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Mateusz Homenda, Pawel Jakubczyk, [Hiroyuki Yamase](https://orcid.org/0000-0003-0328-5657)

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[Generalized Hertz action and quantum criticality of two-dimensional Fermi systems](https://mdr.nims.go.jp/datasets/db2fe2fb-2489-4caf-8013-73d4dc68365a)

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Generalized Hertz action and quantum criticality of two-dimensional Fermi systemsMateusz Homenda∗ and Pawel Jakubczyk†Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, PolandHiroyuki Yamase‡Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0047, Japan(Dated: May 28, 2024)We reassess the structure of the effective action and quantum critical singularities of two-dimensional Fermisystems characterized by the ordering wavevector Q⃗ = 0⃗. By employing infrared cutoffs on all the masslessdegrees of freedom, we derive a generalized form of the Hertz action, which does not suffer from problems ofsingular effective interactions. We demonstrate that the Wilsonian momentum-shell renormalization group (RG)theory capturing the infrared scaling should be formulated keeping Q⃗ as a flowing, scale-dependent quantity.At the quantum critical point, scaling controlled by the dynamical exponent z = 3 is overshadowed by a broadscaling regime characterized by a lower value of z ≈ 2. This in particular offers an explanation of the results ofquantum Monte Carlo simulations pertinent to the electronic nematic quantum critical point.Introduction Quantum criticality in Fermi systems consti-tutes a highly relevant and largely open problem for con-densed matter theory. Its significance stems from the growingexperimental evidence demonstrating non-Fermi-liquid be-havior of thermodynamic as well as transport properties at theonset of different ordered states in a diversity of compounds;the high-Tc cuprate superconductors being the most promi-nent examples[1]. A persistent question concerns the struc-ture of the low-energy effective action to correctly capture thecritical singularities at quantum criticality in Fermi systems.The traditional approach, first developed by Hertz[2] and laterextended by Millis[3], borrowed the spirit of the Wilsoniantheory of classical critical phenomena[4, 5]. It proposed tointegrate out the original degrees of freedom resulting in anexact representation of the problem in terms of an effectiveorder parameter action. In the subsequent step this action wasexpanded in powers of the ordering field, truncating at quar-tic order; the two-point function was replaced by its low mo-mentum/frequency asymptotic form, and the (supposedly ir-relevant) momentum/frequency structure of the bosonic self-interaction was disregarded. This leads to a relatively simpleHertz action[2, 3, 6]S H[ϕ] =∫qϕ−q⃗,−q0[m2 + Zq⃗2 + A|q0||q⃗|]ϕq⃗,q0 +u∫xϕ(x)4 (1)describing the propagation of a damped collective bosonicmode ϕ, where the interaction with fermions is describedby the so-called Landau damping term ∼ |q0|/|q⃗|, q0 beingthe frequency and q⃗ the momentum of the order parameterfield. Here {m2,Z, A, u} are constants, q := (q0, q⃗),∫q :=1/(2π)3∫dq0∫d2q, and∫x :=∫dτ∫d2x encompasses in-tegration over space and the imaginary time τ. The form ∼|q0|/|q⃗| is valid for instabilities occurring at ordering wavevec-tor Q⃗ = 0⃗.