevant for l 0 and Fl 0, as it −1 from above. In the limit F →−1 + 0 , the two poles c(s) 1 allows one to distinguish between the cases of smaller F . χ long l of 1 (s) coalesce into a single double pole at the origin, For the former, the zero-sound pole is present and located χ long ∗ =± − γ > as shown in Fig. 13(b). Had we tried to compute 1 (t )by above the branch cut, at s a ib , where a 1 and integrating along the real axis of s, we would have intersected b < 1. For the latter, the zero-sound pole is located on the χ c(s) a divergence. To eliminate the divergence, one needs to bend unphysical Riemann sheet. In both cases, Im l (s) at real the integration contour and bypass the double pole along =± c(s) ¯γ s has a peak at s a, but for larger Fl its width is b a semicircle above it. The extension of this procedure for ¯ > γ c(s) with b 1, i.e., it is larger than . Accordingly, for smaller < − c(s) ∗ ∗ F1 1 yields the contour shown in Fig. 13(c). χ Fl , the dominant contribution to l (t ) at large t comes ∗ −bγ t ∗ ∗ c(s) from the pole, and χl (t ) ∝ e cos(at ). For larger F , 2. l = 1, transverse channel ∗ ∗ ∗l χl (t ) at large t comes from the branch cut, and χl (t ) ∝ + c(s) −γ t ∗ ∗ − π/ / ∗ 3/2 For small 1 F1 , the pole in the transverse susceptibility e cos(t 4) (t ) . Because this property holds only = χ tr ∗ > for l 1 is on the imaginary axis. The behavior of 1 (t )is for l 0 and in the presence of disorder, it was not discussed then the same as for the l = 0 case. in previous works [9,31–37], which studied collective modes

