Fermi-Liquid Theory and Pomeranchuk Instabilities: Fundamentals and New Developments

Fermi-liquid theory and Pomeranchuk instabilities: fundamentals and new developments

Andrey V. Chubukov1, Avraham Klein1, and Dmitrii L. Maslov2 1Department of Physics, University of Minnesota, 116 Church Street, Minneapolis, MN 55455 2Department of Physics, University of Florida, P. O. Box 118440, Gainesville, FL 32611-8440 (Dated: May 7, 2019) This paper is a short review on the foundations and recent advances in the microscopic Fermi- liquid (FL) theory. We demonstrate that this theory is built on ﬁve identities, which follow from conservation of total charge (particle number), spin, and momentum in a translationally and SU(2)- invariant FL. These identities allows one to express the eﬀective mass and quasiparticle residue in terms of an exact vertex function and also impose constraints on the “quasiparticle” and ”incoherent” (or “low-energy” and “high-energy”) contributions to the observable quantities. Such constraints forbid certain Pomeranchuk instabilities of a FL, e.g., towards phases with order parameters that coincide with charge and spin currents. We provide diagrammatic derivations of these constraints and of the general (Leggett) formula for the susceptibility in arbitrary angular momentum channel, and illustrate the general relations through simple examples treated in the perturbation theory.

It is our great pleasure to write this article for the excitations vF is replaced by the effective Fermi velocity ∗ special volume in celebration of the 85th birthday of Lev vF (or, equivalently, the fermionic mass m = pF /vF is ∗ ∗ Petrovich Pitaevskii, who, in our view, is one the greatest replaced by the effective mass m = pF /vF ) and ii) the physicist of his generation and the model of a scientist wave function of a state near the FS in an interacting and citizen. His seminal volumes on Statistical Physics system is renormalized by a factor of √Z < 1, so that (with E. M. Lifshitz),1 Physical Kinetics2 (also with E. the corresponding probability for the state to be occu- M. Lifshitz), and Quantum Electrodynamics3 (with V. B. pied is renormalized by Z. The factor Z is often called Berestetskii and E. M. Lifshitz), all parts of the Landau quasiparticle residue. Its presence reflects the fact that and Lifshitz Course on Theoretical Physics, as well as even infinitesimally close to the FS, the spectral function on Bose-Einstein condensation and superfluidity4 (with of interacting fermions is not just a δ-function, like for S. Stringari) are not only used by every contemporary free fermions, but also contains incoherent background, physicist but also, as we are positive, will serve the future which extends to energies both above and below EF . The generations of scientists from around the world. fact that the residue Z of the δ functional piece is less The paper we present for this volume is devoted to than unity implies that interactions− moves some spectral the microscopic theory of a Fermi liquid, which was pio- weight into incoherent background. neered by Lev Petrovich in the early 1960s. The general It is customary to consider two groups of fermionic relations he obtained with Landau, which express the ef- states: near the FS and away from it. We will be re- fective mass m∗/m and the quasiparticle residue Z in ferring to the first group as to “high-energy fermions”, terms of the vertex function (the Pitaevskii-Landau rela- or simply as to “high energies”, and to the second one tions), set the gold standard for many-body theory. We as to “low-energy fermions”, or simply to “low energies”. hope that Lev Petrovich and other readers of this volume The conventional wisdom is that the fundamental proper- will find our summary of recent developments in this field ties of a FL, such as its thermodynamic characteristics at interesting. low T , are completely determined by low-energy fermions while high-energy fermions can be safely integrated out, e.g., within the renormalization group formalism.12 On I. INTRODUCTION a technical level, high-energy fermions are believed to determine only the value of Z and the vertex function ω Despite its apparent simplicity, the Fermi-liquid (FL) Γ (pF , qF ), which parametrizes the interaction between theory is one of the most non-trivial theories of inter- low-energy fermions. acting fermions. 1,5–11 In general terms, it states that a The potential instabilities of a FL – superconductiv- system of interacting fermions in dimension D > 1 dis- ity and a spontaneous deformation of the FS (a Pomer- 13 plays behavior which differs from that of free fermions anchuk instability) – are also believed to be fully deter- quantitatively rather than qualitatively. In particular, mined by the interaction between low-energy fermions. the FL theory states that at temperatures much lower In particular, the condition for a Pomeranchuk instabil- than the Fermi energy EF , the inverse lifetime of a ity to occur in the charge or spin channel with orbital c(s) c(s) fermionic state near the Fermi surface (FS) is much momentum l is given by Fl = 1, where Fl are smaller than its energy, so that to first approximation the Landau parameters, which are partial− components of ω these states can be viewed as sharp energy levels with properly normalized Γ (pF , qF ) energy p = vF (p pF ), measured from EF . The only Such wisdom, however, is based on the phenomenolog- two differences with− free fermions are i) the velocity of ical formulation of the FL theory, originally developed by 2

Landau5,6. In this formulation, one deals exclusively with topic has re-surfaced recently in the context of the dis- low-energy fermions. On the contrast, the microscopic cussion about a p wave Pomeranchuk instability in the theory of a FL, developed later by Landau,7 Pitaevskii14 spin channel.18–22− and others,1,10 allows one to express the fundamental In the rest of the paper, we analyze the interplay be- properties of a FL, such as m∗/m, Z, and charge and spin tween the effects from high- and low-energy fermions in susceptibilities, in terms of the exact vertices parameter- some detail. We consider a translationally and rotation- izing the interactions between all states. Depending on ally invariant system of fermions with some dispersion the particular realization of a FL, as well as on the prop- p, which is not necessary parabolic (as it would be for a erty considered, the result may or may not be expressed Galilean-invariant system) but can be an arbitrary func- solely via low-energy fermions. One example is the ef- tion of k . We first review the formulation of the mi- fective mass, which happens to be a low-energy property croscopic| | theory of a FL in terms of the Ward identities only for a Galilean-invariant, or, more generally, Lorentz- associated with conservation laws for total charge, spin, invariant FL,10,15 but contains a high-energy contribu- and momentum. We show that these conservation laws tion otherwise. give rise to five relations. The first two are the origi- The interest to microscopic foundations of the FL the- nal Pitaevskii-Landau relations. They express 1/Z and ∗ ω ory has intensified over the last few decades due to ubiq- m /m in terms of the vertex function Γ (pF , q), in which uitous observations of non-FL behaviors in a wide va- the first fermion is on the FS while the other is, in gen- riety of solid-state systems, such as the cuprate and eral, away from it. The other three relations impose the ω Fe-based high-temperature superconductors, bad metals, constraints on Γ (pF , q), one of which directly relates the and other itinerant-electron systems driven to the vicin- contributions from low-energy and high-energy fermions ity of a quantum phase transition. The hope is that if to each other. We then show how these constrains pre- we understand better the conditions for the FL theory vent a Fermi surface deformation with the structure of to work, we will gain a better insight into its failures in spin current and charge current order order parameters. these and other cases. Another stimulus for such interest Following that, we review a diagrammatic derivation of is that currently there are several real-life examples of the constraints, imposed by conservation laws, and a di- electronic nematic order, which sets in as a result of a agrammatic calculation of the charge and spin suscep- Pomeranchuk instability, e.g., quantum Hall systems,16 tibilities with arbitrary form-factors. We argue that, 16 ? Sr3Ru2O7, and Fe-based superconductors. The the- for Fermi-surface deformations with structures different oretical literature is abundant with proposals for even from those of charge or spin currents, renormalization by more esoteric nematic states, and it is important to un- high-energy fermions reduces the divergence of the cor- derstand which of them are feasible. responding susceptibility at the Pomeranchuk instability In this communication we review earlier and recent but does not eliminate it completely, i.e., a Pomeranchuk work on the microscopic theory of a FL, with spe- instability towards a phase with such order parameter is cial attention paid to the interplay between contribu- not forbidden. Finally, we present the results of per- tions from high- and low-energy fermions. Our cen- turbative calculations to second order in a four-fermion tral message is that conservation laws set up delicate interaction and identify a particular relation involving balances between these contributions, with sometimes particle-hole and particle-particle polarization bubbles. surprising effects, and it is not always possible to re- This relation allows one to re-express the contribution duce the high-energy contributions to mere renormal- from high-energy fermions as the contribution from the izations of the input parameters for the low-energy the- FS, and vice versa. ory. The most spectacular example of this are the susceptibilities of the charge-current and spin-current order parameters: ρˆc (q) = P ∂p c† c and II. MICROSCOPIC THEORY OF A FERMI J p,α ∂p p−q/2,α p+q/2,α LIQUID s P ∂p † z ρˆJ (q) = p,αβ ∂p cp−q/2,ασαβcp+q/2,β. Within the random phase approximation (RPA), which includes only A. Pitaevskii-Landau and Kondratenko relations the low-energy contributions, both susceptibilities be- c(s) ∗ c(s) have as χJ (m /m)/(1 + F1 ) and diverge at Consider a translationally-invariant system of fermions c(s) ∝ F1 = 1, as is expected within the Pomeranchuk with H = Hkin + Hint, where 13− scenario. However, when one includes both high- and X † low-energy contributions and utilizes the continuity equa- Hkin = pcp,αcp,α (2.1) tion associated with conservation of total charge (particle pα number) or total spin, one finds that the divergent piece (with chemical potential included into p) and c(s) c(s) c(s) in χ cancels out, and χ remains finite at F = 1. J J 1 − X † † As a consequence, a Pomeranchuk instability towards the Hint = U(q)ck+q/2,αcp−q/2,βcp+q/2,βck−q/2,α. c s k,p,q,α,β phase with an order parameter ρˆJ (q) or ρˆJ (q) cannot c(s) occur. The absence of divergence of χJ was originally We also assume that rotational invariance is intact, i.e., demonstrated by Leggett back in 1965 (Ref. 17). This that the dispersion p depends on the magnitude of p 3 but not on its direction, and U(q) = U( q ). How- Γω (p, q), where α . . . δ denote the spin projections. | | αβ,γδ ever, we do not assume a specific form of p. It can This vertex describes the interaction between fermions be parabolic, as in 3He and near the Γ-point of the Bril- with incoming D +1-momenta p = (ω, p) and q = (ω0, q) 0 louin zone in cubic materials, or linear in p , as in Dirac and outgoing momenta p1 = (ω1, p1) and q1 = (ω1, q1), and Weyl materials,23 or else quadratic| at| the small- taken in the limit of strictly zero momentum transfer 24 est p and linear at larger p, as in bilayer graphene. and vanishingly small energy transfer, i.e., for p1 = p , 0 0 | | | | For all these cases, we assume that renormalization of q1 = q , ω1 ω, and ω1 ω . To first order in | | |ω | → → the fermionic properties by interaction comes predomi- U(q), Γαβ,γδ = U(0)δαγ δβδ U( p q )δαδδβγ . In gen- ω − | − | nantly from those momenta which are small enough for eral, Γαβ,γδ(p, q) contains contributions only from high- the lattice effects to be irrelevant.25 In all the cases, energy fermions and can be computed by setting both p pF ( p pF )/m near the Fermi momentum, where energy and momentum transfer to zero. ≈ −1 | | − 2 (2m) = ∂p/∂p . |p|=pF ∗ The propagator of free| fermions is The relations between Z and m /m and the vertex Γω follow from the identities for the derivatives of the 1 fermionic Green’s functions. These identities are associ- Gp = , (2.2) ω p + iδsgnp ated with conservation of total charge (or, equivalently, − total number of fermions), total spin, and total momen- where p = (ω, p). For interacting fermions the Landau tum. The set is over-complete in the sense that the iden- FL theory states that tities associated with charge conservation alone allow one to express Z and m∗/m via the vertex function. The re- Z maining identities place constraints on the vertex func- Gp = ∗ + Gp,inc, (2.3) ω p(m /m) + iδsgnp tion (see below). − where Gp,inc describes incoherent background. For ω The identities associated with charge conservation ∗ ≈ p(m/m ), Gp,inc is vanishingly small compared to the were first derived by Pitaevskii and Landau, and it is first term in (2.3). appropriate to call them Pitaevskii-Landau (PL) rela- The microscopic theory of a FL expresses the quasi- tions1,8,14. (For the history of deriving the PL relations, particle residue Z and the effective mass m∗ in Eq. see Ref. 26.) Although these relations were derived orig- (2.3) in terms of the bare mass m in (2.2) and a fully inally for a quadratic dispersion, one can readily obtain renormalized and anti-symmetrized four-fermion vertex, them in a form valid for arbitrary p:

−1 Z D+1 ∂Gp 1 i X ω 2 ω d q = = 1 Γ (pF , q)(G ) , (2.4a) ∂ω Z − 2 αβ,αβ q (2π)D+1 αβ −1 2 2 Z D+1 ∂Gp pF pF i X k pF q ∂q 2 k d q pF = ∗ = + Γαβ,αβ(pF , q) · par (Gq) D+1 , (2.4b) · ∂p −m Z − m 2 m ∂q (2π) αβ

where q = (ω, q), pF = pF pˆ,p ˆ is a unit vector along p, lated to each other by an integral equation par 2 and q = q /(2m) EF with the same m as in (2.2). k ω − Γαβ,αβ(p, q) = Γαβ,αβ(p, q) D−2 2 ∗ Z kF Z m X ω 0 k 0 Γ (p, q )Γ (q , q)dΩq0 . − (2π)D αξ,αη ηβ,ξβ ξ,η (2.5)

where dΩq is the infinitesimally small solid angle around 2ω In Eqs. (2.4a) and (2.4b) the object G is the prod- 2k 2ω q vector q, while Gq is related to Gq by uct of two Green’s functions with the same momenta and infinitesimally close frequencies, and Γk is the vertex 2 ∗ αβ,αβ 2k 2ω 2 2πiZ m in the limit of zero frequency transfer and vanishing mo- Gq Gq δGq = δ(ω)δ( q pF ). ω k 2ω − ≡ − pF | | − mentum transfer. In similarity to Γ and Γ , Gq con- (2.6) tains only contributions from high-energy fermions and Equations (2.4a) and (2.4b) are the most general re- can be replaced by just the square of the Green’s func- sults for Z and m∗/m Each equation contains the inte- 2 2k tion, Gq, while Gq contains an additional contribution grals over the intermediate states with momenta q not from fermions at the FS. The vertices Γk and Γω are re- confined to the FS. Therefore, in general, renormaliza- 4

∗ ∗ tions of both Z and m /m come from high energies. Sub- Ωq, which implies that this contribution to m /m comes stituting Eqs. (2.5) and (2.6) into Eq. (2.4b) we obtain, solely from fermions on the FS. On the other hand, the after some manipulations, factor of Q comes from high-energy fermions. Similarly to Q, renormalization of Z in (2.4a) also comes from high ∗ 2 (D−2) ∗ Z ! m Z p m pF qF energies. = Q 1 F Γω (p , q ) · dΩ , D αβ,αβ F F 2 q For an SU(2) -invariant FL, the vertex function can be m − 2(2π) pF (2.7) decoupled into the density (charge) and spin components where as

i P R ω 2 ω dD+1q Γω (p, q) = δ δ Γc(p, q) + σ σ Γs(p, q), (2.9) 1 Γ (pF , q)(G ) D+1 αβ,γδ αγ βδ αγ βδ − 2 αβ αβ,αβ q (2π) · Q = D+1 . i P R ω 2 ω pF ·q ∂q d q 1 αβ Γαβ,αβ(pF , q)(Gq) 2 par D+1 2 pF ∂q (2π) where σ is a vector of Pauli matrices. Because − P ω c (2.8) αβ Γαβ,αβ(p, q) = 2Γ (p, q), relations (2.4a) and (2.4b) The integral in the round brackets in (2.7) goes only over contain only the charge components of Γ:

−1 Z D+1 ∂Gp 1 c 2 ω d q = = 1 2i Γ (pF , q)(G ) , (2.10a) ∂ω Z − q (2π)D+1 −1 2 2 Z D+1 ∂Gp pF pF c pF q ∂q 2 k d q pF = ∗ = + 2i Γk(pF , q) · par (Gq) D+1 , (2.10b) ∂p −m Z − m m ∂q (2π)

c k where Γk is the charge component of Γ . tion associated with momentum conservation c s Due to rotational invariance, Γ and Γ can be ex- Z D+1 1 i X ω 2 ω pF q d q panded in partial components with diﬀerent angular mo- = 1 Γ (pF , q)(G ) · Z − 2 αβ,αβ q p2 (2π)D+1 menta αβ F Z p q D+1 c(s) X c(s) c 2 ω F d q = 1 2i Γ (pF , q)(G ) · . Γ (p, q) = Γl (p, q)Kl, (2.11) q 2 D+1 − pF (2π) l (2.14) where K are the normalized angular momentum eigen- l It is similar to Eq. (2.4a), but contains an extra functions, which depend on the angle θ between p and momentum-dependent piece in the r.h.s. Although (2.14) q. In 3D, K = K (θ) = (2l + 1)P (θ), where P (θ) are l l l l was derived originally for a Galilean-invariant FL,1,6,8,14 Legendre polynomials. In 2D, Kl(θ) = αl cos lθ, where it holds for arbitrary p (see below). α0 = 1 and αl>0 = 2. For the vertex function on the FS, c(s) Equations (2.4a) and (2.14) show that 1/Z can be ex- the partial components Γl are related to the Landau pressed via Γ in two diﬀerent ways. This obviously places c(s) parameters Fl , introduced in the phenomenological FL a constraint on the vertex function, namely, it must sat- theory, via isfy

D−2 Z D+1 2 ∗ pF q d q c(s) Z pF m c(s) c 2 ω Γ (pF , q)(Gq) 1 · = 0. (2.15) Fl = (D−1) Γl . (2.12) 2 D+1 π − pF (2π)