∗ mateusz.homenda@fuw.edu.pl† pawel.jakubczyk@fuw.edu.pl‡ yamase.hiroyuki@nims.go.jpIn contrast to classical statistical physics systems, this pro-cedure involves integrating out gapless particle-hole excita-tions across the Fermi surface, the consequence of whichbecomes revealed by inspection of the nature of the fre-quency/momentum expansion of the bosonic interaction ver-tices (for example the fermionic box diagram), which turnsout to be singular at T = 0 [7–10]. For this reason quan-tum critical Fermi systems (at least in dimensionality d = 2and temperature T = 0) cannot be adequately described by apurely bosonic action characterized by local interactions.The above issues motivated development of a diversityof approaches that retain the fermionic degrees of freedom,which are coupled to order parameter fluctuations[9, 11–27].These theoretical routes come with their own questions. Oneof these concerns the transition between the microscopic andeffective low-energy action. This is transparent, for instance,in the analysis concerning generation of the damped dynamicsof the bosonic mode. As was emphasized in previous litera-ture (see in particular Ref. 15), appearance of the ∼ |q0|/|q⃗|term requires that the fermions be integrated out down to theFermi level. It is not conceivable to generate the standardLandau damping term by a Wilsonian-type RG flow until thecutoff on fermions is completely removed and therefore thefermionic degrees of freedom become once and for all in-tegrated out of the theory. In consequence, accounting forthe Landau damping (in its standard form) within such ap-proaches requires fully dressing the boson propagator withself-energy before calculating any loops that involve internalboson lines in the coupled Bose-Fermi theory.In the present work we systematically readdress the theoryof quantum criticality in Fermi systems featuring Q⃗ = 0⃗ in-stabilities and develop a Wilsonian RG approach, where boththe bosonic and fermionic propagators become equipped withmomentum cutoffs, and, upon lowering these, generate theRG flow, leading to a generalization of the Hertz action. Ourgoal is to develop an approximate approach fully encompass-ing the Hertz-Millis framework, but at the same time refrain-ing from completely integrating femions out, such that the sin-gular effective bosonic interactions never appear. We demon-strate that the RG flow of the bosonic properties involves a2completely different, not previously recognized contribution,which encodes a structure richer than the conventional Hertz-Millis theory.Generalized Hertz action Our approach relies on the non-perturbative RG framework in the Wetterich formulation[28–30]. This methodology has, in recent years, led to several im-portant new insights concerning key problems of condensedmatter theory, critical systems in particular. Examples includeidentification of strong-coupling fixed points for the Kardar-Parisi-Zhang problem in d > 1 [31], resolution of the prob-lem of dimensional reduction and its breaking for the randomfield Ising model [32], discovery of new multicritical RG fixedpoints for the O(N) models in d = 3 [33], and invalidation [34]of the predictions of perturbative approaches concerning non-analyticity of the critical exponents as function of d and N. Inour approach we integrate the coupled fermionic ( {ψ̄, ψ} ) andbosonic ( ϕ ) fluctuating fields out of the partition functionvia a renormalization group flow. The central object is thescale-dependent effective action ΓΛ[ψ̄, ψ, ϕ], which continu-ously interpolates between the bare effective action S [ψ̄, ψ, ϕ]and the full effective action (free energy), when the infraredcutoff Λ is lowered from the UV scale towards zero. Belowwe suppress the arguments of Γ for readability. The evolutionof Γ upon varying Λ is governed by the exact Wetterich flowequationΓ̇ = βb + β f , (2)whereβb =12Tr{Ṙb(Γ̃(2)ϕϕ)−1[1 − Γ̃(2)ϕψ(Γ̃(2)ψψ)−1Γ̃(2)ψϕ(Γ̃(2)ϕϕ)−1]−1}, (3)β f =12Tr{Ṙ f (Γ̃(2)ψψ)−1[1 − Γ̃(2)ψϕ(Γ̃(2)ϕϕ)−1Γ̃(2)ϕψ(Γ̃(2)ψψ)−1]−1}. (4)The quantity Γ̃ := Γ + ∆S denotes the action Γ supplementedwith the regulator term ∆S = 12Φ(RΦT ), which is quadraticin the fields Φ = (ψ̄, ψ, ϕ) and contains bosonic (Rb) and fer-monic (R f ) components. The quantity Γ̃(2) denotes the second(functional) field derivative of Γ̃ with the relevant fields speci-fied by the subscript in each case. By Ẋ we mean ∂ΛX. Finally,the trace (Tr) sums over the field components, momenta andfrequencies. Our notation is equivalent to that introduced inRef. 