The Conditions for $ L= 1$ Pomeranchuk Instability in a Fermi

The Conditions for l = 1 Pomeranchuk Instability in a Fermi Liquid

Yi-Ming Wu, Avraham Klein, and Andrey V. Chubukov School of Physics and Astronomy, University of Minnesota, MN (Dated: January 23, 2018) We perform a microscropic analysis of how the constraints imposed by conservation laws aﬀect q = 0 Pomeranchuk instabilities in a Fermi liquid. The conventional view is that these instabilities are determined by the static interaction between low-energy quasiparticles near the Fermi surface, in the limit of vanishing momentum transfer q. The condition for a Pomeranchuk instability is set c(s) c(s) by Fl = −1, where Fl (a Landau parameter) is a properly normalized partial component of the anti-symmetrized static interaction F (k, k + q; p, p − q) in a charge (c) or spin (s) sub-channel with angular momentum l. However, it is known that conservation laws for total spin and charge prevent Pomeranchuk instabilities for l = 1 spin- and charge- current order parameters. Our study aims to understand whether this holds only for these special forms of l = 1 order parameters, or is a more generic result. To this end we perform a diagrammatic analysis of spin and charge susceptibilities for charge and spin density order parameters, as well as perturbative calculations to second order in the Hubbard U. We argue that for l = 1 spin-current and charge-current order parameters, certain c(s) vertex functions, which are determined by high-energy fermions, vanish at Fl=1 = −1, preventing a Pomeranchuk instability from taking place. For an order parameter with a generic l = 1 form- c(s) factor, the vertex function is not expressed in terms of Fl=1 , and a Pomeranchuk instability does c(s) occur when F1 = −1. We argue that for other values of l, a Pomeranchuk instability occurs at c(s) Fl = −1 for an order parameter with any form-factor

I. INTRODUCTION mentum (see e.g. Refs 4–13). A Pomeranchuk instability in a given channel occurs when the corresponding interaction exceeds 1/NF , where NF is the density of states This paper is devoted to the analysis of subtle effects at the FS. When WNF 1, a Pomeranchuk instability associated with a Pomeranchuk instability in a Fermi liq- occurs well inside the metallic regime. uid (FL) due to the interplay with conservation laws. A system of interacting fermions is called a Fermi liquid if its properties differ from those of free fermions in A Pomeranchuk instability is generally expressed a quantitative, but not qualitative manner1–3. Specif- as a condition on a Landau parameter. For a rotationally-invariant and SU(2) spin-invariant FL, an ically, the distribution function nk undergoes a finite anti-symmetrized static interaction between fermions jump at the Fermi momentum, kF , with some jump mag- nitude Z < 1; the velocity v∗ of fermionic excitations at the FS and at strictly zero momentum transfer, F ω near the Fermi surface (FS) remains finite; and the life- Γ (k, k; p, p) can be separated into spin and charge com- time of fermionic excitations near a FS is parametrically ponents, and each can be further decomposed into sub- larger than the energy counted from the Fermi level, i.e., components with different angular momenta l. Landau fermions infinitesimally close to the FS can be viewed as parameters are properly normalized dimensionless sub- c(s) infinitely long lived. These three features form the basis components Fl , where c(s) selects charge (spin) chan- for the description of low-energy fermionic states in terms nel, and l = 0, 1, 2, ... 1,2,14,15. Pomeranchuk argued in 16 c(s) of quasiparticles, whose distribution function at T = 0 is his original paper that a static susceptibility χl scales a step function. The validity of FL postulates has been as 1/(1 + F c(s)) and diverges when the corresponding verified in microscopic calculations for realistic interac- l F c(s) = −1. The divergence signals an instability to- tion potentials and was found to hold at small/moderate l couplings in dimensions d > 1. wards a q = 0 density-wave order with angular momentum l. Stronger interactions can destroy a FL. In general, such

arXiv:1801.06571v1 [cond-mat.str-el] 19 Jan 2018 destruction can occur in two ways. One option is the The 1/(1 + F c(s)) form of the susceptibility can be transformation of a metal into a Mott insulator, once the l interaction U becomes comparable to a fermionic band- reproduced diagrammatically by summing up particle- width W . This instability involves fermions located ev- hole bubbles of free fermions within RPA. The momen- erywhere in the Brillouin zone. Another option is an tum/frequency integration within each bubble is confined to the FS, hence the dimensionless interaction between instability driven by fermions only very near the FS, c(s) such as superconductivity and q = 0 instabilities in a the bubbles is exactly Fl . The RPA series are geo- c(s) c(s) particle-hole channel, often called Pomeranchuk instabil- metric, hence χl,RP A = χl,0/(1 + Fl ), where χl,0 is ities. The latter leads to either phase separation, or fer- a free-fermion susceptibility. This agrees with the exact romagnetism, or a deformation of a FS and the develop- forms of the susceptibilities of the l = 0 order parame- ment of a particle-hole order with non-zero angular mo- ters, which correspond to the total charge and the total 2 spin: Some of these issues were addressed by Landau and Pitaevskii (see e.g. Ref. 1) and by Leggett (Ref. 17) m∗/m χc(s) = χ , (1) in the early years of FL theory, by invoking (i) con- l=0 l=0,0 c(s) 1,2 1 + Fl=0 servation laws and the corresponding Ward identities and (ii) the continuity equation and the longitudinal sum ∗ ∗ 17,19 where m = kF /vF . It is tempting to assume that RPA rule . For a conserved order parameter, conservation works for a generic order parameter with angular mo- laws require that the full susceptibility coincides with mentum l. However, corrections to RPA are of order one the coherent term, i.e., the corresponding ΛZ = 1 and c(s) when Fl = O(1), and it is a’priori unclear whether χinc = 0. The l = 0 charge and spin Pomeranchuk c s in the generic case the full susceptibility has the same order parametersρ ˆl=0 and ρˆl=0 with a constant form- functional form as χc(s) . factor are conserved quantities, hence the corresponding l,RP A c(s) c(s) One can actually go beyond RPA and obtain the exact χl=0 = χl=0,qp, as in Eq. (1). This is fully consistent expression for a static χc(s) for a generic order parameter with RPA. For l = 1 charge- or spin-current order param- l c(s) eters with λl=1 (k) = k the continuity equation imposes c X c † ρˆl (q) = λl (k)ck−q/2,αck+q/2,α, (2) the relation k,α c(s) m c(s) s X s † αβ ZΛl=1 = ∗ 1 + Fl=1 (7) ρˆl (q) = λl (k)ck−q/2,ασ ck+q/2,β, (3) m k,αβ c(s) 2 c(s) c(s) such that (ZΛl=1 ) χl,qp ∝ 1 + Fl=1 vanishes at c(s) c(s) with any l and any form-factor λl (k). The exact for- 17 Fl=1 = −1 instead of diverging. In addition, the lon- mula was originally obtained by Leggett , based on ear- gitudinal sum rule yields lier work by Eliashberg18 (we present the diagrammatic derivation in Sec. II). It reads c(s) χl=1 = χ1,0, (8) 2 χc(s) = Λc(s)Z χc(s) + χc(s) . (4) c(s) l l l,qp l,inc i.e. all interaction-induced renormalizations of χl=1 cancel out, implying, Here c(s) m c(s) ∗ m χl=1,inc = χl=1,0 1 − 1 + Fl=1 (9) χc(s) = χc(s) (5) m∗ l,qp m l,RP A We emphasize that this holds even in the presence of ∗ is the RPA result with an extra factor of m /m, often a lattice potential V (r). For a Galilean-invariant FL, called the quasiparticle contribution. Other terms in Eq. m∗/m by itself is expressed via Landau parameters as (4) describe two contributions that incorporate effects ∗ c m /m = 1 + F1 . Then, for spin-current susceptibility, c(s) c(s) 2 s s c s c beyond RPA. First, χl,qp gets multiplied by (Λl Z) , ZΛl=1 = (1 + F1 )/(1 + F1 ) and χl=1,inc = χl=1,0(F1 − c(s) s c where Z is the quasiparticle residue and Λ accounts F1 )/(1 + F1 ), while for charge-current susceptibility, l c c for the renormalization of the vertex containing the form- ZΛl=1 = 1 and χl=1,inc = 0. This last result is con- c(s) sistent with the fact that for a Galilean-invariant FL, factor. Second, there is an extra term χl,inc. These additional terms come from processes in which at least one charge-current coincides with the momentum and is a fermion is located away from the FS. conserved quantity. Although the Z-factor itself comes from fermions away Eqs. (7)+(9) represent a qualitative breakdown of from the FS, its presence in (4) can be easily understood RPA for spin and charge-current order parameters. Within RPA, ΛZ = 1, and so to reproduce Eqs. (7)+(9) because the fermionic propagator near the FS is ∗ c −1 s −1 one must require m /m = (1 + F1 ) = (1 + F1 ) . Z Such behavior comes about naturally if one assumes10 G(ω, k) ≈ ZG (ω, k) = , (6) qp ∗ that the dressed interaction remains a function of k − p, ω − v (k − kF ) + iδsgnω F ω i.e., Γ (k, k; p, p) = Ueff (|k − p|). In this situation the hence there is a Z2 factor in each bubble (Z2 also appears charge component of Γω(k, k; p, p) is (U(0) − U(k − p)/2 c(s) and the spin component is −U(k − p)/2. Then F c = F s, in the normalization of Fl , see below). As long as l l Z is a number 0 < Z < 1, it alone does not change for all l > 0, including l = 1. However, in fact, at or- c(s) der U 2 and higher, the interaction gets renormalized in the functional form of χl compared to the RPA result. The other two terms are potentially more relevant. First, both particle-hole and particle-particle channels, and the renormalized interaction between fermions on the FS de- the vertex function Λc(s) may cancel the 1 + F c(s) term l l pends on both k−p and k+p. The terms which depend c(s) c(s) and, second, either Λl or χl,inc may diverge on their on k − p and on k + p behave differently under antisym- own and give rise to a Pomeranchuk-type instability not metrization, and, as the consequence, spin and charge c(s) ω associated with Fl = −1 (and with the fermions on components of Γ (k, k; p, p) are generally not equivalent the FS). for any l. In this situation, Eqs. (7) and (9) are obeyed 3 not because of some some special relation between Lan- channels, which relates the contribution to this sum com- dau parameters, but rather because ΛZ is expressed via ing from fermions at the FS and the one from fermions the particular combination of Landau parameters, such away from the FS. We use this identity to prove dia- s s that for l = 1 spin current, Λl=1Z cancels out 1 + F1 . grammatically Eqs. (7) and (9). We do the same com- This issue has been recently re-analyzed by Kiselev et putation for l = 0 and verify that for conserved spin and 20 c(s) c(s) al . They discussed how the absence of l = 1 Pomer- charge order parameters χl,inc = 0 and Λl = 1/Z, i.e., anchuk instability for the spin-current order parameter χc(s) = χc(s). places additional constraints on spontaneous generation l l,qp We next consider l = 2 and investigate an argument of spin-orbit coupling21,22, often associated with l = 1 by Kiselev et al 20 that for certain order parameters spin Pomeranchuk order. Kiselev et al also derived a with l = 2, spin and charge susceptibilities again do general formula for the susceptibility of a current of a c(s) conserved order parameter. not diverge when the corresponding Fl=2 = −1 because c(s) −1 c(s) c(s) The purpose of the current work is three-fold. First, (1 + Fl=2 ) in χl=2,qp is canceled out by ZΛl=2 . Our we provide a transparent diagrammatic derivation of Eq. perturbative results do not support this claim. We argue c(s) (4) and extend it to the case when both q and Ω are that ZΛl=2 cannot be expressed solely in terms of Lan- small, but the ratio vF q/Ω is arbitrary. Second, we ana- dau parameters. Of particular interest here is the l = 2 lyze Eqs. (7) and (9) from a microscopic perspective, and charge order parameter in a Galilean-invariant case. It identify what relates the contributions to the susceptibil- is tempting to view this order parameter as a current of ity from fermions near the FS, which determine Landau conserved l = 1 total momentum, and relate the corre- c parameters, and fermions away from the FS, which de- sponding ZΛl=2 to Landau parameters via the continuity termine Z,Λc(s), and χc(s) (and m∗/m in the absence of equation. However, we show that the current operator for l l,inc momentum cannot be expressed solely in terms of bilin- Galilean invariance). Lastly, we investigate how generic ear combination of fermions, and contains an interaction- is the statement about the absence of Pomeranchuk in- induced four-fermion term. As a result, the l = 2 charge stabilities for l = 1 order parameters, and what happens susceptibility is only a portion of the full current-current for other l. correlator, and as such is not determined by the continu- To derive Eqs. (4) and (5) diagrammatically, we use ity equation. the expansion in the number of fermionic loops, and at To understand how generic is the statement about the each loop order separate the contributions from fermions absence of Pomeranchuk instabilities for l = 1, we notice at the FS and away from it. The contributions away from that there exists an infinite set of Pomeranchuk order the FS can be computed by setting q to zero, while for parameters in any channel, including l = 0. These order the contributions from the vicinity of the FS one needs to c(s) keep q small but finite, because each bubble contribution parameters contain form-factors λl (k), which are obtained by multiplying the base form factor (a constant to χc(s) comes from the tiny range near the FS where the l,qp for l = 0, k for l = 1, etc), by an arbitrary function poles of the two Green’s function in a bubble are in dif- f (|k|). When f (|k|) is not a constant, it changes the ferent half-planes of frequency. We then re-arrange the l l contribution to susceptibility from fermions away from perturbation series and evaluate partial contributions to the FS compared to that from fermions at the FS. We ar- the susceptibility with M = 0, 1, 2 etc. cross-sections in c(s) which the contribution comes from the FS. Summing up gue that the identity, which allowed us to express ZΛl=1 c(s) terms with all M we reproduce Eqs. (4) and (5). We then in terms of Landau parameters and cancel 1/(1 + Fl=1 ), extend the analysis and consider the dynamical suscep- does not hold if fl(|k|) is not a constant. As a result, c(s) ZΛs no longer vanishes when F s = −1. We show tibility χl (q, Ω) in the limit when both |q| and Ω are l=1 l=1 ∗ this non-cancellation explicitly to second order in the small, but the ratio vF |q|/Ω is arbitrary. We show that in an arbitrary FL, the form of the dynamical suscep- Hubbard U by comparing susceptibilities for order pa- ˆ ˆ tibility is rather complex, except for special cases when rameters with form-factors k and kF k, where k is a unit ∗ ∗ vector directed along k. We further argue that for a vF |q|/Ω is either small or large, or vF |q|/Ω is arbitrary, but only a few Landau parameters are not small. non-constant fl(|k|), the incoherent contribution to sus- In order to understand Eqs. (7) and (9) for current ceptibility cannot be expressed via Landau parameters order parameters, we explicitly compute low-energy and for any l, including l = 0, and can potentially diverge on high-energy components of charge and spin susceptibili- its own, even if the corresponding Landau parameter is ties for l = 1 for the 2D Hubbard model, to second order still larger than −1. This opens up a possibility for an in Hubbard U. At this order the dressed interaction be- instability of a FL, not associated with the singularity in tween fermions becomes dynamical, and both low-energy the coherent part of the susceptibility. and high-energy contributions are non-zero. For simplic- The paper is organized as follows. In the next Section ity, in this calculation we neglect the lattice potential, we review the diagrammatic formulation of FL theory i.e., consider a Galilean-invariant system. We show that and present our diagrammatic derivation of Eq. (4) for c(s) there exists a particular identity on the sum of dynamical the static susceptibility χl and its extension to finite ∗ polarization bubbles in particle-hole and particle-particle vF |q|/Ω. In Sec. III we discuss the forms of l = 1 sus- 4

