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

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[Quantum criticality and non-Fermi liquids: The Wilsonian renormalization group perspective](https://mdr.nims.go.jp/datasets/1e44bb2b-b258-4a60-a7e6-e62a6e6aa378)

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Quantum criticality and non-Fermi liquids:the Wilsonian renormalization group perspectiveMateusz Homenda,1, ∗ Pawel Jakubczyk,1, † and Hiroyuki Yamase2, ‡1Institute of Theoretical Physics, Faculty of Physics,University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland2Research Center for Materials Nanoarchitectonics,National Institute for Materials Science, Tsukuba 305-0047, Japan(Dated: August 8, 2026)We develop a theoretical framework based on the nonperturvative renormalization group (RG) inthe one-particle irreducible (Wetterich) formulation to tackle the interplay of coupled fermionic andorder-parameter fluctuations at metallic quantum critical points (QCPs) with ordering wavevector®𝑄 = ®0. We consistently treat the dynamical emergence of the Landau damping of the bosonic modeand non-Fermi liquid scaling of fermions upon lowering the cutoff scale. The loop integrals of thepresent theory involve only contributions from fluctuations above the cutoff scale, which protects thesystem from developing singular bosonic interactions. We emphasize the importance of the nontrivialrelative scaling of the bosonic and fermionic cutoffs Λ and Λ 𝑓 , which we fix by analyzing the RGflow of the scale-dependent ordering wave-vector ®𝑄Λ. Upon neglecting Fermi self-energy in the loopintegrals of the functional RG flow, we identify a non-Fermi liquid RG fixed point and recover thefeatures obtained earlier within RPA-type approaches. In a subsequent step, we self-consistentlyinclude the scaling of the self-energy and the Yukawa coupling. We find a generic instability of thenon-Fermi liquid RG fixed point. This implies, at least at this truncation level, absence of the QCPwith ®𝑄 = ®0 and development of a first-order phase transition or a phase characterized by ®𝑄 ≠ ®0.I. INTRODUCTIONQuantum criticality of metallic systems and the asso-ciated breakdown of Landau’s Fermi liquid theory havebeen receiving enormous interest over the last decadesdue to their relevance to high-𝑇𝑐 cuprate superconduc-tors [1] and other strongly-correlated materials.The earliest theoretical investigations of these phenom-ena [2–4] (referred to as Hertz-Millis theory) departedfrom the paradigms of Wilsonian momentum-shell renor-malization group applied to an effective bosonic action.The latter was obtained by completely integrating outthe fermionic degrees of freedom out of the theory, giv-ing rise to a bosonic self-energy involving the so-calledLandau-damping term, which describes damping of thepropagating order-parameter mode due to scattering offthe particle-hole excitations. Later studies revealed inad-equacies of the Hertz-Millis approach in two-dimensionalsystems. The reason for these is understood by notingthat, in addition to the order-parameter mode, the sys-tem hosts electronic excitations across the Fermi level,which are also soft. Integrating out some of the masslessmodes and leaving the others in the effective theory isunjustified from the point of view of Wilsonian RG the-ory. Several studies spread over years [5–11] note thatthe procedure yields singular effective bosonic interac-tions, which cannot be handled within the Hertz-Millisframework.∗ mateusz.homenda@fuw.edu.pl† pawel.jakubczyk@fuw.edu.pl‡ yamase.hiroyuki@nims.go.jpTheoretical developments which followed, led overyears to the consensus that a satisfactory effective theoryof metallic quantum criticality should explicitly keep thelow-energy fermionic degrees of freedom. Several sem-inal papers pursued this path [10, 12–26], implement-ing the tools of field-theoretic RG, a variety of differentregularizations, and taking advantage of affinity to anearlier studied problem of fermions coupled to a 𝑈 (1)gauge field. A common feature of these approaches isthat the bosonic propagator becomes dressed in the sin-gular Landau-damping term before performing any loopintegrals of the coupled Bose-Fermi theory. The approachyields a picture with some surprising features: the stan-dard 1/𝑁 expansion (𝑁 being the number of Fermi fla-vors) is questionable [12] since diagrams of any loop or-der contribute also in the limit 𝑁 → ∞; the one-loopresult for the dynamical exponent of fermions acquires atiny correction only at three-loop level, while the corre-sponding result for the bosonic dynamical exponent 𝑧𝑏 isrobust at two- and three-loop order. However, an unex-pected thing happens at four-loop [18, 27, 28], where anonrenormalizable divergence is found, which, as far aswe are aware, was not clearly interpreted up to now.Studies of coupled Fermi-Bose theories of metallicquantum criticality from the Wilsonian RG perspectiveare more scarce [29–33]. A key question in this ap-proach concerns the way Landau damping should emergein course of the RG flow. As emphasized in Ref. [30], itsstandard form cannot be continuously generated at fi-nite scales upon reducing the cutoff around the Fermisurface, since (as visible even from evaluating the bareFermi loop), the Landau damping term arises in a singu-lar way exclusively from fermions directly at the Fermilevel. From the Wilsonian RG point of view, therefore,2it is illogical to use the standard Landau-damped Bosepropagator in RG theory of coupled Bose-Fermi systemsbecause it appears only after the fermions become com-pletely removed from the theory. On the other hand, ob-servable physical quantities depend on the precise formof the Bose propagator. A notable case that permitsan unbiased Wilsonian RG analysis without addressingthe Landau damping is the chiral non-Fermi liquid [34],where exact critical exponents can be deduced.An approach to this generic problem pertinent to in-stabilities at ordering wavevector ®𝑄 = ®0 was recently pro-posed by us [35] implementing the modern formulation ofWilsonian nonperturbative RG relying on the Wetterichequation [36, 37], applied to a system of coupled fermionicand order-parameter fluctuations. This leads to a gener-alization of the Hertz action, which recovers the standardform only in an undesirable procedure of integrating outfermions before the bosons. On the other hand, it encom-passes the same physics and yields a consistent picture,free of singular effective interactions, where fermions andbosons are treated on equal footing and integrated outin parallel. Ref. [35] concentrated on bosonic propertiesand the emergent order-parameter action neglecting theFermi self-energy terms. The present paper constitutesa continuation of that avenue, accounting, inter alia, forthose effects. We additionally elaborate on the relationbetween the two cutoff scales present in our approachand clarify how, at RPA-type level, it leads to the dy-namical exponent 𝑧𝑏 = 3. This picture however breaksdown, when both the self-energy effects and the flow ofthe Yukawa vertex are taken into account.Throughout the present paper we restrict to the caseof instabilities occurring at zero ordering wavevector ®𝑄.The primary one-loop prediction concerning the bosonicproperties in this case is the mean-field nature of thecritical exponents at 𝑇 = 0. This can be understood[3, 4, 11, 38] by considering that integrating fermionsout generates a bosonic action in effective dimensionality𝑑eff := 𝑑 + 𝑧𝑏, where 𝑑 is the spatial dimension and 𝑧𝑏the dynamical critical exponent that characterizes scal-ing of correlation time with correlation length, 𝜏 ∼ 𝜉𝑧𝑏 .According to the one-loop result, the low-energy order-parameter dynamics is governed by the Landau dampingterm ∼ |𝑞0 |/| ®𝑞 | in the propagator, where 𝑞0 and ®𝑞 de-note bosonic Matsubara frequency and momentum, re-spectively. By comparing the Landau damping term withthe standard quadratic term | ®𝑞 |2 in the boson propagator,one obtains 𝑧𝑏 = 3. In consequence, for two-dimensionalsystems 𝑑eff = 5, which exceeds the upper critical dimen-sion 𝑑up = 4. These are the main one-loop characteristicsof the bosonic sector at 𝑇 = 0. In reference to fermions,we will use the phrases ”one-loop” and ”RPA-type” inter-changeably. Concerning the fermionic properties, we in-voke the broadly known one-loop result [11] stating thatthe Fermi propagator 𝐺 𝑓 for small frequencies 𝑘0 behavesas:𝐺−1𝑓 (𝑘0, | ®𝑘 | = 𝑘 𝑓 ) ∼ −𝑖 sgn(𝑘0) |𝑘0 |𝛼 , (1)where ®𝑘 denotes the fermionic wavevector and 𝑘 𝑓 is theFermi momentum. At one loop level one obtains 𝛼 = 2/3,which implies non-Fermi liquid behavior.We emphasize that the precise values of 𝑧𝑏 and 𝛼 in𝑑 = 2 have never been well established in literature be-yond the robust one-loop result. The situation is in-triguing considering that perturbation theory correctsthe value 𝛼 = 2/3 only at three-loop level [10], while,as already mentioned, a four-loop calculation leads tocontributions to 𝑧𝑏 that have no clear interpretation [18]and rather point towards foundational problems of theapplied approach. The picture provided by the quan-tum Monte Carlo simulations is also far from definitivewith regard to numbers. The study of peculiar electronic-nematic systems shows 𝑧𝑏 ≈ 2 scaling [39] while simula-tions of Ref. [40] yield a bosonic susceptibility inconsis-tent with the Hertz form ∼ |𝑞0 |/| ®𝑞 |. These deviationswere however later attributed to finite size [41] and ther-mal effects [42, 43]. Interestingly, simulations of the 𝑋𝑌model of itinerant ferromagnetism provide evidence for𝑧𝑏 = 2 and 𝛼 = 1/2 [44]. The latter result is also con-sistent with a recent renormalization group analysis ofisotropic Fermi gas interacting with U(1) gauge field,which is expected to belong to the same universality class[33]. Notably, at Hertz-Millis level, these values of 𝑧𝑏and 𝛼 are characteristic to instabilities featuring order-ing wavevector ®𝑄 ≠ ®0. For more recent insights into thisproblem see [45].In the present work we develop an approach based on astarting point, in which the propagators are only gradu-ally dressed upon decreasing the RG cutoff scale and theproblems encountered in the previous studies are absent.The outline of the paper is the following: In Sec. II weintroduce the model and discuss the applied functionalRG approach. In Sec. III we analyze non-self-consistenttruncations of the Wetterich equation leading to a gen-eralization of the Hertz action, flow of the self-energy toa non-Fermi liquid state, and the associated RG fixedpoint. We also demonstrate how the standard Hertz ac-tion is recovered by a dubious procedure of scaling thecutoff on fermionic degrees of freedom to zero much fasterthan the one applied on the bosonic ones. In Sec. IV weself-consistently include the flow of the self-energy andthe Yukawa vertex. We demonstrate non-existence ofthe non-Fermi liquid RG fixed point at the consideredtruncation level. In Sec. IV.C we speculate on the possi-bility of reemergence of the second-order quantum phasetransition with ®𝑄 = ®0 if additionally four-point functionsinvolving both fermions and bosons were taken into ac-count. We summarize the paper in Sec. V, and technicaldetails are presented in the Appendices.II. MODEL AND FUNCTIONALRENORMALIZATION GROUPWe analyze the coupled flows of fermionic and order-parameter fluctuations employing a modern version of3the Wilsonian RG based on the Wetterich equation[36, 37]. The latter is an exact functional flow equationfor the one-particle-irreducible vertex functions equippedwith a (momentum) cutoff at scale Λ. This nonper-turbative framework has, in recent years, allowed forgroundbreaking progress in addressing several problemshardly accessible to traditional approaches based on per-turbation theory, and sometimes completely beyond theirreach. Notable achievements of this method within thefield of condensed matter theory include identifying thestrong-coupling fixed points for the Kardar-Parisi-Zhangproblem in 𝑑 ≥ 1 [46, 47], resolution of the long-standingproblem of dimensional reduction and its breaking in dis-ordered systems [48, 49], finding entirely new multicrit-ical RG fixed points for the 𝑂 (𝑁) models in 𝑑 = 3 [50],and invalidation [51] of the predictions of perturbativeapproaches concerning non-analyticity of the critical ex-ponents as function of 𝑑 and 𝑁. We refer to [52, 53]for reviews. In view of the present problem, the impor-tant characteristic of the functional Wetterich approachis that it allows for efficient analysis of coupled flows ofinteracting degrees of freedom without specifying anypreimposed form of the flowing quantities and consis-tently arranging all of the fluctuations contributing tothe partition function according to the decreasing energyscale.A. The bare action and the flowing actionAs our starting point we consider a two-dimensionalsystem of fermions, described by Grassmann fields {𝜓, 𝜓},coupled to the real scalar bosonic field 𝜙 by a Yukawa-type interaction. The bare action 𝑆[𝜓, 𝜓, 𝜙] is defined ata microscopic scale Λ𝑈𝑉 and involves three terms [11]:𝑆[𝜓, 𝜓, 𝜙] = 𝑆𝑏 + 𝑆 𝑓 + 𝑆 𝑓 𝑏𝑆 𝑓 =∑︁𝜎∫𝐾𝜓𝐾,𝜎(−𝑖𝑘0 + 𝜉 ®𝑘)𝜓𝐾,𝜎𝑆𝑏 =∫𝑄(𝑚2𝑏 + | ®𝑞 |2 + 𝑞20)𝜙𝑄𝜙−𝑄 + 𝑢∫𝑥,𝑡(𝜙(𝑥, 𝑡))4𝑆 𝑓 𝑏 = 𝑔∑︁𝜎∫𝑄,𝐾𝜙𝑄𝜓𝐾,𝜎𝜓𝐾−𝑄,𝜎 ,(2)where 𝑚2𝑏, 𝑢, and 𝑔 > 0 are the bare parameters of thesystem. Here 𝐾 = {𝑘0, ®𝑘} encompasses fermionic fre-quency and momentum, while 𝑄 = {𝑞0, ®𝑞} stands fortheir bosonic counterparts. We also define an abbrevi-ation for integrals,∫𝐾:= ( 12𝜋 )3∫d𝑘0∫d2𝑘. Finally weconsider the spherical Fermi surface 𝜉 ®𝑘 =12𝑚 𝑓( | ®𝑘 |2 − 𝑘2𝑓).The bosonic part 𝑆𝑏 taken alone describes the phasetransition of the 2𝑑 quantum Ising type, which is gov-erned by the RG fixed point of the classical 3𝑑 Isinguniversality class. In the Hertz-Millis framework (emerg-ing after integrating out the fermionic fields {𝜓, 𝜓}) thissituation is different due to appearance of the Landaudamping term ∼ |𝑞0 |/| ®𝑞 |, which effectively describes in-teractions of the bosonic mode with particle-hole exci-tations. The 𝑇 = 0 critical behavior of such a dampedboson field is controlled by the Gaussian fixed point.In order to consistently formulate the theory andin particular describe the gradual generation of Lan-dau damping in course of the flow, we add the scale-dependent cutoff functions 𝑅𝑏 (𝑄,Λ) and 𝑅 𝑓 (𝐾,Λ 𝑓 ) tothe bosonic and fermionic inverse propagators, which de-forms the quadratic part of the action according to:𝑆 𝑓 ,Λ 𝑓 = 𝑆 𝑓 +∑𝜎∫𝐾𝑅 𝑓 𝜓𝐾,𝜎𝜓𝐾,𝜎 (3)𝑆𝑏,Λ = 𝑆𝑏 +∫𝑄𝑅𝑏 𝜙𝑄𝜙−𝑄 . (4)Here Λ is the IR cut-off scale for bosonic momenta andΛ 𝑓 (Λ) describes the fermionic cutoff scale, i.e., the dis-tance to the Fermi surface up to which the fermionicdispersion is modified. The relation between Λ 𝑓 and Λwill be discussed later. The introduction of cut-off func-tions regularizes all infrared divergencies and at the sametime allows for the derivation of the Wetterich equation,which governs the flow of the effective action ΓΛ [𝜓, 𝜓, 𝜙]upon varying the cutoff scale — see the Appendix A.The functional ΓΛ [𝜓, 𝜓, 𝜙] interpolates between the bareaction 𝑆[𝜓, 𝜓, 𝜙], recovered in the limit Λ → Λ𝑈𝑉 , andthe full effective action (i.e. the Gibbs free energy) forΛ → 0. Taking consecutive derivatives of the Wetterichequation with respect to fields yields exact flow equationsfor the (1-particle irreducible) correlation functionsΓ(𝑛)𝛼1...𝛼𝑛:=𝛿𝑛ΓΛ [𝜓, 𝜓, 𝜙]𝛿𝛼1...𝛿𝛼𝑛, 𝛼𝑖 ∈ {𝜙𝑄𝑖 , 𝜓𝐾𝑖 ,𝜎𝑖 , 𝜓𝐾𝑖 ,𝜎𝑖 } .(5)In this work, we focus on the flow of fermionic andbosonic self-energies, Σ(𝐾) and Π(𝑄), directly related toΓ(2)𝛼1 ,𝛼2; the corresponding flow equations are given in theAppendix A — see Eq. (A12) and Eq. (A13), and arerepresented by the Feynman diagrams depicted in Fig. 1.For future reference we also introduce the regularized,scale-dependent propagators𝐺𝑏 (𝑄) :=[Γ(2)𝜙𝜙(𝑄) + 𝑅𝑏 (𝑄)]−1(6)=1𝑚2𝑏+ | ®𝑞 |2 + 𝑞20 + Π(𝑄) + 𝑅𝑏 (𝑄),𝐺 𝑓 (𝐾) :=[Γ(2)𝜓𝜓(𝐾, 𝜎) + 𝑅 𝑓 (𝐾, 𝜎)]−1(7)=1−𝑖𝑘0 + 𝜉 ®𝑘 + Σ(𝐾) + 𝑅 𝑓 (𝐾),as well as the single-scale propagatorsS𝑖 := −(𝜕Λ𝑖𝑅𝑖) (𝐺𝑖)2 , 𝑖 ∈ {𝑏, 𝑓 } . (8)Above we suppressed the scale-dependencies for clarity.Our notation is fully consistent with the one of Ref. [35](see also [54, 55]). For a thorough discussion of the Wet-terich framework we refer to [52, 53, 56, 57].4FIG. 1. Feynman diagrams representing the flow of theself-energies. The wiggly (straight) line denotes the boson(fermion) propagator 𝐺𝑏/ 𝑓 dressed by fluctuations down tothe cutoff scale Λ (or Λ 𝑓 ). The strikethrogh indicates thesingle scale propagator S𝑏/ 𝑓 . Contributions involving 4-pointfunctions with fermionic legs were omitted.III. GENERALIZED HERTZ ACTIONAND THE NON-FERMI LIQUID FIXED POINTIn perturbation theory the standard Landau dampingis generated at one-loop level, and does not require ac-counting for self-energy contributions. In what follows weexplain how that result is recovered in our present frame-work by a questionable procedure of integrating fermionicdegrees of freedom via the flow much faster than thebosonic ones and how improvement can be achieved inthe present methodology. We subsequently present a de-tailed discussion concerning the structure of the regu-larized bubble diagram (second line in Fig. 1) and therelation between the scales Λ and Λ 𝑓 , which, in our ap-proach, is fixed by the magnitude of the flowing orderingwave-vector ®𝑄Λ. This characteristic is crucial for valuesof scaling exponents.Having in mind the goal of recovering the standardone-loop theory from the Wetterich flow and setting upa starting point for more elaborate analysis, we proposea truncation of the Wetterich equation, which is summa-rized as follows:1. We neglect the Fermi self-energy in the loop integralsof the flow equations.2. We disregard the flow of the Yukawa coupling andthe higher order interactions generated by the RG flowexcept for the bosonic self-interaction 𝑢, which is treatedas a flowing local coupling.Truncations reaching beyond this level will be dis-cussed in Sec. IV. Note that there is no preimposedparametrization of the momentum and frequency depen-dencies of the flowing propagators. As we discuss below,the truncation allows for making contact with the Hertz-Millis framework at any time by scaling Λ 𝑓 to zero beforeΛ. Presence of the fermionic cutoff prevents generationof singular bosonic interactions. With the above spec-ified truncation, we obtain the following flow equationsfor the self-energies:𝜕ΛΠ(𝑄′) = −∫𝑄S𝑏 (𝑄)Γ (3)𝜙𝜙𝜙𝐺𝑏 (𝑄 +𝑄′)Γ (3)𝜙𝜙𝜙+ 12∫𝑄S𝑏 (𝑄)Γ (4)𝜙𝜙𝜙𝜙+ 2𝑔2 (𝜕ΛΛ 𝑓 )∫𝐾S 𝑓 ,0(𝐾)𝐺 𝑓 ,0(𝐾 +𝑄′)+ 2𝑔2 (𝜕ΛΛ 𝑓 )∫𝐾S 𝑓 ,0(𝐾)𝐺 𝑓 ,0(𝐾 −𝑄′) ,(9)𝜕ΛΣ(𝐾 ′) = − 𝑔2∫𝑄S𝑏 (𝑄)𝐺 𝑓 ,0(𝐾 ′ −𝑄)− 𝑔2 (𝜕ΛΛ 𝑓 )∫𝑄S 𝑓 ,0(𝐾 ′ −𝑄)𝐺𝑏 (𝑄) ,(10)where𝐺 𝑓 ,0(𝐾) :=1−𝑖𝑘0 + 𝜉 ®𝑘 + 𝑅 𝑓 (𝐾), S 𝑓 ,0 := −(𝜕Λ 𝑓 𝑅 𝑓 ) (𝐺 𝑓 ,0)2 .