35 (for details and derivations see also Ref. 36) with theexception that ϕ is a real scalar in our case. Differentiatingthe flow equation [Eq. (2)] with respect to fields gives rise toan hierarchy of flow equations for the one-particle irreduciblevertex functions. We concentrate on the RG flow equation forthe bosonic two-point function, obtained by taking the secondfunctional derivative of Eq. (2) with respect to ϕ. The result-ing equation [29, 30, 35, 36] involves terms represented viaone-loop Feynman diagrams depicted in Fig. 1.The expressions involve the flowing fermion propagatorG̃ := (Γ̃(2)ψψ)−1 supplemented with a momentum cutoff RΛf (⃗k):G̃k,k′,σ,σ′ =(−ik0 + ξk⃗ + RΛf (⃗k) + ΣΛ(k))−1δk,k′δσ,σ′ (5)FIG. 1. Terms contributing to the flow of the bosonic 2-point func-tion. Dressed (scale dependent) fermion and boson propagators aredepicted as full and dashed lines respectively. Black triangles andrectangles represent the bosonic vertices, while dotted (grey) tria-gles and rectangles stand for fermion-boson interactions. The strokedlines represent the single-scale propagators: S f for fermion and Sbfor boson propagators (see the main text).[with k := (k0, k⃗)]; the flowing regularized boson propagatorG̃b := (Γ̃(2)ϕϕ)−1, interaction vertices, as well as the so-calledsingle-scale propagators defined asS f := −(Γ̃(2)ψψ)−1Ṙ f (Γ̃(2)ψψ)−1 (6)Sb := −(Γ̃(2)ϕϕ)−1Ṙb(Γ̃(2)ϕϕ)−1 . (7)In Fig. 1 the single-scale propagators correspond to strokedlines. The bare (microscopic) action contains only contribu-tions quadratic in fields and a Yukawa-type term coupling thebosonic field with two fermionic variables [6]. In addition,the bare boson propagator carries no momentum/frequencydependence. These dependencies are generated by graduallyintegrating the fermions out via the contribution to the flowgiven by the first diagram in Fig. 1.The flow parameter Λ appearing in the Wetterich equationis identified with the bosonic momentum cutoff (Λb = Λ). Theprecise form of the bosonic cutoff will be specified later. Wewill use the following form of the cutoff function on fermions:R f (⃗k) =(ξkF+ΛF − ξk⃗)θ(ΛF − (|⃗k| − kF))for |⃗k| ≥ kF(ξkF−ΛF − ξk⃗)θ(ΛF − (kF − |⃗k|))for |⃗k| < kF .(8)The quantity ΛF = ΛF(Λ) is a function of Λ. The effect ofadding R f (⃗k) to the dispersion ξk⃗ amounts to deforming it in asliver of extension 2ΛF around the Fermi level, as depicted inFig. 2. We expect that our conclusions are completely insen-sitive to the precise choice of the momentum cutoff functionR f (⃗k).The Hertz-like approach corresponds in our framework tosending ΛF to zero before Λ. This can be realized e.g. bytakingΛF = (Λ − Λ0)θ(Λ − Λ0) (9)withΛ0 > 0, such thatΛF becomes zero at positiveΛ. In whatfollows, we will perform a detailed comparison between thepictures emergent for Λ0 = 0 and Λ0 > 0 (Hertz-Millis case).3kF0kxky2 ΛFFIG. 2. A schematic plot of the regularized dispersion ξk⃗ + R f (⃗k).Including the regulator introduces a deformation of the dispersion ina strip of extension 2ΛF(Λ) around the Fermi level. In the inset theblack line represents the Fermi surface and gray shell designates thearea of the deformation.The flow equation represented by the terms depicted inFig. 1 is exact, but can be solved only approximately. Itspresent truncation is devised such that it encompasses theHertz-Millis theory if ΛF is scaled to zero first [e.g when onetakes Λ0 > 0 in Eq. (9)], but does not require this in any way.The gradual generation of the dynamics of the boson propa-gator can be followed upon reducing Λ towards zero. The keypresent approximation amounts to disregarding the Fermi selfenergy [ΣΛ(k) = 0] and the flow of the Yukawa coupling gas well as other fermionic interactions generated by the flow.This allows us to write the contribution to the flow of the bo-son propagator represented by the first diagram in Fig. 1 as:X(q,ΛF) = −2g2∫k∂ΛR f (⃗k)G̃0(k)2(G̃0(k + q) + G̃0(k − q)),(10)where the scale-dependent (regularized) fermion propagatoris given by G̃0(k)−1 = [−ik0 + ξk⃗ + R f (⃗k)]. To simplify thecalculations and highlight the new theoritical insight clearly,we employ the standard quadratic dispersion ξk⃗ = (⃗k2 −k2F)/2m f . We then evaluate the integrals in Eq. (10) and sub-sequently integrate over the cutoff scale, which results in thefrequency/momentum structure of the boson propagator (gen-erated from integrating the fermions from the UV cutoff scaleΛu down to the scale Λ). ComputingB(q⃗, q0,ΛF(Λ)):=∫ ΛΛudΛ′X , (11)we obtain:B(q⃗, q0,ΛF) = B<θ(−|q⃗| + ΛF) + B>θ(|q⃗| − ΛF) , (12)whereB< = −N<|q⃗|ΛFq20 + 4v2FΛ2F, (13)B> ≈ −N<q⃗2q20 + 4v2F q⃗2+N>q0|q⃗|[arctan2vF |q⃗|q0− arctan2vFΛFq0],(14)and N< = N>v3F = 4g2kFvF/π2. The above expressions areessential for the present work. Eq. (13) follows from exactlyevaluating the integrals of Eq. (10) for |q⃗| < ΛF and subse-quently integrating over the cutoff scale according to Eq. (11).Eq. (14) results from evaluating Eq. (10) for |q⃗| ≫ ΛF retain-ing the terms, which generate the standard Landau damping∼ |q0|/|q⃗| if we first take ΛF → 0 and subsequently consider|q⃗|q0→ ±∞; the dropped terms are regular in q in the limitΛF → 0 and we made no assumptions concerning the rela-tive magnitude of |q0| and vF |q⃗|. Concerning the structure ofB(q⃗, q0,ΛF), we emphasize that: (i) for ΛF → 0 it recovers,via B>, the standard Landau damping term of the Hertz action;(ii) it takes minimum at (q0, |q⃗|) = (0,ΛF), which indicate thatthe ordering wavevector depends on the cutoff scale and fallsat |Q⃗Λ| = QΛ = ΛF , thus scaling to zero under RG. Note inparticular that artificially putting QΛ = 0 suppresses the flowof the mass generated from fermionic bubble. This explains(and evades) the unwelcome features of the mass flow underWilsonian RG, discussed in Ref. 15. Observe that the massflow is generated from B< [evaluated at (q0, |q⃗|) = (0,ΛF)].In the present generalization of the Hertz-Millis approach wewill parametrize the flowing inverse boson propagator asΓ(2)Λ= Z(|q⃗| − QΛ)2 + Aq20 + m2Λ + B(q⃗, q0,ΛF(Λ)) . (15)The essential modification of the standard Hertz actionamounts to replacing the term ∼ |q0|/|q⃗| occurring in Eq. (1)with the formula B(q⃗, q0,ΛF) obtained above, such thatfermionic fluctuations are included only down to the scaleΛF(Λ) (which is sent to zero as Λ → 0). We emphasize thatthe RG flow of the boson propagator will be strongly influ-enced by the first term in Eq. (12), corresponding to |q⃗| small.The dynamical exponent We now examine the conse-quences of the term B< for the dynamical exponent z of theorder parameter field. If we first take ΛF → 0 setting Λ0 , 0in Eq. (9) [thus removing the term B< from B in Eq. (15)], andsubsequently consider the limit vF |q⃗|/q0 → ±∞, we recoverthe Hertz result z = zH = 3, proceeding along the standardpath [6].The situation radically changes, if we instead integrate bothbosons and fermions in parallel by considering Eq. (9) withΛ0 = 0 (Λ f = Λ ) , in which case B< plays a prominent role.The anticipated value of z resulting from the q0 dependenceof B< can be deduced by putting |q⃗| = Λ in B< and expandingfor q0 ≪ 2vFΛ. The leading term renormalizes the mass m2Λin Eq. (15) and the second term is proportional to q20/Λ2. Wefind that the B< term in Γ(2)Λscales as Λ2 (thus leading to scaleinvariant propagator) provided q0 ∼ |q⃗|2 which corresponds tothe dynamical exponent z = z< = 2. We also note that thechoice ΛF ∼ Λ is the only one, which allows to write Γ(2)Λgiven by Eq. (15) (keeping either B< or B>) in a scaling form.From this above heuristic picture one anticipates a competi-tion between two scaling behaviors governed by z ≈ 2 andz ≈ 3. This is checked and confirmed by solving the RG equa-tions as described below.RG flow A convenient way to extract the dynamical expo-nent z from the RG flow, capturing possible crossovers, is