� ceptibilities for the currents of conserved fermionic charge � − and spin, and also discuss the relation between the l = 2 2 charge order parameter (bilinear in fermions) and the current of a total fermionic momentum. In Sec. IV we present the results of numerical and analytical calcula- � � + tions to second order in the Hubbard U. Our key em- 2 phasis here is to understand why contributions to the l = 1 susceptibilities from fermions at the FS and away from it are related. In Sec. V we discuss the implica- FIG. 1. Free fermion susceptibility, where k = (k, ωk) and q = (q, Ω). The black dots on the two sides represent form tions for susceptibilities of order parameters which con- factors λ (k). tain an additional dependence on k beyond symmetry l related overall factors. We summarize our results in Sec. VI A. Perturbation theory

The free fermion Hamiltonian is

Z X −∇2 H = dr c† (r) − µ c (r) kin α 2m α α II. FL THEORY. DIAGRAMMATIC APPROACH X † = ξkck,αck,α (11) kα

In this section we brieﬂy review the diagrammatic ap- 2 proach to a FL and present the diagrammatic derivation where ξk = k /(2m) − µ, and we set ~ = 1 through- of Eq. (4). We also obtain a more general expression for out this paper. The corresponding free-fermion Green’s c(s) function is χl (q, Ω) when both q and Ω are small, but the ratio ∗ vF q/Ω is arbitrary. The full formula is 1 G0(k) = (12) ωk − ξk + iδω

2 + c(s) c(s) c(s) c(s) where δω = δsgnω and δ = 0 . The free-fermion suscep- χl (q, Ω) = Λl Z χl,qp (q, Ω) + χl,inc (10) c(s) tibility χl,0 (q) is diagrammatically represented as the bubble made out of two fermionic propagators (Fig II A) c(s) with form-factors λl in the vertices: ∗ c(s) For arbitrary vF q/Ω, χl,qp (q, Ω) is a complex function of all Landau parameters. For deﬁniteness and to Z d3k 2 q q χc(s)(q) = −2 λc(s)(k) G (k + )G (k − ), make computational steps less involved, we consider two- l,0 (2π)3 l 0 2 0 2 dimensional (2D) Galilean-invariant systems. (13) where the factor 2 comes from spin summation. In 2D, Our goal is to distinguish between high-energy and low-energy contributions to the susceptibility and re- c(s) l c(s) c(s) c(s) c(s) λl (k) = cos lφk|k| × fl (|k|) (14) late χl,qp , χl,inc, Z, and Λl to particular sets of diagrams. We express diﬀerent contributions to the sus- where φk is the angle between k and q. One may verify ω ceptibility via the vertex function Γ (k, p). Here and that the frequency integral in (13) is non-zero only if below k denotes a fermionic 3-vector, k = (k, ω ) and k ξk+q/2 and ξk−q/2 have opposite signs, i.e., it comes from q = (q, Ω) denotes a bosonic 3-vector. We show that the tiny range near the FS of width O(q). In explicit form c(s) ω χl,qp is expressed via Γ (k, p) in which both k and p are we have, after integrating over frequency c(s) ω on the FS, χl,inc is expressed via Γ (k, p) in which both 2 Z d k nF (ξk− q ) − nF (ξk+ q ) 2 c(s) χc(s)(q) = −2 2 2 λc(s)(k) k and p are away from the FS, and Λl is expressed via l,0 2 q q l (2π) Ω + ξk− − ξk+ + iδΩ Γω(k, p) in which k is on the FS and p is away from it, 2 2 (15) or vice versa. We combine our analysis with the Landau- where n (ξ) = Θ(−ξ) is a unit step function in zero Pitaevskii equations1,2 which relate the inverse quasipar- F temperature limit. In the case of vanishingly small |q| ticle residue 1/Z to Γω(k, p) in which k is on the FS and c(s) one can integrate over k and obtain, p is away from it, similarly to Λl . The contributions from away from the FS are insensitive to the ratio Ω/|q| c(s) m 2 χ (q) = − kl f c(s)(k ) and can be computed at q = 0 and Ω = 0. The quasi- l,0 π F l F c(s) Z particle part χl,qp (q) depends on how the limit q, Ω → 0 dφk 2 vF |q| cos φk (cos lφk) (16) is taken. 2π Ω − vF |q| cos φk + iδΩ 5

We next include the interaction term � � � − � − � � 2 Z � − � � − 2 1 0 X † 0 † 0 0 2 � − 2 Hint = drdr c (r)cα(r)U (|r − r |) c (r )cβ(r ) 2 2 α β α,β 2× � 1 X † † � + � � = U(|q|)c c c c δ δ , 2 � + � + k+q/2,α p−q/2,β p+q/2,δ k−q/2,γ αγ βδ 2 2 2V (a) (b) (18) � � � − � − 2 2 where the summation is over all momenta and all spin indices.