Substituting this relation into Eq. (2.7), we obtain Equation (2.14) allows one to re-write Q in Eq. (2.8) in a more transparent way, as ∗ m c D+1 = (1 + F1 )Q. (2.13) R c 2 ω pF ·q d q m 1 2i Γ (pF , q)(Gq) p2 (2π)D+1 − F Q = D+1 . (2.16) R c 2 ω pF ·q ∂q d q c 1 2i Γ (pF , q)(Gq) p2 ∂par (2π)D+1 In the phenomenological FL theory, F1 is considered as − F q an input, and Eq. (2.13) relates m∗/m to this parame- c We immediately see that if the dispersion is parabolic ter. In the microscopic FL theory, F1 is obtained from par the vertex function and by itself contains m∗ via (2.12). within the domain of integration over q, i.e., p = p , Q = 1. In this case, mass renormalization comes solely Equation (2.13) then should be viewed as an equation for 1,6,8 m∗/m, which one has to solve, if the goal is to express from fermions at the FS : ∗ ω m /m in terms of Γ . m∗ = 1 + F c. (2.17) Pitaevskii and Landau derived also an additional rela- m 1 5

This result was originally derived in the phenomenolog- are small. Combining Eqs. (2.4a), (2.7), and (2.14), one 6,8 ical FL theory with the help of Galilean boost. Later can construct the self-energy to ﬁrst order in ω and p. on, it was shown to be also valid for a Lorentz-invariant After some manipulations, we obtain relativistic FL.15 A Green’s function at arbitrary ω and p can expressed −1 via the self-energy as G (ω, p) = ω p + ΣFL(ω, p). − ΣFL(ω, p) = Q1 (ω p) + Q2p, (2.18) Near the FS, the Green’s function must reduce to the ﬁrst − term in Eq. (2.3). This implies that the self-energy must scale linearly with ω and p when both these variables where

Z D+1 1 c 2ω d q Q1 = 1 = 2i Γ (pF , q) G , Z − − q (2π)D+1  −1 2 (D−2) ∗ Z ! 1 m 1 1 Z pF m ω pF qF Q2 = 1 ∗ = 1 1 D Γαβ,αβ(pF , qF ) ·2 dΩq  . (2.19) Z − m Z − Q − 2(2π) pF

Later on, Kondratenko27,28 derived the relations be- of total spin. The relations are the same as Eqs. (2.4a) tween 1/Z, m∗/m and Γω associated with conservation and (2.4b) but contain extra Pauli matrices, which select the spin components of Γω and Γk

−1 Z D+1 ∂Gp 1 s 2 ω d q = = 1 2i Γ (pF , q)(G ) , (2.20a) ∂ω Z − q (2π)D+1 −1 2 2 Z D+1 ∂Gp pF pF s pF q ∂q 2 k d q pF = ∗ = + 2i Γk(pF , q) · par (Gq) D+1 . (2.20b) ∂p −m Z − m m ∂q (2π)

s k where Γk is the spin component of Γ . Combining general formalism of Ward identities, in which conser- Eqs. (2.10a) and (2.20a), and Eqs. (2.10b) and (2.20b), vation laws are expressed as relations between certain we obtain two additional constraints on the vertex func- vertex functions and Green’s functions. To obtain these tion, which relate the charge and spin components of Γω relations, we introduce three momentum and frequency- and Γk to each other via29,30 dependent operators, bilinear in fermions, which we as- D+1 sociate with conserved ”charges”. In our case these con- Z ω d q c s 2 served charges are charge, spin, and momentum densities, (Γ (pF , q) Γ (pF , q)) Gq D+1 = 0, − (2π) which are deﬁned as Z D+1 k,c k,s pF q ∂q 2 k d q c X † Γ (pF , q) Γ (pF , q) · par (Gq) D+1 . ρˆ (q) = cp−q/2,αcp+q/2,α, (2.22a) − m ∂q (2π) p,α = 0 X s q † z (2.21) ρˆ ( ) = cp−q/2,ασαβcp+q/2,β, (2.22b) p,αβ We emphasize that the integrals in (2.15) and (2.21) are mom X † ρˆ (q) = pc c . (2.22c) determined by high-energy fermions. These equations p−q/2,α p+q/2,α p,α set the conditions on the input parameters of the phenomenological FL theory. Due to spin-rotational invariance, we can consider only one component of the spin density, e.g., along the z- axis. For each conserved quantity there exists a conti- B. Pitaevskii-Landau and Kondratenko relations as nuity equation of the form Ward identities ∂ρˆc,s(q) = iq ρˆc(s)(q), ∂t − · J The PL relations, Eqs. (2.10a), (2.10b), and (2.14), mom ∂ρˆl (q) X mom and the Kondratenko relations, Eqs. (2.20a) and (2.20b), = i qkρˆ (q), (2.23) ∂t − J,lk can be recast into a more compact form by adopting the k 6

where ρˆJc,s are the operators of charge and spin currents, and its corresponding current with fully renormalized mom c(s) mom c(s) andρ ˆJ,lk (q) is the momentum current , i.e., the energy- three-leg vertices Λ (p, q), Λ , ΛJ (p, q), and momentum tensor. Both relations in Eq. (2.23) are exact mom c(s) ΛJ,ij , as shown in Fig. 1. We define ΛJ (p, q) and for a quadratic dispersion, i.e., for a Galilean-invariant mom Λ without overall form-factors of ∂p/∂p and p, re- system, and valid to lowest order in q /pF otherwise. | | spectively. With this definition, all vertices are equal to Using Heisenberg equations of motion for the opera- unity for free fermions. tors of charge and spin densities, one can verify that the To derive the Ward identities connecting the three-leg corresponding currents are also bilinear in fermions: vertices for charges and currents, we follow Engelsberg and Schrieffer32,33 and compute the time derivative of a X ∂p ρˆc (q) = c† c , time-ordered combination J ∂p p−q/2,α p+q/2,α p,α q † q n Ttc(p + , t1)c (p , t2)ˆρ (q, t) , (2.25) s X ∂p † h 2 − 2 i ρˆ (q) = c σz c . (2.24) J ∂p p−q/2,α αβ p+q/2,β p,αβ where n = c(s). It is easy to see that the derivative ∂/∂t n n yields a term proportional to ∂ρˆ /∂t = iq ρˆ and addi- − · J The situation with the the energy-momentum tensor tional terms proportional to δ(t t1) and δ(t t2), which is more subtle. In the phenomenological FL theory, originate from differentiating the− time-ordering− operator mom this tensor has the usual hydrodynamic form ρJ,ij = Tt. Using standard manipulations and Fourier transform- R D D 32,33 pi(∂p/∂pj)npd p/(2π) , where np is the quasiparti- ing in time, one finds cle occupation number.10 This might suggest that the second-quantized form of the energy-momentum tensor n ∂p n −1 −1 ωΛ (p, q) q ΛJ (p, q) = G G , (2.26) is also a bilinear, similar to those in Eq. (2.24), but with − · ∂p p+q/2 − p−q/2 ∂p/∂p replaced by pi(∂p/∂pj). An explicit calcula- mom n n ρˆ q where ΛJ pˆ ΛJ and q = (ω, q). tion indeed shows that the commutator [Hkin, ( )] ≡ · ω k does yield a bilinear part of the energy-momentum ten- Similarly to Γ and Γ in Eqs. (2.4a) and sor. However, at q = 0 there is also another, quartic in (2.4b), it is convenient to define the ver- 6 mom tices Λω(p) = lim Λ(p, q) and fermions part which comes from [Hint, ρˆ (q)] (Refs. 22 ω→0,|q|→0,|q|/ω→0 k and 31). Λ (p) = limω→0,|q|→0,ω/|q|→0 Λ(p, q). In analogy Because of the latter part, the energy-momentum ten- with Γk and Γω, renormalization of Λω comes from sor cannot be expressed via a purely bilinear combination high-energy fermions while renormalization of Λk comes of fermions. In what follows, we will use the continuity from both high- and low-energy fermions. The relation k ω equation for the momentum density only at q = 0, when between the vertices Λn,Jn and Λn,Jn follows from Fig. 1, the complications due to the quartic part of the energy- and is similar to the one which relates Γk and Γω in momentum tensor do not arise. Eqs. (2.5) and (2.6). For example, the charge and spin Graphically, we can associate each conserved charge vertices satisfy