toinspect the behavior of the flowing order parameter expecta-4FIG. 3. The renormalized value of ρ = ϕ20/2 plotted as function ofthe control parameter τ (bare mass) at T = 0 for a sequence of valuesof the boson-fermion coupling g (N>vF ∝ g2). The uppermost curve(blue) exhibits crossover between scaling pertinent to the classical3d Ising universality class and mean-field scaling outside the truecritical region. Upon gradually switching on g the asymptotic criticalscaling corresponding to efffective dimensionality D = d+ z ≥ 4 setsin and the system again crosses over to mean-field behavior. Thebehavior exhibited in the inset corresponds to Λ0 > 0 (integratingout fermions first). There is no qualitative difference between thesetwo cases, which demonstrates that including B< has no impact onthe critical singularities of the order parameter.tion value, which follows ϕΛ0 ∼ Λz/2. This is, for d = 2,implied from m2Λ∼ Λ2, uΛ ∼ Λ4−(d+z) and the relation m2Λ=uΛ(ϕΛ0 )2 (see e.g. Refs. 37 and 38). Equivalently, one mayinvoke the scaling dimension of the ϕ field [ϕ] = d + z− 2+ η,which gives z/2 for d = 2. Here we neglect the anomalousdimension η.We evaluate the flow of the boson order parameter ϕΛ0 andquartic coupling uΛ within a simple truncation of the Wet-terich equation, where the bosonic propagator is dressed asdictated by Eq. (15). We include[38, 39] the renormalizationof u via bosonic fluctuations of order ∼ u2, which allows forcapturing also the 3d Wilson-Fisher fixed point. The lattergoverns the critical behavior in the absence of the Fermi-Bosecoupling g and gives rise to an intermediate scaling regimedescribed by the dynamical exponent z = 1, as observed inRef. 15 and also clearly captured in our approach (see Fig. 3).Within our present framework, the flow equations for ϕΛ0 andthe quartic coupling uΛ are derived following the standard pro-cedure described for example in Refs. 37 and 38. We choosethe Litim cutoff[40] on bosonsRb(q⃗) = Z[Λ2 − (|q⃗| − QΛ)2]θ[Λ2 − (|q⃗| − QΛ)2] . (16)We verified, that implementing the Wetterich cutoff[37] in-stead, does not change any of our results. Our major conclu-sion concerning z is best summarized by comparing the RGflows of the order parameter depicted in Fig. 4 and Fig. 5.Before discussing the scale dependencies arising in the RGflow, we inspect the T = 0 phase diagram - see Fig. 3. WeFIG. 4. The RG flows of ρΛ = (ϕΛ0 )2/2 for a sequence of valuesof τ progressively tuning the system towards the QCP in a situation,where fermions are integrated first [Λ0 > 0 in Eq. (9)]. The value ofz can be read off by fitting the power law (see the main text). Thecrossover between z = 1 and z = 3 is clearly visible both as functionof g ( N>vF ∝ g2) and the cutoff scale Λ.FIG. 5. The RG flows of ρΛ = (ϕΛ0 )2/2 for a sequence of values ofτ progressively tuning the system towards the QCP in the case whenΛF = Λ. The value of z can be read off by fitting the power law (seethe main text). The value z ≈ 2 following from B< is clearly visiblefor nonzero values of the boson fermion coupling g ( N>vF ∝ g2).observe hardly any difference between the situations corre-sponding to Λ0 = 0 and Λ0 > 0 that would be visible in thescaling of the order parameter as a function of the non-thermalcontrol parameter τ. A similar situation may be anticipatedfor T > 0 in the behavior of the critical temperature Tc(τ).This is because (at least in the Hertz-Millis framework) onehas Tc ∼ τψ with ψ = z/(d + z − 2), which for d = 2 yieldsψ = 1 irrespective of the value of z. As far as the behavior ofTc is concerned, the distinction between z = 2 and z = 3 isonly revealed at the level of logarithmic corrections[3]. In arealistic (experimental or simulation) situation the value of z5may, for example, be read off from the scaling behavior of thecorrelation function above the quantum critical point (see e.g[41]).While presence of the term B< has no impact on the