� � � + � + 2 2 1. First order in U(|q|) (c)

c(s) To ﬁrst order in U(q), there are three interaction- FIG. 2. Corrections to χ to ﬁrst order in U. l induced corrections to the bubble diagram for the susceptibility. They are shown in Fig. 2. Diagram 2a repre- sents a self-energy correction. The self-energy is purely static (because U(|q|) is static) and gives rise to mass ∗ renormalization m /m = 1 − (1/vF )dΣ/d|k|. One can easily verify that the integral for Σ(k) for k near the FS is determined by q connecting points on the FS. A simple calculation yields

∗ Z In the static limit Ω = 0, q → 0 we have m m dθ θ = 1 − U 2kF sin cos θ (19) m 2π 2π 2 and m∗ χc(s)(q) = − 1 χc(s)(q) (20) l,2a m l,0

Diagram 2b contains two cross-sections with internal k and p. Because the interaction U(k−p) is static, in each cross-section the frequency integral is again non-zero only 2 if the dispersions have opposite signs. The result is that c(s) m c(s) χ = f (kF ) the integration is again conﬁned to a narrow region near l=0,0 π l=0 m 2 the FS. Evaluating frequency and momentum integrals, χc(s) = kl f c(s)(k ) , l > 0 (17) we obtain l,0 2π F l F

2 2 Z c(s) 1 m l c(s) dφk dφp φk − φp χ (q) = k f (kF ) cos lφk cos lφpU 2kF sin × l,2b 2 π F l 2π 2π 2 v |q| cos φ v |q| cos φ F k F p (21) Ω − vF |q| cos φk + iδΩ Ω − vF |q| cos φp + iδΩ

In the static limit Ω = 0, q → 0, Finally, diagram 2c contains U(0) and is non-zero only for charge susceptibility at l = 0. It gives

c c 2 c −2 χl=0,2c(q) = U(0) χl=0,0(q) (fl=0(kF )) (23)

2 2 c(s) 1 m c(s) The sum of the three diagrams can be cast into a known χ = kl f (k ) l,2b 2 π F l F FL form by re-expressing the results in terms of the Lan- Z 2 ∗ ω dφk dφp φk − φp dau function Fαβ,γδ(k, p) = (Z m /π)Γαβ,γδ(k, k; p, p), cos lφk cos lφpU 2kF | sin | (22) ω 2π 2π 2 where Γαβ,γδ(k, k; p, p) is the fully renormalized antisym- 6

� � � �

� � � � −

� � � � � � � � c(s) FIG. 4. Example of a higher order contribution to χl . At this order, the static interaction acquires dynamics due to ω FIG. 3. The vertex Γαβ,γδ(k, k; p, p) to first order in U. particle-hole screening. The diagram’s computation is split into three (see Sec. II A 2). It belongs to the M = 0 sector when both bubbles are evaluated away from the FS, to M = 1 metrized static interaction between fermions on the FS, when one is evaluated on the FS and one away from it, and to M = 2 when both are evaluated at the FS. taken in the limit of zero momentum transfer. The an- tisymmetrized interaction to first order in U is shown 2 ∗ graphically in Fig. 3. To this order, Z m /π = m/π. from the real frequency axis. A simple analysis shows Combining the diagrams from this figure, we obtain that the series is geometric and its sum yields m Fαβ,γδ(k, p) = [U(0)δαγ δβδ − U(k − p)δαδδβγ ] (24) 1 + F c π c(s) l=1 χl,RP A = χl,0 (27) m 1 1 1 + F c(s) = U(0) − U(k − p) δ δ − U(k − p)σ σ l π 2 αγ βδ 2 αγ βδ The RPA susceptibility obviously diverges when F c(s) = The two terms in the last line in (24) are charge and l −1, except for the special case of F c(s) = F c, as occurs spin components of the Landau function F (k, p) = l 1 αβ,γδ e.g. for l = 1 if we require that the interaction is purely F c(k, p)δ δ + F s(k, p)σ σ . Each component can αγ βδ αγ βδ static, see the previous section and our comments in the be further expanded in partial harmonics with different Introduction. l as c(s) We next go beyond RPA. A diagram for χl at any c(s) c(s) X c(s) loop order is represented by a series of ladder segments F (k, p) = F0 + 2 Fl cos lφ, (25) l>0 separated by interactions. In each of these ladders there is an integration over both high-energy and low-energy where φ = φ − φ is the angle between k and p (|k| = c(s) k p frequencies and momenta. To obtain χl , we follow 18,23,24 |p| = kF ). Using this expansion, one may easily check earlier diagrammatic studies and and re-arrange that the sum of zero-order and first-order contributions c(s) perturbation series by assembling contributions to χl to the static susceptibility can be cast into from diagrams with a given number M of ladder segments with poles shifted into different directions from the real c(s) c c(s) χl = χl,0 1 + Fl=1 − Fl frequency axis, and then sum up contributions from the sub-sets with different M = 0, 1, 2, etc. c c(s) ≈ χl,0 (1 + Fl=1) 1 − Fl (26) We start with M = 0. The corresponding contribu- 2 tions to the susceptibility contain products of G (k, ωk). This formula is valid for all l, including l = 0. Eq. (26) Taken alone, each such term will vanish after integra- trivially fulfils the constraints of Eqs (7) and (9) for the tion over frequency. The total M = 0 contribution then c s simple reason that to this order, Fl = Fl for all l > 0. vanishes to first order in U(q) because the static interaction does not affect the frequency integration. How- ever, at second and higher orders in U(q), the interac- 2. Higher orders in U(q), static limit tion gets screened by particle-hole bubbles and becomes a dynamical one. An example of second-order suscep- We now move to higher orders in U, still considering tibility diagram with screened interaction inserted into the static limit Ω = 0, q → 0. Within RPA, higher- the bubble is shown in Fig. 4. This screened dynami- order diagrams are treated as series of ladder graphs (l > cal interaction contains a Landau damping term, which 0) or ladder and bubble graphs (l = 0), Each element is non-analytic in both half-planes of complex frequency. 2 of the ladder/bubble series contains the product of two As a result, the product of G (k, ωk) and the dressed fermionic Green’s functions, dressed by static self-energy. interaction at order U 2 and higher has both a double The two Green’s functions have the same frequency and pole and a branch cut. A pole can be avoided by closing their momenta differ by q. Within this approximation, the integration contour in the appropriate frequency half- a non-zero contribution to susceptibility from each cross- plane, but the branch cut is unavoidable, and its presence section comes from the states very near the FS, where the renders the frequency integral finite. Since there is no poles in the two fermionic Green’s functions, viewed as splitting, relevant fermionic ωk and k are not confined functions of frequency, are shifted in different directions to the FS and are generally of order EF (or bandwidth). 7

c(s) beled M = 2. It contains fully dressed side vertices Λl and a fully dressed anti-symmetrized static interaction = between fermions on the FS. One can easily verify that this interaction appears with the prefactor Z2(m∗/m), � = 0 i.e., the extra factor in the M = 2 sector compared to M = 1 is the product of χl,0 and the corresponding component of the Landau function. Using (25) we then ob- + + Γ + ⋯ tain 2 m∗ χc(s) +χc(s) = ZΛc(s) χc(s) 1 − F c(s) (29) � = 1 � = 2 l,M=1 l,M=2 l m l,0 l (the minus sign comes from the number of fermion bub- FIG. 5. The ladder series of diagrams for the static sus- bles.) A simple bookkeeping analysis shows that con- c(s) tributions from sectors with larger M form a geometric ceptibility χl . The exact χl is represented as a series M = 0, 1, 2,... of bubbles comprised of Green’s functions with c(s) c(s) series, which transform 1 − Fl into 1/(1 + Fl ). Col- poles on opposite halves of the complex frequency plane, i.e. lecting all contributions, we reproduce Eq. (4). whose contributions are computed close to the FS.

3. The susceptibility χc(s)(q, Ω) at finite Ω/v∗ |q|. Fermions at such high energies have a finite damping, i.e., l F are not fully coherent quasiparticles. By this reason, the c(s) We now extend the analysis to the case when both M = 0 contribution to χl is labeled as an incoherent c(s) c(s) transferred momentum q and transferred frequency Ω are one, χ = χ (although at small U fermions with ∗ l,M=0 l,inc vanishingly small, but the ratio Ω/vF |q| is finite. The energies of order EF are still mostly coherent). computational steps are the same as for static suscep- We next move to the M = 1 sector. Here we select tibility. The contribution to χc(s)(q) from the M = 0 the subset of diagrams with one cross-section, in which l sector and the vertex function Λc(s) do not depend on we pick up the contribution from G(k, ω )G(k + q, ω ) l k k the ratio of Ω/(v∗ |q|) and remain the same as in the from the range where the poles in the two Green’s F functions are in different half-planes of complex fre- static case. However, the integrand in the expression for quency. The sum of such diagrams can be graphi- χl,0(q), Eq. (16), now contains a non-trivial angular de- cally represented by the skeleton diagram in Fig. 5 la- pendence via vF |q| cos φk/(Ω − vF |q| cos φk + iδΩ). This beled M = 1. The internal part of this diagram gives makes the computation of series with M = 1, 2,... more 2 ∗ involved. Z (m /m)χl,0(q), where χl,0(q) is given by (16). The c(s) c(s) Consider first the limit Ω vF |q|. For even l, the side vertices contain Λ1 λl (kF ), i.e. the product free-fermion susceptibility is of the bare form-factor (which we already incorporated 2 c(s) m c(s) iΩ into χl,0(q)), and the contributions from all other cross- l χl,0 (q) = kF fl (kF ) 1 + αl sections, in which G(k, ωk)G(k + q, ωk) is approximated αlπ vF |q| 2 by G (k, ωk). These contributions would vanish if we c(s) iΩ used a static U(|q|) for the interaction, but again be- = χl,0 1 + αl (30) vF |q| come non-zero once we include dynamical screening at 2 where αl = 1 if l = 0 and αl = 2 if l = 2m, m > 0. order U and higher. Similarly to the M = 0 sector, the 2 c(s) For odd l, the expansion in Ω starts with Ω . The total difference Λl − 1 is determined by fermions with en- contribution from the M = 1 sector still is proportional ergies of order EF . Note, however, that in the M = 0 to χl,0: sector, all internal energies are of order E . In the M = 1 F ∗ 2 2 c(s) c(s) m c(s) l c(s) sector, internal energies for the vertices Λl are of order χl,M=1(q) ≈ ZΛl kF fl (kF ) αlπ EF , but external ωk are infinitesimally small, and ex- ∗ ternal k are on the FS. Overall, the contribution to the m iΩ 1 + αl static susceptibility from the M = 1 sector is m vF |q| 2 ∗ ∗ 2 ∗ c(s) m c(s) m iΩ c(s) c(s) m c(s) χ = ZΛ χ (28) = ZΛl χl,0 1 + αl (31) l,M=1 l m l,0 m m vF |q| In the contribution from the M = 2 sector, the iΩ/v∗ |q| Sectors with M = 2, M = 3 are the subsets of diagrams F term can be taken from the cross-section on the right or with 2, 3,... cross-sections in which we split the poles on the left. This gives a combinatoric factor of 2. Then of the Green’s functions with equal frequencies and mo- 2 ∗ menta separated by q. In the cross-sections in between c(s) c(s) c(s) m c(s) χl,M=1(q) + χl,M=2(q) ≈ ZΛl χl,0 the selected ones G(k, ωk)G(k + q, ωk) is again approxi- m 2 ∗ mated by G (k, ωk). The contribution from the M = 2 c(s) c(s) m iΩ 1 − Fl + (1 − 2Fl ) αl (32) sector is represented by the skeleton diagram in Fig. 5 la- m vF |q| 8