033134-21 KLEIN, MASLOV, PITAEVSKII, AND CHUBUKOV PHYSICAL REVIEW RESEARCH 1, 033134 (2019) of a 2D Fermi liquid either in the l = 0 channel or in the Furthermore, in the Galilean-invariant case the static l = 1 absence of disorder. charge and spin susceptibilities are not renormalized at all by the electron-electron interaction [29]. On the other hand, ∗/ = + c(s) = V. SPECIAL CASES OF CHARGE- AND SPIN-CURRENT m m 1 F1 does vanish at the transition in the l 1 ORDER PARAMETERS charge channel. We believe that the vanishing mass indicates a global instability of a non-Pomeranchuk type, which is not = A. Ward identities and static susceptibility in the l 1 channel associated with the l = 1 deformation of the FS. In previous sections, we assumed that the behavior of the full susceptibility is at least qualitatively the same as that B. Dynamical susceptibility in the l = 1 channel of the quasiparticle susceptibility, i.e., the collective modes, Now, let us look at the dynamics. Consider for definiteness present in χ c(s)(q,ω), are also present in the full χ c(s)(q,ω). qp,l l the l = 1 longitudinal susceptibility. Near 1 + F c(s) = 0, the The full and quasiparticle susceptibilities differ by the factor 1 quasiparticle susceptibility has poles given by Eq. (33). One ( c(s))2 [see Eq. (3)], which accounts for renormalizations l of the poles moves into the upper frequency half-plane when from high-energy fermions. For a generic order parameter 1 + F c(s) becomes negative. To relate the full and quasiparti- with a form factor f c(s)(k), the vertex c(s) isassumedtobe 1 l l cle dynamical susceptibilities, we need to know c(s) in the finite for all F c(s), including F c(s) =−1. The pole structure of 1 l l dynamical case. The FL theory assumes that c(s) can be χ c(s)(q,ω) is then fully determined by that of χ c(s)(q,ω). 1 l qp,l computed by setting both ω and q to zero. The argument is that We now consider the special case of order parameters with c(s) is renormalized only by high-energy fermions, hence, its = c(s) = cos θ ∂ /∂ 1 l 1, for which f1 (k) (sin θ ) k k, up to an overall frequency and momentum dependencies come in a form of factor. These order parameters correspond to charge or spin regular functions of q/kF and ω/EF . If so, then the relation currents. The special behavior of a FL under perturbations of c(s) + c(s) between 1 and 1 F1 ,Eq.(116), holds in the dynamical this form has been discussed in recent studies of the static case as well. We will verify this statement explicitly via a susceptibility in the l = 1 channel [26–28]. Namely, for the perturbative calculation in Sec. VD. spin- or charge-current order parameter, the vertices c(s) c(s) 1 Taking 1 from Eq. (116) and substituting it along with satisfy the Ward identities which follow from conservation the dynamical quasiparticle susceptibility into Eq. (3), we of the total number of fermions (the total “charge”) and total c(s) ≈− obtain the full susceptibility for F1 1: spin. In the static limit, the Ward identities read as [28,52] + c(s) 2 ∗ 2 ∗ long m 1 F1 (vF q) m χ (q,ω) ≈−NF + χinc,1, Z c(s) = 1 + F c(s). (116) 1 m∗ ω2 − 1 + F c(s) (v∗ q)2/2 m 1 1 1 F Under certain