phasediagram (see Fig. 3), switching on g leads to scaling of theorder parameter characterized by z ≈ 3 in absence of B< andz ≈ 2 when B< is present. At vanishingly low cutoff scale,deviation from the z = 2 scaling towards higher values ofz is clearly visible in the right plot in Fig. 5. Despite highnumerical accuracy, for the considered parameter values, wewere however not able to obtain a scaling regime, which couldbe trustfully interpreted as capturing the behavior correspond-ing to z = 3. This emergent picture seems to be consistentwith, and offer a potential explanation to results of quantumMonte Carlo simulations of certain models of the electronicnematic[41] as well as the itinerant ferromagnetic[42] quan-tum critical points which provide evidence for z = 2-type scal-ing behavior, rather than the conventionally anticipated behav-ior corresponding to z = 3.Conclusion and perspective Within the nonperturbative RGframework, we have derived a generalization of the Hertz ac-tion pertinent to fermionic quantum critical systems in d = 2characterized by the ordering wavevector Q⃗ = 0⃗. In our ap-proach both the fermionic and bosonic degrees of freedom areregularized and integrated out of the partition function in par-allel, which uncovers a new term in the corresponding order-parameter action and gives rise to a broad scaling regime char-acterized by the dynamical exponent z ≈ 2. Our results in-dicate that a consistent formulation of the momentum-shellWilsonian RG approach to this problem necessarily requirestreating the ordering wavevector as a flowing quantity, whichscales to zero exclusively in the infrared limit. The presentstudy addresses only properties pertinent to the order param-eter degrees of freedom and the structure of the bosonic ef-fective action. 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Sun, A. V. Chubukov,and Z. Y. Meng, Phys. Rev. B 105, L041111 (2022).Generalized Hertz action and quantum criticality of two-dimensional Fermi systemsMateusz Homenda∗ and Pawel Jakubczyk†Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, PolandHiroyuki Yamase‡Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0047, JapanSupplemental MaterialEVALUATION OF X(q, ΛF)In this section we present the calculation of X(q, ΛF) - see Eq.(10) in the main text. We start with an explicit formula:X(q, ΛF) = −2g2(2π)3∫ ∞−∞dk0∫d2k∂ΛR f (⃗k)(− ik0 + f (⃗k))2[1−i(k0 + q0) + f (⃗k + q⃗)+1−i(k0 − q0) + f (⃗k − q⃗)], (1)where we introduced the regularized fermionic dispersion f (⃗k) = ξk⃗ + R f (⃗k). The term ∂ΛR f restricts the region of integrationover momenta to the shell |⃗k| ∈ (kF − ΛF , kF + ΛF), such that:X(q, q0,ΛF) =2 i g2vF(2π)3{(1 +ΛFkF) ∫ ∞−∞dk0∫ kF+ΛFkFdk∫ 2π0dϕk(k0 + i ξkF+ΛF)2[1k0 + q0 + i f (⃗k + q⃗)+1k0 − q0 + i f (⃗k + q⃗)]+(2)−(1 −ΛFkF) ∫ ∞−∞dk0∫ kFkF−ΛFdk∫ 2π0dϕk(k0 + i ξkF−ΛF)2[1k0 + q0 + i f (⃗k + q⃗)+1k0 − q0 + i f (⃗k + q⃗)]}.We perform the frequency integration using the residue theorem:X(q, q0,ΛF) = −2g2vF(2π)2{(1 +ΛFkF) ∫ kF+ΛFkFdk∫ 2π0dϕ k[Θ[− f (⃗k + q⃗)](q0 + i ξkF+ΛF − i f (⃗k + q⃗))2+ (q0 → −q0)]+ (3)+(1 −ΛFkF) ∫ kFkF−ΛFdk∫ 2π0dϕ k[Θ[f (⃗k + q⃗)](q0 + i f (⃗k + q⃗) − i ξkF−ΛF)2+ (q0 → −q0)]}.The schematic plot illustrating the geometry of the momentum integration is presented in Fig. 1.Momentum space geometryThe geometry of the integration region depends on the relative ratio |q⃗|/ΛF , because the regularized fermionic dispersion f (⃗k)is constant for certain values of momenta. In Fig. 2 we present the three possible cases: (a) q > 2ΛF , (b) q ∈ (ΛF , 2ΛF), and(c) q < ΛF . From now on q B |q⃗|. Green color denotes the region for which the integrand in Eq.(3) becomes independent of k⃗and red color denotes the region, where the integrand depends on k⃗. We marked characteristic points of the geometric setup. We∗ mateusz.homenda@fuw.edu.pl † pawel.jakubczyk@fuw.edu.pl‡ yamase.hiroyuki@nims.go.jp2FIG. 1. Left:The gray shell around the Fermi surface marks the region of momenta integration restricted by the factor ∂ΛRF . The Fermi surfaceshifted by the vector −q⃗ is represented by green solid line. Right: the ultimate momentum integration region with nonzero contribution (seeEq.