For the contribution from the M = 3 sector the same The presence of |q|2/Ω2 in the susceptibility for l = 0 is reasoning yields the combinatoric factor of 3 and so on. a natural consequence of the fact that the total fermionic Using charge and spin are conserved quantities, i.e., they don’t change when we probe the system at diﬀerent times. 1 1 − 2F c(s) + 3 F c(s) + ... = (33) For free fermions, this holds for all l because all partial l l c(s) 2 (1 + Fl ) fermionic densities at a given direction of k are separately conserved, hence χl,0(q = 0, Ω) must vanish for an any we obtain angle-dependent form-factor. The contribution from the 2 M = 1 sector is, c(s) c(s) c(s) c(s) χl (q) = ZΛl χl,qp (q) + χl,inc (34) where to order Ω/|q|, for even l, 2  2 m 2 m v |q| 2 ∗ ∗ ! c(s) c(s) F l m /m iΩ m /m χ (q) ≈ − ZΛ kF fl(kF ) × χc(s)(q) = χc(s) + α l,M=1 π l m∗ Ω l,qp l,0  c(s) l c(s)  vF |q| (1 + Fl ) 1 + Fl Z dφ k (cos lφ )2(cos φ )2. (37) (35) 2π k k For l = 0 this result has been obtained before25. ∗ In the opposite limit Ω vF |q| we have

2 m v |q| 2 F l ∗ ∗ χl,0(q) ≈ − kF fl(kF ) The overall m/m factor is due to one m /m factor from π Ω ∗ 2 Z the integration over momentum and an (m/m ) from dφk 2 2 the expansion to second order in v∗ |q|/Ω. From the M = (cos lφk) (cos φk) (36) F 2π 2 sector we have, at order |q|2/Ω2

v |q|2 m 2 m 2 ZZ dφ dφ χc(s) = − F ZΛc(s) kl f c(s)(k ) k p (cos lφ )(cos lφ )(cos φ )(cos φ )F c(s)(φ −φ ) l,M=2 Ω π l m∗ F l F 2π 2π k p k p k p (38) Substituting F c(s) from Eq. (25), we obtain

1 v |q|2 m 2 2 m χc(s) = − F ZΛc(s) f c(s)(k ) F c(s) l=0,M=2 2 Ω π l l=0 F m∗ 1 1 v |q|2 m 2 2 m χc(s) = − F ZΛc(s) k f c(s)(k ) 2F c(s) + F c(s) l=1,M=2 8 Ω π l F l=1 F m∗ 0 2 1 v |q|2 m 2 2 m χc(s) = − F ZΛc(s) kl f c(s)(k ) F c(s) + F c(s) l>1,M=2 8 Ω π l F l F m∗ l−1 l+1 (39) The contribution from the sectors with M > 2 contains higher power of |q|/Ω. Hence, to order |q|2/Ω2, the full result for the dynamical susceptibility is

1 v |q|2 2 m χc(s)(q) = − F χc(s) ZΛc(s) 1 + F c(s) + χc(s) l=0 2 Ω l=0,0 l m∗ 1 l=0,inc 3 v |q|2 2 m 2 1 χc(s)(q) = − F χc(s) ZΛc(s) 1 + F c(s) + F c(s) + χc(s) l=1 4 Ω l=1,0 l m∗ 3 0 3 2 l=1,inc 1 v |q|2 2 m 1 χc(s)(q) = − F χc(s)(q) ZΛc(s) 1 + F c(s) + F c(s) + χc(s) l>1 2 Ω l,0 l m∗ 2 l−1 l+1 l,inc (40)

For l = 0 this result has been obtained in Ref. 17. is rather involved for all l, including l = 0. As an il- c(s) lustration, consider the seemingly simplest case l = 0 For a generic Ω/vF |q|, the full expression for χl (q) 9 and set f0(|k|) = 1 (i.e., consider susceptibilities for spin volved, because the interaction between the bubbles with and charge order parameters). Due to spin/charge con- internal momenta k and p is expressed via the Landau c(s) c(s) c(s) function F c(s)(k, p), Eq. (25), and the latter dependens servation ZΛl=0 = 1 and χl=0,inc = 0, so χl=0 (q) = on φ = φ − φ . It is suﬃcient to analyze the ﬁrst few χc(s) (q). k p l=0,qp orders in the expansion in powers of F c(s)(k, p) to understand that the full result is The full dynamical χc(s) (q) is given by series of bub- l=0,qp m∗ χ¯(q) bles, each is determined by fermions in the vicinity of the χc(s) (q) = (41) FS. The integration over frequency and over fermionic l=0,qp π c(s) 1 + Fl=0 χ¯(q) dispersion can be performed independently in each bubble, but angular integration is, in general, rather in- whereχ ¯(q) is given by series of terms

X c(s) χ¯(q) = K0 − 2 Fn KnKm × n,m>0 " " ## X X δ − Q F c(s) δ − Q F c(s) (δ − ...) (42) n,m n,m1 m1 m1,m m1,m2 m2 m2,m m1>0 m2>0

c(s) where δn,m is Kroneker symbol and χl=0,qp(q) can be obtained if only a few Landau parameters are sizable, e.g., if we assume that |Fl| |F0|, |F1| Qn,m = Kn+m + Kn−m. (43a) for all l > 1. In this situation, only one term in each sum in (45) and (46) survives, and these two equations Here simplify to Z ∗ c(s) dθ vF |q| cos θ 2 1 K (q) = − cos nθ χ¯(q) = K0 − 2F1 K1 S1 (47) n 2π Ω − v∗ |q| cos θ + iδ F Ω and α p 2 |n| = δn,0 − √ (α − α − 1) , (43b) 2 1 c(s) α − 1 + iδ S1 1 + Q1,1Γ1 = 1 (48)

∗ 1 and α = Ω/vF |q|. In explicit form Using Q1,1 = K0 + K2 we ﬁnd S1 = 1/(1 + (K0 + K )F c(s)). Substituting this into (47) and then substi- Ω 2 1 K0(q) = 1 − tuting (47) into (41), we obtain pΩ2 − (v∗ |q|)2 + iδ F c(s) 2 2F1 K1 ! ∗ K0 − c(s) Ω Ω c(s) m 1+F1 (K0+K2) K1(q) = 1 − χ (q) = (49) ∗ p 2 ∗ 2 l=0,qp c(s) 2F c(s)F c(s)K2 vF |q| Ω − (v |q|) + iδ π 0 1 1 F 1 + F K0 − c(s) 0 1+F (K +K ) 2 1 0 2 Ω 26 K2(q) = 2 ∗ The same result has been obtained previously using a vF |q| ∗ Boltzmann equation approach. At Ω/vF |q| 1, we have 2! 2 2 2 Ω Ω K0(q) ≈ −(1/2)(vF |q|/Ω) , K1 (q) ≈ (1/4)(vF |q|/Ω) , + 1 − 2 (44) 2 p 2 ∗ 2 ∗ K2(q) ≈ −(3/8)(vF |q|/Ω) . Substituting into (49) we Ω − (v |q|) + iδ vF |q| F c(s) 2 c(s) obtain χl=0 (q) = −(1/2)(vF |q|/Ω) (1 + F1 ), as in Eq. Eq. (42) can be equivalently re-expressed as (40).

X c(s) m χ¯(q) = K0 − 2 F KnKmS (45) n n III. SUSCEPTIBILITIES OF THE CURRENTS n,m>0 OF CONSERVED ORDER PARAMETERS m where Sn is the solution of the matrix equation In this section we discuss the relationship between or- X Sm + Q F c(s)Sm = δ (46) der parameters associated with conserved “charges” (to n n,m1 m1 m1 n,m m1>0 be distinguished from the speciﬁc electric charge) and their currents. We review the derivation of the conti- In the static limit K0 = 1, Kn>0 = 0. Thenχ ¯(q) = 1, nuity equation for susceptibilities of these order param- and Eq. (41) reduces to Eq. (1) for the static suscepti- eters (Refs. 17 and 20 and show that this equation ex- ∗ bility. For a generic Ω/vF |q| a closed-form expression for plicitly connects high energy properties of a FL, namely 10

c(s) c(s) χ , Λ ,Z, with low-energy properties, namely χl,qp. � l,inc l � � We discuss the implications for the l = 0, 1 channels and � � � � obtain Eqs. (7)-(9). Finally we discuss the implications Λ = + Γ of the continuity equation for the l ≥ 1, 2 channels. Our � � � focus here is to identify the constraints placed by the � � � � conservation law on high- and low- energy FL properties. We will then analyze these constraints microscopically in Sec. IV. FIG. 6. Relation between a 3-leg vertex Λ and a 4-leg vertex Γ, for a conserved charge density.