assumptions, these identities allow one to (118) decide which of three factors on the left vanishes at the where we recall that the last term represents the contribution instability. First, we assume that the Z factor, being a high- from high-energy fermions. The static limit of the first term + c(s) / ∗ energy property of the system, remains finite at the instability. in the equation above, i.e., NF (1 F1 )(m m ), is indeed ∗/ c(s) c(s) → Therefore, the product (m m) 1 should vanish at F1 nonsingular at the transition, in agreement with the conclu- −1. Next, we divide the versions of Eq. (116) for the charge sions of the previous section. Nevertheless, one of the poles of χ long ,ω and spin channels by each other and obtain 1 (q ) moves into the upper frequency half-plane when c c + c(s) 1 + F 1 F1 becomes negative, i.e., a dynamical perturbation 1 = 1 . (117) s 1 + F s with the structure of spin or charge current grows exponen- 1 1 tially with time, which is an indication of a Pomeranchuk c s We then rule out a very special case, when both F1 and F1 instability. The peculiarity of the l = 1 case in that the residue reach the critical value of −1 simultaneously, and also assume of the pole vanishes right at the transition, but it is finite both the charge (spin) vertex remains finite at an instability in above and below the transition. c(s) + c(s) the spin (charge) channel. Then, 1 vanishes as 1 F1 , ∗/ which implies that m m remains finite. C. Case of more than one nonzero Landau parameter ∗/ c(s) 2 Given that m m remains finite while ( 1 ) vanishes c(s) c(s) It is instructive to derive an analog of Eq. (118)fora as (1 + F )2, the full static susceptibility χ (q,ω = 0) = 1 1 more general case of several nonzero Landau parameters. We c(s) 2χ c(s) ,ω = + χ c(s) c(s) = ( 1 ) qp,1(q 0) inc,1 does not diverge at F1 − remind the reader that in this situation the pole structure of 1, despite the fact that the quasiparticle susceptibility χ long ,ω c(s) χ c(s) ,ω = / + c(s) 1 (q ) is more complex than when only F1 is present qp,1(q 0) diverges as 1 (1 F1 ). c(s) c(s) What was said above does not apply to the special case of [see Eq. (65) for the case when F1 and F0 are nonzero]. χ long ,ω a Galilean-invariant system. In this case, the charge current is The issue we address is whether 1 (q ) for charge and c(s) 2 ∝ + c(s) 2 equivalent to the momentum and thus is conserved. The Ward spin currents still has ( 1 ) (1 F1 ) as the overall c = c identity for the momentum implies that Z 1 1, i.e., 1 factor. We argue that it does. + c = remains finite at 1 F1 0. Equation (116) then implies that To demonstrate this, we need to express the full suscep- ∗/ = + c ¯ c(s) ,ω m m 1 F1 , which is the standard result for a Galilean- tibility via vertices (q ), which include both high- invariant FL. The static susceptibility still remains finite at and low-energy renormalizations. Vertices ¯ c(s)(q,ω) can be + c → ∗/ ¯ c(s) ,ω = 1 F1 0, this time because the factor of m m in the expanded into a series of partial harmonics: (q ) χ c + c c(s) θ θ numerator of qp,1 cancels out with 1 F1 in its denominator. l al l (s) cos l , where is the angle between the