D−2 2 ∗ Z k,n n ω,n n kF Z m X k,n 0 n ω 0 Λ (p)σ = Λ (p)σ Λ (p )σ Γ (p , p)dΩp0 , (2.27) ββ ββ − (2π)D F ξη ηβ,ξβ F F ξ,η

with n = (c, s) and similarly for the vertices of corre- pressed as a bilinear combination of fermions, hence it sponding currents. Here, σc denotes the identity matrix cannot be expressed graphically as in Fig. 1. Neverthe- s z and σ = σ . Projecting p onto the FS, using (2.27) to less, we can still use Eq. (2.26) for ρˆmom at q = 0, when express Λk in terms of Λω and Γω, and taking separately the momentum-energy tensor does not contribute. Tak- the limits ω = 0 and q = 0, we obtain the following four ing this limit, we obtain the fifth identity identities ΛmomZ = 1, (2.29) ΛcZ = 1, ΛsZ = 1; (2.28a) mom ω m∗ m∗ where Λ =p ˆ Λ . Λc Z = 1 + F c, Λs Z = 1 + F s, (2.28b) · m J 1 m J 1 We now go back to the PL relations. We easily identify the first two PL relations (2.4a) and (2.4b), as Eqs. n ω,n n ω,n c where we re-defined Λ Λ and ΛJ ΛJ for (2.28a) and (2.28b) for Λc and ΛJ . Indeed, the r.h.s. brevity. ≡ ≡ of Eqs. (2.4a) and (2.4b) are just the expressions for the c As we said, the energy-momentum tensor is not ex- three-leg vertices Λc and ΛJ , as can be seen directly from 7

transition is a Pomeranchuk instability, while an antifer- romagnetic transition is not. k ω k ω Λ = Λ + Λ Γ The order parameters associated with Pomeranchuk instabilities are bilinear in fermions. Examples of such order parameters were already presented in Eqs. (2.22a) FIG. 1. Graphical representation of the relation between and (2.24). More general order parameters with angu- three-leg vertices Λk and Λω. lar momentum l in the charge and spin channels can be defined as X ρˆc(q) = λc(p)c† c , (2.33) Fig. 1. Explicitly, l l p−q/2,α p+q/2,α p,α Z D+1 X † c c 2 ω d q s s z ρˆl (q) = λl (p)cp−q/2,ασαβcp+q/2,β. (2.34) Λ = 1 2i Γ (pF , q)(Gq) D+1 , (2.30) − (2π) p,αβ Z D+1 c c 2 ω pF q ∂q d q c(s) ΛJ = 1 2i Γ (pF , q)(Gq) 2· par D+1 . where λ (p) is a form-factor, which transforms under − p ∂q (2π) l F rotations according to its angular momentum channel The factor of Q in Eq. (2.8), which incorporates high- (i.e., as 1 for l = 0, as p for l = 1, etc.). In 2D energy contributions to m∗/m, is then equal to (Λc Z)−1. J c(s) l c(s) Similar considerations apply to the Kondratenko rela- λ (p) = cos(lφp) p f ( p ), (2.35) l | | × l | | tions in the spin channel, Eqs. (2.20a) and (2.20b). In p this case, we have or equivalently with sin instead or cos. In (2.35), fl( ) can be any function. | | Z D+1 s c 2 ω d q A Pomeranchuk instability is usually expressed as a c(s) Λ = 1 2i Γ (pF , q)(Gq) D+1 , (2.31) − (2π) condition on the Landau parameter Fl , defined in 13 Z D+1 Eq. (2.12). Pomeranchuk’s original argument was that s s 2 ω pF q ∂q d q ΛJ = 1 2i Γ (pF , q)(Gq) 2· par D+1 . the prefactor of the term in the ground state energy, − pF ∂q (2π) quadratic in the variation of the shape of a FS with given c(s) c(s) The additional PL relation, Eq. (2.14), is identical to l, c(s), scales as 1 + F and vanishes when F l l → Eq. (2.29), i.e., the r.h.s. of (2.14) is just the definition c(s) 1. The corresponding susceptibility χl then scales as of Λmom: − c(s) c(s) 1/(1 + Fl ) and diverges at Fl = 1. Z D+1 c(s) − pF q d q The susceptibility χ computed within the random Λmom = 1 2i Γc(p , q)(G2)ω · . (2.32) l F q 2 D+1 phase approximation (RPA) shows just this behavior, − pF (2π) i.e., Note that Eq. (2.14) is valid both for Galilean- and non- 1 Galilean-invariant systems, as long as the total momen- c(s) χl,RPA = χl,0 c(s) , (2.36) tum is a conserved. 1 + Fl We re-iterate that Eqs. (2.28a) and (2.29) express Z in three different ways, and thus place constraints on where χl,0 is the susceptibility of free fermions, given by 2 the vertex functions, Eqs. (2.30) and (2.32). Equation c(s) m c(s) c(s) ∗ χl=0,0 = fl=0 (pF ) , (2.28b) relates the product of ΛJ , Z, and m /m to π c(s) 2 1 + F1 . For a non-Galilean-invariant system, all the c(s) m l c(s) χ = p f (pF ) . (2.37) quantities on the l.h.s. come, at least partially, from l>0 2π F l c(s) high-energy fermions, while F1 is proportional to the The integral in the free particle-hole bubble is confined to l = 1 component of the interaction vertex between low- an infinitesimal region around the FS (see the derivation energy fermions. around Eq. (3.2) below). Since the interaction between c(s) fermions on the FS is parameterized Fl , the appear- c(s) C. Implication of conservation laws for ance of the denominator 1+Fl in (2.36) is the result of Pomeranchuk instabilities in a Fermi liquid summing up the geometric series of particle-hole bubbles. Note that the free-fermion χl,0 is finite in the static A Pomeranchuk instability is a spontaneous develop- limit Ω = 0, q 0 (which is the case in Eq. (2.37)), but vanishes, for any→ l, in the opposite limit of q = 0, Ω 0, ment of a long-range order in the spin or charge chan- → nel, which occurs when the fermion-fermion interaction given that the system is SU(2)-symmetric. This vanish- reaches a critical value. A distinctive feature of a Pomer- ing is the consequence of the fact that for free fermions anchuk instability is that it breaks either rotational sym- any particle-hole order parameter, bilinear in fermions, metry of the FS or its topology leaving translational is a conserved quantity. At small but finite vF q/Ω, c(s) 2 symmetry intact. For example, a ferromagnetic (Stoner) χl,0 (q, Ω) scales as (vF q/Ω) . 8

The RPA expression, however, is not the full result for cide with the RPA expressions, modulo a factor of m∗/m χc(s). An exact formula for the static susceptibility was l ∗ ∗ obtained by Legget.17 It reads: c m 1 s m 1 χ = c , χ = s , π 1 + F0 π 1 + F0 ∗ 2 2 ∗ m kF 1 c(s) c(s) m c(s) c(s) χ = (2.39) χ = Λ Z χ + χ . (2.38) mom c l l m l,RPA l,inc 2π 1 + F1

(for definiteness, we use the explicit forms of χl,0 in 2D). c(s) The l = 0 instability in the charge channel corresponds to Here Λl is the same three-leg vertex as before but now for arbitrary order parameter with angular momentum l. phase separation, the one in the spin channel corresponds The first term in (2.38) is often called the “quasiparticle to ferromagnetism, and the one at l = 1 signals the emer- contribution” because it is finite in the static limit Ω = gence of a charge nematic order. In a Galilean-invariant ∗ c 0, q 0, but vanishes at q = 0, Ω 0 regardless of system, m /m = 1 + F1 , and an l = 1 Pomeranchuk wether→ the corresponding order parameter→ is conserved instability in the charge channel does not occur. or not Next, we consider the susceptibilities of charge and c(s) ∗ spin currents, i.e., for order parameters with l = 1 and Still, we recall that Λl ,Z and m /m (for a non- c(s) form factor λl=1(p) = ∂p/∂px. We label the corre- parabolic spectrum) in the first term are the three in- c(s) put parameters which come, at least partially, from high- sponding susceptibilities as χJ . Using (2.28b), we can c(s) c(s) energy fermions. The second term, χl,inc, is the contri- re-express Eq. (2.38) for χJ as bution only from high-energy fermions. In a generic case, 2 this term is not described at all within the FL theory, and c(s) mp m c(s) c(s) χ = F (1 + F ) + χ . (2.40) its value does not depend on the order of limits Ω 0 J 2π m∗ 1 J,inc and q 0. → | | → We see that the quasiparticle part of the susceptibility We will review a diagrammatic derivation of Eq. (2.38) of either charge-current or spin-current order parameter in the next Section. Here, we focus on the implications actually vanishes when the corresponding Landau param- of Eq. (2.38) for Pomeranchuk instabilities of a FL. eter reaches 1, i.e., a Pomeranchuk instability does not − First, Eq. (2.38) shows that there is more than one show up if one probes it by analyzing particular suscep- scenario for the divergence of the susceptibility in a given tibilities as specified above.21 channel. In addition to the Pomeranchuk scenario (the Equation (2.40) can be also derived explicitly, by c(s) c(s) vanishing of 1 + F ), the susceptibility χc(s) can also expressing the susceptibilities χ via DM the time- l l 0 diverge if contributions from high-energy fermions give ordered correlators ofρ ˆ at times t and t , differentiating 0 c(s) c(s) ∗ over t and t , and using the continuity equation, (2.23). rise to a divergence of ZΛl or χl,inc. Finally, m /m 17,21 for a non-parabolic spectrum may also diverge due to This yields singular contributions from high-energy fermions. These c(s) c(s) 34 ω/ q )2χc(s)(q, ω) = χ (q, ω) χ (q, 0). (2.41) three scenarios are outside the FL theory. | | J − J c(s) ∗ Second, Eq. (2.38) shows that the divergence of χl,RPA Now we take the limit ω v q, keeping both ω and q F may, in principle, be canceled by the vanishing of its pref- infinitesimally small. The incoherent parts of χc(s)(q, ω) c(s) ∗ J actor, Λl Z(m /m). If this happens, the corresponding c(s) c(s) and χJ (q, 0) cancel each other, while the quasiparticle susceptibility remains finite at F = 1. It will be c(s) l part of χ (q, ω) vanishes. As a consequence, the r.h.s. shown below that this is the case for order− parameters J of (2.41) reduces to which coincide with the momentum density, and charge 21,22 and spin currents. ∗ 2 ! c(s) m vF c(s) 2 1 To see this, we now systematically analyze the implica- χJ (q, 0) = (ZΛJ ) . (2.42) − − 2π c(s) tions of the conservation laws for the relation between the 1 + F1 two terms in (2.38). We first consider the susceptibilities ∗ of three conserved order parameters - total charge (par- The l.h.s. of (2.41) tends to a constant at ω vF q ∗ c(s) ticle number), total spin, and total momentum. (Here because χc(s)(q, ω) = m χ (q, ω) scales as ( q /ω)2 m RPA | | and thereafter, a susceptibility of any vector quantity and cancels out the factor (ω/ q )2. The expression for will be understood as a longitudinal part of the corre- c(s) | | χRPA(q, ω) in this limit, obtained by Leggett in Ref. 35, sponding tensor.) For the first two order parameters reads c(s) p l = 0 and λl=0( ) = 1, while for the third one l = 1 and 2 c(s) 2 c(s) m vF q m c(s) λ (p) = px, where we choose the x-axis to be along q. l=1 χRPA(q, ω) = | | ∗ 1 + F1 . In all three cases, ΛZ = 1 and χinc = 0. Consequently, −2π ω m the susceptibilities of the three conserved quantities coin- (2.43) 9