A. The continuity equation for charge and current susceptibilities B. Implication of conservation laws for the susceptibilities A conserved “charge” is an operatorρ ˆ(q, t) that com- mutes with the Hamiltonian at q = 0, so that it does not For a conserved charge (55) yields evolve in the Heisenberg picture, χρ(q = 0, Ω) = 0. (58) ∂ρˆ 1 = [ˆρ, H] = 0. (50) We also we recall that the coherent part of χρ, which ∂t i corresponds to the M = 1, 2,... diagrams of Fig. 5, Examples of such charges are the number (or electric vanishes at q = 0. Thus, Eq. (58) also implies c charge) and spin density in the model of Sec. II:ρ ˆl=0 s χρ,inc = 0. (59) and ρˆl=0 from Eq. (2) with constant form-factors. The continuity equation for a conserved chargeρ ˆ can be de- Finally, the relation ΛρZ = 1 follows from the fact that rived in the Heisenberg picture: ω Λρ and 1/Z are identically expressed via the vertex Γ , ∂ρˆ(q, t) 1 Z 3 = [ˆρ(q, t),H = Hkin + Hint] ≡ −iq · Jˆ. (51) i X d k ω 2 ω λρ(k) ∂t i Λρ = 1 − 3 Γαβ,αβ(kF p,ˆ k)(Gk) 2kF (2π) λρ(kF ) The continuity equation relates the susceptibilities of or- αβ (60a) ˆ der parameters associated withρ ˆ and J, Z 3 1 i X d k ω 2 ω λρ(k) 0 = 1 − 3 Γαβ,αβ(kF p,ˆ k)(Gk) χρ = h[ˆρ(q, t), ρˆ(−q, t )]i (52) Z 2kF (2π) λρ(kF ) αβ i j 0 χJ = h[(Jˆ) (q, t), (Jˆ) (−q, t )]i (53) (60b)

2 ω 2 Taking the derivative ∂t∂t0 χρ and transforming to the where (Gq) = limΩ→0G(q, ω)G(q, ω + Ω) = G (q, ω) is frequency domain we obtain the regular part of the product of two Green’s functions, For the vertex, Eq. (60a) follows from Fig. 6 (and is 2 X mn mn Ω χρ(q) = qm [χJ (q) − χJ (q, 0)] qn. (54) valid for a conserved ”charge” in both charge and spin m,n channels, while for 1/Z the relation (60b) is the Ward identity for a conserved charge with form-factor λ (k). Here, the sum is over spatial indices m, n = {x, y}. ρ We recall that Λ is defined without the factor λ (k ). Equivalently we may write, ρ ρ F We plug these results into Eq. (55), take the limit Ω 2 k k vF q → 0, and obtain, (Ω/q) χρ(q) = χJ (q) − χJ (q, 0). (55) 2 Here we have defined the longitudinal component of the 2 Ω q (ΛJ Z) χJ,qp(q → 0, 0) = − 2 χρ,qp( → 0). (61) susceptibilityq ˆ · χJ · qˆ. Note, that the RHS of Eqs. q Ω (54)+(55) includes only the time dependent part of χ . J We showed in Sec. II that for any l, χc(s)( q → 0) This is an automatic consequence of taking the time l,qp Ω 2 2 derivative of χρ and going to the Fourier domain. scales as q /Ω , and the prefactor is expressed in terms Let’s assume that bothρ ˆ and J are expressed via bi- of Landau parameters and is not singular. Assuming linear combinations of fermions with some given l. We that this holds for the conserved charge, we find that 2 then can use Eq. (10) and write (ΛJ Z) χJ,qp(q → 0, 0) remains finite when Landau parameters change and pass through −1. Eq. (61) then 2 χρ(q, Ω) = (ΛρZ) χρ,qp(q, Ω) + χρ,inc, (56) implies that there is no Pomeranchuk instability in the J 2 channel. It also explicitly connects Λ , Λ , Z, m∗/m and χJ (q, Ω) = (ΛJ Z) χJ,qp(q, Ω) + χJ,inc, (57) ρ J χρ,qp, χJ,qp via Eqs. (56)+(57). This is the essence of Combining these expressions and Eq. (55) we express the our argument that the continuity equation implies con- current susceptibility via the susceptibility of a conserved straints that connect low- and high- energy properties of charge. the FL. 11

For the speciﬁc case of spin and charge density order density by the symbol ρ ≡ ρi, and to the energy tensor c s parameters, one can easily verify thatρ ˆ (q) andρ ˆ (q) by J ≡ Jij where i, j denote spatial indices. commute with Hint so the current density is bilinear in In Sec. III A we did not specify the nature of charge the creation and annihilation operators: density and current. Thus, eq. (54) is equally valid for the momentum densities and currents, the only change c 1 X † ij Jˆ (q, t) = kc c , (62) being that χρ = χ (q, Ω) is a rank-2 symmetric tensor, m k−q/2,α k+q/2,α ρ k,α k ij ij and so is (χJ ) = (ˆq · χJ (q, Ω) · qˆ) . In the same man- ˆs 1 X αβ † ner, all arguments relating high frequency behavior of χρ Jm(q, t) = σm kck−q/2,αck+q/2,β. (63) m with the static behavior of χJ go through, leading to Eq. k,αβ 2 (61). Thus (ΛJ Z) χJ,qp is fully determined by χρ and In this case, the susceptibilities ofρ ˆc(s), Jˆc(s) correspond furthermore is always ﬁnite. However, we now demonstrate that Jij cannot, in gen- precisely to χl=0 and χl=1: eral, be expressed as a bilinear operator in c†, c. As a c(s) c(s) c(s) c(s) k χρ = χl=0 , χJ = χl=1 . (64) result, χJ does not have a simple relationship with χl, e.g. with χl=2. To see this, it is enough to examine the Eq. (55) then implies Hubbard model, i.e. take U(|q|) = U in Eq. (18). The current operator Eq. (62) has the following equation of c(s) 1 h c(s) c(s) i (Ω/q)2χ (q) = qˆ · χ (q) − χ (q, 0) · q.ˆ (65) motion, l=0 m2 l=1 l=1 Taking the Ω v∗ q limit, we obtain ∂ρˆ(q, t) F = −iq · Jˆ (70) ∂t ∗ 2 c(s) vF vF |q| c(s) 4 4 χl=0 = −χl=0,0 ∗ (1 + F1 ) + O(|q| /Ω ) where vF Ω (66) ˆ ˆ ˆ Plugging the result into Eq. (61) yields, J = Jkin + Jint, (71)

∗ c,s 2 vF 1 vF c(s) with (Λl=1Z) ∗ c,s = (1 + F1 ), (67) vF 1 + F1 vF q · Jˆkin = [ρ, Hfree] , q · Jînt = [ρ, Hint] (72) i.e., ∗ which gives, c(s) vF c(s) Λl=1 Z = (1 + F1 ), (68) vF 1 X Jîj = k k c† c , (73) kin m2 i j k−q/2 k+q/2 which is Eq. (7). k For the currents of conserved charge and spin there ex- U X Jîj = δ n(k)n(q − k), (74) ists another constraint imposed by the longitudinal sum int ij m2 rule17,19: k k P † ˆ χJ (q, 0) = n/m (69) where n(k) = p cp−k/2cp+k/2. If we had had Jint = 0, then indeed Eq. (61) could be used to constrain the l = where n is the number density. The longitudinal sum rule ˆ is analogous to the longitudinal f-sum rule for the imag- 0, l = 2 channels, both of which appear in Jkin. However, inary part of the inverse dielectric function27 and can be as it is, while Eq. (61) does constraint χJ to be finite, derived from the gauge-invariance of the electromagnetic by itself it does not constrain any specific l channels. field19. It is exact for a system where the electric current is proportional to the momentum density (with or without Galilean invariance), which is the case for any model IV. PERTURBATIVE CALCULATIONS FOR of the form of Eqs. (11), (18) with or without external THE HUBBARD MODEL: CHARGE-CURRENT AD SPIN-CURRENT ORDER PARAMETERS. potential V (r). In effective low energy models (e.g. on a lattice), it is only approximately correct20. In either In this section we perform perturbative analysis of Eq. case, its implication is that the total χJ is also finite. (10) for l = 1 and Eq. (65). We have three goals in our calculation: the first is to show how one can derive C. Conservation of momentum and l = 2 the continuity equation diagrammatically, the second is susceptibility c(s) c(s) c(s) to verify the relations between Λl=1 Z, χl=1,inc and Fl=1 , Eqs. (7) and (9), in direct expansion in the interaction, Finally, we address the issue of the implication of the and the third goal is to clarify the origin of the relation continuity equation for momentum in a Galilean invari- between high- and low- energy contributions to Eqs. (7) ant system. In this section we will refer to the momentum and (9). 12

We proceed in three steps. First, we derive Eq. (65) and obtain diagrammatically to first-order in U(q). We will see that Z 2 although there are no dynamical corrections to this order 2 c(s) 2 d k q χ (q) = 2m [nF (ξk− q ) − nF (ξk+ q )] c(s) l=1,0 (2π)2 2 2 (i.e. Z, Λl = 1), nevertheless self-energy corrections are crucial, indicating one should go beyond RPA. Then, 1 Ω2 × −Ω − k · q + we perform a combined analytical and numerical analysis 1 m Ω − m k · q + iδΩ c(s) 2 of χl=1 at order U for the Hubbard model, and explic- (77) itly verify Eqs. (7), (9). Going to to second order in U is essential, because only at this order do contributions The Ω term vanishes after integration over k. The away from the FS begin to accumulate, see Sec. II A 2. 2 c(s) other two terms are easily identified as q χl=1,0(q, 0) and Finally, we demonstrate that the high-energy contribu- 2 c(s) c(s) Ω χl=0,0(q), so that: tions to χl=1 can be re-expressed as low-energy ones, due to a special property of the sum of particle-hole and q2 particle-particle bubbles. χc(s) (q) − χc(s) (q, 0) = Ω2χc(s) (q). (78) m2 l=1,0 l=1,0 l=0,0 We now use the same tactics for first order corrections A. Diagrammatic derivation of the continuity c(s) to χl=1 . The corresponding diagrams are given in Fig. equation 2. Diagram 2c, the RPA correction, gives

2 In this subsection we show how Eq. (65) can be re- 2 Z 2 q q q c(s) d k nF (ξk− ) − nF (ξk+ ) produced in a diagrammatic calculation. Already at this χ = 2 2 2 k · q × m2 l=1,2c (2π)2 Ω − 1 k · q + iδ order we will see that one needs to treat self-energy and m Ω vertex corrections on equal footings because the conti- U(q) nuity equation emerges due to a particular cancellations (79) between these two types of corrections. To begin with, we re-write Eq. (15) for free- + By making use of fermion susceptibility for a current order parameter with c(s) k · q mΩ λl=1 (k) = k · qˆ as 1 = 1 − m (80) Ω − m k · q + iδΩ Ω − m k · q + iδΩ 2 Z d k nF (ξk− q ) − nF (ξk+ q ) 2 c(s) 2 2 2 we ﬁnd q χl=1,0(q) = −2 2 1 (k · q) (2π) Ω − m k · q + iδΩ q2 (75) χc(s) (q) = Ω2χc(s) (q) (81) m2 l=1,2c l=0,2c Here and later on we omit the k symbol for clarity. We then rewrite the form factor as: Note that there is no need to subtract the static part c(s) because χl=1,2c(q, 0) vanishes. (k · q)2 = (k · q + mΩ)(k · q − mΩ) + m2Ω2 (76) For the remaining two diagrams in Fig. 2 we obtain