033134-22 COLLECTIVE MODES NEAR A POMERANCHUK … PHYSICAL REVIEW RESEARCH 1, 033134 (2019) momentum√ of the incoming fermion and q, a0 = 1, and where K0,1,2 are defined by Eq. (66). Substituting the last two al =0 = 2. With this definition, the full longitudinal suscep- equations into Eq. (119), we obtain tibility in the l = 1 channel can be written as c(s) 2F K2 K + K − 0 1 0 2 1+F c(s)K θ χ long = ν c(s) 2 0 0 . c(s),long d c(s) (s) F c(s) c(s) χ =−ν ¯ θ 1 1 c(s) 2F F K2 1 (s) F al l (s) cos l 1 + F (K + K ) − 0 1 1 π 1 0 2 1+F c(s)K l 0 0 cos2 θ (123) × c(s). (119) − θ + δ 1 χ long s cos i Comparing the last result with qp,1 in Eq. (65), we see that χ long(s) = ( c(s))2χ long(s), exactly as in the static case. This The vertex ¯ c(s)(q,ω) is given by a series of diagrams which 1 1 qp,1 l result implies that the residue of the pole is proportional to contain momentum and frequency integrals of the product c(s) 2 ∝ + c(s) 2 ( 1 ) (1 F1 ) and thus vanishes at the Pomeranchuk Gp+ q ,ω + ω Gp− q ,ω − ω , convoluted with fully renormalized 2 p 2 2 p 2 instability also for the case of two nonzero Landau param- four-fermion vertices. The diagrammatic series can be rep- eters, when the pole structure of the susceptibility becomes resented as the sum of subsets of diagrams, each with a more involved. Still, like in the case when only F c(s) is = , , , ... 1 fixed number n 0 1 2 3 of cross sections which contain c(s) 2 = ω/ ∗ ω nonzero, ( 1 ) is independent of s (vF q), and it does contributions from the regions where the poles of Gp+ q ,ω + 2 p 2 χ long + c(s) < and G − q ,ω − ω are in the opposite half-planes of complex not cancel the poles in qp,1 .At1 F1 0, one pole moves p 2 p 2 frequency. This constraint binds the internal p and ωp to the into the upper frequency half-plane, signaling a Pomeranchuk FS. The subset with n = 0 is nonzero only for l = 1 and gives instability. c(s) > 1 , while the sum of contributions with different n 0 ¯ c(s) gives l (s). Combining the contributions from all n, we find D. Perturbation theory for the vertex in the dynamical case ¯ c(s) c(s) that the vertex l (s) satisfies an integral equation with 1 We now return to the case of a single Landau parameter as the source term: c(s) c(s) 2 F1 and verify by a perturbative calculation that ( 1 ) does 2 ∗ not cancel the dynamical poles in the full l = 1 susceptibility. c(s) c(s) Z m c(s) ¯ (s) = δ , + ¯ (s)a We perform the calculation to second order in the Hubbard l 1 l 1 4π 3 l l l (pointlike) interaction U. For simplicity we limit our attention 2π 2π to the spin channel and also consider a Galilean-invariant × θ θ θ θ ¯ s d d cos l cos l system. We will show that the dynamical vertex 1(s) has 0 0 c(s) the pole structure of Eq. (121) with s independent 1 . cos θ s × c(s) θ,θ , To demonstrate this, it suffices to show that the vertex 1, ( ) (120) ¯ s s − cos θ + iδ which acts as a source for the dynamical vertex 1(s)in Eq. (120), does not vanish at s which corresponds to the pole where c(s)(θ,θ ) is the four-fermion (four-leg) vertex with s of the dynamical vertex. Instead of calculating 1 directly, we external fermions right on the FS. By construction, c(s)(θ,θ ) s compute the product 1Z for reasons that will become clear contains only renormalizations from high-energy fermions ω later in the section. Using the Ward identity associated with (in the FL theory, such a vertex is called ,seeRef.[7]). the Galilean invariance, we express quasiparticle Z as [7] c(s) Landau parameters Fl are related to the angular harmonics c s c(s) ∗ c(s) c(s) 3 of ( )(θ,θ )viaF = (Z2m /π ) . When only F is 1 i d p ω 2 ω l l 1 = 1 − αβ,αβ (k, p) G (kˆ · p), ∗ c(s) 3 p 2 /π c(s) θ,θ = θ θ Z 2kF (2π ) nonzero, i.e., (Z m ) ( ) 2F1 cos cos , only αβ ¯ c(s) 1 (s) is nonzero as well. Then, (124) where the (2 + 1)-momentum p = (p,ωp) is not necessarily c s ( ) close to the FS, and k = (k kˆ, 0) + is infinitesimally close ¯ c(s)(s) = 1 F 1 − c(s) 1 2π θ cos3 θ to the FS, i.e., = (q,ω) with both |q| and ω being infinites- 1 F π d − θ+ δ 1 0 s cos i imally small. The direction of kˆ in Eq. (124) is arbitrary. The c(s) ω is the dressed four-fermion vertex and (G2 )ω(kˆ · p) = 1 . (121) αβ,αβ p 1 + F c(s)χ long (s) is the regular part of the product Gp+ /2Gp− /2 of two exact 1 free,1 Green’s functions, whose arguments differ by . (Note that the c(s) q and ω in the definition of the Z factor do not need to coincide Substituting Eq. (121)into(119) and expanding near F = 1 with the corresponding variables describing the collective − F c(s) F c(s) 1, we reproduce Eq. (118). When, e.g., 0 and 1 are mode, but we choose them to be the same for simplicity.) The nonzero, the solution of Eq. (120)is product Gp+ /2Gp− /2 can be written as the sum of a regular 1 part and a singular contribution from the FS [7]: ¯ c(s)(s) = c(s) , 1 1 2F c(s) )F c(s)K2 2 + c(s) + − 0 1 1 ω 2πiZ pˆ · qˆ 1 F1 (K0 K2 ) + c(s) = 2 + 1 F0 K0 Gp+ /2Gp− /2 Gp ∗ √ v s − pˆ · qˆ + iδ sgnωp c(s) F c(s) c(s) 2F0 K1 ¯ (s) =− , (122) × δ(ωp)δ(|p|−kF ), (125) 0 1 2F c(s)F c(s)K2 + c(s) + − 0 1 1 1 F (K0 K2 ) c(s) ∗ 1 1+F K0 = /| | = ω/ | | 0 wherep ˆ p p and, as before, s vF q .