(11)Throughout the paper we concentrate on the quantumcritical state, which is achieved by tuning the bare bo-son mass such that it scales to zero in the limit Λ → 0.The flow equation for the mass 𝑚2𝑏= Π(𝑄 = 0) includescontributions from the fermion loop, boson bubble, andboson tadpole. In fact, we checked that disregarding thepurely bosonic contributions does not affect our results.The present framework is different from the approachesrelying on perturbation theory. In the latter, theone-loop boson self-energy is computed using the barefermion propagator, leading to the |𝑞0 |/| ®𝑞 | Landau damp-ing term. This is subsequently inserted into the one-loopfermion self-energy diagram, which yields the |𝑘0 |2/3 con-tribution. The scheme implemented here is entirely dif-ferent: both the propagators are equipped with their cut-offs and the corresponding scales are sent to zero ”in par-allel”. The perturbative procedure can be mimicked bysending the cutoff on fermions (Λ 𝑓 ) to zero first, whichfully dresses the Bose propagator and therefore shouldrecover the standard Landau damping via the last twoterms in Eq. (9). If the limit Λ → 0 is considered onlysubsequently, one anticipates to recover the Hertz-Millistype flow of bosons from the first two terms in Eq. (9)and also obtain the non-Fermi liquid from Eq. (10). Aswe discuss in Sec. III.A.1, this is indeed what happens.The point is that taking the limit Λ 𝑓 → 0 first leadsto the known problem of singular Bose interactions andshould therefore be avoided.A. Regularized fermionic bubbbleWe begin with a discussion of the fermionic contribu-tion to the bosonic self-energy flow as represented by the5last two terms of Eq. (9), the corresponding expressionis defined asX0 (𝑄,Λ 𝑓 ) := 2𝑔2∫𝐾S 𝑓 ,0(𝐾)(𝐺 𝑓 ,0(𝐾 +𝑄) + 𝐺 𝑓 ,0(𝐾 −𝑄)).(12)An analogous expression, retaining dressed propagatorsinstead of 𝐺 𝑓 ,0 and S 𝑓 ,0 will be denoted as X(𝑄,Λ 𝑓 ). Apractical calculation requires specifying the form of thefermionic cut-off function, which is taken as [35]:𝑅 𝑓 ( ®𝑘) ={(𝜉𝑘 𝑓 +Λ 𝑓 − 𝜉 ®𝑘)𝜃[Λ 𝑓 − (| ®𝑘 | − 𝑘 𝑓 )]for | ®𝑘 | ≥ 𝑘 𝑓(𝜉𝑘 𝑓 −Λ 𝑓 − 𝜉 ®𝑘)𝜃[Λ 𝑓 − (𝑘 𝑓 − |®𝑘 |)]for | ®𝑘 | < 𝑘 𝑓 .(13)Evaluation of the integrals involved in Eq. (12) is pre-sented in Appendix B. We also introduce𝐵(𝑄,Λ 𝑓):=∫ ΛΛ𝑈𝑉𝑑Λ′ (𝜕Λ′Λ 𝑓 )X(𝑄,Λ 𝑓 (Λ′)) , (14)which represents the contribution to the bosonic self-energy from integrating fermions down to the scale Λ.Note again the presence of two cutoff scales Λ and Λ 𝑓 .The former plays the role of the bosonic cutoff and theflow parameter at the same time. The latter defines thesupport of the fermionic cutoff function [see Eq. (13)].By changing the integration variable we get rid of ex-plicit dependence on Λ. At the present truncation levelwe obtain:𝐵0(𝑄,Λ 𝑓):=∫ Λ 𝑓Λ𝑈𝑉, 𝑓𝑑Λ′ X0 (𝑄,Λ′) . (15)The precise form of the relation Λ 𝑓 (Λ) is not specified atthis point and will be discussed in the following sections,while an improved truncation involving fermionic self-energy feedback is presented in section IV. The essentialimprovement over the Hertz-Millis theory, proposed inRef. [35] amounts to using𝐺𝑏 (𝑄,Λ 𝑓 ) :=1𝑚2𝑏+ | ®𝑞 |2 + 𝑞20 + 𝐵0 (𝑄,Λ 𝑓 )(16)for the bosonic propagator instead of the fully-dressedHertz-Millis form. We refer to the bosonic action involv-ing contributions from fermions described by 𝐵0 (𝑄,Λ 𝑓 )as the generalized Hertz action [35].For every Λ > 0 the function 𝐺−1𝑏exhibits a mini-mum at 𝑞0 = 0 and | ®𝑞 | = 𝑄Λ > 0, such that the order-ing wavevector remains finite and scales to zero only forΛ 𝑓 → 0. A sequence of plots of 𝐺−1𝑏for different valuesof Λ 𝑓 is presented in Fig. 2.We also observe that X0 (𝑄,Λ 𝑓 ) is identically zero for| ®𝑞 | = 0 such that no mass flow from fermions arises if oneattempts to artificially constrain the ordering wavevec-tor to zero, which is an unwelcome property encounteredand discussed in Ref. [30]. This feature is absent in ourpresent framework, where 𝑄Λ is a flowing quantity.Λf=0.5Λf=0.25Λf=0.125Λf=0.0625Λf=0.03125Λf=0.015625Λf=0.00781250.1 0.2 0.3 0.4 0.5 0.6 0.7q-1.-2.-3.14Gb-1(q0= 0, q, Λf )FIG. 2. The inverse propagator 𝐺−1𝑏given by Eq. (16), plot-ted as a function of | ®𝑞 | for 𝑞0 = 0 and a sequence of valuesof Λ 𝑓 . The minimum defining the ordering wavevector 𝑄Λapproaches zero for Λ 𝑓 → 0. The convergence of 𝐺−1𝑏is non-uniform in this limit. The dashed line marks the ultimatebosonic mass shift limΛ→0 Δ𝑚2𝑏= −𝜋A = −𝑔2𝑘 𝑓 /(𝜋𝑣 𝑓 ). Inthis calculation we set the initial bosonic mass 𝑚2𝑏= 0 andA = 1.1. Recovering the Hertz-Millis theoryAs demonstrated in Ref. [35], the Hertz-Millis frame-work can be recovered from our present theory in a ques-tionable procedure of sending Λ 𝑓 to zero much faster thanΛ. This can be realized for example by imposing the fol-lowing relation Λ 𝑓 (Λ):Λ 𝑓 = (Λ − Λ0)𝜃 (Λ − Λ0) (17)with Λ0 > 0, such that Λ 𝑓 becomes zero for a positivevalue of Λ. The standard Landau damping term is thengenerated from a contribution to the asymptotic form of𝐵0 (𝑄,Λ 𝑓 ) valid for | ®𝑞 | ≫ Λ 𝑓 , which has the form:𝐵> := A 𝑞0𝑣 𝑓 | ®𝑞 |[arctan2𝑣 𝑓 | ®𝑞 |𝑞0− arctan2𝑣 𝑓Λ 𝑓𝑞0], (18)where A = 𝑔2𝑘 𝑓 /(𝜋2𝑣 𝑓 ), for the derivation see the Sup-plemental material to Ref. [35] or Appendix C in thepresent manuscript. By taking Λ 𝑓 → 0 and consider-ing the static limit |𝑞0 | ≪ 𝑣 𝑓 | ®𝑞 |, ignoring the generatedsingular bosonic interactions, one recovers a generaliza-tion [58, 59] of the Hertz-Millis framework, in which theRG flow is governed exclusively by bosonic fluctuations.The parameter Λ0 may then be interpreted as the uppercutoff of the bosonic effective theory.It is clear that the above procedure relying on Eq. (17)and leading to the Landau-damping term is theoreti-cally unsatisfactory. In fact, the structure of the coupledFermi-Bose theory developed below is entirely distinct, inparticular it involves a different form of the Bose propaga-tor and contributions to the loop integrals from momentawith | ®𝑞 | < Λ 𝑓 , where the Hertz-Millis parametrization isinvalid.6FIG. 3. Plot of the function ℎ(𝑡) and the function ℎ𝑘2 (𝑡) (forthe latter see Sec. IVB and Appendix B 2 ).2. Flowing ordering wavevectorWe now analyze the structure of the regularizedFermi bubble for 𝑞0 = 0. In the scaling limit, where𝑚𝑎𝑥(Λ,Λ 𝑓 , 𝑞) ≪ 𝑘 𝑓 , we obtain𝐵0 (0, 𝑞,Λ 𝑓 ) = A∫ Λ 𝑓Λ 𝑓 ,𝑈𝑉d𝜆ℎ(𝜆/𝑞)𝑞= A∫ Λ 𝑓 /𝑞Λ 𝑓 ,𝑈𝑉/𝑞d𝑡ℎ(𝑡) .(19)Here we denote | ®𝑞 | = 𝑞 and the function ℎ(𝑡) is given inthe Appendix B.2 and plotted in Fig. 3 . We concentrateon the scaling regime, where Λ 𝑓 ,𝑈𝑉 is sent to infinity.We identify the scale-dependent ordering wavevector𝑄Λ by searching for minima of 𝐵0 (0, 𝑞,Λ 𝑓 ) + 𝑞2 with re-spect to 𝑞, which leads to2𝑄Λ − AΛ 𝑓𝑄2Λℎ(Λ 𝑓𝑄Λ) = 0 . (20)Eq. (20) can be analytically solved in the asymptoticregimes of Λ 𝑓 ≫ 𝑄Λ and Λ 𝑓 ≪ 𝑄Λ, which yields𝑄Λ ≈ A2Λ 𝑓, and 𝑄Λ ≈(3A2Λ 𝑓)1/3(21)respectively. We are however presently interested onlyin the limit where both Λ 𝑓 and 𝑄Λ vanish, which corre-sponds to the latter of the two above solutions. Compar-ison of the obtained asymptotic expressions with directnumerical evaluation of 𝑄Λ is given in Fig. 4.We point out that resolving the full momentum profileof the function 𝐵0 (0, 𝑞,Λ 𝑓 ) is necessary for obtaining thecorrect scaling of 𝑄Λ. Neglecting analytical terms in X0as was proposed in Ref. [35] leads to 𝑄Λ ∼ Λ 𝑓 instead of𝑄Λ ∼ Λ1/3𝑓. The obtained result for 𝑄Λ implies that 𝜖Λ :=Λ 𝑓 /𝑄Λ may be considered as a small parameter. Thisoffers an avenue for simplifications relying on expansionof 𝐵0 in 𝜖Λ, as an alternative to the analysis of Ref. [35]which used the full form 𝐵0 (𝑄,Λ 𝑓 ).We proceed by evaluating the contribution to the flowof the boson mass from fermions. We obtain:𝐵0 (0, 𝑄Λ,Λ 𝑓 ) ≈ Δ𝑚2𝑏,0 + 2𝑄2Λ , (22)FIG. 4. Comparison of 𝑄Λ obtained numerically from Eq. (20)(black points) and the asymptotic expressions of Eq. (21)(solid lines). The numbers indicate slope fits to the numericaldata. We set A = 1.valid for 𝑄Λ ≫ Λ 𝑓 , see Appendix C. The constant massshift is given as Δ𝑚2𝑏,0= −𝜋A. The present result sug-gests that at RG fixed point Λ 𝑓 ∼ Λ3, which is requiredto obtain the same scaling of the two terms in Eq. (22)and the conventional scaling 𝑚2𝑏∼ Λ2 of the boson mass.We elaborate further on this point below.3. Frequency dependence and Landau dampingWe are now ready to explore the frequency dependenceof the flowing fermion bubble diagram. One of our aims isto extract the exponent 𝑧𝑏 determined by the character-istic frequency 𝑞0 ∼ Λ𝑧𝑏 . We evaluate 𝐵0 at the flowingordering wavevector 𝑄Λ. The corresponding integrandcan, in analogy to the static case 𝑞0 = 0, be expressed asX0 (𝑞0, 𝑞,Λ 𝑓 ) =A𝑞ℎ(Λ 𝑓𝑞,1𝑣 𝑓𝑞0𝑞)(23)— see Appendix B. We introduce𝑤 =𝑞0𝑣 𝑓𝑄Λ(24)and obtain:𝐵0 (𝑞0, 𝑄Λ,Λ 𝑓 ) = A∫ 12∞d𝑡 ℎ(𝑡, 𝑤)+∫ Λ 𝑓𝑄Λ12d𝑡 ℎ(𝑡, 𝑤) .(25)The ℎ(𝑡) function used previously corresponds to thestatic limit ℎ(𝑡, 𝑤 = 0). The first term is expandablein 𝑤 and yields, for 𝑤 ≪ 1, the leading contribution oforder 𝑤2. Terms of a different type arise from the vicinityof the upper integration limit in the second term, whichis of order 𝜖Λ (see Appendix C for details). Note that,by virtue of the result of the previous section, we have7FIG. 5. Dynamical part of the function 𝐵0 (𝑞0, 𝑞,Λ 𝑓 ). Lin-ear scaling is clearly visible in the frequency range between2𝑣 𝑓Λ 𝑓 and 𝑣 𝑓𝑄Λ (see the main text). Quadratic dependenceon frequency sets in for 𝑞0 < 2𝑣 𝑓Λ 𝑓 . The inset shows thesame data in linear scale.𝜖Λ = Λ 𝑓 /𝑄Λ ≪ 1. We obtain:𝐵0 (𝑞0, 𝑄Λ,Λ 𝑓 ) = −A[32−3Λ 𝑓𝑄Λ+ 12𝑞20𝑞20 + 𝑣2𝑓𝑄2Λ−Λ 𝑓𝑄Λ𝑞20𝑞20 + 4𝑣2𝑓Λ2𝑓+ 2𝑞0𝑣 𝑓𝑄Λ(arctan(2𝑣 𝑓Λ 𝑓𝑞0)− arctan( 𝑣 𝑓𝑄Λ𝑞0) )]+ O(𝑤2) .