2 2 Z d k d p [nF (ξp− q ) − nF (ξp+ q )][nF (ξk− q ) − nF (ξk+ q )] q2χc(s) (q) = −2 2 2 2 2 U(|p − k|)(k · q)2 (82) l=1,2a 2 2 1 2 (2π) (2π) (Ω − m k · q + iδΩ) 2 2 Z [nF (ξ q ) − nF (ξ q )][nF (ξ q ) − nF (ξ q )] 2 c(s) d k d p p− 2 p+ 2 k− 2 k+ 2 q χl=1,2b(q) = 2 2 2 1 1 U(p − k)(k · q)(p · q) (83) (2π) (2π) (Ω − m p · q + iδΩ)(Ω − m k · q + iδΩ)

Applying again (80) we ﬁnd, (65), and only the sum of the two terms obeys (84). This is an indication that, within diagrammatics, the conti- q2 nuity equation emerges due to ﬁne cancellations between (χc(s) (q)+χc(s) (q)) = Ω2(χc(s) (q)+χc(s) (q)) m2 l=1,2a l=1,2b l=0,2a l=0,2b self-energy and vertex corrections, and one should go be- (84) yond RPA at each order in U to reproduce it. The static part of the sum of the two contributions cancel out. Eqs. (81) and (84) verify Eq. (65) to order U. c(s) c(s) We emphasize that χl=1,2a(q) and χl=1,2b(q), when taken separately, do not satisfy the continuity equation 13

� �′ − 2 � − � � � � � � � � − �′ − � − �′ + �′ − � � − �′ − 2 2 2 2 2 2 � � � � � � � − � − 2 � − � 2 � − � �′ − � � − � �′ + � 2× � � � � � � � � + � + �′ + � �′ + 2 2 2 � � � � � + �′ + � + �′ − 2 2 2 2 (�) (�) (�) (�)

�′ − � � � � − � � �′ − 2 � − �′ − � � 2 2 2 � − �′ − � � 2 � 2 � � � � � − � − � − � − � − � � − 2 � 2 2 2 2 �′ 2 � � + � − �′ � − � � + � − �′ 2× 2× 2×

� � � � � �′ � � + �′ + � + �′ + � + � + 2 2 2 2 � � 2 2 � + � + (�) (�) (�) 2 (ℎ) 2 (�)

FIG. 7. Diagrams in second order of U. For constant interaction, (e) and (f), (h) and (i) cancels out. (g) half cancels (a) and (b) half cancels (d). What remains are half of (a), (c), and half of (d).

c(s) 2 B. Evaluation of χl=1 to order U ferences between the charge and spin channels emerge, in the form of the Aslamazov-Larkin (AL) diagrams, Figs. We now present the results of explicit calculations of 7c,d. The AL diagrams contribute in the charge channel the static susceptibilities to order U 2. We identify contri- and vanish in the spin channel, as can be seen from direct 2 spin summation. butions to χc(s) and Λc(s)Z , and χc(s) from each l=1,inc l=1 l,qp Consider the charge channel first. It is straightforward diagram, and compute them by a combination of ana- to identify the diagrams in Figs. 7, which give equal lytical and numerical methods. We also independently contributions, up to overall factor. One can easily verify c(s) 2 compute the vertex renormalization Λl=1 to order U . that χ6a = −2χ6g, χ6d = −2χ6b, and χ6e = −χ6f . In There are nine different diagrams for the current sus- addition, using the relation ceptibility to second order in U(q), see Fig. 7. To simplify Z Z the numerics, we approximate U(q) by a constant U i.e., 0 3 0 1 0 0 2 2 dωp(G q ) G q = − dωp(G q G q ) , (87) consider U renormalizations in the Hubbard model. For p− 2 p+ 2 p− 2 p+ 2 c(s) 2 a constant U, Landau parameters Fl also only emerge 2 at order U , i.e., the incoherent part of the susceptibility, we find χ6h = −χ6i. In Eq. (87) and throughout this sec- 0 vertex renormalizion, renormalization of the quasiparti- tion we denote G0(k) ≡ G for compactness. Summing 2 k cle Z, and Landau parameters are all of order U . We up the contributions to the charge-current susceptibility, 28 make use of previously known results we obtain at order U 2,

2 2 2 2 c s m U m U F = −F = ,Z = 1 − 1.39 , (85) c 1 1 1 1 2 2 δχ = χ + χ + χ (88) 8π 8π l=1 2 6a 6c 2 6d and A similar consideration for the spin susceptibility yields ∗ 2 2 m c m U = 1 + F1 = 1 + (86) 1 m 8π2 δχs = (χ − χ ) (89) l=1 2 6a 6d which holds for a Galilean-invariant system1,2. The order U 2 is the ﬁrst one in perturbative expansion at which dif- In explicit form

Z 3 3 0 3 2 d kd k d p 2 0 2 0 0 0 0 χ6a = 8U (p · qˆ) (Gp− q ) Gp+ q Gp−kGk0−kGk0− q , (2π)9 2 2 2 Z 3 3 0 3 2 d kd k d p 0 0 0 0 0 0 0 χ6c = 4U (p · qˆ)(k · qˆ)Gp− q Gp+ q Gk0− q Gk0+ q Gp−kGk0−k, (90) (2π)9 2 2 2 2 Z 3 3 0 3 2 d kd k d p 0 0 0 0 0 0 0 χ6d = 4U (p · qˆ)(k · qˆ)Gp− q Gp+ q Gk0− q Gk0+ q Gp−kGk0+k. (2π)9 2 2 2 2 14

H M L We set Ω = 0 and take q to be small but ﬁnite. After e.g. χ6a = χ6a+χ6a +χ6a. In this computational scheme, integration over frequency, we split each diagram into AL diagrams contain “H”, “M”, and “L” parts, while the three parts: “high”, “middle”, and “low” (which we label diagram with self-energy renormalization contains “H” “H”, “M”, and “L”), depending on whether zero, one, or and “M” parts. In explicit form we have two internal fermionic momenta are conﬁned to the FS,

Z 2 2 0 2 H 2 d kd k d p 2 [nF (ξk0 ) − nF (ξk0−k)][nF (ξp) − nF (ξp−k)][nB(ξk0 − ξk0−k) − nB(ξp − ξp−k)] χ6a = −8U 6 (p · qˆ) 3 (2π) (ξk0 − ξk0−k − ξp + ξp−k) (91a) Z 2 2 0 2 H 2 d kd k d p 0 [nF (ξk0 ) − nF (ξk0−k)][nF (ξp) − nF (ξp−k)][nB(ξk0 − ξk0−k) − nB(ξp − ξp−k)] χ6c = +8U 6 (p · qˆ)(k · qˆ) 3 (2π) (ξk0 − ξk0−k − ξp + ξp−k) (91b) Z 2 2 0 2 H 2 d kd k d p 0 [nF (ξk0 ) − nF (ξk0+k)][nF (ξp) − nF (ξp−k)][nB(−ξk0 + ξk0+k) − nB(ξp − ξp−k)] χ6d = −8U 6 (p · qˆ)(k · qˆ) 3 (2π) (ξk0 − ξk0+k + ξp − ξp−k) (91c) Z 2 2 0 2 0 M 2 d kd k d p 2 |k | cos φk0 0 [nF (ξk0 ) − nF (ξk0−k)][nB(ξk0 − ξk0−k) − nB(ξp − ξp−k)] χ6a = −4U 6 (p · qˆ) 1 + nF (ξp) 2 (2π) |p| cos φp (ξk0 − ξk0−k − ξp + ξp−k) (91d) Z 2 2 0 2 M 2 d kd k d p 0 0 [nF (ξk0 ) − nF (ξk0−k)][nB(ξk0 − ξk0−k) − nB(ξp − ξp−k)] χ6c = +8U 6 (p · qˆ)(k · qˆ)nF (ξp) 2 (91e) (2π) (ξk0 − ξk0−k − ξp + ξp−k) Z 2 2 0 2 M 2 d kd k d p 0 0 [nF (ξk0 ) − nF (ξk0+k)][nB(−ξk0 + ξk0+k) − nB(ξp − ξp−k)] χ6d = +8U 6 (p · qˆ)(k · qˆ)nF (ξp) 2 (91f) (2π) (ξk0 − ξk0+k + ξp − ξp−k) Z 2 2 0 2 0 L 2 d kd k d p 2 |k | cos φk0 0 0 nB(ξk0 − ξk0−k) − nB(ξp − ξp−k) 0 χ6a = +4U 6 (p · qˆ) nF (ξp)nF (ξk ) (91g) (2π) |p| cos φp ξk0 − ξk0−k − ξp + ξp−k Z 2 2 0 2 L 2 d kd k d p 0 0 0 nB(ξk0 − ξk0−k) − nB(ξp − ξp−k) 0 χ6c = −4U 6 (p · qˆ)(k · qˆ)nF (ξp)nF (ξk ) (91h) (2π) ξk0 − ξk0−k − ξp + ξp−k Z 2 2 0 2 L 2 d kd k d p 0 0 0 nB(−ξk0 + ξk0+k) − nB(ξp − ξp−k) 0 χ2d = −4U 6 (p · qˆ)(k · qˆ)nF (ξp)nF (ξk ) (91i) (2π) ξk0 − ξk0+k + ξp − ξp−k