∗ Hence, at ω vF q, The current susceptibilities correspond to the choice (see (2.24)) 2 2 c(s) m vF q m c(s) c(s) (ω/ q ) χ (q, ω) = | | ∗ 1 + F1 . 1 ∂p | | −2π ω m fl ( p ) = par . (2.51) (2.44) | | m ∂p

Substituting this into (2.41) we reproduce Eq. (2.28b): For order parameters with fl=1 different from the equa- c(s) c(s) m c(s) tion above, there are no general reasons to expect Λl=1Z c(s) ZΛJ = 1 + F1 . (2.45) m∗ to be proportional to 1 + F1 , hence a Pomeranchuk c(s) instability is expected to occur at F1 = 1 (Ref. 22). Substituting this further into (2.38) we reproduce (2.40). Interestingly enough, we can interpret this− instability Note that Eq. (2.45) can be re-written as in two different ways, depending on how we write the c(s) c c susceptibility χ . If we use the original Eq. (2.38) ΛJ 1 + F1 l=1 s = s . (2.46) ΛJ 1 + F1 2 ∗ c(s) 2 c(s) mpf m 2 (Λl=1 ) c(s) χ = Z c(s) + χ , (2.52) c(s) l=1 l=1,inc 2π m 1 + F1 We see that the vanishing of 1+F1 is always associated c(s) (2.53) with the vanishing of the corresponding ΛJ , excluding c s an unlikely case when 1 + F1 and 1 + F1 vanish simulta- we would conclude that the instability is determined by neously. the condition on the interaction between fermions on the 17 c(s) Leggett showed that there exists another, even FS: F = 1. However, re-writing χc(s) as stronger constraint on the susceptibilities of charge and 1 − l=1 spin currents. Namely, the longitudinal sum rule implies 2 c(s) 2 c(s) mpf (Λ ) c(s) that χc and χs are not renormalized by the interaction, χ = Z l=1 + χ (2.54) J J l=1 2π c(s) l=1,inc i.e., ΛJ (2.55) mp2 χc = χs = F . (2.47) J J 2π we would conclude that the Pomeranchuk instability is c(s) driven by the vanishing of ΛJ , which is determined by The longitudinal sum rule is analogous to the longitu- high-energy fermions. This dual interpretation is yet an- dinal f-sum rule for the imaginary part of the inverse other consequence of the fact that conservation of charge dielectric function36 and is the consequence of the gauge- 37 and spin imposes the relations between the properties of invariance of the electromagnetic field. low- and high-energy fermions. Constraint (2.47) relates χJ,inc in (2.40) to the Landau c(s) parameter F1 : III. DIAGRAMMATIC DERIVATION OF 2 LEGGETT’S RESULT FOR THE STATIC c(s) mpF m c(s) χ = 1 (1 + F ) . (2.48) SUSCEPTIBILITY J,inc 2π − m∗ 1

This is yet another condition on the contribution coming In this section we review a diagrammatic derivation of from high-energy fermions. Eq. (2.38), closely following the presentation in Ref. 22. ∗ c c(s) In a Galilean-invariant FL, m /m = 1 + F1 , and Eqs. The purpose of this derivation is to show that Λl and (2.40)-(2.47) reduce to c(s) χl,inc arise from high-energy contributions. For definite- c c ness, we consider the 2D case. To simplify notations, ZΛJ = 1, χJ,inc = 0 (2.49) here and in the next section we suppress the superscript s 2 c s ω ω 1 + F1 mp F1 F1 2 2 2 s s F ω in Gq , i.e., replace Gq just by Gq. ZΛJ = c , χJ,inc = − c . (2.50) 1 + F1 2π 1 + F1 Let’s start with the free-fermion susceptibility for an c(s) order parameter with form-factor λ , as in Eq. (2.35). c(s) l The fact that χJ is constrained by Eqs. (2.40) and The diagrammatic representation of the free-fermion sus- (2.47) does not imply that a Pomeranchuk transition in c(s) ceptibility χl,0 (q) is a bubble composed of two fermionic the l = 1 channel can never occur. Indeed, these con- c(s) straints do not preclude the system from developing an propagators (Fig. 2) with form-factors λl at the ver- instability towards a phase described by an order param- tices: Z 3 2 eter with a form-factor, which has the same symmetry c(s) d p c(s) χ (q) = 2i λ (p) Gp+ q Gp− q . (3.1) as the charge- or spin-current order parameter but de- l,0 − (2π)3 l 2 2 pends differently on p . In Eq. (2.35) we defined an infi- | | nite family of order parameters with a given angular mo- Here, Gk stands for a free-fermion Green’s function, mentum l, specified by an overall scalar function f( p ). given by Eq. (2.2), and the factor of 2 comes from spin | | 10

tibility diagram with screened interaction inserted into the bubble is shown in Fig. 4. This screened dynamical interaction contains a Landau damping term, which is non-analytic in both half-planes of complex frequency. 2 As a result, the product of Gp and the dressed interaction at order U 2 and higher has both the double pole and a branch cut. A pole can be avoided by closing the integration contour in the appropriate frequency half-plane, but the branch cut is unavoidable, and its presence renders FIG. 2. The free fermion susceptibility the frequency integral finite. Since one does not have to make sure that the poles of the Green’s functions are in the opposite half-planes, relevant p are not confined to summation. The frequency integral in (3.1) is non-zero the FS, and both ω and p are generally of order EF (or only if p+q/2 and p−q/2 have opposite signs which, for bandwidth). Fermions at such high energies are strongly q pF , implies that the integral over p comes from damped, i.e., they are incoherent quasiparticles. By this a| narrow| region near the FS. At T = 0, we| | have c(s) reason, the M = 0 contribution to χl is labeled as an c(s) c(s) 2 c(s) m l c(s) incoherent one, χl,M=0 = χl,inc χ (q) = p f (pF ) l,0 − π F l Next, we next move to the M = 1 sector. Here we se- Z dφp 2 vF q cos φp lect a subset of diagrams with just one low-energy contri- (cos lφp) | | . (3.2) × 2π ω vF q cos φp + iδsgnω bution from some ladder segment. The sum of such dia- − | | grams can be graphically represented by the skeleton dia- In the static limit we reproduce Eq. (2.37). gram in Fig. 3, labeled M = 1. The ladder segment gives c(s) c(s) 2 The 1/(1 + F ) dependence of χ can be repro- R 3 c(s) l l d p λl (p) Gp+q/2Gp−q/2 at ω = 0 and q 0, duced diagrammatically within RPA. Because the mo- FS | | → where R denotes an integral taken close to the FS. Each mentum/frequency integration within each bubble is con- FS fined to the FS, the dimensionless interaction between of the Green’s function in this integral can be replaced by c(s) its quasiparticle form, given by the first term in Eq. (2.3), the bubbles is exactly Fl . Re-summing the geometric and the integral gives the static free-fermion susceptibil- series, we reproduce Eq. (2.36). ity in Eq. (2.37) multiplied by a factor of Z2m∗/m. The To obtain an exact expression, we need to go beyond c(s) c(s) side vertex, Λ , is the sum of high-energy contributions RPA. To this end, we note that a diagram for χ at l l from all other cross-sections either to the right or to the any loop order can be represented by a series of “ladder left of the one in which we select the low-energy piece. segments” separated by interactions. By “ladder seg- c(s) ment” we mean the product G G with vanish- [We remind that Λl is defined without the form-factor p+q/2 p−q/2 c(s) ingly small but still finite q. Each ladder segment con- λl (pF ), which was already incorporated into χl,0(q).] tains integration over both high- and low-energy states. In all these other cross-sections we can set q = 0, i.e., 2 We define a high-energy contribution as the one where replace Gp+q/2Gp−q/2 by Gp. These contributions would the p is larger than vF q , such that the the poles of vanish for a static interaction, but again become non- | | | | 2 Gp+q/2 and Gp−q/2 are located on the same side of the zero once we include dynamical screening at order U real frequency axis. This contribution can be evaluated and higher. Similarly to the M = 0 sector, the difference right at q = 0. A low-energy contribution is the one Λc(s) 1 is determined by fermions with energies of or- l − where p is smaller than vF q , and the poles of G der E (or bandwidth). Overall, the contribution to the | | | | p+q/2 F and Gp−q/2 are on different sides of the real frequency static susceptibility from the M = 1 sector is c(s) axis. To obtain χl , we re-arrange the perturbation se- 2 m∗ ries by assembling contributions from diagrams with a χc(s) = ZΛc(s) χc(s). (3.3) given number M of low-energy contributions from lad- l,M=1 l m l,0 der segments, and then sum up contributions from the sub-sets with different M = 0, 1, 2, etc (see Refs. 38–40). The sectors with M = 2, M = 3 ... are the subsets of This procedure is demonstrated graphically in Fig. 3. diagrams with two, three ... low-energy parts from ladder We start with the M = 0 sector. The correspond- segments. The contribution from the M = 2 sector is ing contributions to the susceptibility contain products represented by the skeleton diagram in Fig. 3 labeled 2 of Gp. Taken alone, each such term would vanish on in- M = 2. A new element, compared to the M = 1 is tegration over frequency. The total M = 0 contribution sector, is a fully dressed vertex Γω between fermions on then vanishes to first order in U(q) because the static in- the FS. One can easily verify that this vertex appears teraction does not affect the frequency integration. How- with a prefactor of Z2(m∗/m), i.e., an extra factor in ever, at second and higher orders in U(q), the interac- the M = 2 sector compared to M = 1 is the product tion gets screened by particle-hole bubbles and becomes a of χl,0 and the corresponding component of the Landau dynamical one. An example of the second-order suscep- function, as defined by Eq. (2.12). Using (3.3), we then 11