(3).FIG. 2. The regions of momentum space where the integrand in Eq. (3) is nonzero. The integrand is constant for the momentum vector k⃗which lies in the green region due to cut-off function which regularizes the fermi dispersion. For a momentum vector which lies in the redregion, the cut-off function becomes inactive. Black points define the geometry of the region.oriented the axis kx along q⃗. As we reduce q , the points ”1” and ”6” get closer to the kx axis and merge with analogous pointsfrom the lower half plane at value q = 2ΛF . The situation is analogous for intermediate momenta with the points ”2”, ”3”, ”4”and ”5”. In the last case, q < ΛF only the point ”0” survives.The points are described by the intersections of two circles of radius r ∈ {kF − ΛF , kF , kF + ΛF}. It is convenient to proceed inthe reference frame shifted by the vector −q⃗. We introduce a function u(ϕ, r, q), which describes the distance from the origin ofthe coordinate system to a point on the circle of radius r, centered at point (q, 0).u(ϕ, r, q) = q cos(q) +√r2 − q2 sin2(ϕ) . (4)Below we define the angles which describes the characteristic points in the new frame of reference. The angles are depicted inthe Fig. 3 and the corresponding expressions are:3FIG. 3. The angles ϕ1, ϕ2, ϕ3, ϕ4, ϕ5, ϕ6 -see the text and Eq.(5).4ϕ0 B arccos(q2kF)kF≫q,ΛF−→ ϕ0 ≈ arccos(π2)ϕ1 B arccos((kF + ΛF)2 − (kF − ΛF)2 + q22q(kF + ΛF))kF≫q,ΛF−→ ϕ1 ≈ arccos(2ΛFq)ϕ2 B arccos((kF + ΛF)2 − (kF)2 + q22q(kF + ΛF))kF≫q,ΛF−→ ϕ2 ≈ arccos(ΛFq)ϕ3 B arccos((kF)2 − (kF − ΛF)2 + q22q kF)kF≫q,ΛF−→ ϕ3 ≈ arccos(ΛFq)(5)ϕ4 B arccos((kF − ΛF)2 − (kF)2 + q22q(kF − ΛF))kF≫q,ΛF−→ ϕ4 ≈ arccos(−ΛFq)ϕ5 B arccos((kF)2 − (kF + ΛF)2 + q22q kF)kF≫q,ΛF−→ ϕ5 ≈ arccos(−ΛFq)ϕ6 B arccos((kF − ΛF)2 − (kF + ΛF)2 + q22q(kF − ΛF))kF≫q,ΛF−→ ϕ6 ≈ arccos(−2ΛFq).Small bosonic momenta: X< B X(|q⃗| < ΛF)In this section we evaluate the expression for X<. For small magnitudes of the bosonic momentum q < ΛF the integrals inEq. (3) simplify because kF − ΛF < |⃗k + q⃗| < kF + ΛF :X<(|q⃗|, q0,ΛF) = −2g2vF(2π)2{(1 +ΛFkF) ∫ kF+ΛFkFdk∫ 2π0dϕ k[Θ[kF − |⃗k + q⃗|](q0 + i ξkF+ΛF − i ξkF−ΛF)2 + (q0 → −q0)]+ (6)+(1 −ΛFkF) ∫ kFkF−ΛFdk∫ 2π0dϕ k[Θ[|⃗k + q⃗| − kF](q0 + i ξkF+ΛF − i ξkF−ΛF)2 + (q0 → −q0)]}.Further calculations can be done analytically. The integrals represent the area of the restricted region of integration (the greenregion in right plot in Fig. 2), because of no k⃗ dependence. We obtain:X<(|q⃗|, q0,ΛF) = −2g2vFπ2q20 − 4v2FΛ2F(q20 + 4v2FΛ2F)2(2k2F arcsin(q2kF) +q2√4k2F − q2). (7)For kF ≫ q and kF ≫ ΛF Eq.(7) simplifies to:X<(|q⃗|, q0,ΛF) ≈ −N< qq20 − 4v2FΛ2F(q20 + 4v2FΛ2F)2 , (8)whereN< B4g2vF kFπ2 . We now integrate Eq. (8) over the cutoff scale from the upper cutoff Λu to ΛF and get a contribution to thebosonic self energy:B<(q, q0,ΛF ,Λu) = −N< q ΛFq20 + 4v2FΛ2F−Λuq20 + 4v2FΛ2u Λu→∞−→ −N< q ΛFq20 + 4v2FΛ2F . (9)Intemediate bosonic momenta: XM B X(|q⃗| ∈ (ΛF , 2ΛF))Now we restrict to |q⃗| ∈ (ΛF , 2ΛF). We start from the Eq.(3). The axis kx is oriented along q⃗ and centered in the middle of theFermi surface. We introduce a new frame of reference: k′x = kx + qx, k′y = ky and use the angles defined in Eq.(5) and shown inFig. 3. We can write the expression for XM(|q⃗|, q0,ΛF) as follows:5XM(|q⃗|, q0,ΛF) = −g2vFπ2{(1 +ΛFkF) ∫ πϕ4dϕ∫ kF−ΛFu(ϕ,kF ,q)dk[k(q0 + i k2−(kF+ΛF )22mF)2+ (q0 → −q0)]++(1 −ΛFkF) ∫ ϕ20dϕ∫ u(ϕ,kF ,q)kF+ΛFdk[k(q0 + i k2−(kF−ΛF )22mF)2+ (q0 → −q0)]+ (10)+(1 +ΛFkF)2(q20 − 4v2FΛ2F)(q20 + 4v2FΛ2F)2(∫ ϕ4ϕ0dϕ∫ kFu(ϕ,kF ,q)dk k +∫ ϕ5ϕ4dϕ∫ kFkF−ΛFdk k +∫ πϕ5dϕ∫ u(ϕ,kF+ΛF ,q)kF−ΛFdk k)++(1 −ΛFkF)2(q20 − 4v2FΛ2F)(q20 + 4v2FΛ2F)2(∫ ϕ20dϕ∫ kF+ΛFu(ϕ,kF−ΛF ,q)dk k +∫ ϕ3ϕ2dϕ∫ u(ϕ,kF ,q)u(ϕ,kF−ΛF ,q)dk k +∫ ϕ0ϕ3dϕ∫ u(ϕ,kF ,q)kFdk k)}.Here u and the angles ϕi are defined in Eq.