Here nB(ξ) = −Θ(−ξ) at T = 0. and The “H” contributions can be evaluated by just setting m2U 2 Ω = 0 and q = 0 in Eq.(90), e.g. χL = − χ (96) 6d 4π2 l=1,0 Z 3 3 0 3 H 2 d kd k d p 2 0 3 0 0 0 where χl,0 is a free-fermion susceptibility, given by (17). χ = 8U p (G ) G G 0 G 0 (92) 6a (2π)9 p p−k k −k k c(s),L c(s) c(s) Using (88), (89) and (85), we ﬁnd δχl=1 = −χl=1,0Fl=1 as in (94). The “M” and “H” terms in Eq. (91) are high The sum of “H” parts is then the incoherent part of the dimensional principal value integrals, which we evaluate susceptibility numerically. Details of our numerics can be found in the Appendix. δχc(s),H = χc(s) (93) l=1 l=1,inc According to Eqs. (8) and (9), the total “H” contributions to charge-current susceptibility, δχc should The “M” and “L” parts determine l=1,H vanish, while other contributions should obey, to order ∗ 2 2 c(s),M c(s) m c(s) U , δχl=1 = χl=1,0 Λl=1 Z − 1 m c m c 2 c δχl=1,M = (1 + F1 ) − 1 χl=1,0 ≈ F1 χl=1,0 c(s),L c(s) c(s) m∗ δχl=1 = −χl=1,0Fl=1 (94) s m s 2 s c δχl=1,M = (1 + F1 ) − 1 χl=1,0 ≈ (2F1 − F1 )χl=1,0 The “L” part can be computed analytically and yields m∗ ∗ s m s c s L L δχl=1,H = − 1 − Fl χl=1,0 ≈ (F1 − F1 )χl=1,0 (97) χ6a = χ6c = 0, (95) m 15

In explicit form � � 2 Z 3 3 � − � � − � 2U d kd p 0 2 0 0 � − � � + � Λ = − (p · qˆ)(G ) G G 7a 6 p p−k kF nˆ−k kF (2π) � (100) � 2 Z 3 3 2U d kd p 0 2 0 0 (�) Λ = − (p · qˆ)(G ) G G (�) 7b 6 p p−k kF nˆ+k kF (2π) FIG. 8. The two AL vertex correction diagrams for three-leg wheren ˆ is a unit vector. We evaluated the integrals in the vertex. RHS of (100) numerically and the results are presented in Table II. From (97) and (85) we expect

� � � � c s Λl=1 − 1 = 1.39Λ, Λl=1 − 1 = −0.61Λ (101) � � � � 2 2 where Λ = m U . We see that these relations are satis- − 8π2 ﬁed, as they should be. � � � � � � � �

� � � � � � � � � C. Microscopic explanation for the absence of l = 1 � � � � � � Pomerachuk instabilities + + − � � � � � � We now present microscopic arguments as to why ΛZ � � + � − � � � � + � − � � � � + � − � � and χl=1,inc, for charge-current and spin-current susceptibilities are expressed via Landau parameters. We will analyze Eq. (68) for the spin channel, where ΛZ = ω s c FIG. 9. Diagrams for Γ to second order, for a constant U. 1 + F1 − F1 . The quasiparticle residue Z can be expressed via ω Γαβ,αβ using a Ward identity for any conserved c(s) 2 Using Eq. (85) for F1 and χl=1,0 = mkF /(2π) (re- ”charge”1,2,29. For our purpose it is best to use the Ward c(s) call that for spin and charge currents fl=1 (kF ) = 1), we identity associated with conservation of total momen- obtain tum (recall that we consider a Galilean invariant system).

c c Substituting λρ(k = k into (60b) we obtain δχl=1,M = χ, δχl=1,H = 0, s s (98) Z 3 δχl=1,M = −3χ, δχl=1,H = 2χ, 1 i X d q ω 2 ω = 1 − 3 Γαβ,αβ(kF p,ˆ q)(Gq) pˆ · q Z 2kF (2π) αβ 3 2 2 3 c(s) where χ = m U kF /16π . In Table I we list δχl=1,H , (102) c(s) c(s) δχl=1,M and δχl=1,L in units of χ. We also computed The renormalization of the spin-current vertex can be c(s) written as Λl=1 independently, by collecting vertex correction diagrams, keeping external particles at the FS. Applying the Λs σz = σz − same tactics as before, i.e., identifying equivalent contri- l=1 ββ ββ i X Z d3q butions to reduce the number of diagrams, we ﬁnd that Γω (k p,ˆ q)(G2)ωpˆ · qσz k (2π)3 αβ,αβ F q αα 1 F α Λc = 1 + Λ + Λ l=1 7a 2 7b (103) s 1 Λl=1 = 1 − Λ7b (99) wherep ˆ · q is now simply the form-factor for the cur- 2 rent. The vertex function Γω to order U 2 is given by the where Λ7a and Λ7b are two vertex corrections in Fig.8. diagrams in Fig. 9. In explicit form

1 Z d3k Γω (p = (k p,ˆ 0), q) = δ δ U + iU 2 (2G G + G G ) αβ,γδ F 2 αγ βδ (2π)3 k q−p+k k q+p−k 1 Z d3k − σ · σ U + iU 2 G G . (104) 2 αγ βδ (2π)3 k q+p−k

Summing up contributions from both Z and Γ we obtain, to order U 2, s Λl=1Z = 1− 2 Z 3 3 U d kd q 2 ω 6 (GkGq−p+k + GkGq+p−k)p ˆ · q(Gq) kF (2π) (105) 16

channel δχl=1,L δχl=1,M δχl=1,H δχl=1 channel from Eq.(100) from δχl=1,M charge −1 +0.99 ± 0.02 +0.01 ± 0.04 0.01 ± 0.04 charge +1.39 ± 0.02 +1.38 ± 0.01 spin +1 −2.97 ± 0.02 +1.98 ± 0.02 0.01 ± 0.02 spin −0.604 ± 0.008 −0.60 ± 0.01

TABLE I. The contributions to susceptibilities for c(s) TABLE II. Numerical results for Λl=1 − 1 for the case charge-current and spin-current order parameters (l = 1 c(s) c(s) 2 when the form factor is λl=1 (p) = p. The results are in orders with form factor λ (p) = p) at order U , from 2 2 l=1 units of Λ = m U . The ﬁrst column is obtained from fermions at high (“H”), middle (“M”) and low (“L”) en- 8π2 a direct evaluation of Eq.(100) and the second one is ex- ergies. The “L” contribution was obtained analytically, c(s) and the “M” and “H” contributions were obtained numer- tracted from our calculation of δχl=1,M , via Eq.(94) . The 3 2 2 3 results are in agreement with Eq.(101) and, hence, with ically. The numbers are in units of χ = m U kF /16π . The results agree with Eq. (98) and, hence, with Eqs. (8) Eq.(7). and (9).

2 k 2 k As written, the integral in the RHS of Eq. (105) is not where (Gq) = limk→0G(q + k, Ω)G(q, Ω). This (Gq) 2 ω conﬁned to the FS. However, the sum can be re-expressed has a regular piece, equal to (Gq) , and an extra piece as an integral over the FS. The reason for this is the which comes from the FS. Using the known relation1 identity28,

Z d3kd3q (G G + G G )(G − G ) = 0, (2π)6 k q−p+k k q+p−k q+p q (106) 2πiZ2 (G2)k = (G2)ω − δ(ω)δ(|q| − k ), (108) where p = (k p,ˆ Ω) and → 0. This identity can be q q ∗ F F vF proven by a simple relabeling of indices on the p-h bubble. Choosing Ω = 0 and expanding to ﬁrst order in , we obtain Z d3kd3q (G G + G G )(G2)kpˆ · q = 0, (2π)6 k q−p+k k q+p−k q substituting into (107), and using Eq. (104) to extract (107) the Landau parameters, we obtain

2 Z 3 3 Z U d kd q 2 ω dθ c s c s 6 (GkGq−p+k + GkGq+p−k)(Gq) pˆ · q = (F (θ) − F (θ)) cos θ = Fl=1 − Fl=1 (109) kF (2π) 2π

Substituting into (105), we recover Eq. (101). fl=1(|k|) 6= 1. We argue that in this case there is no re- We emphasize that only the product Λs Z is ex- c(s) c(s) l=1 lation Λl Z ∝ (1 + F1 ) and therefore a Pomeranchuk pressed via the integral over the FS. Taken separately, c(s) s instability does occur when Fl = −1. Λl=1 and Z are determined by integrals which are not c(s) conﬁned to the FS. We also note that the same Eq. (109) The argument is quite straightforward – ρl=1 with allows one to express the eﬀective mass, computed to or- fl=1(|k|) 6= 1 is not a current of a conserved quantity, der U 2 in a direct perturbation theory, as the integral hence it is not related by a continuity equation to a quan- over the FS in Eq. (86) (see Ref. 28 for details). tity, such as a conserved charge, whose susceptibility is expressed in terms of Landau parameters. Rather, it has two pieces and is of the form,

c(s) c(s) c(s) c(s) χ =χ ˜ + δχ , (110) V. ARBITRARY FORM-FACTOR λl=1 (k) AND l=1 l=1 l=1 OTHER VALUES OF l c(s) whereχ ˜l=1 is finite and can be expressed in terms of c(s) The purpose of this final section is to clarify how Landau parameters at Ω/vF |q| → 0, but δχl=1 cannot. c(s) c(s) generic are the constraints imposed by Eqs. (7), (8), As a result,whileχ ˜l=1 remains finite when Fl=1 = −1, which prevent a Pomeranchuk instability for charge- c(s) δχl=1 diverges, signaling a Pomeranchuk instabilitiy. current and spin-current order parameters. In this sec- An indication of this appears already at first order in tion we first study the case of an order parameter U. To see this, we evaluate the diagrams of Fig. 2 in Sec. c(s) c(s) ρl=1 with form factor λl=1(k) = kfl=1 (|k|) for which IV A for the more general case fl(|k|) 6= 1. Then we find 17

channel δχl=1,L δχl=1,M δχl=1,H δχl=1 channel from Eq.(100) from δχl=1,M charge −1 +0.36 ± 0.04 +0.12 ± 0.04 −0.52 ± 0.06 charge +1.07 ± 0.01 +1.07 ± 0.01 spin +1 −0.86 ± 0.02 +1.50 ± 0.04 −0.36 ± 0.04 spin −0.548 ± 0.008 −0.538 ± 0.008

TABLE III. Numerical results for high- and middle- energy c(s) TABLE IV. Numerical results of Λl=1 − 1 for the case when contributions to charge and spin susceptibilities for l = 1 c(s) c(s) the form factor is λl=1 (p) = kF p/|p|. The numbers are in order parameters with the form factor λ (p) = k p/|p| 2 2 l=1 F units of Λ = m U . The first column is obtained from a di- (“M” and “H” terms), together with the analytical result for 8π2 the low-energy “L” contribution. The numbers are in units rect evaluation using Eq.(100)(in which p is replaced with 3 2 2 3 kF p/|p|, and the second column is extracted from our calcu- of χ = m U kF /16π . The results clearly deviate from those in Table I and do not satisfy Eqs. (8) and (9). lation of δχl=1,M via Eq.(94). The results show that Eq.(7) is not satisfied if the form-factor is different from k.