033134-23 KLEIN, MASLOV, PITAEVSKII, AND CHUBUKOV PHYSICAL REVIEW RESEARCH 1, 033134 (2019)

p p p k p p p p p k

p k k k k k p k k k

p p p k p k p k

k p k k k p k p

ω FIG. 15. Diagrams for the four-fermion vertex αβ,γδ to second order in the Hubbard-type interaction. One of external (2 + 1)-momenta, k = (k,ωk ), is chosen to be on the FS, i.e., |k|=kF and ωk = 0, while the other, p, is generically away from the FS. All internal momenta are ω away from the FS. In this sense, renormalization of αβ,γδ comes from high-energy fermions.

ω To second order in U, αβ,αβ is given by the diagrams and shown in Fig. 15. Explicitly, 2i d3 p ω s = 1 − s(k, p) G2 (kˆ · p). (129) 3 1 3 p ω 1 2 d k kF (2π ) αβ,γδ(k, p) = δαγ δβδ U + iU 2 (2π )3 s To second order in U, the product 1Z can then be written as 3 × 2G G − + + G G + − 2i d p k p k k k p k k s Z = 1 − 1 3 kF (2π ) 3 1 2 d k c s 2 ω − σαγ · σβδ U + iU G G + − × [ (k, p) − (k, p)]kˆ · p G . (130) 2 (2π )3 k p k k p s c s We now show that the product Z does not depend on ≡ δαγ δβδ (k, p) + σαγ · σβδ (k, p), (126) 1 = ω/ ∗ | | ˆ · s vF q . To see this, we add a term kF s to k p on the where at the last step we defined the charge and spin parts of right-hand side of Eq. (130) and then subtract off the same c s the four-fermion vertex (k, p) and (k, p), respectively. term. Equation (130) then goes over to The renormalized spin-current vertex can be written as (see s = − − , Fig. 16) 1Z 1 Q1 Q2 (131) i d3 p where s σ z = σ z − ω (k, p) 1 ββ ββ π 3 αβ,αβ 3 kF α (2 ) d p c s 2 ω Q1 = 2is [ (k, p) − (k, p)] G (132) ω π 3 p × 2 ˆ · σ z , (2 ) Gp (k p) αα (127) and where the internal momentum p = (p,ωp) is again not con- 3 fined to the FS. It is to be understood that s is a function d p p ω 1 Q = 2i [ c(k, p) − s(k, p)] kˆ · − s G2 . + s = = 2 π 3 p of the (2 1)-momentum , and 1 0evenfor 0. (2 ) kF Substituting the last formula in Eq. (126)into(124) and (129) (133) and summing over spin indices, we obtain We now use the fact that conservation of charge and 1 2i d3 p ω = 1 − c(k, p) G2 (kˆ · p) (128) spin allows one to derive two independent relations for Z π 3 p Z kF (2 ) (Refs. [3,4,7,28]): 1 i d3 p ω = 1 − c(k, p) G2 (charge); 3 p kk kk Z 2kF (2π ) kk αβ pp 1 i d3 p ω = + = 1 − s(k, p) G2 (spin). (134) p π 3 p p Z 2kF αβ (2 ) kk kk kk = s = − Combining the two, we find that Q1 0, i.e., 1Z 1 Q2. We now analyze Q2.TofirstorderinU, the vertex remains FIG. 16. Diagrammatic representation of the high-energy triple static. Then, O(U )termsin c(s) in Eq. (133) vanish because vertex s . The shaded box is ω . 2 ω O 2 1 αβ,γδ of the double pole of (Gp) . Let us focus on the (U )terms.