(26)The range of variation of the above expression, treatedas a function of 𝑞0 spans the interval 𝑞0 ∈ (2𝑣 𝑓Λ 𝑓 , 𝑣 𝑓𝑄Λ),which contains frequencies for which 𝑤 ∼ 𝑄2Λ. In thisregime we expand the first arctan(𝑥) function around𝑥 = 0 and the second one around infinity. This leads to acontribution of order |𝑤 | and yields a simple parametriza-tion of the flowing bubble and the bosonic propagator:𝐵0 (𝑞0, 𝑄Λ,Λ 𝑓 ) ≈ −3A2+ 2𝑄2Λ + A 𝜋|𝑞0 |𝑣 𝑓𝑄Λ, (27)𝐺𝑏 ≈ 1Δ𝑚2𝑏+ 3𝑄2Λ+ (| ®𝑞 | −𝑄Λ)2 + A𝜋 |𝑞0 |𝑣 𝑓𝑄Λ. (28)For an illustration see Fig. 5. We note affinity of theabove expression to the Hertz-Millis form pertinent tothe antiferromagnetic-type transition where the orderingwavevector ®𝑄 ≠ ®0 and the (one-loop) dynamical exponentof the Bose field 𝑧𝑏 = 2. However, taking note of theanalysis of the previous section and requiring that thedistinct terms in 𝐺−1𝑏all scale as Λ2, we obtain|𝑞0 | ∼ 𝑄3Λ ∼ Λ 𝑓 , (29)which leads to 𝑧𝑏 = 3, in agreement with the Hertz-Millistheory. Note that, in spite of the very different forms ofthe Bose propagators, the same value of 𝑧𝑏 is obtainedin our approach if fermions are scaled out much fasterthan bosons - see Sec. III.A.1, resulting in the standardHertz-Millis propagator instead of the form of Eq. (28).The key point of the above analysis is the identificationof the relation 𝑄Λ ∼ Λ ∼ Λ1/3𝑓, establishing the relativerate at which the distinct degrees of freedom need to bescaled. The earlier reasoning of Ref. [35] disregarded an-alytical contributions to the regularized static part of thebubble, which led to the flawed result Λ 𝑓 ∼ Λ ∼ 𝑄Λ and𝑧𝑏 = 2. The other point to note is that the expandedform of Eq. (28) is valid only for small 𝑤, but the com-plete formula for 𝐵0 (𝑄,Λ 𝑓 ), derived and implementedin Ref. [35] necessarily spanned the entire range of mo-menta and frequencies, although it neglected analitycalcontributions in 𝑤.We emphasize once again that the above calculationentirely evades the step of integrating the fermionic de-grees of freedom before the bosons and constitutes a ro-bust starting point for further developments accounting,in particular, for self-energy effects — see Sec. IV.B. Fermion self-energyWe now explain how the one-loop result for the Fermiself-energy and the associated non-Fermi liquid state isrecovered in our approach relying on the generalizedHertz action of the previous sections. We define:Σ(𝑘0) := Σ(𝑘0, ®𝑘 = 𝑘 𝑓 𝑒𝑥) (30)and concentrate for now on the imaginary part:Σ̃(𝑘0) := 𝑘0 − Im [Σ(𝑘0)] . (31)The non-Fermi liquid propagator is generated from theFermi-Bose loop [Eq. (10)], which reduces to the form:𝜕ΛΣ̃(𝑘0) = 2𝑔2∫𝑄(𝜕ΛΛ 𝑓 ) 𝜕Λ 𝑓 𝑅 𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞)𝑚2𝑏+ | ®𝑞 |2 + 𝐵0 (𝑞0, 𝑞,Λ 𝑓 ) + 𝑅𝑏 ( | ®𝑞 |)(𝑞0 − 𝑘0) 𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞)[(𝑞0 − 𝑘0)2 +(𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞))2]2+𝑔2∫𝑄𝜕Λ𝑅𝑏 ( | ®𝑞 |)(𝑚2𝑏+ Λ2𝑏+ 𝐵0 (𝑞0, 𝑞,Λ 𝑓 ))2𝑞0 − 𝑘0(𝑞0 − 𝑘0)2 +(𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞))2 ,(32)where we introduced 𝑓 ( ®𝑘) = 𝜉 ®𝑘 + 𝑅 𝑓 ( ®𝑘). The term in-volving 𝜕Λ 𝑓 𝑅 𝑓 is crucial, dropping the contribution with𝜕Λ𝑅𝑏 has no important impact on the results of this sec-tion. The bosonic propagator introduces a characteristicscale 𝑞0 ∼ Λ𝑧𝑏 , effectively acting as a cut-off for frequen-cies above 𝑞0. In the analysis to follow, we pursue twostrategies. In the first calculation, we directly numeri-cally integrate Eq. (32) using bosonic propagator given8in Eq. (28), which yields the entire profile of Σ̃(𝑘0) includ-ing all the relevant regimes both as function of frequency𝑘0 and cutoff scale Λ 𝑓 (one can use Λ equivalently). Wedefine the flowing exponent:𝛼Λ (𝑘0) :=𝜕 ln(Σ̃(𝑘0))𝜕 ln(𝑘0), (33)which encodes the scale dependence of the function. Thecorresponding results demonstrating the flow of 𝛼Λ (𝑘0)are given in Fig. 6. Due to the presence of the cutoff at fi-nite Λ, the bosonic field acquires non-zero mass 𝑚2𝑏∼ Λ2,which suppresses the non-Fermi liquid scaling and re-stores the Fermi liquid in deep infrared as long as Λ > 0.Directly at the QCP, the frequency range of this scalingwill shrink and ultimately vanish as Λ is reduces towardszero. On the other hand, if the system becomes detunedfrom criticality the bosonic mass would freeze at a certainscale Λ̃ and stay finite till the end of the flow. We clearlysee that for 𝑘0 > 𝑘0 we recover the anticipated scaling ofthe self-energy governed by the exponent 𝛼 ≈ 2/3, whilefor 𝑘0 < 𝑘0 the self-energy crosses back to the Fermi liq-uid behavior (𝛼 = 1). In the UV regime we recover lineardependence of Σ on frequencies. The scale dependence ofthe crossover frequency and the frequency profile of 𝐺−1𝑓at the QCP are presented in Figs. 7 and 8. The crossoverfrequency coincides with the frequency scale imposed bya bosonic propagator 𝑘0 ∼ Λ 𝑓 ∼ Λ3 ∼ 𝑞0.In the other, simplified variant of the calculation, wetake advantage of the fact that for 𝑘0 ≪ 𝑞0, we canexpand the RHS of Eq. (32) in powers of 𝑘0. As a resultwe find that 𝜕ΛΣ̃(𝑘0) ∼ 𝑘0. This allows us to adopt acommonly applied parametrization:Σ̃(𝑘0) ≈ 𝑍 𝑓 𝑘0 for 𝑘0 < Λ𝑧𝑏 , (34)where 𝑍 𝑓 depends on the scale Λ 𝑓 . In this regime, thefrequency dependence of the Fermi self-energy can be en-tirely determined from its low-energy expansion:𝜕ΛΣ̃(𝑘0) = 2𝑔2∫𝑄(𝜕ΛΛ 𝑓 ) 𝜕Λ 𝑓 𝑅 𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞)𝑚2𝑏+ | ®𝑞 |2 + 𝐵(𝑞0, 𝑞,Λ 𝑓 )𝑍 𝑓 (𝑞0 − 𝑘0) 𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞)[𝑍2𝑓(𝑞0 − 𝑘0)2 +(𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞))2]2+𝑔2∫𝑄𝜕Λ𝑅𝑏 ( | ®𝑞 |)(𝑚2𝑏+ Λ2𝑏+ 𝐵0 (𝑞0, 𝑞,Λ 𝑓 ))2𝑍 𝑓 (𝑞0 − 𝑘0)𝑍2𝑓(𝑞0 − 𝑘0)2 +(𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞))2 . (35)Note that in this approach we may easily include theself energy in the loop, which amounts to multiplyingthe frequencies by 𝑍 𝑓 (as done in the above equation).The flow of 𝑍 𝑓 is extracted from Eq. (35) by taking thederivative with respect to frequency at 𝑘0 = 0, whichleads to:𝜕Λ𝑍 𝑓 = −2𝑔2𝑍 𝑓∫𝑄(𝜕ΛΛ 𝑓 ) 𝜕Λ 𝑓 𝑅 𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞)𝑚2𝑏+ | ®𝑞 |2 + 𝐵(𝑞0, 𝑞,Λ 𝑓 )𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞)[𝑍2𝑓𝑞20 +(𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞))2]2 ©­«1 −4𝑍2𝑓𝑞20𝑍2𝑓𝑞20 +(𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞))2 ª®¬+− 𝑔2𝑍 𝑓∫𝑄𝜕Λ𝑅𝑏 ( | ®𝑞 |)(𝑚2𝑏+ Λ2𝑏+ 𝐵0 (𝑞0, 𝑞,Λ 𝑓 ))21𝑍2𝑓𝑞20 +(𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞))2 ©­«1 −2𝑍2𝑓𝑞20𝑍2𝑓𝑞20 +(𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞))2 ª®¬ .(36)We then examine the power counting dictating thescaling of 𝑍 𝑓 . On the RHS of Eq. (36) the bosonic prop-agator scales as 𝐺𝑏 ∼ 𝑄−2Λ∼ Λ−2/3𝑓, and the fermionicpart as ∼ 𝑣−4𝑓Λ−4𝑓. The frequency integral yields an ad-ditional factor 𝑣 𝑓Λ 𝑓 /𝑍 𝑓 , while the momentum integralscontributes 𝑄ΛΛ 𝑓 ∼ Λ4/3𝑓. In total we obtain:𝜕Λ 𝑓 𝑍 𝑓 ∼ −𝑔2𝑍 𝑓 𝑣2𝑓Λ 𝑓2𝜋3𝑣 𝑓Λ 𝑓𝑍 𝑓︸︷︷︸d𝑞01𝑣4𝑓Λ4𝑓︸︷︷︸𝑓 𝑒𝑟𝑚𝑖𝑜𝑛𝑖𝑐1Λ2/3𝑓︸︷︷︸𝑏𝑜𝑠𝑜𝑛𝑖𝑐Λ4/3𝑓︸︷︷︸d𝑞2= − 𝑔2𝑣 𝑓Λ4/3𝑓∫𝑑Λ 𝑓=====⇒ 𝜂 𝑓 =13, (37)where the exponent 𝜂 𝑓 is defined by𝜂 𝑓 := − 1𝑍 𝑓𝑑𝑍 𝑓𝑑 lnΛ 𝑓. (38)Interestingly 𝑍 𝑓 cancels out in Eq. (37), which sug-gests that the result 𝜂 𝑓 = 1/3 may remain unchanged nomatter if we retain or neglect the fermionic self-energyon the RHS of the loop integral. This is reminiscent ofthe behavior obtained at RPA [60], where results fromthe self-consistent and non self-consistent variants of theanalysis yield the same behavior of the self-energy.The relation between 𝛼 and 𝜂 𝑓 can be derivedfrom continuity of Σ̃(𝑘0) at 𝑘0 = 𝑘0. Within thepresent parametrization, for small enough frequencies thefermionic propagator can be written as:Σ̃(𝑘0 < 𝑘0) ∼ Λ 𝑓−𝜂 𝑓 𝑘0𝑘0→𝑘−0−−−−−−→ Λ 𝑓−𝜂 𝑓Λ 𝑓 , (39)while for frequencies from the IR non-Fermi liquid regionwe get :Σ̃(𝑘0 < 𝑘0 < 𝑘𝑈𝑉0 ) ∼ 𝑘𝛼0𝑘0→𝑘+0−−−−−→ (Λ 𝑓 )𝛼 . (40)Comparing the two limits, we obtain the relation:𝛼 = 1 − 𝜂 𝑓 , (41)9FIG. 6. Evolution of 𝛼Λ (𝑘0) upon reducing the cutoff scale Λ 𝑓 . In the calculation presented in the left panel we disregardedthe term involving 𝜕Λ𝑅𝑏. In both cases we reach the anticipated non-Fermi liquid scaling with 𝛼 ≈ 2/3. For low frequenciesand finite Λ we observe the crossover back to the Fermi liquid behavior. The frequency 𝑘0 (Λ 𝑓 ), identified with this crossover,is plotted in Fig. 7. The initial boson mass was tuned so that the system reaches quantum critical scaling.FIG. 7. The crossover frequency estimated via 𝛼Λ (𝑘0) = 𝛼∗ +0.1 ≈ 0.77 (𝛼∗ is a fixed point value). There is no visibledifference between the two variants of the calculation. Wededuce that 𝑘0 ∼ Λ 𝑓 ∼ Λ3.FIG. 8. Frequency dependence of 𝐺−1𝑓from integrating theflow down to Λ 𝑓 ≈ 0. At low frequencies we recover the RPA-type scaling ∼ 𝑘2/30 . The inset shows the same data in loga-rithmic scale. There is no visible difference between the twovariants of the calculation.thus leading to the RPA value 𝛼 = 2/3 at the presentapproximation level.The present section demonstrated how the frameworkof the Wetterich equation recovers the previously estab-lished one-loop results for bosonic and fermionic prop-erties at the ®𝑄 = ®0 quantum critical point of itinerantFermi systems. We contrasted the flawed procedure (sim-ilar in spirit to the Hertz-Millis theory) of scaling Λ 𝑓 tozero before the bosonic cutoff Λ (Sec. III.A.1) with themore satisfactory one, where fermions and bosons are in-tegrated out in parallel leading, for the bosonic sector, tothe generalized Hertz action. The obtained fixed pointgoverning the critical singularities