(4) and Eq.(5) respectively. For kF ≫ q, kF ≫ ΛF , the result simplifies as follows:XM(|q⃗|, q0,ΛF) = −g2vFπ2{q20 − 4v2FΛ2F(q20 + 4v2FΛ2F)2[4kFΛF(π − ϕ5 + ϕ2) + 4kFq(1 − sin(ϕ2) − sin(ϕ5))+ 2qΛF(sin(ϕ2) − sin(ϕ5))]+−∫ πϕ4dϕ2mFvF(ΛF − q cos(ϕ) + q22kFsin2(ϕ))q20 + v2F(ΛF − q cos(ϕ) + q22kFsin2(ϕ))2−∫ ϕ20dϕ2mFvF(ΛF + q cos(ϕ) − q22kFsin2(ϕ))q20 + v2F(ΛF + q cos(ϕ) − q22kFsin2(ϕ))2++(π − ϕ4 + ϕ2) 4mFvFΛFq20 + 4v2FΛ2F}. (11)In fact XM does not contribute to the Landau damping, as it is explained below.Large bosonic momenta regime: X> B X(|q⃗| > 2ΛF)In this section we address the topic of large bosonic momenta: |q⃗| > 2ΛF . Following the same procedure as above we obtain:X>(|q⃗|, q0,ΛF) = −g2vFπ2{(1 +ΛFkF)[ ∫ πϕ6dϕ∫ u(ϕ,kF+ΛF ,q)u(ϕ,kF ,q)dk k(q0 + i (kF+ΛF )2−k22mF)2+ (q0 → −q0)++∫ ϕ6ϕ4dϕ∫ kF−ΛFu(ϕ,kF ,q)dk k(q0 + i (kF+ΛF )2−k22mF)2+ (q0 → −q0)] ++(1 −ΛFkF)[ ∫ ϕ10dϕ∫ u(ϕ,kF ,q)u(ϕ,kF−ΛF ,q)dk k(q0 + i k2−(kF−ΛF )22mF)2+ (q0 → −q0)+ (12)+∫ ϕ2ϕ1dϕ∫ u(ϕ,kF ,q)kF+ΛFdk k(q0 + i k2−(kF−ΛF )22mF)2+ (q0 → −q0)] ++(1 +ΛFkF) 2(q20 − 4v2FΛ2F)(q20 + 4v2FΛ2F)2(∫ ϕ4ϕ0dϕ∫ kFu(ϕ,kF ,q)dk k +∫ ϕ5ϕ4dϕ∫ kFkF−ΛFdk k +∫ ϕ6ϕ5dϕ∫ u(ϕ,kF+ΛF ,q)kF−ΛFdk k)++(1 −ΛFkF) 2(q20 − 4v2FΛ2F)(q20 + 4v2FΛ2F)2(∫ ϕ2ϕ1dϕ∫ kF+ΛFu(ϕ,kF−ΛF ,q)dk k +∫ ϕ3ϕ2dϕ∫ u(ϕ,kF ,q)u(ϕ,kF−ΛF ,q)dk k +∫ ϕ0ϕ3dϕ∫ u(ϕ,kF ,q)kFdk k)}.For kF ≫ q, kF ≫ ΛF this reduces to:6X>(|q⃗|, q0,ΛF) = −N<2{q∫ πϕ6dϕcos(ϕ) − q2kFsin2(ϕ)q20 + v2Fq2( cos(ϕ) − q2kFsin2(ϕ))2 − q∫ πϕ5dϕcos(ϕ) − q2kFsin2(ϕ) − ΛFqq20 + v2Fq2( cos(ϕ) − q2kFsin2(ϕ) − ΛFq)2+− q∫ ϕ10dϕcos(ϕ) − q2kFsin2(ϕ)q20 + v2Fq2( cos(ϕ) − q2kFsin2(ϕ))2 + q∫ ϕ20dϕcos(ϕ) − q2kFsin2(ϕ) + ΛFqq20 + v2Fq2( cos(ϕ) − q2kFsin2(ϕ) + ΛFq)2++q20 − 4v2FΛ2F(q20 + 4v2FΛ2F)2[2ΛF(ϕ2 + ϕ6 − ϕ1 − ϕ5)− q(2 sin(ϕ2) + 2 sin(ϕ5) − sin(ϕ1) − sin(ϕ6) − 2 sin(ϕ0))]++2ΛF (ϕ1 + ϕ5 − ϕ2 − ϕ6)q20 + 4v2FΛ2F}. (13)Landau dampingThe aim of the present section is to explain which parts of X contribute to a non-analytic Landau term after integrating theflow down to ΛF = 0. The Landau damping term appears as a result of integrating out all fermions around the Fermi surface andexpansion of the bubble diagram around |q0 |q → 0. For this reason we focus on situation, where q ≫ ΛF . We find:B(q > 2ΛF , q0,ΛF) =∫ ΛF∞dΛ′X(|q⃗|, q0,Λ′) =∫ ΛFq/2dΛ′X>(|q⃗|, q0,Λ′) +∫ q/2qdΛ′XM(q, q0,Λ′) +∫ q∞dΛ′X<(|q⃗|, q0,Λ′) .(14)The last two integrals do not contribute to the Landau damping term, which is clear from equations and can also be easilychecked numerically. The entire Landau damping contribution is included in X>.We consider Eq. (13) for q ≫ ΛF , which yields:X>(|q⃗|, q0,ΛF) = −N<2{q∫ πϕ6dϕcos(ϕ) − q2kFsin2(ϕ)q20 + v2Fq2( cos(ϕ) − q2kFsin2(ϕ))2 − q∫ πϕ5dϕcos(ϕ) − q2kFsin2(ϕ) − ΛFqq20 + v2Fq2( cos(ϕ) − q2kFsin2(ϕ) − ΛFq)2+− q∫ ϕ10dϕcos(ϕ) − q2kFsin2(ϕ)q20 + v2Fq2( cos(ϕ) − q2kFsin2(ϕ))2 + q∫ ϕ20dϕcos(ϕ) − q2kFsin2(ϕ) + ΛFqq20 + v2Fq2( cos(ϕ) − q2kFsin2(ϕ) + ΛFq)2++ 2Λ2Fq 4v2FΛ2F − q20(q20 + 4v2FΛ2F)2+2q20 + 4v2FΛ2F} . (15)The first two lines only yield analytical contributions. The integration over Λ according to Eq.(14) can be applied to the last linein Eq. (15) leading to:B> ≈ −N<4v3FvFΛFq(3 +q20q20 + 4v2FΛ2F)−vF2(3 +q20q20 + v2Fq2)+ 2q0q(arctan(2vFΛFq0)− arctan(vFqq0)) . (16)After taking ΛF → 0 and q0/q→ 0 one obtains the Landau damping term with the correct coefficient. This conclusion was alsochecked numerically.FLOW EQUATIONSBelow we present the flow equations for the scale-dependent order parameter ρΛ0 B12 (ϕΛ0 )2 and the bosonic self-interactionvertex constant uΛ.7∂ΛρΛ0 =32(2π)3∫ ∞−∞dq0∫d2q∂ΛRb(m2Λ+ Z(|q⃗| − ΛF(Λ))2+ Aq20 + B(q⃗, q0,ΛF(Λ))+ Rb(q⃗))2 (17)∂ΛuΛ =9(2π)3∫ ∞−∞dq0∫d2q∂ΛRb(m2Λ+ Z(|q⃗| − ΛF(Λ))2+ Aq20 + B(q⃗, q0,ΛF(Λ))+ Rb(q⃗))3 . (18)For Λ0 = 0 we chose the Litim cut-off function as a regulator:Rb(q⃗) = Z(Λ2 − (|q⃗| − ΛF(Λ))2)Θ[Λ2 − (|q⃗| − ΛF(Λ))2], (19)while for Λ0 > 0 we used:Rb(q⃗) = Z(Λ2 − |q⃗|2)Θ[Λ2 − |q⃗|2]. (20)