c(s) the contribution of diagrams 2a,b is: Here,χ ˜l=0 is the susceptibility of a channel with l = 0 symmetry, but with fl=1(|k|) in the form-factor, and m2Ω2 χc(s) + χc(s) = χ˜c(s) +χ ˜c(s) + δχc(s), l=1,2a l=1,2b q2 l=0,2a l=0,2b l=1 (111)

2 2 Z d kd p nF ξ − nF ξ nF ξ − nF ξ δχc(s) = 2 p−q/2 p+q/2 k−q/2 k+q/2 l=1 (2π)2 (m−1q)2 2Ω U(p − k) f (|k|)f (|p|) − f 2 (|k|) 1 − , (112) l=1 l=1 l=1 Ω − m−1k · q

c(s) is an additional term which is exactly zero for fl = 1. The not cancel 1/(1 + F1 ) in the quasiparticle part of the results to order U are somewhat special because each of susceptibility. Since there is no cancellation of the di- the three terms in Eq. (111) has an additional q2 factor verging part, a Pomeranchuk instability does occur when in the Ω/q → 0 limit. c(s) c(s) F1 = −1 for any order parameter with fl=1 6= 1. Nevertheless, the appearance of δχl=1 already at this We also explicitly calculated “L”, “M”, and “H” con- c(s) c(s) c(s) order indicates that δχl=1 is not expressed via δχl=0 , tributions to susceptibility in l = 2 with fl=2 (k) = 1 and taken in the q/Ω → 0 limit, as it was the case for a k2 f = F . For l = 2, F c = −F s = χ/2, such that the current of a conserved order parameter. l=2 |k|2 2 2 low-energy contributions to the l = 2 charge and spin To see explicitly that for fl 6= 1 Eqs. (7) and (9) are susceptibilities are δχc = −δχs = −χ0/2, where no longer valid we perform the same calculations as in l=2,L l=2,L χ0 = χk2 = m3U 2k4 /16π3. We show the results in Ta- Sec. IV C for f c,s (|k|) 6= constant. For definiteness, we F F l=1 ble V. We didn’t find any relation between “M” and “H” c(s) c(s) ˆ consider fl=1 (|k|) = kF /|k|, i.e., λl=1 (k) = kF k · qˆ. The contributions to both spin and charge susceptibilities and cancellation between different diagrams for susceptibility c(s) s s 1 + Fl=2 . In particular, we checked the expressions for still holds, and the results for δχl=1 and δχl=1 to order 2 l = 2 case presented in Ref. 20 and did not reproduce U are still given by Eqs (88) and (89), and the contribu- them. This can be also seen by comparing the results tion from each diagram can again be split into “H”, “M”, in Ref. 20 with our expressions for susceptibility to first and “L” parts. However, now each contribution has to be order in momentum-dependent U(q), Eq. (26). computed with different prefactors. This does not affect c(s) the “L” contribution as, by construction, fl=1 (kF ) = 1, c(s) but the modification of fl=1 (k) does affect “M” and “H” VI. SUMMARY contributions. In Table I we present the results for “H”, “M”, and “L” contributions to δχc and δχs in units l=1 l=1 In this paper we studied the constraints placed by of χ. We also computed Λc(s) by evaluating the renor- l=1 conservation laws on Pomeranchuk transitions, particu- malization of the three-leg vertex. We show the results larly the role of the continuity equation and longitudinal in Table IV, again in units of Λ. We see that neither the sum rule. This issue has been previously considered by constraints on the components of the susceptibilities, Eq. 17 c(s) Leggett back in 1965, and was re-analyzed recently by 20 (98), nor the conditions on Λl=1 , Eq. 101, are obeyed. Kiselev et al . The continuity equation and the sum c(s) c(s) Therefore, Λl=1 does not scale with (1 + F1 ) and does rule reveal interesting properties of susceptibilities of cur- 18

channel δχl=2,L δχl=2,M δχl=2,H δχl=2 1 charge, fl=2 = 1 + 2 −1.48 ± 0.02 +6.40 ± 0.04 +4.92 ± 0.04 1 spin, fl=2 = 1 − 2 −1.50 ± 0.02 +1.32 ± 0.04 −0.18 ± 0.04 2 kF 1 charge, fl=2 = |k|2 + 2 −1.46 ± 0.02 +1.18 ± 0.04 −0.28 ± 0.04 2 kF 1 spin, fl=2 = |k|2 − 2 −1.66 ± 0.02 +0.70 ± 0.04 −0.96 ± 0.04

TABLE V. Charge and spin susceptibilities in the quadrupolar l = 2 channel, calculated from Eqs.(91) using two different 0 3 2 4 3 form factors. The numbers are in units of χ = m U kF /16π . The results show no connection between high-energy and middle-energy contributions and the low-energy contribution. Different form factors depend on fl=2 through our definition in Eq.(14).

c(s) rents of conserved total charge and spin. Namely, high factor λl=1 (k) = k. Such an order parameter describes c(s) currents of the fermionic number and spin - both of energy features of a system, such as Λl=1 Z, and the in- c(s) which are conserved quantities. For any form factor coherent piece of the susceptibility, χl=1,inc, can be ex- c(s) with l = 1 symmetry, but different functional behav- c(s) c(s) pressed in terms of the Landau parameters Fl , which ior, λl=1 (k) = fl=1 (|k|)k with f(|k|) 6= 1, high-energy describe the interaction between fermions on the FS. In and low-energy contributions to the susceptibility are not particular, Λc(s)Z scales as (1 + F c(s)) and vanishes at l=1 1 correlated. The same is true for other values of l. As a c(s) F1 = −1, when the quasiparticle contribution to sus- result, the susceptibility for any other order parameter c(s) c(s) ceptibility diverges as 1/(1 + F1 ). The vanishing of with either l = 1 or other l diverges when Fl = −1, c(s) i.e., the Pomeranchuk instability does occur. Λl=1 Z cancels out the divergence, and, as a result, the system does not undergo a p-wave Pomeranchuk instability. Our aim was to verify this in diagrammatic perturbation theory, present a microscopic explanation why VII. ACKNOWLEDGEMENTS high-energy and low-energy contributions to susceptibility are related, and check how general such constraints We thank J. Schmalian, P. Woelfle, and particularly are. D. Maslov for valuable discussions. The work was sup- ported by NSF DMR-1523036. AVC is thankful to KITP We showed that the constraints work only for l = 1 at UCSB where part of the work was done. KITP is sup- and for the specific l = 1 order parameter with form- ported by NSF grant PHY-1125915.

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APPENDIX: DETAILS OF THE NUMERICAL (3kF , 6kF ) and (3kF , 15kF ). Every subregion was sam- EVALUATION pled using a maximum of 109 points. We evaluated each subregion 10 times to ensure the convergence of the nu- c(s) c(s) For our numerical evaluation of the high energy and merical sums. The various χl , Λl Z we needed are middle energy contributions to the second order dia- readily found from the numerical expressions for the “H” grams, Eqs. (91) and (100) we used Mathematica 11.1.1 and “M” diagrams as detailed in the text. The deviation with the built-in algorithm NIntegrate, using the Monte of these 10 evaluations are the basis for computing the Carlo integration strategy. In our evaluation of diagrams error brackets of Tables I-V. we used polar coordinates and cut off the momentum at As a check of the reliability of our numerical scheme we computed the quasiparticle residue Z, which is known 15kF , e.g. {|p|, 0, 15kF }. The UV divergence in Eqs. m2U 2 (91) is avoided by the symmetry factor cos lφ and this to be Z = 1 − 1.39 8π2 (see text). Our calculation for 15kF truncation is large enough to obtain our results ac- Z is based on Pitaevskii-Landau relations, Eq. (102) of curately. Since only the angle differences of three the the text and momenta(p, k and k0) enter our integrals, one can inte- 1 i X Z d3q grate out one of these three angles by hand to achieve = 1 − Γω (k p,ˆ q)(G2)ω (113) Z 2 (2π)3 αβ,βα F q higher accuracy. αβ H In evaluations of the high energy contributions(χl,6a, H H Eq. (113) and Eq. (102) must give the same result. χl,6c and χl,6d), the integral region {|p|, 0, 15kF } × Numerically we found, 0 {|k|, 0, 15kF } × {|k |, 0, 15kF } is divided into 8 parts: 2 2 every dimension of momentum is divided into (0, 3kF ) m U Z = 1 − (1.389 ± 0.045) based on Equation (113), and (3kF , 15kF ), e.g. {|p|, 0, 15kF } = {|p|, 0, 3kF } + 8π2 {|p|, 3kF , 15kF } . In evaluations of mixed energy 2 2 M M M m U contributions(χl,6a, χl,6c and χl,6d), the integral region Z = 1 − (1.390 ± 0.028) 2 based on Equation (102), 0 8π {|k|, 0, 15kF } × {|k |, 0, 15kF } is divided into 9 parts instead. Each momentum dimension is divided as (0, 3kF ), which gives us confidence our integrals are accurate. 21

TABLE VI. Numerical results of high energy and mixed energy contributions for diﬀerent form factors. The unit here are 2 m3U2 4 m3U2 kF 8π3 for the ﬁrst two lines and kF 8π3 for the last two lines.

1 H H 1 H 1 M M 1 M Form factor 2 χl,6a χl,6c 2 χl,6d 2 χl,6a χl,6c 2 χl,6d λp = |p| cos φp 0.613 ± 0.008 −0.226 ± 0.017 −0.379 ± 0.005 −0.879 ± 0.008 0.767 ± 0.006 0.608 ± 0.005 λp = kF cos φp 0.500 ± 0.013 −0.193 ± 0.008 −0.250 ± 0.008 −0.879 ± 0.008 0.508 ± 0.013 0.548 ± 0.005 2 λp = |p| cos 2φp 1.956 ± 0.015 −0.052 ± 0.008 1.292 ± 0.014 −0.879 ± 0.008 0.266 ± 0.006 −0.379 ± 0.006 2 λp = kF cos 2φp 0.500 ± 0.013 −0.058 ± 0.006 0.147 ± 0.012 −0.879 ± 0.008 0.202 ± 0.008 −0.303 ± 0.005