Λc(s) Λc(s) Λc(s) ω Λc(s) + l l + l Γ l +

M = 0 M = 1 M = 2

Λc(s) ω ω Λc(s) l Γ Γ l + ··· M = 3

c(s) FIG. 3. The ladder series of diagrams for the static susceptibility χl . The exact χl is represented as a series of M = 0, 1, 2,... bubbles comprised of Green’s functions with poles in the opposite half-planes of complex frequency, whose contributions are computed close to the FS. Gray shading denotes contributions from high energy fermions, for which the poles in the Green’s functions are in the same half-planes of complex frequency. These include the M = 0 bubble (on the far left), as well as the c(s) ω vertices Λl and Γ .

spectively), while the ratio

Ω β = , (3.5) v∗ q F | | ∗ ∗ ∗ is arbitrary (here, vF = pF /m = vF (m/m )). Because q and ω are small, we may still split momentum and frequency integrals of Gk−q/2Gk+q/2 into the low- and high-energy contributions. At the same time, the vertices c(s) ω c(s) Λl and Γ can be still taken at q = 0 and ω = 0 because FIG. 4. Example of a higher-order contribution to χl . At this order, the static interaction acquires dynamics due to they are not sensitive to the order in which these two screening by particle-hole pairs. The diagram contributes to limits are taken. The decomposition of the perturbation the M = 0 sector if both Green’s functions adjacent to one series into the M = 0, 1, 2,... sectors, depicted in Fig. 3, of the external vertices are evaluated away from the FS; to remains unchanged. However, the RPA susceptibility in the M = 1 sector, if one of them is evaluated on the FS and Eq. (2.38) now has a nontrivial dependence on β and another one away from it; and to the M = 2 sector, if both are evaluated on the FS. 2 c(s) c(s) c(s) c(s) χl (q) = ZΛl χl,RPA(β) + χl,inc. (3.6) obtain c(s) The computation of χl,RPA(β) is more technically in- c(s) volved than that of static χ (0) because different an- 2 ∗ l,RPA c(s) c(s) c(s) m c(s) c(s) gular momentum channels no longer decouple. χl,M=1 + χl,M=2 = ZΛl χl,0 1 Fl m − Consider first the limit ω v q . For even l, the (3.4) F quasiparticle contribution fromM =| 1| sector is (the minus sign in the second term becomes evident if one compares the number of the fermionic loops in the M = ∗ c(s) m c(s) 1 and M = 2 sectors). A simple bookkeeping analysis χl,M=1,RPA(q) χl,0 (1 + iαlβ) , (3.7) shows that the contributions from sectors with larger M ≈ m form a geometric series, which is re-summed into 1/(1 + c(s) where αl = 1 if l = 0 and αl = 2 if l = 2n, n > 0. For odd Fl ). Collecting all contributions, we obtain Eq. (2.38). l, the expansion starts with β2 –this means that Landau The diagrammatic approach can be extended to the damping is suppressed in odd momentum channels.41 For case when both external momentum q and frequency ω M > 1, the contribution proportional to β can come from are finite (but still much smaller than pF and EF , re- any of the M cross-sections, yielding a combinatorial fac- 12 tor of M. Summing up the series one finds 22 limit of the charge and spin susceptibilities,9,10 obtained by solving the kinetic equation for a FL. ! m∗ 1 α β χc(s) (q) = χc(s) + i l , l,RPA l,0 m c(s) c(s) 2 1 + Fl (1 + Fl ) In the opposite limit of β 1, only the M = 0, 1, 2 (3.8) elements of the bubble series need to be included, be- where l = 2n is an even number. For l = 0, the equation cause higher order terms are small in 1/β. Some further above reproduces the known result for the quasi-static analysis then yields17,22

1 m∗ χc(s) (q) = β2χc(s) 1 + F c(s) , l=0,RPA −2 l=0,0 m 1 3 m∗ 2 1 χc(s)(q) = β2χc(s) 1 + F c(s) + F c(s) , l=1 −4 l=1,0 m 3 0 3 2 1 m∗ 1 χc(s)(q) = β2χc(s) 1 + F c(s) + F c(s) , (3.9) l>1 −2 l,0 m 2 l−1 l+1

c(s) where χl,0 is the free-fermion static susceptibility in the We stress that the analysis above concerns only the quasi- c(s) corresponding channel. particle (RPA) contribution, χl,RPA. The incoherent part c(s) For generic β, the form of χ (q) is rather involved c(s) l of χ does not depend on β, as long as q pF and for all l, including l = 0. We illustrate the behavior l | | ω EF . c(s) of χl (q) for the simplest case of the charge and spin susceptibilities (l = 0 and f0( p ) = 1). Analyzing the series of bubbles, we find | | m∗ χ¯(q) χc(s) (q) = , (3.10) l=0,RPA π c(s) 1 + Fl=0 χ¯(q) IV. PERTURBATION THEORY whereχ ¯(q) is given by X χ¯(q) = K 2 F c(s)K K Sm, (3.11) 0 n n m n In this section, we use the perturbation theory to verify − n,m>0 the results derived from the conservation laws in the pre- ω ∗ Z ∗ vious sections. Specifically, we compute Γ , Z, m /m, dθ vF q cos θ Kn(q) = cos nθ ∗ | | and ΛJc(s) to second order in the interaction U(q). The − 2π Ω v q cos θ + iδΩ − F | | purpose of this calculation is to demonstrate how the β p 2 |n| interplay between the contributions from high and low = δn,0 (β (β) 1) p 2 ∗ c(s) − (β) 1 + iδ − − energies works both for m /m and Λl . − (3.12) In the earlier days of the FL theory, perturbative cal- m 43 and Sn is a solution of the linear system culations were used as a check of general FL relations . m X c(s) m However, several subtle issues, e.g., whether in a di- S + Qn,m F S = δn,m (3.13) n 1 m1 m1 rect perturbative calculation mass renormalization comes m1>0 solely from low energies even in the Galilean-invariant in which Qn,m = Kn+m + Kn−m. In the static limit case, as in Eq. (2.17), were not verified till fairly recently. K0 = 1 and Kn>0 = 0. Thenχ ¯(q) = 1, and Eq. (3.10) reduces to Eq. (2.39) for the static susceptibility. As an additional simplification, we consider the case when all Landau parameters with l 2 can be neglected compared c(s) c(s) ≥ to F0 and F1 . In that case, the infinite set of linear equations in (3.13) is reduced to a 2 2 system. After A. The vertex function Γω some further analysis, we obtain22,42 ×

c(s) 2 2F1 K1 ∗ K0 c(s) ω c(s) m − 1+F1 (K0+K2) Diagrams for Γ to second order in U( k ) are pre- χl=0,RPA(q) = c(s) c(s) 2 . | | π c(s) 2F0 F1 K1 sented in Fig. 5. The most frequently studied case is of 1 + F0 K0 c(s) k − 1+F1 (K0+K2) the Hubbard (contact) interaction: U( ) = const U. (3.14) In this case we have | | ≡ 13

Z D+1 Z D+1 ω 2 d l 2 d l Γ (pF , q) = δαγ δβδ U + iU (GlGq−p +l + GlGq+p −l) δαδδβγ U + iU GlGq+p −l . αβ,γδ F F (2π)D+1 − F (2π)D+1 (4.1)