is of Gaussian natureand leads, after correct account of the scaling of the flow-ing ordering wavevector ®𝑄Λ, to the value 𝑧𝑏 = 3 for thedynamical exponent of the Bose field. It also recovers theRPA result for the Fermi self-energy exponent 𝛼 = 2/3.We emphasize that in spite of the fact that, at this ap-proximation level, the two paths lead to the same valuesof the scaling exponents, the latter one evades the prob-lem of generating singular effective Bose interactions andthe singular Landau-damping term. In the sections tofollow, we address the question of stability of the fixedpoint obtained above, upon taking account of the scalingof the Yukawa coupling 𝑔 and inclusion of self-energy intothe Fermi loop governing the flow of the Bose propagator.Before proceeding we briefly comment on the nu-merical procedures implemented in this section. Weevaluate Σ̃(𝑘0) on a logarithmic frequency grid with120 points. The resulting set of flow equations is in-tegrated using an adaptive version of the Dormand-Price method. The internal momentum and fre-quency integrals are calculated numerically at eachtime step for each of the external frequencies 𝑘0 us-ing the Gauss-Legendre quadrature. If not specifiedotherwise, the flows exhibited in the paper correspondto the following numerical values of the parameters:𝑘 𝑓 = 10, 𝑣 𝑓 = 1, 𝑔 = 2, Λ𝑈𝑉 = 1,Λ 𝑓 ,0 ≈ 10−15. Furtheranalysis of the momentum integrals in Eq. (32) is given10in Appendix D. Technical details are further discussed inAppendix E.IV. SELF-CONSISTENT TREATMENTThe following discussion concerns stability of the non-Fermi liquid RG fixed point of Sec. III upon elevating thetruncation level. It is performed in two steps. We firstinclude (Sec. IV.A) scaling of the Yukawa coupling 𝑔 andthe imaginary part of the self energy Σ̃(𝑘0) in the bub-ble diagram governing the flow of the boson propagator.We obtain a result indicating instability of the RPA-typenon-Fermi liquid and no possibility of a sensible alterna-tive RG fixed point. In the second step (Sec. IV.B) weaccount also for the flow of the real part of the self energy,which is parametrized by the flowing Fermi velocity 𝑣 𝑓 .Our results indicate that the non-Fermi liquid RG fixedpoint is not restored at this truncation level.One possible simple explanation of these findingswould invoke a generic instability of the ®𝑄 = ®0 QCP,which was demonstrated in previous literature for a va-riety of systems (see e.g. [5, 7, 61]). In this interpreta-tion, the one-loop fixed point discussed in Sec. III wouldappear as a mere approximation artifact. The other pos-sibility, which we discuss in Sec. IV.C is that the cor-rect account of the presently discussed non-Fermi liquidstate requires a significantly higher level of truncation,in particular inclusion of terms absent in the bare effec-tive action of Eq. (2), but generated by the RG flow. Wespeculate on how inclusion of the vertex of type Γ(4)𝜙𝜙𝜓𝜓might potentially lead to reappearance of a non-Fermiliquid RG fixed point with ®𝑄 = ®0.A. Yukawa coupling and Fermi self-energy scalingThe logic of our reasoning is the following: we will as-sume conventional fixed point scaling of both the fermionself-energy and the Yukawa coupling and subsequentlyshow that this leads to inconsistencies in the flow equa-tions for the self-energies irrespective of the values of thescaling exponents. Our argument does not require anal-ysis of flow equation for 𝑔.We concentrate on the low-energy regime and reexam-ine the flow of the self-energies assuming the followingfixed-point scaling of 𝑍 𝑓 ,Λ and 𝑔Λ:𝑍 𝑓 = 𝑍∗𝑓 Λ−𝜂 𝑓𝑓, 𝑔Λ = 𝑔∗ Λ−𝜂𝑔𝑓. (42)The regularized fermionic bubble (denoted at this ap-proximation level as X𝑠) takes the following form:X𝑠 (𝑄,Λ 𝑓 ) = −2𝑔2Λ(2𝜋)3𝑍 𝑓∫ ∞−∞d𝑘0∫d2𝑘𝜕Λ 𝑓 𝑅 𝑓 ( ®𝑘)(− 𝑖𝑘0 + 𝑓 ( ®𝑘))2[1−𝑖(𝑘0 + 𝑍 𝑓 𝑞0) + 𝑓 ( ®𝑘 + ®𝑞)+ 1−𝑖(𝑘0 − 𝑍 𝑓 𝑞0) + 𝑓 ( ®𝑘 − ®𝑞)].(43)In analogy with the scaling form of Eq. (23), we canrewrite this expression asX𝑠 (𝑄,Λ 𝑓 ) =A∗Λ𝑦𝑓𝑞ℎ(Λ 𝑓𝑞,𝑍∗𝑓𝑣 𝑓𝑞0𝑞1+𝜂 𝑓), (44)where we defined the exponent:𝑦 = −2𝜂𝑔 + 𝜂 𝑓 . (45)The function ℎ(𝑡, 𝑤) has the same form as in the approx-imation discussed in Sec. III — see Appendix B. Regard-ing the static part of the integrated bubble, the modifi-cation as compared to the previous case occurs via the 𝑦exponent. We obtain:𝐵𝑠 (0, 𝑞,Λ 𝑓 ) = A∗∫ Λ 𝑓Λ 𝑓 ,𝑈𝑉d𝜆 𝜆𝑦ℎ(𝜆/𝑞)𝑞, (46)where A∗ =(𝑔∗ )2𝑘 𝑓𝜋2𝑣 𝑓 𝑍∗𝑓. The quantity X𝑠 shares all the fea-tures of X0 analyzed in the previous section, however theproperties of 𝐵𝑠 now crucially depend on 𝑦. Let us exam-ine the equation for the minimum of the inverse bosonicpropagator 𝐺−1𝑏,𝑠:𝑑𝑑𝑞(𝑞2 + 𝐵𝑠 (0, 𝑞,Λ 𝑓 ))𝑞=𝑄Λ=2𝑄Λ + 𝑦𝑄Λ𝐵𝑠 (0, 𝑄Λ,Λ 𝑓 ) − A∗Λ1+𝑦𝑓𝑄2Λℎ(Λ 𝑓𝑄Λ)= 0 .(47)In Fig. 9, we present numerical solutions obtained forthree exemplary values of 𝑦: 𝑦 ∈ {−0.05, 0, 0.05}. Forsufficiently large Λ 𝑓 , we observe only minor deviationsfrom the solution discussed in Sec. III. However, for pos-itive 𝑦, the flowing order parameter 𝑄Λ saturates for Λ 𝑓small. This behavior is generic for 𝑦 > 0 and indicates in-stability towards a phase with broken translational sym-metry, where limΛ 𝑓→0𝑄Λ ≠ 0. In contrast, for 𝑦 < 0,the scaling of the momentum wavevector deviates fromthe previously identified asymptotic solution and we ob-tain 𝑄Λ ∼ Λ 𝑓 . This behavior leads to an inconsistency inthe mass flow, which is described below. In consequence,𝑦 = 0 appears at first sight as a necessary condition foroccurrence of a second-order phase transition with ®𝑄 = ®0at this level of truncation. This however yields in turn anunphysical behavior of the self-energy flow as we discussin what follows.To explain the problem, we first analyze the limit:limΛ→0𝐵𝑠 (0, | ®𝑞 | = Λ 𝑓 ,Λ 𝑓 ) ∼ Δ𝑚2𝑏 . (48)11FIG. 9. Numerical solution of Eq.(47). The numbers indicateslope fits to the numerical data. In this calculation we setA∗ = 1.A straightforward calculation yields:𝐵𝑠 (0, | ®𝑞 | = Λ 𝑓 ,Λ 𝑓 ) = A∗∫ Λ 𝑓∞d𝜆 𝜆𝑦Λ 𝑓𝜆2=A∗ 1𝑦 − 1Λ𝑦𝑓Λ 𝑓→0−−−−−→0 for 𝑦 ∈ (0, 1)𝑐𝑜𝑛𝑠𝑡 for 𝑦 = 0−∞ for 𝑦 < 0. (49)As already noted, for 𝑦 > 0 the flow converges to a statewith 𝑄Λ→0 > 0, while for 𝑦 < 0 we obtain an unphysicalbehavior of the mass. It follows that occurrence of acritical state (within the present truncation level) shouldrequire that𝑦 = −2𝜂𝑔 + 𝜂 𝑓 = 0 . (50)One may demonstrate that the present procedure thenleads to the superficially appealing result𝑧𝑏 = 3𝑥 = 3(1 + 𝜂 𝑓 ) . (51)However, inspection of the Fermi self-energy flow revealsa problem. We evaluate the flow of 𝑍 𝑓 , and take note ofthe fact that, at the fixed point:𝑦 = −2𝜂𝑔 + 𝜂 𝑓 = 0 =⇒𝑔2Λ𝑍 𝑓→ 𝑐𝑜𝑛𝑠𝑡 , (52)which leads to𝜕Λ 𝑓 𝑍 𝑓 ∼ −𝑔2Λ𝑍 𝑓 𝑣2𝑓Λ 𝑓2𝜋3𝑣 𝑓Λ 𝑓𝑍 𝑓︸︷︷︸d𝑞01𝑣4𝑓Λ4𝑓︸︷︷︸𝑓 𝑒𝑟𝑚𝑖𝑜𝑛𝑖𝑐1Λ2/3𝑓︸︷︷︸𝑏𝑜𝑠𝑜𝑛𝑖𝑐Λ4/3𝑓︸︷︷︸d𝑞2∼−𝑔2Λ𝑣 𝑓Λ4/3𝑓∼ −𝐴∗𝑍 𝑓𝑘 𝑓Λ4/3𝑓. (53)This behavior is incompatible with the scaling assumedin Eq. (42) for any finite value of 𝜂 𝑓 .In conclusion to this section: we proposed a stepbeyond the treatment discussed in Sec. III, by self-consistently introducing the scaling of frequency depen-dence of the self-energy as well as the Yukawa interactioncoupling. At the same time we kept the momentum de-pendencies in the Fermi self-energy unrenormalized. Theobtained result indicates no possibility of realizing a sen-sible non-Fermi liquid fixed point within this procedure.In what follows, we extend the truncation by addition-ally allowing for RG scaling of the Fermi velocity anddemonstrate that this does not remove the inconsistency.B. Scaling of 𝑣 𝑓We now extend the truncation by additionally allowingfor the flow of the Fermi velocity. We impose𝜉𝑘,Λ =𝑣 𝑓 ,Λ2𝑘 𝑓( | ®𝑘 |2 − 𝑘2𝑓 ) (54)and the fixed point scaling 𝑣 𝑓 ,Λ = 𝑣∗𝑓Λ−𝜂𝑣 𝑓𝑓, which defines𝜂𝑣 𝑓 . We also introduce𝑦 := −2𝜂𝑔 + 𝜂 𝑓 + 𝜂𝑣 𝑓 , 𝑥 := 1 + 𝜂 𝑓 − 𝜂𝑣 𝑓 . (55)The flow of 𝑣 𝑓 ,Λ is reflected in the flowing bubble diagramvia the redefinition of the regularized dispersion 𝑓Λ ( ®𝑘) =𝜉 ®𝑘,Λ + 𝑅 𝑓 ( ®𝑘) and via an entirely new term appearing in𝜕Λ𝑅 𝑓 . The scaling form of the regularized bubble, heredenoted by X𝑠𝑐, is given by:X𝑠𝑐 (𝑞0, 𝑞,Λ 𝑓 ) =A∗Λ𝑦𝑓𝑞[(1 − 𝜂𝑣 𝑓 ) ℎ(𝑡, 𝑤)+ 𝜂𝑣 𝑓 ℎ𝑘2(𝑡, 𝑤) ],(56)where 𝑡 = Λ 𝑓 /𝑞 and 𝑤 =𝑍∗𝑓𝑣∗𝑓𝑞0𝑞𝑥. The derivation of theabove formula and the form of the function ℎ𝑘2(𝑡, 𝑤)arepresented in Appendix B (for a plot of ℎ𝑘2 (𝑡) see alsoFig. 3). Following the path described in the previoussections, we analyze the minimum of the inverse propa-gator and extract its behavior for Λ 𝑓 → 0. We obtaindd𝑞(𝑞2 + 𝐵(0, 𝑞,Λ 𝑓 ))𝑞=𝑄Λ== 2𝑄Λ + 𝑦𝑄Λ𝐵𝑠𝑐 (0, 𝑄Λ,Λ 𝑓 ) −A∗Λ1+𝑦𝑓𝑄2Λ[(1 − 𝜂𝑣 𝑓 ) ℎ(Λ 𝑓𝑄Λ)+ 𝜂𝑣 𝑓 ℎ𝑘2(Λ 𝑓𝑄Λ)]= 0 . (57)We have performed a detailed analysis of the solutionsto Eq. (57) depending on 𝑦, 𝜂𝑣 𝑓 , and Λ 𝑓 . Our analysisindicates a generic lack of possibility of obtaining an RGfixed point with properties reminiscent of the RPA-typebehavior discussed in Sec. III. Our conclusion is summa-rized in the diagram of Fig. 10. In the physically mostinteresting range of 𝑦 > 0 and 0 < 𝜂𝑣 𝑓 < 1 the flowingwavevector saturates for Λ 𝑓 small at a positive value. In12FIG. 10. Schematic illustration of the behavior of 𝑄Λ de-pending on 𝑦 and 𝜂𝑣 𝑓 . The solid blue line separating the tworegimes follows from the condition 𝑦 𝑏0 (𝑦, 𝜂𝑣 𝑓 ) = 0; see themain text for a discussion.the other regime (where 𝜂𝑣 𝑓 > 1) we obtain a linear scal-ing of 𝑄Λ as a function of Λ 𝑓 , which however is accom-panied by appearance of another competing minimum of𝐺−1𝑏directly at momentum 𝑞 = 0. This picture points ata possibility of a first-order transition and fragility of thesystem upon further elevating the truncation level. Thesetwo regimes are separated in the diagram of Fig. 10 by aline located at𝑦 𝑏0 (𝑦, 𝜂𝑣 𝑓 ) := 𝑦∫ 0∞d𝑡 𝑡𝑦[(1 − 𝜂𝑣 𝑓 )ℎ(𝑡) + 𝜂𝑣 𝑓 ℎ𝑘2 (𝑡)]= 0 ,(58)where we find 𝑄Λ ∼ Λ(1+𝑦)/3𝑓; in this case 𝐺−1𝑏also fea-tures two minima as function of 𝑞. The scaling of 𝑄Λis exemplified in Fig. 11 for the three cases discussedabove. An entirely separate case occurs for 𝑦 = 0, wherewe identified a physical inconsistency in the flow of 𝑍 𝑓 ,analogous to the one discussed in Sec. IV.A for 𝜂𝑣 𝑓 = 0.To supplement the above numerical analysis we studiedthe asymptotic properties of Eq. (57). Let us begin withthe case 𝑄Λ ≪ Λ 𝑓 . It is easy to show that Eq. (57)features a solution given by𝑄Λ = −4(1 − 𝜂𝑣 𝑓 ) (2 − 𝑦)𝜋 𝜂𝑣 𝑓 (1 − 𝑦)Λ 𝑓 , (59)the positivity of which imposes constraints on 𝜂𝑣 𝑓 and 𝑦.It turns out that this solution corresponds to a maximumof 𝐺−1𝑏, which is visible in the curves plotted in Fig. 12.Another solution is identified by imposing𝑦 𝑏0 (𝑦, 𝜂𝑣 𝑓 ) = 0. In this case we find𝑄Λ =(3(1 − 𝜂𝑣 𝑓2)A∗2(1 + 𝑦)) 13Λ1+𝑦3𝑓, (60)which fully agrees with the numerical data (see Figs. 10and 11). For 𝑦 = 0 we retrieve the result of Sec. IV.A,where 𝑄Λ ∼ Λ1/3𝑓. Finally, by expanding Eq. (57) for theFIG. 