033134-24 COLLECTIVE MODES NEAR A POMERANCHUK … PHYSICAL REVIEW RESEARCH 1, 033134 (2019)

c(s) c(s) Substituting Eq. (125)into(133)forQ2 and choosing the h1 1 , and the total susceptibility is identical to the quasi- direction of k to be along q, we obtain particle susceptibility. For the charge- and spin-current order, c(s)c(s) d3 p p the coupling is still proportional to h1 1 term, but the con- Q = 2i [ c(k, p) − s(k, p)] kˆ · − s 2 3 tributions from high-energy fermions to the proportionality (2π ) kF coefficient cannot be neglected, as with these contributions the π 2 ˆ · fully dressed coupling vanishes at the transition. To see this, × − 2 iZ k pˆ Gp+ /2Gp− /2 ∗ v s − kˆ · pˆ + iδ we explicitly separate the four-fermion interaction into the F components coming from the states near and away from the × δ(ω )δ(|p|−p ) (135) FS. Such an approach has been used in statistical FL theory p F and in renormalization group studies of FLs [54–60]. d3 p Our point of departure is the effective, antisymmetrized in- = 2i [ c(k, p) − s(k, p)] 3 teraction between fermions, expressed via the vertex function (2π ) αβ;γδ(k, k ; q), where q is a small momentum transfer: × ∗ | | −1 − [(vF q ) (Gp− /2 Gp+ /2 ) H = , † † . π 2 int αβ;γδ(k k ; q)a + q ,αa − q ,γ a − q ,β a + q ,δ 2 iZ k 2 k 2 k 2 k 2 + kˆ · pˆδ(ω )δ(|p|−p )]. (136) k,k,q,α,β,γ ,δ v∗ p F F (140) One can now verify that the first term in the right-hand side of = , Eq. (136) is zero, i.e., The generic form of the l 1 component of αβ;γδ(k k ; q) is 3 d p c s = [ (k, p) − (k, p)](Gp− /2 − Gp+ /2 ) = 0. l 1 , =−˜ · ˜ c c |˜ |, |˜ | δ δ π 3 αβ;γδ(k k ; q) k k U1 f ( k k ) αγ βδ (2 ) + s s |˜ |, |˜ | σ · σ , (137) U1 f ( k k ) αγ βδ (141) This can be done by substituting the explicit expressions for where k˜ = k/kF and k˜ = k /kF , and kF should be treated c(s) from Eq. (126) and changing integration variables [53]. here as just a normalization constant, which we choose for We are then left with a contribution coming solely from the convenience to match the Fermi wave number of the quasi- FS, particles. For charge- and spin-current orders, we replace the c,s f |k˜ |, |k˜ | Z2 dθ form factors ( ) by constants and incorporate them Q = θ c θ,θ − s θ,θ into U c(s). An instability in the l = 1 channel can occur if 2 π ∗ π cos [ ( ) ( )] 1 2 vF c(s) > U1 0. Below, we approximate the full vertex function = c − s, = F1 F1 (138) αβ;γδ by its l 1 component. Other components are not necessarily small, but we assume they are irrelevant for the low- where θ and θ are the azimuthal angles of kˆ andp ˆ, respec- energy theory near the l = 1 Pomeranchuk instability. In this tively. This term is independent of s and only contributes to approximation, the effective interaction is separable into two the static vertex [26]. Going back to Eq. (131), we obtain parts that depend on k˜ and k˜ , and can be written as the sum H = Hc + Hs s c s m s of the charge and spin components int int int, where Z = 1 − F + F = (1 + F ), (139) ⎛ ⎞ 1 1 1 m∗ 1 whereweusedtherelationm∗/m = 1 + F c valid for a Hc(s) =− c(s) ⎝ ˜ † c(s) ⎠ 1 int U1 ka + q ,αtαγ a − q ,γ k 2 k 2 Galilean-invariant system. This result agrees with the analysis q k,α,γ in the previous section. ⎛ ⎞ × ⎝ ˜ † c(s) ⎠, k ak− q ,βtβδ ak+ q ,δ (142) E. Charge- and spin-current order parameter: 2 2 k,β,δ Ginzburg-Landau functional and time evolution c = δ s = σ z We now analyze the structure of the Landau functional that and, as before, tμν μν and tμν μν . We next rewrite the describes the l = 1 Pomeranchuk transition. Our purpose is sums over the fermionic momenta as to reconcile the apparent contradiction that on one hand the † c(s) c(s) ak+ q ,αtαγ ak− q ,γ FL ground state becomes unstable for F < −1, while on 2 2 1 k the other hand the static l = 1 susceptibility remains finite at c(s) =− = † c(s) δ δ F1 1. The Ginzburg-Landau functional can be derived a + q ,αtαγ a − q ,γ |k+ q |,k |k− q |,k k 2 k 2 2 F 2 F from the Hamiltonian of interacting fermions, coupled to k an infinitesimal external perturbation hc(s), via a Hubbard- 1 + † c(s) − δ δ c(s) ak+ q ,αtαγ ak− q ,γ (1 |k+ q |,k |k− q |,k ) (143) Stratonovich (HS) transformation with an auxiliary field . 2 2 2 F 2 F 1 k For a generic l = 1 order parameter, the vertex remains finite δ at the Pomeranchuk transition. In this case, it is sufficient to (and the same for the sum over k ). Here, a,b is nonzero consider only the quasiparticle part of the Hamiltonian and only for |a − b| < , and is small compared to kF and will neglect the contributions from high-energy fermions. Then, be taken to zero at the end of the calculation. The purpose δ the coupling to the external field is given by a bilinear term of the projectors a,b is to split the fermions into those near