Here Gk stands for a free-fermion Green’s function, with Eq. (2.2), and pF stands for a D + 1-momentum with zero frequency and the spatial part equal in magnitude to the Fermi momentum and directed along p. The ﬁrst term in Eq. (4.1) is the renormalized interaction with zero momentum transfer, the second term is obtained by antisymmetrization. We see that the ﬁrst (“direct”) Z D+1 c U 2 1 d l Γ = + iU GlGq−p +l + GlGq+p −l , term contains contributions from both the particle-hole 2 F 2 F (2π)D+1 and particle-particle channels, while the second (“ex- 2 Z D+1 change”) term contains only a contribution from the s U iU d l Γ = GlGq+pF −l . (4.4) particle-particle channel. Using the relation − 2 − 2 (2π)D+1 1 δαδδβγ = (δαγ δβδ + σαγ σβδ) , (4.2) 2 · we re-write Eq. (4.1) as the sum of the density (charge) and spin parts: P ω c Substituting αβ Γαβ,αβ(pF , q) = 4Γ (pF , q) into the FL ω c s 2 Γ (pF , q) = δαγ δβδΓ + σαγ σβδΓ (4.3) form of the self-energy, Eq. (2.18), we obtain to order U αβ,γδ ·

Z 2 2 ΣFL(ω, p) = (ω p) U (2GlGq−p +l + GlGq+p −l) G dql − F F q Z 2 pF q ∂q 2 + p U (2GlGq−pF +l + GlGq+pF −l) 1 2· par Gqdql − pF ∂q Z 2 pF q 2 U (2GlGq−pF +l + GlGq+pF −l) 2· δGqdql , (4.5) − pF

D+1 D+1 2(D+1) where we labeled dql d qd l/(2π) . The Labeling the momenta as shown in this diagram and in- O(U) term in Γω gives≡ only a constant term in the self- tegrating over the internal D+1-momentum l, we express energy (a shift of the chemical potential), which is omit- Σpert via a particle-hole bubble. ted in the equation above. Subtracting from Σpert(ω, p) its value at ω, p = 0, we ﬁnd Z B. Direct perturbative calculation of the 2 Σpert(ω, p) Σ(0, 0) = U GlGk−pF +l (Gk+ Gk) dkl, self-energy − − − (4.6) where We now compare Eq. (4.5) with the self-energy obtained in the diagrammatic perturbation theory. As we mpF just said, the term of order U does not depend on ω and = ω, p 2 (4.7) pF p and is therefore irrelevant for our purposes. We focus 2 on the U terms. The second-order self-energy diagrams parameterizes the (small) external D + 1-momentum. are shown in panels b, c, and d of Fig. 6. Only diagrams c The same self-energy can also be computed in a dif- and d give rise to ω- and p-dependent terms in the self- ferent way, by combing internal fermions into a particle- energy, the diagram b just adds another constant term hole bubble. Re-labeling the momenta in the diagram c to the chemical potential. Relabeling the fermionic mo- as shown in panel e of Fig. 6, we obtain menta for U = const, it is easy to see that diagram d is Z equal to 1/2 of diagram c, so we only need to consider ˜ ˜ 2 − Σpert(ω, p) Σ(0, 0) = U GlGk+pF −l (Gk− Gk) dkl. diagram c. This diagram contains three Green’s func- − − − tions, two of which share a common internal momentum. (4.8) 14

a) b)

p p p p p

ω c) l d) e) Γ = = − l+k qq q q q p p+k p p p FIG. 6. First- and second-order diagrams for the fermionic self-energy Σpert(ω, p). For a momentum-independent inter- + − action U(|q|) = U, only second-order diagrams renormalize the mass and Z. For a momentum-dependent interaction, mass renormalization starts already at the ﬁrst order, while renormalization of Z still starts at the second order. Diagram qqq e is the same as c, except for internal fermions are combined into a particle-particle rather than particle-hole pair. Repro- FIG. 5. First and second order diagrams for the Fermi-liquid duced from Ref. 44. ω vertex Γαβ,γδ(p, q). The initial four-momenta p and q are associated with spin projections α and β, respectively. The ﬁnal four-momenta p and q are associated with spin projections γ and δ, respectively. Reproduced from Ref. 44.

We denote the self-energy obtained in this way as Σ˜ pert Σ˜ pert(ω, p) in Eq. (4.8), we obtain just to distinguish it from the self-energy Σpert(ω, p) in the particle-hole form. Σ˜ pert(ω, p) Σ˜ pert(0, 0) (4.11) Since the two expressions for the self-energy must be Z − 2 equal, the Green’s functions must satisfy the following = U (2GlGk−pF +l + GlGq+pF −l)(Gk− Gk) dkl. − identity

To first order in , the difference Gk+ Gk can be rep- Z resented as − GlGk−p +l (Gk+ Gk) dkl F − Z pF k ∂q 2 Gk+ Gk = ω p G = GlGk+pF −l (Gk− Gk) dkl. (4.9) 2 par k − − − − pF ∂q pF k 2 +p 2 δGk. (4.12) Indeed, this identity can be proven explicitly by relabel- pF ing the fermionic momenta.44 To first order in , the difference Gk+ Gk in the first line of (4.9) can be replaced The first (second) term in the equation above is a − 2 by Gk Gk− +O( ). To order , therefore, identity (4.9) high (low)-energy contribution. Substituting (4.12) into can be− written as (4.11) and comparing the result with (4.5), we see that Σ˜ pert(ω, p) becomes equivalent to ΣFL. This means that Z the expressions for m∗/m and Z, obtained from the self- (GlGk−p +l + GlGk+p −l)(Gk− Gk) dkl = 0. F F − energy to order U 2, are exactly the same as in the FL (4.10) theory. Using the same trick, one can also show that 2 Multiplying Eq. (4.10) by 2U and adding the result to Σpert(ω, p) and ΣFL are identical. 15

C. Momentum-dependent interaction

The results for the self-energy can be readily extended to the case of a momentum-dependent interaction. The vertex function to order U 2 is

Z D+1 ω d l 2 Γ (pF , q) = δαγ δβδ U(0) + i U ( pF l )(GlGq−p +l + GlGq+p −l) δαδδβγ αβ,γδ (2π)D+1 | − | F F − Z dD+1l U( q pF ) i [(2U( q pF ) 2U( q pF )U( pF l )) GlGl+q−p U( pF l )U( l q )GlGq+p −l] × | − | − (2π)D+1 | − | − | − | | − | F − | − | | − | F

Mass renormalization now occurs already at the ﬁrst or- from second-order diagrams. To second order in U, per- der in U( q ). To this order, perturbative and FL self- turbative self-energy becomes equivalent to ΣFL with the energies just| | coincide. Renormalization of Z still comes help of an analog to (4.10):

Z 2 dqlU ( pF l )(GlGk−p +l + GlGq+p −l)(Gk+ Gk) . (4.13) | − | F F −

The rest of the calculations proceeds in the same way as contributions. We note in this regard that (4.10) estab- for the case of constant U. lishes a relation between these two contributions. Indeed, extracting the linear-in-ω and p terms from (4.10) and using (4.12), we ﬁnd that Eq. (4.10) is equivalent to two D. Where does mass renormalization come from in equations the perturbation theory?

The issue we consider in this section is the separation of the perturbative self-energy into the low- and high-energy

Z 2 dkl (GlGk−pF +l + GlGk+pF −l) Gk = 0, (4.14a) Z Z 2 pF k 2 pF k dkl (GlGk−pF +l + GlGk+pF −l) Gk 2· = dkl (GlGk−pF +l + GlGk+pF −l) δGk 2· . (4.14b) pF − pF

Equation (4.14a) shows that a certain integral over high- shows that mass renormalization comes solely from low- energy states vanishes, while Eq. (4.14b) shows that an- energy fermions. We argue that this is not the case if we ∗ other integral over high-energy states can be expressed extract m /m from either Σpert(ω, p) or Σ˜ pert(ω, p). 2 as an integral over the FS. (We remind that δGk is a Below we present the results of the calculations sepa- projector on the FS, see (2.6).) rately for D = 3 and D = 2. Using (4.14b) and adding identity (4.10) to either Σpert(ω, p) or Σ˜ pert(ω, p), we can redistribute the weights of low- and high-energy contributions in the ﬁ- nal result. This implies that the same result for mass renormalization, computed either from Σpert(ω, p) or 1. 3D Galilean-invariant FL Σ˜ pert(ω, p), does not have to come from the same states. This observation is most relevant to a Galilean- In 3D, explicit expressions for Γc and Γs for fermions invariant FL, where the phenomenological FL theory on the FS, i.e, for p = q = pF , and to second order in | | | | 16

U read8,45 we ﬁnd 2 mUpF δΣ1(ω, p) = 8 ln 2 (ω p) (4.20a) 2 4π2 − c U mU pF cos θ 1 + sin θ/2 Γ (θ) = + 2 2 + log + ... 2 2 4π 2 sin θ/2 1 sin θ/2 4 mUpF + p (4 ln 2 1) , 2 − 2 s U mU pF sin θ/2 1 + sin θ/2 3 − 4π Γ (θ) = 2 1 log + ... 2 − 2 − 4π − 2 1 sin θ/2 4 mUpF − δΣ2(ω, p) = p (2 ln 2 1) .(4.20b) (4.15) −5 − 4π2 Adding up the two parts, we obtain 2 where θ is the angle between pF and qF and dots stand mUpF 2 Σpert(ω, p) Σ(0, 0) = (4.21) for the angle-independent U terms. − 4π2 Substituting Γω from Eq. (4.1) into Eqs. (2.4a) and 8 43,45 8 ln 2 (ω p) + (7 log 2 1) p . (2.13), and evaluating the integrals, we obtain × − 15 − Using Eq. (2.18), we ﬁnd that the perturbative self- ∗ 2 ∗ m 8 mUpF energy indeed gives the same results for m /m and Z, = 1 + (7 ln 2 1) (4.16) as in the FL theory, Eqs. (4.17) and (4.16). We note, m 15 − 4π2 however, that mass renormalization is determined by the prefactor of the total p term, and, according to and Eqs. (4.20a) and (4.20b), this prefactor comes from both high and low energies. Only the sum of the two contributions recovers the FL formula for m∗/m. On the other 2 mUp hand, renormalization of Z comes only from δΣ1, i.e., Z = 1 8 ln 2 F . (4.17) − 4π2 only from high energies.

