Vertex Functions at Finite Momentum: Application to Antiferromagnetic

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CONSERVATION LAW AND CONTINUITY identities.16,17 These identities are usually derived by EQUATION: VERTEX FUNCTION IN LINEAR considering global symmetry transformations effected by RESPONSE the application of unitary operators (gauge transformations, rotations, etc.). Typically, a Ward identity relates We consider systems of identical fermions interacting the structure of the single-particle Green’s function to a via local interactions. To be concrete, we may take the three-point vertex function Λ(q,ν). A standard way18 of Hubbard model, whose interaction term is of the form constructing a Ward identity is to identify a conservation law that follows from a symmetry of the Hamiltonian and † † then using it to simplify the equation of motion for a two- Hint = U drΨ↑(r)Ψ↑(r)Ψ↓(r)Ψ↓(r) (2) particle Green’s function that contains the three-point Z vertex. Since we are primarily interested in an incom- † where Ψα(r), Ψα(r) creates or annihilates an electron of mensurate antiferromagnetic QCP in the framework of spin α =↑, ↓ at location r. the strong-coupling theory developed in Refs. 11–13, in As a practical example, we consider the response of the what follows, we shall concentrate on the consequences system to an external magnetic field H~ (r) that couples of spin-rotational invariance. † to the spin density ~ρs(r) = α,β Ψα(r)~σαβ Ψβ(r) via an The Ward identities are usually applied for the limit interaction term P of vanishing wavevector and frequency of an applied test field. They are therefore of limited use in characterizing H′ = λ drH~ (r) · ~ρ (r). (3) fluctuations at non-zero wave vector, such as antiferro- Z s magnetic fluctuations or charge density wave fluctuations in a metal. Before showing how the spin-density vertex function, the expectation value of ~ρs(r, t) (in Heisenberg representa- However, the local conservation laws are valid on all tion), controls the response to H′, we remind that when spatial and temporal scales and give rise to generalized a density operator commutes with the interaction term Ward identities even at non-zero wavevector. This has 19 in the Hamiltonian, then it obeys a continuity equation, already been recognized and implemented by U. Behn, derived from ∂Ψ/∂t = i[H, Ψ]. Since the spin density who used the procedure described above and which we operator commutes with Eq. (2), only the kinetic energy elaborate in Sec. III. Although these identities may be enters the commutator and we have the familiar local less stringent because, as we shall see, they could involve conservation law (repeated greek indices are summed): two vertex functions, one of a density type, the other of a current-density type, they nonetheless may be used to infer qualitative information. This is of particular inter- ∂~ρs(r, t) i † † − [∇~ · (∇~ Ψ ~σ Ψ − Ψ ~σ ∇~ Ψ ]=0. (4) est if single-particle properties, such as the quasiparticle ∂t 2m α αβ β α αβ β effective mass at the Fermi surface, are singular. This The spatial derivatives in Eq. (4) come from the commu- may happen at a quantum critical point. In metallic tator of ρs with the kinetic energy, which in momentum compounds, quantum critical points are often found to † be of antiferromagnetic or charge-density wave character, space is k,α ǫkck,αck,α For later convenience, it is useful which involve fluctuations of spin or charge at non-zero to reexpressP the continuity equation in momentum space: wave vector. Specific heat data in the neighborhood of † i∂~ρq/∂t = (εkq )ck+,α~σαβck−,β, (5) such critical points often indicate a divergent quasipar- X ticle effective mass. Examples are many of the heavy- k fermion compounds, some of the iron-based supercon- † where ~ρq = c ~σαβck−,β, with k± = k ± q/2 and ductors and possibly the cuprate superconductors. The k k+,α ε = ǫ − ǫP− In compact form, it is question becomes: How does a singularity in the single- kq k+ k particle properties affect the two-particle vertex functions † D(k, q; t) c ~σαβ ck−,β =0, (6) at non-zero wavevector? Here the generalized Ward iden- k+α Xk tities may be useful. In the present paper, the answer will be given as where the operator D(k, q; t) is ∂ D(k, q; t)= − (εkq) (7) −1 −1 ∂t Λ(Q,ν → 0) ∼ Z (p+)+ Z (p−), (1) The considerations above apply also to the charge response and lead to the usual charge continuity equa- where Z−1 =1 − ∂Σ/∂ω is proportional to the quasipar- tion. For our general purposes, we want to discuss the ticle effective mass and p± = p±Q/2 are the momenta of vertex function that describes the coupling of an elec- the incoming and outgoing legs of the three-point vertex tron (spin or charge) density to an external perturba- describing momentum transfer Q and energy transfer ν tion such as a charge or spin density fluctuation boson 3