11. Exemplary illustration of the scaling of the minimum𝑄Λ ∼ Λ𝑎𝑓for different values of 𝑏0 (𝑦, 𝜂𝑣 𝑓 ), keeping 𝑦 = 0.6fixed. The numerical results are plotted as black dots, whilethe lines represent fits in the region of small Λ 𝑓 . The threecurves illustrate the three types of behavior described in thetext: convergence to 𝑄Λ→0 > 0, 𝑄Λ ∼ Λ(1+𝑦)/3𝑓, and 𝑄Λ ∼ Λ 𝑓 .FIG. 12. An exemplary plot of 𝐺−1𝑏(𝑞, 𝑞0 = 0,Λ 𝑓 ) for a se-quence of values of Λ 𝑓 in a situation involving the 𝑄Λ ∼Λ(1+𝑦)/3𝑓scaling. The plot parameters are 𝑦 = 0.6 and𝜂𝑣 𝑓 = 1.29568, which leads to 𝑏0 (𝑦, 𝜂𝑣 𝑓 ) ≈ 0.other scaling limit, 𝑄Λ ≫ Λ 𝑓 , we obtain𝑄𝑦+1Λ≈3(1 − 𝜂𝑣 𝑓2)𝑦(𝑦 + 1) 𝑏0 (𝑦, 𝜂𝑣 𝑓 )Λ𝑦+1𝑓, (61)leading to the scaling 𝑄Λ ∼ Λ 𝑓 as found in the numericalanalysis. Note that the condition 𝑄Λ ≫ Λ 𝑓 remains ful-filled by Eq. (61) due to the large value of the coefficienton the RHS.We summarize this section by the conclusion that con-sistent account of scaling of the momentum and fre-quency dependencies of the self-energy (parametrized byflowing 𝑍 𝑓 and 𝑣 𝑓 ) as well as the Yukawa coupling 𝑔 doesnot allow for obtaining a sensible non-Fermi liquid fixedpoint with ordering wavevector ®𝑄 = ®0. Instead, we iden-tified a generic tendency towards renormalizing the or-dering wavevector to a non-vanishing value or an inverse13FIG. 13. Exemplary diagrams which contribute to𝜕Λ𝛾Λ (𝑄, 𝐾).propagator featuring two minima (pointing towards pos-sible first order transition) accompanied by a large valueof the scaling exponent 𝜂𝑣 𝑓 .Existence of a non-Fermi liquid fixed point within theself-consistent treatment is however not ruled out uponincluding higher order terms, which are absent in theinitial action of Eq. (2), but become generated by the RGflow. We speculate on this possibility below in Sec. IV.C.C. Fermion-boson 4-point vertexLet us now consider the following contribution to theflow of the boson self-energyX𝛾 (𝑄,Λ 𝑓 ) = −2∫𝐾𝑆 𝑓 (𝐾) 𝛾Λ (𝑄, 𝐾) , (62)which involves the 4-point fermion-boson vertex 𝛾Λ𝛾Λ (𝑄, 𝐾) :=𝛿4ΓΛ [𝜓, 𝜓, 𝜙]𝛿𝜙𝑄𝛿𝜙−𝑄𝛿𝜓𝐾,↑𝛿𝜓𝐾,↑. (63)In standard treatments (including our own analysis thusfar), this contribution is neglected for several reasons.First, the 4-point vertex is absent in the bare action.Furthermore, evaluating it strictly at the minimum ofthe inverse propagator, 𝐾 = {𝑘0 = 0, | ®𝑘 | = 𝑘 𝑓 }, yieldszero, since Eq. (62) involves an integral over a second-order pole. Therefore, any non-trivial result can onlyarise from the frequency-dependent part of 𝛾Λ (𝑄, 𝐾) thatis generated during the integration of the flow.Fig. 13 depicts representative diagrams that contributeto 𝜕Λ𝛾Λ (𝑄, 𝐾). Only the first of these — the box diagram— can be constructed from vertices present already inthe bare action. Below we consider only this diagram.While we retain the scaling of the Yukawa coupling andthe fermionic self-energy, for simplicity, we omit the flowof the Fermi velocity. Consequently, Eq. (62) can berewritten as follows:X𝛾 (𝑄,Λ 𝑓 ) = 2∫𝐾1𝑆 𝑓 (𝐾1,Λ 𝑓 )∫ Λ 𝑓Λ 𝑓 ,𝑈𝑉d𝜆{𝑔4𝜆∫𝐾2𝐺2𝑓 (𝐾2, 𝜆)𝐺𝑏 (𝐾1 − 𝐾2, 𝜆)[𝑆 𝑓 (𝐾2 +𝑄, 𝜆) + 𝑆 𝑓 (𝐾2 −𝑄, 𝜆)]++ 2𝑔4𝜆∫𝐾2𝑆 𝑓 (𝐾2, 𝜆)𝐺 𝑓 (𝐾2, 𝜆)𝐺𝑏 (𝐾1 − 𝐾2, 𝜆)[𝐺 𝑓 (𝐾2 +𝑄, 𝜆) − 𝐺 𝑓 (𝐾2 −𝑄, 𝜆)]}.(64)By inserting the box diagram into the tadpole diagramcontributing to the flow of the boson self-energy, we intro-duced two distinct flowing scales, Λ 𝑓 and 𝜆. We distin-guish the scale dependence of the propagators explicitly.As in the previous section, we take the limit Λ 𝑓 ,𝑈𝑉 → ∞.Collecting all terms that depend on the integration vari-able 𝐾1, we define:𝑊 (𝐾, 𝜆,Λ 𝑓 ) := −𝑔2𝜆∫𝐾1𝑆 𝑓 (𝐾1,Λ 𝑓 )𝐺𝑏 (𝐾1 − 𝐾, 𝜆) . (65)Notice that for 𝜆 = Λ 𝑓 , this function resembles the flow ofthe fermionic self-energy [see Eq. (10)]. We can thereforeapproximate this term as:𝑊 (𝐾, 𝜆,Λ 𝑓 ) ≈ −𝑖𝜕Λ 𝑓 𝑍 𝑓 ,Λ 𝑓 𝑘0 = 𝑖 𝜂 𝑓𝑍 𝑓 ,Λ 𝑓Λ 𝑓 𝑍 𝑓 ,𝜆𝑍 𝑓 ,𝜆𝑘0 . (66)Substituting this result and taking the limit 𝑘 𝑓 ≫𝑚𝑎𝑥(Λ,Λ 𝑓 , 𝑞) we obtain the static expression:X𝛾 (0, 𝑞,Λ 𝑓 ) ≈ 2𝜂 𝑓𝑍 𝑓 ,Λ 𝑓Λ 𝑓∫ Λ 𝑓∞d𝜆A𝜆𝑍 𝑓 ,𝜆𝑞𝑓(𝜆𝑞). (67)For the definition and details of the scaling function 𝑓 (𝑡),see Appendix F. The contribution from this vertex to thebosonic propagator is obtained by integrating over thescale:𝐵𝛾 (0, 𝑞,Λ 𝑓 ) =∫ Λ 𝑓+∞d𝜆X𝛾 (𝑞0 = 0, 𝑞, 𝜆) ≈ (68)≈ 2A∗𝑞𝑦∫ Λ 𝑓 /𝑞+∞d𝑡 𝑡𝑦 𝑓 (𝑡)(1 −( 𝑡𝑞Λ 𝑓) 𝜂 𝑓 ).(69)As before, we determine the flowing ordering wavevectorby solving for the extremum of the bosonic propagator:dd𝑞(𝑞2 + 𝐵𝑠 (0, 𝑞,Λ 𝑓 ) + 𝐵𝛾 (0, 𝑞,Λ 𝑓 ))𝑞=𝑄Λ= 0 . (70)Inspection of the structure of 𝐵𝛾 (0, 𝑞,Λ 𝑓 ) reveals that ittends to reduce the magnitude of 𝑄Λ towards zero. Thenumerical results of solution of Eq. (70) are summarizedin Fig. 14. For small values of the exponent, 𝜂 𝑓 < 2/3(1−14FIG. 14. Scaling of the global minima 𝑄Λ for different valuesof 𝜂 𝑓 and 𝑦. The most promising scaling from the point ofview of restoring the QCP with ®𝑄 = ®0 is marked by a light-blue region (𝑄Λ ∼ Λ 𝑓 ) and dark blue line (𝑄Λ = 0). The latterdiffers from the white region in momentum dependence of thebosonic propagator around minimum. The upper right cornerof this plane (orange region) exhibits strong dependence onthe UV cut-off. We do not explore this region in the presentwork.𝑦), we find a linear dependence of 𝑄Λ on the fermioniccutoff scale Λ 𝑓 :𝑄Λ = 𝐶1Λ 𝑓 where 𝐶1 > 2 . (71)At the boundary of the blue region (for 𝜂 𝑓 = 2/3(1− 𝑦)),the global minimum is located at 𝑞 = 0. For 𝜂 𝑓 ≥ 1 − 𝑦,the procedure of sending Λ 𝑓 ,𝑈𝑉 to infinity is no longervalid. Starting from the RPA calculation, we focus on theregion where 𝑦 is small and 𝜂 𝑓 ≈ 1/3. We observe thatthe predicted scaling of 𝑄Λ departs from the previouslyobtained result, 𝑄Λ ∼ Λ1/3𝑓, thereby avoiding the prob-lems encountered in the flow of the fermionic self-energypresented in Eq. (53). Within the present analysis, thevalue of 𝑧𝑏 remains undetermined, as it depends heavilyon the precise values of 𝑦 and 𝜂 𝑓 . The role of the omittedcontributions to 𝛾Λ (𝑄, 𝐾) is also out of present controland is relegated to future studies. In Fig. 15, we de-pict the full inverse bosonic propagator 𝐺−1𝑏in the staticlimit, accounting for mass renormalization.As demonstrated in Sec. III, flow of 𝑄Λ appears es-sential for a correct account of Landau damping withinWilsonian-type RG. On the other hand it gives rise to theinstability of the non-Fermi liquid fixed point in the im-proved truncations discussed in Sec. IV.A and B, which isreflected in the flow of 𝑄Λ towards positive values. Thismay however, perhaps, change when also quartic inter-action vertices (including their frequency dependencies)are considered, as signaled in the present section.FIG. 15. The plots show 𝐺−1𝑏(0, 𝑞,Λ 𝑓 ) for different valuesof Λ 𝑓 . The black dashed line demonstrates scaling of theminimum of the inverse propagator with Λ 𝑓 . For 𝑦 > 0 wefind a tendency towards 0 (lower panel), while for 𝑦 = 0 itflows to a constant value (upper panel). The log-log plots of𝑄Λ (Λ 𝑓 ) are shown in the insets.V. SUMMARYQuantum criticality of two-dimensional Fermi systemshas since long remained a topic of top relevance and inter-est to condensed matter theory. Despite substantial ef-forts over many years, a fully satisfactory theory of thesephenomena has not emerged up to this point. In thepresent paper we have proposed a consistent approachto this problem from the Wilsonian RG point of view,adapting the Wetterich equation as the point of depar-ture. In the first place, remaining at the functional RGlevel, we have shown how a relatively simple, non-self-consistent truncation of the Wetterich equation allowsfor obtaining a non-Fermi liquid RG fixed point withthe features anticipated from earlier one-loop-level stud-ies. We also tested the validity of simplified treatments,which abandon the computation of the full frequency de-pendence of the self-energy in favor of a commonly im-plemented simplified parametrization capturing only thelow energy asymptotics. Our approach regularizes all thedegrees of freedom present in the theory and evades theproblem of singular effective bosonic interactions. In par-ticular, the boson propagator in the loop integrals is onlydressed by fluctuations above the cutoff scale, leading toa structure entirely different as compared to the singularfully dressed Hertz-Millis form commonly used in manyearlier