2. 2D Galilean-invariant FL We now turn to the perturbative self-energy in the particle-hole representation, Σpert(ω, p). Using (4.12) Two-dimensional analogs of Eqs. (4.15) are46 we split Σpert(ω, p) into two parts as U mU 2 Γc = + (2 + log cos θ/2) + ... 2 2π Σpert(ω, p) Σ(0, 0) = δΣ1(ω, p) + δΣ2(ω, p), (4.18) U mU 2 − Γs = log cos θ/2 + ... (4.22) − 2 − 2π where Evaluating m∗/m and Z with the help of Eqs. (2.4a) and (2.13), we obtain46

2 Z p k m∗ 1 mU 2 2 F = 1 + . (4.23) δΣ1(ω, p) = U GlGk−pF +lGk ω p 2· dlk, − pF m 2 2π

(4.19a) 44 Z For fermionic Z, numerical integration yields pF k δΣ (ω, ) = U 2 · δG2 G G d . 2 p p 2 k k−pF +l l lk 2 − pF mU Z 1 C , (4.24) (4.19b) ≈ − 2π

where C = 0.6931.... To high numerical accuracy, C is equal to ln 2. The ﬁrst (second) term in (4.18) is a high (low)-energy We now turn to perturbative self-energy. We again contribution. The low-energy contribution cannot be ob- split Σ into the high- and low-energy contributions, tained by expanding Σ in before doing the inte- pert pert p δΣ (ω, ) and δΣ (ω, ), as in Eq. (4.18). The particle- grals. 1 p 2 p hole bubble for 2D fermions can be obtained analytically Evaluating the integrals in Eqs. (4.19a) and (4.19b), for any ω and k : | | 17     m  √2˜ω  Πph(ω, k) = 1 + i  , (4.25) −2π  s r   2  k˜2 k˜4 ω˜2 + k˜2 k˜4 ω˜2 4˜ω2k˜4 − − − − −

2 whereω ˜ = 2ωm/p and k˜ = k /2pF . To calculate δΣ1, E. Direct perturbation theory for static F | | one needs to know the entire bubble, while δΣ2 is deter- susceptibility mined by the static bubble Πph(0, k ). Performing the angular integral in δΣ analytically| and| remaining inte- 1 We now show that the same subtle interplay be- grals numerically, and all integrals in δΣ analytically, 2 tween the low- and high contributions also occurs for we obtain the charge/spin spin susceptibility in the channel with 2 2 c(s) mU p mU angular momentum l, χl . δΣ1(ω, p) =C˜ (ω p) + (4.26a) 2π − 2 2π Rather than going through an exhaustive analysis, we δΣ2(ω, p) =0, (4.26b) consider a single illustrative example, namely the l = 1 spin channel in a Galilean-invariant system. Our goal is ˜ ˜ s ∗ s with C = 0.6931 ... To high numerical accuracy, C = C. to reproduce the relation ΛJ Z(m /m) = 1 + F1 . Explic- The vanishing of δΣ2 = 0 in 2D is due to the fact that itly, we have it is expressed via a static particle-hole bubble:

2 Z 2pF Z pU 1 c (pF , p) 2 = 1 2i dkΓ (pF , p) G (4.29a) δΣ2(ω, p) = 2 d k Πph(ω = 0, k ) 2 k −2π vF 0 | | | | Z − pF 2 2 Z 1 k /2p (pF , p) F s s p p 2 q − | | . (4.27) ΛJ = 1 2i dkΓ ( F , )Gk 2 (4.29b) × 2 − pF 1 ( k /2pF ) − | |

Because Πph(ω = 0, k ) is independent of k for k c s 2 | | | | | ≤ The vertex functions Γ and Γ to order U i are given 2pF , the integral over k vanishes. c(s) | | by (4.4). Combining the contributions from Z and Γ Casting the result into the form of Eq. (2.19), we again we obtain, to order U 2, reproduce the FL results for m∗/m and Z, Eqs. (4.23) ∗ and (4.24). However, we see that now m /m comes solely Z from the high-energy part of the self-energy. s 2 ΛJ Z 1 = 2U dkl (GlGk−pF +l + GkGk+pF −l) If we compute the perturbative self-energy by combin- − − ing two internal fermions into a particle-particle bubble (pF p) 2 ˜ 2· Gk (Σpert(ω, p) in our notations), and again split it into pF high-energy and low-energy contributions, δΣ˜ 1(ω, p) (4.30) 44 and δΣ˜ 2(ω, p), we obtain

mU 2 mU 2 As written, the integral on the r.h.s. of Eq. (4.30) is δΣ˜ 1(ω, p)= C¯ (ω p) + p , 2π − 2π not conﬁned to the FS. However, it can be converted into a FS contribution using identity (4.14b), which expresses (4.28a) 2 the r.h.s. of (4.30) via the integral over δGk. We then 2 p mU obtain δΣ˜ 2(ω, p)= , (4.28b) − 2 2π Z s 2 2 pF k where, as before, C¯ = 0.6931 ... . Comparing with ΛJ Z 1 = 2U dkl (GlGk−pF +l + GlGk+pF −l) δGk. · − p2 Eq. (4.23), we see that now the low-energy contribution F (4.31) to mass renormalization in the particle-particle case is ﬁnite but opposite in sign that to mass renormalization in the FL theory. The correct sign is reproduced once we Finally, we use Eq. (4.4) and re-write the r.h.s. of (4.31) add the low- and high-energy contributions. as

Z Z 2 2 pF k dθ s c s c 2U dkl (GlGk−pF +l + GlGk+pF −l) δGk 2· = 2 (F (θ) F (θ)) cos θ = F1 F1 . (4.32) pF 2π − − 18

s s Substituting this into (4.31), we obtain ZΛJ = (1 + F1 grating out contributions from high-energy fermions. For c 2 s − F1 ), which, to order U , is equivalent to ZΛJ = (1 + example, the low- and high-energy contributions to the s c F1 )/(1 + F1 ), as in Eq. (2.50). susceptibilities of charge and spin currents cancel each s We emphasize that only the product ΛJ Z can be ex- other, so that Pomeranchuk instabilities towards phases s pressed via an integral over the FS. Taken separately, ΛJ with spontaneously generated charge and spin currents and Z are determined by integrals which are not confined are impossible.17,21 Even more so, the corresponding sus- to the FS. ceptibilities are not renormalized at all by the interaction.17 On the other hand, an instability towards a phase with the order parameter, which has either the same sym- V. CONCLUSIONS metry as the charge or spin current but a different form- factor, or a different symmetry, is not forbidden by con- In conclusion, we reviewed certain aspects of the mi- servation laws.22 croscopic FL theory. We argued that this theory is We also demonstrated how the constrains imposed by based on five Ward identities, which follow from con- conservation laws can be derived diagrammatically, and servation laws. The first two identities (the Pitaevskii- along the same lines, provided a diagrammatic derivation Landau relations14) follow from U(1) symmetry and of the Leggett formula17 for the charge and spin suscep- reflect charge conservation. The next two (the Kon- tibility in a channel with arbitrary angular momentum. dratenko relations27,28) follow from SU(2) symmetry and Finally, we illustrated the interplay between the low- and reflect spin conservation. The last, fifth relation, fol- high-energy contributions by calculating the FL inter- lows from translational symmetry and reflects momen- action vertex, effective mass, quasiparticle residue, and tum conservation. This last identity was derived origi- susceptibility to second-order in interaction. nally for a Galilean-invariant system,1 but is generalized here for any translationally invariant system and thus can be attributed to momentum conservation. These identi- ACKNOWLEDGMENTS ties express quasiparticle Z and the effective mass m∗ in terms of the vertex function. In addition, they im- We thank J. Schmalian, P. Woelfle, and Y. Wu for valu- pose certain constraints on the interplay between low- able discussions. The work was supported by NSF DMR- and high-energy contributions to observable quantities. 1523036 (A.V.C. and A.K.) and NSF DMR-1720816 These constraints imply that extra care is needed in inte- (D.L.M.).

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