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 pZ < 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 ! 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 ! 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 croscopicj j 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 ∗ ! 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 ! 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.