033134-25 KLEIN, MASLOV, PITAEVSKII, AND CHUBUKOV PHYSICAL REVIEW RESEARCH 1, 033134 (2019) the FS, which form the FL of quasiparticles, and those away perturbation. Below, we denote fermions near the FS as from the FS, whose role is to renormalize the interaction ψ†(ψ), and fermions away from the FS as ψ˜ †(ψ˜ ). Using between quasiparticles and their coupling to an external (143), we rewrite (142)as

⎡ ⎤⎡ ⎤ H =− c(s) ⎣ ˜ ψ† c(s)ψ + ψ˜ † c(s)ψ˜ ⎦⎣ ˜ ψ† c(s)ψ + ψ˜ † c(s)ψ˜ ⎦. int U1 k + q ,αtαγ − q ,γ + q ,αtαγ − q ,γ k − q ,βtβδ + q ,δ − q ,βtβδ + q ,δ k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 q k,α,γ k,β,δ (144)

The coupling of the charge- and spin-current order parameter c(s) , to the external field acquires a factor of c(s); and (iii) to a weak external field h1 (q t ) (which may be time depen- 1 dent) can be split into the low- and high-energy parts in the the coupling constant of the interaction between low-energy − c(s) /ν same way: fermions is renormalized from U1 to F1 F . With these modifications, the action for properly normalized low-energy H = hc(s)(q, t ) ψ† ψ h 1 fermions ( k and k) becomes q † −1 ∗ S[ψ] =− ψ Z [ωk − v (|k|−kF )]ψk × ˜ ψ† c(s)ψ + ψ˜ † c(s)ψ˜ . k F k + q ,αtαγ − q ,γ + q ,αtαγ − q ,γ k k 2 k 2 k 2 k 2 k,α,γ c(s) c(s) † c(s) + θ ψ ψ q 1 h1 (q) cos k k+ q ,αtαβ k− ,β ψ ψ˜ 2 2 Note that the field couples to both and . q k,α,β The interaction in Eq. (144) contains one term involving 1 c(s) † c s four fermions near the FS, one term involving four fermions + θ θ ψ ( )ψ q F1 cos k cos k + q ,γ tαγ k− ,α ν k 2 2 away from the FS, and two mixed terms involving two F k,k,q,α,β,γ ,δ fermions at the FS and two away from the FS. We decouple † c(s) × ψ ψ q , the quartic term with fermions away from the FS by going − q ,δtβδ k + ,β (147) k 2 2 from a Hamiltonian representation to a Lagrangian one, upon † where the summation over k is confined to the vicinity of the which ψ and ψ with n = k, p, q become Grassman fields, c(s) n n FS. We see that the factor only changes the response which depend on the (2 + 1)-momentum n = (n,ω ). Next, 1 n function to an external perturbation, but does not affect the we introduce a momentum- and time-dependent HS vector thermodynamic stability of the FL state. If we compute the ˜ c(s) , field 1 (q t ) to decouple only the term in Hint that in- response function by differentiating the partition function volves fermions away from the FS, leaving the term involving c(s) twice with respect to h1 (q), we find the same expression as fermions near the FS and the two mixed terms untouched. in Eq. (3): We then integrate out the high-energy fermions and obtain the c(s) ∂2Z effective action for ˜ . The action is given by χ c(s) = 1 1 (q) ∂hc(s)(q) 2 ˜ c(s) 2 1 c(s) 1 (q) c(s) ⎛ ⎞ S ˜ = + , |β=α, = 2 1 c(s) ln Mα,β (k k ) k k q U1 ,α ⎝ c(s) † ⎠ c(s) k = ψ ψ − / ,α cos θ +χ (145) 1 k+q/2,α k q 2 k inc,1 k,α with S[ψ] ˜ + ˜ = c(s) 2χ c(s) + χ c(s) , c(s) −1 c(s) k k 1 qp,1(q) inc,1 (148) M (k, k ) = G (k)δ , δαβ − t αβ 0 k k αβ 2 where Z is the partition function, ...S[ψ] denotes averaging S ψ χ · c(s) − ˜ c(s) + c(s) with action [ ], and inc,1 is obtained by differentiating h1 (q) 2 1 (q) U1 −S ˜ c(s) c(s) Z˜ = [ 1 ] e twice with respect to h1 without taking into account the contribution from low-energy fermions [the ψ†ψ † c(s) × ψ ψ q | , p˜ − q ,γ tγδ p+ ,δ q=k −k (146) term in (146)]. We recall that the static susceptibility does not p 2 2 ∗ ,γ ,δ c(s) =− c(s) = / + c(s) p diverge at F1 1 because 1 (m m Z)(1 F1 ) vanishes at F c(s) =−1. where G0(k) is the free-fermion propagator. 1 c(s) Corrections to the low-energy theory from high-energy We now introduce a low-energy HS field 1 (q) to de- fermions are obtained by expanding S[˜ c(s)] up to first order couple the quartic term in S[ψ]. Integrating out low-energy 1 c(s) c(s) ψ†ψ fermions, we obtain the effective action for (q)inthe in h1 (q) and up to second order in . For definite- 1 = form ness, we consider the longitudinal l 1 channel and restrict the external field to a longitudinal component, i.e., we set S c(s) = a|c(s)(q)|2 + b|c(s)(q)|4 c(s) = c(s) 1 1 1 h1 (q) qhˆ 1 (q). The new terms generated by integration q c(s) over ˜ (q) affect the action for low-energy fermions in 1 + c(s)hc(s)(q)c(s)(q)χ c(s) (q) + c.c.) (149) three ways: (i) the propagator of low-energy fermions acquires 1 1 1 1,free ∗ a Z factor and vF gets renormalized into vF ; (ii) the coupling plus higher-order terms.

033134-26 COLLECTIVE MODES NEAR A POMERANCHUK … PHYSICAL REVIEW RESEARCH 1, 033134 (2019)

