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 inter- action 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 liq- uid 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 ! 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 momen- tum 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 ^l=0 and ρ^l=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 y ρ^l (q) = λl (k)ck−q=2,αck+q=2,α; (2) the relation k,α c(s) m c(s) s X s y αβ ZΛl=1 = ∗ 1 + Fl=1 (7) ρ^l (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 can- cel 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 ad- ditional 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 ) .