field or a magnetic field. For example, in linear response, the magnetic field perturbation H′ of Eq. (3) gives rise to a correction to the single-particle Green’s Λ(k, q)=1+ Γ(k,p; q)G(p−)G(p+) (16) † Xp function Gαβ (1, 2) = −hT Ψα(1)Ψβ(2)i, where T is the time-ordering operator. In the above, Γ(k,p; q) is the four-point vertex (without external legs) and Λ(k, q) is the three-point vertex am- δG (1, 2) = d3 G(2) (1, 2;3, 3+)λH~ (3) · ~σ . (8) αβ Z αβδγ γδ plitude that enters the Ward identity, We use Eq. (15) in Eq. (13), divide out G(k+)G(k−) and find Here, the two-particle Green’s function is (ν − εpq)Γ(k,p; q)G(p−)G(p+) (2) † † Xp Gαβγδ(1, 2;3, 4) = −hT Ψα(1)Ψβ(2)Ψγ(3)Ψδ(4)i (9) = ν − εkq − Σ(k+)+Σ(k−), (17) + + and in Eq. (8), 3 means (r3,t3 +0 ). It is seen that the RHS of Eq. (8) contains ~ρs(3); where Σ(k) is the self energy part as in Eq. (14). This result has already been anticipated in Ref. 19; we make δG (1, 2) = d3hT Ψ (1)Ψ† (2)~ρ (3)i · λH~ (3). (10) use of it in what follows. αβ Z α β s We take the derivative of Eq. (17) with respect to ν and as we are interested in the behavior of Λ(k,ω; q,ν) This shows how the vertex function controls the response. for ν → 0 and arbitrary q, we then take ν = 0 and obtain Making use of the conservation law we may now derive identities relating the response function to the ∂ q single-particle Green’s function or its self-energy. To Λ(k; , 0) − εpq[ Γ(k,p; q)G(p−)G(p+)]ν=0 X ∂ν achieve this we re-express the two-particle Green’s func- p tion in Eq. (10) in momentum space and let the operator 1 −1 −1 = [Z (p+,ω)+ Z (p−,ω)], (18) D(p, q; t3) that appears in the conservation law, Eq. (6) 2 act on it. where we used the quasiparticle weight factor at (p,ω) (2) † defined as G → hTck+,α(t1)ck−,β(t2)~ρq(t3)i (2) −1 ∂ = G (k,p; q)~σγδ, (11) Z (p,ω)=1 − Σ(p,ω) (19) αβδγ ∂ω Xp The second term on the LHS of the key result Eq. (18) where is the ν = 0 derivative of the spin-current density vertex. (2) † † + The spin (and charge) continuity equations, as in G (k,p; q)= −hTc (t )c (t )c − (t )c (t )i. αβδγ k+,α 1 k−,β 2 p ,δ 3 p+γ 3 Eq. (4) are a consequence of invariance under certain uni- (12) tary transformations (e.g. gauge, rotation). However, as Here, k± = k ± q/2,p± = p ± q/2 are the momenta emphasized by Nambu,18 there may also be non-unitary of the particle-hole pairs entering and leaving the two- transformations under which the local spin and charge particle Green’s function The action of the operator on (2) densities and/or the Hamiltonian are invariant (perhaps the ρq part of G gives zero because of the conservation up to a shift in the chemical potential µ0). These can lead law, whereas the time derivative in D acts on the step to “pseudo-conservation laws” that are operator identi- functions defined by the time ordering. The result is ties for charge and spin density and current and to Ward- 20 (2) like relations similar to Eq. (18). For example, carrying [ν − εpq]G (k,p; q)= G(k−) − G(k+). (13) out the steps above, using the pseudo-conservation law, Xp † ∂c ∂c − Here we have Fourier transformed in time (t ) and on k+,α † k ,β 3 ~σαβ [i ck−,β − ic ] ∂t k+,α ∂t the RHS of Eq. (13), k and q mean (k,ω) and (q,ν) Xk respectively. That is, k± → (k ± q/2,ω ± ν/2) and here † + [εk + εk− − 2µ0]c ck−,β =0, (20) the sum on p includes dω. For the charge response, + k+,α

Eq. (13) holds, as the spinR indices are irrelevant. We one may derive shall restore them later, when necessary. The next steps involve the use of well-known relations 1 ′ ∂ ′ q ′ ′ q among the Green’s functions and associated amplitudes. Λ(k; , 0)+ [2ω − ǫp+ − ǫp− +2µ0] Λ(p,p ; , 0) 2 ′ ∂µ0 Xp −1 −1 G (k,ω)= ω − ǫk − Σk,ω)= G k,ω) − Σ(k,ω) (14) 1 0 = [Z−1(p ,ω)+ Z−1(p ,ω)] (21) 2 µ + µ −