∝ + c(s) with time, i.e., the system becomes unstable toward sponta- In Eq. (149), a 1 F1 changes sign at the critical c(s) neous development of a uniform order parameter, bilinear in point, i.e., fluctuations of the order parameter 1 diverge at the critical point, like for any other order parameter. In this fermions. c(s) > sense, Pomeranchuk order with the structure of spin/charge We also discussed the evolution of the poles with Fl + c(s) −1 both for infinitesimally small and for finite fermionic current does develop when 1 F1 becomes negative. What makes the case of spin/charge current special is that the damping rate. For infinitesimally small damping, we found response to an external field gets critically reduced because that in some channels, the poles are located very close to a real of destructive interference from high-energy fermions. frequency axis and outside particle-hole continuum already c(s) ∝ + c(s) for negative (attractive) F c(s). This result is at a first glance an The presence of 1 1 F1 in the response function l c(s) ∗ unexpected one as naively one would expect the poles to be changes the time evolution of 1 (t ) after an instant pertur- c(s) ∗ = c(s)δ ∗ + c(s) > located inside the continuum. We found that these poles are bation h1 (t ) h1 (t ). For 1 F1 0, we have located below the branch cut and cannot be gradually moved c(s) ∗ c(s) c(s) 2 ∗ −(1+F c(s) )t ∗/4 (t ) ∝ h 1 + F t e 1 to real axis without simultaneously moving from the physical 1 1 1 Riemann sheet to an unphysical one. As the consequence, 1+F c(s) ∗ sin 1 t these poles are hidden in the sense that, although they do × 2 . (150) exist infinitesimally close to the real axis, they are not visible 1+F c(s) c(s) 1 ∗ in Im χ (ω) for real ω. Besides, we found that for l > 2 t l 0, zero-sound poles for positive F c(s) exist only if F c(s) is c(s) ∗ l l c(s) The functional form of 1 (t ) is the same as for a generic below a certain value. For larger F , the poles move from = 1 l 1 order parameter, when high-energy renormalizations the physical Riemann sheet to an unphysical one. This does + can be neglected, just the amplitude is smaller. For 1 not eliminate the zero-sound peak in Im χ c(s)(ω) for real c(s) < l F1 0, a deviation from the normal state grows as ω, but the width of the peak becomes larger than fermionic γ |1+F c(s)| ∗ damping . We argued that in this situation the behavior of sinh 1 t χ c(s) c(s) ∗ ∝ c(s) + c(s) 2 ∗ 2 . time-dependent susceptibility l (t ) at large t is determined 1 (t ) h1 1 F1 t (151) |1+F c(s)| by the end point of the branch cut (ω =±vF q) rather than by 1 t ∗ 2 the zero-sound peak. The functional form is again the same as for a generic l = 1 We next showed that the situation is somewhat different order parameter. The presence of the overall small factor (1 + for l = 1 order parameters with the same form factors as that c(s) 2 of spin or charge currents. In these two cases, the bosonic F1 ) just implies that it takes a longer time for a deviation to develop. In particular, the ratio of c(s)(t ∗)in(151) and response has a zero-sound pole that crosses to the upper 1 half-plane at F c(s) < −1, but its residue vanishes precisely the initial perturbation hc(s) becomes O(1) only after c(s)(t ∗) 1 1 1 c(s) =− begins to grow exponentially. at F1 1. We argued that in this situation, the static c(s) =− uniform susceptibility does not diverge at F1 1, yet + c(s) < VI. CONCLUSIONS at 1 F1 0 the system still develops long-range Pomer- anchuk order, and the shape of the FS gets modified. It just In this paper, we analyzed zero-sound collective bosonic takes more time for the system to reach the steady ordered excitations in different angular momentum channels in a metal state. with an isotropic, but not necessarily parabolic, dispersion k. We explicitly computed the longitudinal and transverse dynamical susceptibility χ c(s)(q,ω) in charge and spin channels l ACKNOWLEDGMENTS for l = 0, l = 1, and l = 2, and extracted zero-sound modes ω = ∗ χ c(s) ,ω at svF q from the poles of l (q ). We also presented We thank L. Levitov, G. Orso, J. Schmalian, P. Woelfle, the generic structure of zero-sound excitations for arbitrary and Y-M. Wu for stimulating discussions. This work was frequency. Our key goal was to identify, in each case, the supported by the NSF Grants No. DMR-1523036 (A.K. and mode, whose frequency moves from the lower to the upper A.V.C.), No. NSF-DMR-1720816 (D.L.M.), and UF DSR half-plane as the system undergoes a Pomeranchuk instability, Opportunity Fund No. OR-DRPD-ROF2017 (D.L.M.). L.P.P. c(s) =− when the corresponding Landau parameter Fl 1. Right acknowledges financial support from the A. V. Humboldt at the transition, the mode is located at ω = 0, i.e., the static foundation. A.V.C. is thankful to the Aspen Center for Physics c(s) < − susceptibility diverges. At Fl 1, the mode moves to (ACP) for hospitality during the completion of this work. the upper frequency half-plane, and a perturbation around ACP is supported by National Science Foundation grant PHY- a state with no Pomeranchuk order grows exponentially 1607611.

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