(2) −1 G (k,p; q)= G(k+)G(k−)Γ(k,p; q)G(p−)G(p+) (15) where Zµ (p,ω)=1 − ∂Σ(p,ω)/∂µ0. 4

Now, a situation of particular interest arises if the spectrum of spin fluctuations and their the coupling to quasiparticle weight factor Z(p,ω) happens to be small, quasiparticles in just the way that was assumed in the or even tends to vanish, implying that the effective quasi- theory of critical quasiparticles. ∗ −1 particle mass m /m = Z (p,ω) is large either in cer- There are two ways in which the vertex function λQ tain regions on the Fermi surface (so-called hot spots) enters the theory: first, the spin-fluctuation spectrum is or all over the Fermi surface. This will be the case in affected in the Landau damping term; it acquires a factor 2 the critical regime near a quantum phase transition to λQ, from the renormalization of the particle-hole bubble e.g. an antiferromagnetic phase. We may then conclude diagram of Landau damping at each end. Thus from the key equation (18) that the three-point vertex 2 N0λ ν is enhanced approximately proportional to the effective q Q Imχ( ,ν)= 2 2 2 2 mass enhancement. This follows from the fact that both (r + (q − Q) ) + (λQν) the spin density and the spin-current density vertices are Here N is the bare density of states, r is the dimension- given by integrals of ΓG(p+)G(p−) multiplied by two dif- 0 less tuning parameter (r → 0 at the QCP) and wavevec- ferent weight factors, 1 and εpq(∂/∂ν), respectively. Al- though the effective mass enhancement occurs for a state tor q and frequency ν are in units of kF and ǫF , respec- near the Fermi surface, we emphasize the new result that tively. Second, since the coupling of the spin fluctuations the vertex enhancement takes place if at least one of the to the quasiparticles also involves a factor λQ, each end −1 partners of the particle-hole pair, with momenta p+ or of a spin fluctuation line receives a factor λQN0 . p− , is on the Fermi surface. In order to demonstrate The large momentum transfer involved in a scatter- that Eq. (18) does indeed imply a proportionality of Λ ing process of quasiparticles off AFM spin fluctuations to Z−1 we consider the limit of small, but finite q and usually takes quasiparticles into final states far from the ν = 0, when Fermi liquid theory applies. In this case the Fermi surface, except for momenta at “hot spots” on the −1 vertex function is given as Λ = Z /(1 + Fa), where Fa Fermi surface. The consequences of these limitations of is the Landau parameter in the spin channel. In order critical scattering are often not compatible with what is for the Ward identity Eq. (18) to be satisfied, the current observed experimentally. It was therefore suggested In −1 density term has to amount to Z Fa/(1 + Fa). The two Ref. 13 that simultaneous scattering off two spin fluc- contributions add up to Z−1, as required by the Ward tuations with opposite momenta, leading to small total 21 identity. In other words, for any non-zero Fermi liquid momentum transfer would be a more relevant process. interaction Fa both terms on the l.h.s. of Eq. (18) are Two spin fluctuations may be thought of as an (exchange) proportional to Z−1, and will therefore diverge whenever energy fluctuation χE(q,ν). The corresponding spec- Z−1 diverges. trum was calculated in Ref. 13, Eq. (3) to be

3 3 5/2 (N0) λ ν q Q IV. VERTEX FUNCTION AT LARGE Q Im χE( ,ν)= 2 2 2 2 (22) (r + q ) + (λQν)

We now calculate the irreducible spin-density vertex The corresponding self energy due to energy fluctuation function Λ(k,ω; Q,ν = 0) (called λQ in Ref. 13) in exchange is given by the framework of the theory of critical quasiparticles near an antiferromagnetic critical point as developed in Σ(k,ω) ∼ dq G(k + q,ω + ν) χ (q,ν) (23) Z E Refs. 11–13. There it was shown that for the case of three-dimensional AFM spin fluctuations, when conven- and leads to Σ ∝ ω3/4 (and hence Z(ω) ∝ ω1/4) tional spin density wave theory is supposed to work, a The first vertex correction diagram that corresponds new strong-coupling regime may be accessible under cer- to the dressing of the spin-density vertex λQ by energy tain conditions. This regime is characterized by a power- fluctuations has one energy fluctuation that bridges the law divergence of the effective mass as a function of en- vertex: ergy, and hyperscaling with critical exponents z = 4 and ν =1/3. The theory requires the particle-hole irreducible (1) 4 2 4 λQ = λQλvu dq G(p + q + Q)G(p + q)χE(q) (24) spin density vertex function at wavevector Q to diverge Z like the effective mass. As shown in Sec. III, this can be Here λv ∝ 1/Z is a vertex correction at small momentum a consequence of the Ward identities. However, since the q governed by the usual Ward identity at (q = 0,ν → Ward identities relate the full vertex functions Λ to the −1 0) and u ∝ N0 . For generic p one of the momenta, effective mass (or the inverse quasiparticle weight factor p + q + Q will be far from the Fermi momentum, while 1/Z), one may ask how the irreducible vertex, which is the other, p + q, is close to it (or vice versa). We may the quantity needed in the strong coupling theory11–13 then put G(p + q + Q) ≈ 1/ǫF . What remains is the depends on Z. We therefore show in the following that self-energy expression, Eq. (23), so that a certain diagram contributing to the irreducible vertex correction is indeed proportional to 1/Z, provided one as- (1) Σ(p,ω) λQ (p,ω; Q,ν = 0) ≈ (25) sumes that this very vertex correction renormalizes the ǫF 5

(1) We see that λQ → 0 as ω → 0 and is not singular. How- V. CONCLUSION ever, singular diagrams do occur if at least three spin fluctuation lines in parallel are internal in a contribution to λQ(p,ω; Q,ν = 0) (any odd number will do). Two of these combine into an energy fluctuation. The resulting The Ward-Takahashi identities are based on conserva- diagram has a spin fluctuation and an energy fluctuation laws; in interacting electron systems they relate the tion in the intermediate state, similar to the Azlamasov- single-(quasi)particle renormalizations to vertex ampli- Larkin diagram in the theory of superconducting fluctu- tudes that describe response to external probes and/or ations. coupling to collective modes. While these identities are usually given, both in quantum field theory and in the many-body problem, for the limit of momentum ~q → 0 λ(3) = A dq G(p − q)T (q; Q)χ(Q + q)χ (q), Q Z E and frequency ν → 0, there are cases for which ~q 6= 0 is of interest, as when quasiparticles interact with non- 6 2 6 where A = λQλvu and we defined the triangle loop uniform external fields or collective fluctuations. Of particular interest is the situation in the neighborhood of a quantum critical point associated with an ordered phase T (q; Q)= dp′ G(p′ + q)G(p′)G(p′ + q + Q) Z having a non-uniform order parameter. Examples are an antiferromagnetic or charge density wave state with a The quantity T (q; Q) is noncritical and may be replaced non-zero ordering vector. Here, quasiparticles couple to by T (q; Q) ≈ N0/ǫF . It is convenient to first calculate nonzero ~q order parameter fluctuations and the quasipar- (3) the imaginary part of λQ at temperature T <<ω: ticle renormalization may be singular. The generalized Ward identity shows how the vertex amplitude for this ω (3) coupling also acquires singular behavior, following the ImλQ ≈ A1 dν d~q Imχ(Q + q)ImχE(q)ImG(p − q) renormalized quasiparticle mass. A new feature is that Z0 Z (26) it is sufficient that (at least) one of the external lines to 6 2 6 Here A1 = λQλvu (N0/ǫF ). The result of the integration the (three-point) vertex is on the Fermi surface, rather over the solid angle of ~q is ∝ 1/q. We restrict ourselves than requiring both to be. Therefore, the identity holds to the critical regime, r = 0 in χE. The q-integration even when ~q does not connect two points on the Fermi 2 2 surface (“hot spots”). Our results are of use in analyzing may then be performed for λQ|ν| < q < ∞ behavior of metals near quantum critical points.

3 ω (3) kF 11 2 7/2 1 ImλQ ∝ λQ λv dν|ν| qdq 4 2 2 2 N0ǫF Z0 Z [q + (λQ|ν|) ] 5 −2 3/2 ∝ λQZ |ω|

−1 where we used λv ∝ Z as stated above. We now iden- (3) tify λQ = λQ , and solve the resulting equation for λQ:

1/2 −3/8 VI. ACKNOWLEDGEMENTS λQ ∝ Z |ω| (27)

This result may be combined with the result for Z in the strong-coupling regime which was obtained in Ref. 13, We acknowledge useful discussions with A. V. Eq. (4): Chubukov, G. Kotliar, D. Maslov, Q. Si, C. M. Varma, and especially J. Schmalian. P.W. thanks the Depart- Z ∝ λ5 |ω|3/2 (28) ment of Physics at the University of Wisconsin-Madison Q for hospitality during several stays as a visiting professor Combining Eqs. (27,28), we ﬁnd and acknowledges an ICAM senior scientist fellowship. Part of this work was performed during the summers of 2012-14 at the Aspen Center for Physics, which is sup- −1 −1/4 λQ ∝ Z ∝ |ω| , ported by NSF Grant No. PHY-1066293. P.W. acknowledges ﬁnancial support by the Deutsche Forschungsge- which is precisely what has been postulated in Ref. 13 on meinschaft through Grant No. SCHM 1031/4-1 and by the basis of phenomenological arguments. the research unit FOR960 ‘Quantum Phase Transitions’.

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