approaches. An unavoidable feature emerging inour scheme is the RG flow of the ordering wavevector15®𝑄Λ, a property which cures problems of the mass flowencountered in earlier Wilsonian RG studies. The fixed-point scaling of 𝑄Λ determines the value of the bosonicdynamical exponent 𝑧𝑏 = 3 and the Fermi self-energy ex-ponent 𝛼 = 2/3. Previous perturbative approaches cor-rect the value of 𝛼 only at the 3-loop level and yieldunusual and very unclear predictions concerning 𝑧𝑏 at 4loops.In the second part of our paper (Sec. IV) we set out toinvestigate how inclusion of nontrivial scaling of the self-energy and the Yukawa coupling impact the RG flow. Wehave demonstrated an instability of the non-Fermi liquidRG fixed point towards either a phase featuring ®𝑄 ≠ ®0 ora behavior typical to first-order transitions. The latteris expected to generically occur in the case of magnetictransitions [7, 61], and, at least at mean-field level, is alsomost common for electronic nematics [62]. Despite thesefindings, occurrence of a QCP featuring ®𝑄 = ®0 is notexcluded, but requires accounting for higher-order termsand would presumably imply critical singularities quitedistinct from those characteristic to one-loop level. InSec. IV.C we presented a plausible, but, at the presentpoint, speculative picture of how critical scaling mightreemerge by including terms involving boson-fermion in-teraction vertices, which are entirely absent in the bareaction, but become generated by the RG flow.The role of the flow of 𝑄Λ, which was disregarded inprevious Wilsonian RG approaches to this problem, isstriking. On one hand, it appears as an indispensablepart of the procedure and its behavior for Λ → 0 de-termines the scaling exponents at the leading order ofthe calculation (Sec. III). On the other hand, within themost natural, lowest-order self-consistent extensions ofthe truncation (Sec. IV) its generic flow to a not-zerovalue gives an indication of an instability of the RPA-type non-Fermi liquid state. It remains open whetherthe instabilities obtained in Sec. 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Metzner, Fermi surface fluctuationsand single electron excitations near pomeranchuk insta-bility in two dimensions, Phys. Rev. B 73, 045127 (2006).[61] T. R. Kirkpatrick and D. Belitz, Generic long-range17order-parameter correlations in metallic quantum mag-nets, Phys. Rev. B 113, 214414 (2026).[62] H. Yamase, V. Oganesyan, and W. Metzner, Mean-fieldtheory for symmetry-breaking fermi surface deformationson a square lattice, Phys. Rev. B 72, 035114 (2005).[63] W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden,and K. Schönhammer, Functional renormalization groupapproach to correlated fermion systems, Rev. Mod. Phys.84, 299 (2012).18Appendix A: Derivation of the Wetterich equationWe summarize the derivation of the Wetterich equation for a mixed fermion-boson system folowing Ref. [55]. Analternative but fully equivalent supertrace formalism can be found in Refs. [52, 63].We consider a general form of the bare action:𝑆[𝜓, 𝜙] = 12∫K ,K′𝜓KG−1𝑓 (K,K ′)𝜓K′ + 12∫Q,Q′𝜙QG−1𝑏 (Q,Q′)𝜙Q′ +𝑉 [𝜓, 𝜙] , (A1)where we have extended the notation for the fermionic and bosonic variables to collect frequencies (𝑘0 or 𝑞0), momenta(®𝑘 or ®𝑞), the spin index 𝜎, and the Grassmann conjugation index 𝑐 for fermions (or complex conjugation for bosons),such that K = {𝐾, 𝜎, 𝑐} and Q = {𝑄, 𝑐}. In this notation, the fermionic field is represented by the vector 𝜓K =[𝜓𝐾,↑, 𝜓𝐾,↓, 𝜓𝐾,↑, 𝜓𝐾,↓], while the bosonic field is given by 𝜙Q = [𝜙𝑄 𝜙−𝑄], since the order parameter in the presentwork is represented by a real field. We introduce scale-dependent cutoff functions parameterized by Λ:Δ𝑆Λ [𝜓, 𝜙] =12∫K ,K′𝜓KR−1𝑓 (K,K ′)𝜓K′ + 12∫Q,Q′𝜙QR−1𝑏 (Q,Q′)𝜙Q′ . (A2)Adding these regulator terms along with source terms for the external fields 𝐽Q and 𝜂K , we define the scale-dependentpartition function:𝑍Λ [𝜂, 𝐽] ≔∫D𝜓D𝜙 exp(−𝑆[𝜓, 𝜙] − Δ𝑆Λ [𝜓, 𝜙] +∫Q𝐽Q𝜙Q +∫𝐾𝜂K𝜓K). (A3)The logarithm of the partition function𝑊Λ [𝜂, 𝐽] = ln (𝑍Λ [𝜂, 𝐽]) (A4)serves as the generating functional for connected correlation functions, the expectation values of the fields are obtainedvia:𝜙Q = ⟨𝜙Q⟩ =𝛿𝑊Λ [𝜂, 𝐽]𝛿𝐽Q𝜓K = ⟨𝜓K⟩ =𝛿𝑊Λ [𝜂, 𝐽]𝛿𝜂K. (A5)We perform a Legendre transform of 𝑊Λ to define the effective action:Γ̃Λ [𝜓, 𝜙] ≔ −𝑊Λ [𝜓, 𝜙] +∫Q𝐽Q𝜙𝑄 +∫K𝜂K𝜓K . (A6)This new functional serves as the generating functional for one-particle-irreducible (1PI) vertex functions, such that:𝜂K = −𝛿Γ̃Λ [𝜓, 𝜙]𝛿𝜓K𝐽Q =𝛿Γ̃Λ [𝜓, 𝜙]𝛿𝜙Q. (A7)We can straightforwardly derive the flow equation for the effective action by taking the total derivative with respectto the scale Λ:ddΛΓ̃Λ [𝜓, 𝜙] = −𝜕Λ𝑊Λ [𝜂, 𝐽] =12∫Q,Q′𝜕ΛR𝑏 (Q,Q′)⟨𝜙Q𝜙Q′⟩ + 12∫K ,K′𝜕ΛR 𝑓 (K,K ′)⟨𝜓K𝜓K′⟩ . (A8)Modifying the Legendre transform by subtracting the regulator term, ΓΛ [𝜓, 𝜙] ≔ Γ̃Λ [𝜓, 𝜙] − Δ𝑆[𝜓, 𝜙], allows us torewrite the flow equation in the form:ddΛΓΛ [𝜓, 𝜙] =12∫Q,Q′𝜕ΛR𝑏 (Q,Q′) 𝛿2𝑊Λ [𝜂, 𝐽]𝛿𝐽Q𝛿𝐽Q′+ 12∫K ,K′𝜕ΛR 𝑓 (K,K ′) 𝛿2𝑊Λ [𝜂, 𝐽]𝛿𝜂K𝛿𝜂K′. (A9)As a final step, we express the second derivatives of 𝑊Λ in terms of the derivatives of the effective action Γ̃Λ. This isachieved by differentiating Eq. (A5) with respect to the fields 𝜓 and 𝜙, and substituting the relations from Eq. (A7).As a result, we obtain the exact flow equation:19ddΛΓΛ [𝜓, 𝜙] =12Tr[ (𝜕ΛR𝑏) (Γ̃(2)𝜙𝜙)−1 (1 − 𝐸1)−1]+ 12Tr[ (𝜕ΛR 𝑓) (Γ̃(2)𝜓𝜓)−1 (1 − 𝐸2)−1]. (A10)Here we used a compat matrix notation, Γ (𝑛)𝛼1...𝛼𝑛:=𝛿𝑛ΓΛ [𝜓, 𝜙]𝛿𝛼1...𝛿𝛼𝑛. The mixing between the bosonic and fermionic sectorsis contained in the quantities 𝐸1 and 𝐸2:𝐸1 = Γ̃(2)𝜙𝜓(Γ̃(2)𝜓𝜓)−1Γ̃(2)𝜓𝜙(Γ̃(2)𝜙𝜙)−1𝐸2 = Γ̃(2)𝜓𝜙(Γ̃(2)𝜙𝜙)−1Γ̃(2)𝜙𝜓(Γ̃(2)𝜓𝜓)−1. (A11)The flow equation for the effective action, presented in Eq. (A10) as a generating functional for 1PI vertices, wasfirst derived by Wetterich [36] for a purely bosonic system. The flow of specific vertices can be obtained by takingappropriate functional derivatives with respect to the fields 𝜓 or 𝜙. By construction, the resulting flow equationscontain only a single internal momentum integration and can therefore be represented by one-loop diagrams. Allinformation regarding the flow is encoded in the structure of the fully dressed, scale-dependent propagators andvertices. Finally, we provide the flow equations for self-energies:ddΛΓ(2)𝜙𝑄𝜙−𝑄=𝜕Π(𝑄)𝜕Λ=12Tr[ (𝜕ΛR𝑏) (Γ̃(2)𝜙𝜙)−1 (Γ̃(3)𝜙𝑄𝜙𝜙) (Γ̃(2)𝜙𝜙)−1 (Γ̃(3)𝜙−𝑄𝜙𝜙) (Γ̃(2)𝜙𝜙)−1+ (𝑄 → −𝑄)]+− 12Tr[ (𝜕ΛR𝑏) (Γ̃(2)𝜙𝜙)−1 (Γ̃(4)𝜙𝑄𝜙−𝑄𝜙𝜙) (Γ̃(2)𝜙𝜙)−1]++ 12Tr[ (𝜕ΛR 𝑓) (Γ̃(2)𝜓𝜓)−1 (Γ̃(3)𝜙𝑄𝜓𝜓) (Γ̃(2)𝜓𝜓)−1 (Γ̃(3)𝜙−𝑄𝜓𝜓) (Γ̃(2)𝜓𝜓)−1+ (𝑄 → −𝑄)]− 12Tr[ (𝜕ΛR 𝑓) (Γ̃(2)𝜓𝜓)−1 (Γ̃(4)𝜙𝑄𝜙−𝑄𝜓𝜓) (Γ̃(2)𝜓𝜓)−1](A12)andddΛΓ(2)𝜓𝐾,↑𝜓𝐾,↑=𝜕Σ(𝐾)𝜕Λ=12Tr[ (𝜕ΛR𝑏) (Γ̃(2)𝜙𝜙)−1 (Γ̃(3)𝜓𝐾,↑𝜙𝜓) (Γ̃(2)𝜓𝜓)−1 (Γ̃(3)𝜓𝐾,↑𝜓𝜙) (Γ̃(2)𝜙𝜙)−1− (𝜓𝐾,↑ ↔ 𝜓𝐾,↑)]+− 12Tr[ (𝜕ΛR𝑏) (Γ̃(2)𝜙𝜙)−1 (Γ̃(4)𝜓𝐾,↑𝜓𝐾,↑𝜙𝜙) (Γ̃(2)𝜙𝜙)−1]++ 12Tr[ (𝜕ΛR 𝑓) (Γ̃(2)𝜓𝜓)−1 (Γ̃(3)𝜓𝐾,↑ ,𝜓,𝜙) (Γ̃(2)𝜙𝜙)−1 (Γ̃(3)𝜓𝐾,↑𝜙𝜓) (Γ̃(2)𝜓𝜓)−1− (𝜓𝐾,↑ ↔ 𝜓𝐾,↑)]− 12Tr[ (𝜕ΛR 𝑓) (Γ̃(2)𝜓𝜓)−1 (Γ̃(4)𝜓𝐾,↑𝜙𝐾,↑𝜓𝜓) (Γ̃(2)𝜓𝜓)−1]. (A13)The above equations are general and exact. In the implementation to the present problem, we specify the explicitforms of the fermionic matrices in given basis as:R 𝑓 = 𝑅 𝑓 (𝐾1) 𝛿(𝐾1 − 𝐾2)0 0 −1 00 0 0 −11 0 0 00 1 0 0 Γ̃(2)𝜓𝜓= −𝐺−1𝑓 (𝐾1) 𝛿(𝐾1 − 𝐾2)0 0 −1 00 0 0 −11 0 0 00 1 0 0 (A14)Γ̃(3)𝜙𝑄𝜓𝜓=0 0 Γ̃(3)𝜙𝑄𝜓𝐾2 ,↑𝜓𝐾1 ,↑00 0 0 Γ̃(3)𝜙𝑄𝜓𝐾2 ,↓𝜓𝐾1 ,↓−Γ̃ (3)𝜙𝑄𝜓𝐾1 ,↑𝜓𝐾2 ,↑0 0 00 −Γ̃ (3)𝜙𝑄𝜓𝐾1 ,↓𝜓𝐾2 ,↓0 0(A15)Γ̃(4)𝜙𝑄𝜙−𝑄𝜓𝜓= Γ̃(4)𝜙𝑄𝜙−𝑄𝜓𝐾2 ,↑𝜓𝐾1 ,↑𝛿(𝐾1 − 𝐾2)0 0 1 00 0 0 1−1 0 0 00 −1 0 0 Γ̃(3)𝜓𝐾,↑𝜙𝜓=Γ̃(3)𝜙𝑄𝜓𝐾,↑𝜓𝐾1 ,↑0 0 0Γ̃(3)𝜙−𝑄𝜓𝐾,↑𝜓𝐾1 ,↑0 0 0 . (A16)20Appendix B: The flowing bubble diagram and the scaling functions ℎ(𝑡, 𝑤) and ℎ𝑘2 (𝑡, 𝑤)This appendix provides the definitions, key derivation steps, and final forms of the scaling functions ℎ(𝑡, 𝑤) andℎ𝑘2 (𝑡, 𝑤). These functions govern the flow of the fermionic bubble diagram X(𝑄,Λ 𝑓 ) in the limit where the Fermimomentum is large compared to all other momentum scales.The derivation presented below applies to the most general approximation discussed in this paper — X𝑠𝑐 (𝑄,Λ 𝑓 ) insection IV.B. The function X𝑠 (𝑄,Λ 𝑓 ) used in section IV.A. and IV.C can be retrieved by replacing the flowing velocity𝑣 𝑓 ,Λ with a constant 𝑣 𝑓 and setting 𝜂𝑣 𝑓 = 0. Additionally, making the substitutions 𝑍 𝑓 → 1 and 𝑔Λ → 𝑔 (whichconsequently sets 𝑦 = 0) recovers the regularized fermionic bubble in the non-self-consistent calculation X0 (𝑄,Λ 𝑓 )used in section III.Our starting point is the full expression for the fermionic bubble diagram:X(𝑄,Λ 𝑓 ) = −2𝑔2Λ(2𝜋)3∫ ∞−∞d𝑘0∫d2𝑘 𝜕Λ 𝑓 𝑅 𝑓 ( ®𝑘)(𝐺 𝑓 (𝐾))2 [𝐺 𝑓 (𝐾 +𝑄) + 𝐺 𝑓 (𝐾 −𝑄)], (B1)where the fermionic self-energy is expanded around Fermi surface as𝐺−1𝑓 = −𝑖𝑘0 + 𝜉 ®𝑘 + Σ( ®𝑘, 𝑘0) ≈ −𝑖𝑍 𝑓 𝑘0 + 𝜉 ®𝑘,Λ . (B2)Here 𝜉 ®𝑘,Λ describes the spherical Fermi surface with flowing Fermi velocity 𝑣 𝑓 ,Λ. Explicitly, the self-consistent bubblediagram is given by:X𝑠𝑐 (𝑄, Λ 𝑓 ) = −2𝑔2Λ(2𝜋)3∫ ∞−∞d𝑘0∫d2𝑘𝜕Λ 𝑓 𝑅 𝑓 ( ®𝑘)(− 𝑖𝑍 𝑓 𝑘0 + 𝑓 ( ®𝑘))2 [1−𝑖𝑍 𝑓 (𝑘0 + 𝑞0) + 𝑓 ( ®𝑘 + ®𝑞)+ 1−𝑖𝑍 𝑓 (𝑘0 − 𝑞0) + 𝑓 ( ®𝑘 − ®𝑞)],(B3)where the regularized dispersion 𝑓 ( ®𝑘) = 𝜉 ®𝑘,Λ+𝑅 𝑓 ( ®𝑘) incorporates the fermionic regulator. While the frequency integralcan be straightforwardly evaluated using the residue theorem, computing the momentum integrals is more involved.The phase space for momentum integration is tightly constrained by the scale derivative of the regulator, which reads:𝜕Λ 𝑓 𝑅 𝑓 ( ®𝑘) ≈ 𝑣 𝑓{(1 − 𝜂𝑣 𝑓 )(𝜃[𝑝 𝑖𝑛 𝑠ℎ𝑒𝑙𝑙]− 𝜃[ℎ 𝑖𝑛 𝑠ℎ𝑒𝑙𝑙] )+ 𝜂𝑣 𝑓| ®𝑘 |2 − 𝑘2𝑓2𝑘 𝑓Λ 𝑓(𝜃[𝑝 𝑖𝑛 𝑠ℎ𝑒𝑙𝑙]+ 𝜃[ℎ 𝑖𝑛 𝑠ℎ𝑒𝑙𝑙] )}. (B4)We neglect contributions of order O(Λ 𝑓 /𝑘 𝑓 ) and we define the specific momentum-space regions”p in shell” and ”hin shell” to denote particles and holes, respectively, located within a shell of width Λ 𝑓 around the Fermi surface.Mathematically, these regions are defined via Heaviside step functions:𝜃[𝑝 𝑖𝑛 𝑠ℎ𝑒𝑙𝑙]:= 𝜃[Λ 𝑓 − (| ®𝑘 | − 𝑘 𝑓 )]𝜃[| ®𝑘 | − 𝑘 𝑓](B5)𝜃[ℎ 𝑖𝑛 𝑠ℎ𝑒𝑙𝑙]:= 𝜃[Λ 𝑓 − (𝑘 𝑓 − |®𝑘 |)]𝜃[𝑘 𝑓 − |®𝑘 |]. (B6)Changing variables 𝑘0 → 𝑘0/𝑍 𝑓 and evaluating the frequency integral via the residue theorem yields:X𝑠𝑐 (𝑄,Λ 𝑓 ) = − AΛ𝑣2𝑓 ,Λ2𝑘 𝑓(1 − 𝜂𝑣 𝑓 ){ ∫ 2𝜋0d𝜙∫ 𝑘 𝑓 +Λ 𝑓𝑘 𝑓d𝑘 𝑘(𝜃[− 𝑓Λ ( ®𝑘 + ®𝑞)](𝑍 𝑓 𝑞0 + 𝑖𝜉𝑘 𝑓 +Λ − 𝑖 𝑓Λ ( ®𝑘 + ®𝑞))2 + (𝑞0 → −𝑞0))++∫ 2𝜋0d𝜙∫ 𝑘 𝑓𝑘 𝑓 −Λ 𝑓d𝑘 𝑘(𝜃[𝑓Λ ( ®𝑘 + ®𝑞)](𝑍 𝑓 𝑞0 + 𝑖 𝑓Λ ( ®𝑘 + ®𝑞) − 𝑖𝜉𝑘 𝑓 −Λ)2 + (𝑞0 → −𝑞0))}+− AΛ𝑣2𝑓 ,Λ2𝑘 𝑓𝜂𝑣 𝑓2𝑘 𝑓Λ 𝑓{ ∫ 2𝜋0d𝜙∫ 𝑘 𝑓 +Λ 𝑓𝑘 𝑓d𝑘 𝑘 (𝑘2 − 𝑘2𝑓 )(𝜃[− 𝑓Λ ( ®𝑘 + ®𝑞)](𝑍 𝑓 𝑞0 + 𝑖𝜉𝑘 𝑓 +Λ − 𝑖 𝑓Λ ( ®𝑘 + ®𝑞))2 + (𝑞0 → −𝑞0))+−∫ 2𝜋0d𝜙∫ 𝑘 𝑓𝑘 𝑓 −Λ 𝑓d𝑘 𝑘 (𝑘2 − 𝑘2𝑓 )(𝜃[𝑓Λ ( ®𝑘 + ®𝑞)](𝑍 𝑓 𝑞0 + 𝑖 𝑓Λ ( ®𝑘 + ®𝑞) − 𝑖𝜉𝑘 𝑓 −Λ)2 + (𝑞0 → −𝑞0))}(B7)with a multiplication factor:AΛ =𝑔2Λ𝑘 𝑓𝜋2𝑣 𝑓 ,Λ𝑍 𝑓= A∗ Λ−2𝜂𝑔+𝜂𝑣 𝑓 +𝜂 𝑓𝑓. (B8)21For convenience, we have not absorbed the factor 𝑣2𝑓 ,Λ/𝑘 𝑓 (appearing in the first part of the expression) into thedefinition of AΛ. In the static limit (𝑞0 = 0), the denominators in the integrand are proportional to 𝑣 𝑓 ,ΛΛ 𝑓 , since𝑓Λ ( ®𝑘 + ®𝑞) − 𝜉𝑘 𝑓 ±Λ 𝑓 ∼ 𝑣 𝑓 ,ΛΛ 𝑓 ; consequently, the Fermi velocity cancels out. The additional factor of 𝑘 𝑓 in the firstpart arises from the momentum integration. For momenta ®𝑘 + ®𝑞 lying inside the Λ 𝑓 -shell around the Fermi surface,the integrals in the first two lines represent the areas of the particle-hole and hole-particle transfer regions (see, e.g.,Fig. B.1 for the specific case where 𝑞 < Λ 𝑓 ). A rough estimate of this area yields ∼ 𝑘 𝑓Λ 𝑓 .kx 'ky '0q < ΛFFIG. B.1. Regions of integration in momentum space for | ®𝑞 | < Λ 𝑓 . Particles to holes transfers are shaded with lighter greencolor and holes to particles with darker green.The second part of the expression (proportional to 𝜂𝑣 𝑓 ) involves a factor of 𝑣2𝑓/𝑘2𝑓. At first glance, this terms appearsto be of higher order in the Λ 𝑓 /𝑘 𝑓 ≪ 1 limit because the third and fourth lines enter with opposite signs. However,estimating the magnitude of the momentum integrals for the particle-hole and hole-particle transfers inside the Λ 𝑓 -shellyields: ∫ 𝑘 𝑓 +Λ 𝑓𝑘 𝑓d𝑘 (𝑘3 − 𝑘 𝑘2𝑓 ) −∫ 𝑘 𝑓𝑘 𝑓 −Λ 𝑓d𝑘 (𝑘3 − 𝑘 𝑘2𝑓 ) ≈ 2𝑘2𝑓Λ2𝑓 . (B9)We therefore conclude that both parts — the one proportional to 1 − 𝜂𝑣 𝑓 and the one proportional to 𝜂𝑣 𝑓 — are ofthe same order in the limit 𝑘 𝑓 ≫ 𝑞 ∈ {Λ,Λ 𝑓 }. By introducing dimensionless variables, the flowing bubble diagramcan be rewritten as follows:X𝑠𝑐 (𝑄,Λ 𝑓 ) =A∗Λ𝑦𝑓𝑞[(1 − 𝜂𝑣 𝑓 ) ℎ(𝑡, 𝑤, 𝜖)+ 𝜂𝑣 𝑓 ℎ𝑘2(𝑡, 𝑤, 𝜖) ](B10)𝑡 :=Λ 𝑓𝑞, 𝑤 :=𝑍∗𝑓𝑞0𝑣∗𝑓𝑞𝑥, 𝜖 :=𝑞𝑘 𝑓(B11)Since we are interested in the limit of a large Fermi surface, we take 𝜖 → 0 to recover Eq. (56). This allows us todefine the scaling functions as:ℎ(𝑡, 𝑤) := − lim𝜖→0𝑞 𝑣2𝑓 ,Λ2𝑘 𝑓{ ∫ 2𝜋0d𝜙∫ 𝑘 𝑓 +Λ 𝑓𝑘 𝑓d𝑘 𝑘(𝜃[− 𝑓Λ ( ®𝑘 + ®𝑞)](𝑍 𝑓 𝑞0 + 𝑖𝜉𝑘 𝑓 +Λ − 𝑖 𝑓Λ ( ®𝑘 + ®𝑞))2 + (𝑞0 → −𝑞0))++∫ 2𝜋0d𝜙∫ 𝑘 𝑓𝑘 𝑓 −Λ 𝑓d𝑘 𝑘(𝜃[𝑓Λ ( ®𝑘 + ®𝑞)](𝑍 𝑓 𝑞0 + 𝑖 𝑓Λ ( ®𝑘 + ®𝑞) − 𝑖𝜉𝑘 𝑓 −Λ)2 + (𝑞0 → −𝑞0))},(B12)22andℎ𝑘2 (𝑡, 𝑤) := − lim𝜖→0𝑞 𝑣2𝑓 ,Λ4𝑘2𝑓Λ 𝑓{ ∫ 2𝜋0d𝜙∫ 𝑘 𝑓 +Λ 𝑓𝑘 𝑓d𝑘 𝑘 (𝑘2 − 𝑘2𝑓 )(𝜃[− 𝑓Λ ( ®𝑘 + ®𝑞)](𝑍 𝑓 𝑞0 + 𝑖𝜉𝑘 𝑓 +Λ − 𝑖 𝑓Λ ( ®𝑘 + ®𝑞))2 + (𝑞0 → −𝑞0))+−∫ 2𝜋0d𝜙∫ 𝑘 𝑓𝑘 𝑓 −Λ 𝑓d𝑘 𝑘 (𝑘2 − 𝑘2𝑓 )(𝜃[𝑓Λ ( ®𝑘 + ®𝑞)](𝑍 𝑓 𝑞0 + 𝑖 𝑓Λ ( ®𝑘 + ®𝑞) − 𝑖𝜉𝑘 𝑓 −Λ)2 + (𝑞0 → −𝑞0))}.(B13)Finally, we point out that due to the non-analytic nature of the regulator 𝑅 𝑓 , the integrals must be evaluated separatelyfor three distinct cases depending on the geometry shown in Fig. B.2. A detailed calculation for the non-self-consistentvariant X0 can be found in the Supplemental Material of Ref. [35].X𝑠𝑐 (𝑄,Λ 𝑓 ) =X< ( | ®𝑞 |, 𝑞0,Λ 𝑓 ), for | ®𝑞 | ≤ Λ 𝑓X𝑀 ( | ®𝑞 |, 𝑞0,Λ 𝑓 ), for | ®𝑞 | ∈ (Λ 𝑓 , 2Λ 𝑓 )X> ( | ®𝑞 |, 𝑞0,Λ 𝑓 ), for | ®𝑞 | ≥ 2Λ 𝑓. (B14)FIG. B.2. The regions of momentum space integration, which have non-zero contribution to the integrals. The integrand isconstant for the momentum vector ®𝑘 which lies in the green region due to cut-off function which regularizes the fermi dispersion.For a momentum vector which lies in the red region, the cut-off function becomes inactive. Black points define the geometryof the region. On these plots we shifted the origin of the axis by ®𝑞, which is convenient for calculations.1. ℎ(𝑡, 𝑤) and ℎ𝑘2 (𝑡, 𝑤)We present the scaling function after evaluating the momentum integration according to the notation in Eq. (B14):ℎ< (𝑡, 𝑤) =1𝑡216 − 4𝑤2𝑡−2𝑥(4 + 𝑤2𝑡−2𝑥)2 (B15)ℎ𝑀 (𝑡, 𝑤) = 1𝑡216 − 4𝑤2𝑡−2𝑥(4 + 𝑤2𝑡−2𝑥)2(1 − 2√1 − 𝑡2 + 2𝑡 arccos(𝑡))− 4𝑡∫ arccos(𝑡 )0d𝜙( 1 + 𝑡−1 cos(𝜙)𝑤2𝑡−2𝑥 + (1 + 𝑡−1 cos(𝜙))2+− 24 + 𝑤2𝑡−2𝑥) (B16)ℎ> (𝑡, 𝑤) =1𝑡216 − 4𝑤2𝑡−2𝑥(4 + 𝑤2𝑡−2𝑥)2(1 +√1 − 4𝑡2 − 2√1 − 𝑡2 − 2𝑡 (arccos(2𝑡) − arccos(𝑡)))+ 8𝑡arccos(𝑡) − arccos(2𝑡)4 + 𝑤2𝑡−2𝑥++ 4√1 + 𝑤2𝑡2−2𝑥arctan(√︂1 − 4𝑡21 + 𝑤2𝑡2−2𝑥)− 4𝑡2∫ arccos(𝑡 )0d𝜙𝑡 + cos(𝜙)𝑤2𝑡−2𝑥 + (1 + 𝑡−1 cos(𝜙))2(B17)23andℎ𝑘2 ,< (𝑡, 𝑤) =𝜋2𝑡34 − 𝑤2𝑡−2𝑥(4 + 𝑤2𝑡−2𝑥)2 (B18)ℎ𝑘2 ,𝑀 (𝑡, 𝑤) = 𝜋 + 4𝑡√1 − 𝑡2 − 4 arccos(𝑡)2𝑡34 − 𝑤2𝑡−2𝑥(4 + 𝑤2𝑡−2𝑥)2 − 2𝑡∫ arccos(𝑡 )0d𝜙 ln(𝑤2𝑡2−2𝑥 + (𝑡 + cos(𝜙))2𝑤2𝑡2−2𝑥 + 4𝑡2)++ 8𝑡2√1 − 𝑡2 − 𝑡 arccos(𝑡)4 + 𝑤2𝑡−2𝑥(B19)ℎ𝑘2 ,> (𝑡, 𝑤) =1𝑡34 − 𝑤2𝑡−2𝑥(4 + 𝑤2𝑡−2𝑥)2 (2𝑡 (1 − 𝑡2 − √1 − 5𝑡2 + 4𝑡4)√1 − 𝑡2+ 2 arcsin(𝑡) − arcsin(2𝑡))+−2𝑡∫ arccos(2𝑡 )0d𝜙[ln(𝑤2𝑡2−2𝑥 +(𝑡 + cos(𝜙))2𝑤2𝑡2−2𝑥 + cos2 (𝜙))− 2𝑡 cos(𝜙)𝑤2𝑡2−2𝑥 + cos2 (𝜙)]+−2𝑡∫ arccos(𝑡 )arccos(2𝑡 )d𝜙[ln(𝑤2𝑡−2𝑥 +(1 + 𝑡−1𝑐𝑜𝑠(𝜙))24 + 𝑤2𝑡−2𝑥)+ 4(𝑡 − cos(𝜙))𝑡 (4 + 𝑤2𝑡−2𝑥)].(B20)2. Static limit: ℎ𝑘2 (𝑡) and ℎ𝑘2 (𝑡)Below we present static limit of the scaling functions, ℎ(𝑡) ≔ ℎ(𝑡, 𝑤 = 0) and ℎ𝑘2 ≔ ℎ𝑘2 (𝑡, 𝑤 = 0):ℎ(𝑡) =1𝑡2, for 𝑡 ≥ 11𝑡2(1 − 2√1 − 𝑡2) + 4𝑡arccos(𝑡) + 4ln(𝑡)√1 − 𝑡2, for 𝑡 ∈ ( 12 , 1)4arctanh(√1 − 𝑡2) + 4ln(𝑡)√1 − 𝑡2+ 4(arccos(𝑡) − arccos(2𝑡))− 1𝑡2(2√1 − 𝑡2 −√1 − 4𝑡2 − 1), for 𝑡 ≤ 12,(B21)andℎ𝑘2 (𝑡) =𝜋8𝑡3, for 𝑡 ≥ 118𝑡3(𝜋 + 4𝑡√1 − 𝑡2 − 4 arccos(𝑡))+ 2𝑡2(√1 − 𝑡2 − 𝑡 arccos(𝑡))− 4𝑡∫ arccos(𝑡 )0d𝜙 ln( 𝑡 + cos(𝜙)2𝑡), for 𝑡 ∈ ( 12 , 1)5(1 − 𝑡2 −√1 − 5𝑡2 + 4𝑡4)2𝑡2√1 − 𝑡2+(arcsin(𝑡) − arcsin(2𝑡)) 1 + 8𝑡2 − 16𝑡2 ln(2𝑡)4𝑡3+ arcsin(𝑡)4𝑡3++4 arctanh(√1 − 4𝑡2) − 4𝑡∫ arccos(𝑡 )0d𝜙 ln(𝑡 + cos(𝜙))+ 4𝑡∫ arccos(2𝑡 )0d𝜙 ln(cos(𝜙)), for 𝑡 ≤ 12.(B22)Both functions have finite 𝑡 → 0 (𝑞 ≫ Λ 𝑓 ) limit:ℎ(𝑡 ≈ 0) = 3 + O(𝑡2) (B23)ℎ𝑘2 (𝑡 ≈ 0) = 32+ O(𝑡2) . (B24)Appendix C: 𝐵0 (𝑞0, 𝑞,Λ 𝑓 ) for small 𝜖Λ.We start from the Eq. (25). Since the function ℎ(𝑡, 𝑤) is an analytical function for any 𝑡 > 0, then 𝐵0 (𝑞0, 𝑞,Λ 𝑓 ) isalso an analytical function of 𝑞0 unless we take 𝜖Λ → 0 (𝑄Λ ≫ Λ 𝑓 ) in the integration limit. Therefore we study thebehavior of ℎ> (𝑡, 𝑤) for 𝑡 → 0. We recognize that the relevant term as the first line in Eq. (B20), which is factorizedinto the static and dynamical part. For 𝑡 ≪ 1 the static part can be expanded in 𝑡 and the subsequent terms take the24form:1𝑡216 − 4𝑤2𝑡−2(4 + 𝑤2𝑡−2)2(1 +√1 − 4𝑡2 − 2√1 − 𝑡2 − 2𝑡 (arccos(2𝑡) − arccos(𝑡)))≈ 16 − 4𝑤2𝑡−2(4 + 𝑤2𝑡−2)2 (1 + O(𝑡2)) (C1)8𝑡arccos(𝑡) − arccos(2𝑡)4 + 𝑤2𝑡−2𝑥≈ 84 + 𝑤2𝑡−2𝑥(1 + O(𝑡2)) . (C2)We approximate the scaling function by:ℎ> (𝑡, 𝑤) ≈16 − 4𝑤2𝑡−2(4 + 𝑤2𝑡−2)2 + 84 + 𝑤2𝑡−2𝑥, (C3)which posses a discontinuity at {𝑡 = 0, 𝑤 = 0}:lim𝑡→0lim𝑤→0ℎ(𝑡, 𝑤) = 3lim𝑤→0lim𝑡→0ℎ(𝑡, 𝑤) = 0 .(C4)The higher order terms in 𝑡 do not play any role. Therefore we plug the expanded ℎ> (𝑡, 𝑤) function into Eq. (25) andafter integration over 𝑡 we obtain the expression in Eq. (26):∫ Λ 𝑓𝑄Λ1/2d𝑡(16 − 4𝑤2𝑡−2(4 + 𝑤2𝑡−2)2 + 84 + 𝑤2𝑡−2𝑥)=[3𝑡 + 𝑡 𝑤24𝑡2 + 𝑤2− 2𝑤 arctan(2𝑡𝑤)] 𝑡=Λ 𝑓 /𝑄Λ𝑡=1/2. (C5)Using Eq. (B23) we can easily retrieve Eq. (22):𝐵0 (0, 𝑄Λ,Λ) = A∫ Λ 𝑓∞d𝜆ℎ(𝜆/𝑄Λ)𝑄Λ= A ©­«∫ 0∞d𝑡 ℎ(𝑡) +∫ Λ 𝑓𝑄Λ0d𝑡 ℎ(𝑡)ª®¬ ≈ Δ𝑚2𝑏,0 + 3A Λ 𝑓𝑄Λ= Δ𝑚2𝑏,0 + 2𝑄2Λ. (C6)To recover the Hertz-Millis theory [see Eq. (18)], we use a finite 𝑞 instead of 𝑄Λ, keeping in mind that Λ 𝑓 → 0 muchfaster than 𝑞. This validates the expansion of the static factors in ℎ> (𝑡, 𝑤) with respect to 𝑡. By construction, theHertz-Millis framework encounters the problem of a vanishing flowing mass term, since Λ 𝑓 = 0 while Λ ≠ 0.Appendix D: Flowing Fock diagramWe calculate the momentum integrals occurring in Eq. (32). We focus on the self-energy at the Fermi surface,Σ(𝑘0) = Σ(𝑘0, ®𝑘 = 𝑘 𝑓 𝑒𝑥):𝜕Λ 𝑓 Σ(𝑘0) =𝑔28𝜋3( ∫𝑄𝜕Λ 𝑓 𝑅𝑏 (𝑞) 𝐺2𝑏 (𝑄)𝐺 𝑓 (𝐾𝑄) +∫𝑄𝐺𝑏 (𝑄)𝜕Λ 𝑓 𝑅 𝑓 (𝑘 𝑓 𝑒𝑥 + ®𝑞) 𝐺2𝑓 (𝐾𝑄)), (D1)where 𝐾𝑄 = {𝑘0 − 𝑞0, 𝑘 𝑓 𝑒𝑥 + ®𝑞}. After applying the fermionic cut-off function introduced in Eq. (13) and bosonicregulator 𝑅𝑏 = (Λ2 − (𝑞 −𝑄Λ)2)𝜃 [Λ2 − (𝑞 −𝑄Λ)2] we find:𝜕Λ 𝑓 Σ(𝑘0) =𝑔28𝜋3∫ ∞−∞d𝑞0( ∫ Λ+𝑄Λ0d𝑞2𝑞(𝜕Λ 𝑓Λ)(Λ + 𝜕Λ𝑄Λ (𝑞 −𝑄Λ))(𝑚2𝑏+ Λ2 + (𝑞0 + 𝑘0)2 + 𝐵(𝑞0 + 𝑘0, 𝑞,Λ 𝑓 ))2 ∫ 2𝜋0d𝜙1𝑖𝑞0 + 𝑓 (𝑘 𝑓 + ®𝑞) + Σ(−𝑞0)+( ∫𝑞𝑜𝑢𝑡1𝑚2𝑏+ (| ®𝑞 | −𝑄Λ)2 + (𝑞0 + 𝑘0)2 + 𝐵(𝑞0 + 𝑘0, | ®𝑞 |,Λ 𝑓 ) + 𝑅𝑏 ( | ®𝑞 |)) 𝑣 𝑓(1 + Λ 𝑓𝑘 𝑓)(𝑖𝑞0 + 𝑣 𝑓Λ 𝑓 (1 +Λ 𝑓𝑘 𝑓) + Σ(−𝑞0))2−( ∫𝑞𝑖𝑛1𝑚2𝑏+ (| ®𝑞 | −𝑄Λ)2 + (𝑞0 + 𝑘0)2 + 𝐵(𝑞0 + 𝑘0, | ®𝑞 |,Λ 𝑓 ) + 𝑅𝑏 ( | ®𝑞 |)) 𝑣 𝑓(1 − Λ 𝑓𝑘 𝑓)(𝑖𝑞0 + 𝑣 𝑓Λ 𝑓 (1 −Λ 𝑓𝑘 𝑓) + Σ(−𝑞0))2 ),(D2)25where 𝑓 ( ®𝑘) = 𝜉 ®𝑘 + 𝑅 𝑓 ( ®𝑘) denotes the regularized fermionic dispersion. We also shifted the frequency domain by −𝑘0.The integral subscripts in the second and third lines denote the integration regions in the ®𝑞 space, which are shownin Fig. B.3. The vector ®𝑞𝑖𝑛 denotes the momentum for which fermionic momentum 𝑘 𝑓 𝑒𝑥 + ®𝑞 lies inside the Fermi sea,while for ®𝑞𝑜𝑢𝑡 fermionic momentum exceeds 𝑘 𝑓 . We neglect the real part of the fermionic self-energy and rewrite theflow equation for the imaginary part using Σ̃(𝑘0) := 𝑘0 − Im(Σ(𝑘0)):𝜕Λ 𝑓 Im(Σ(𝑘0)) = − 𝑔28𝜋3∫ ∞−∞d𝑞0( ∫ Λ+𝑄Λ0d𝑞2𝑞(𝜕Λ 𝑓Λ)(Λ + 𝜕Λ𝑄Λ (𝑞 −𝑄Λ))(𝑚2𝑏+ Λ2 + (𝑞0 + 𝑘0)2 + 𝐵(𝑞0 + 𝑘0, 𝑞,Λ 𝑓 ))2 ∫ 2𝜋0d𝜙Σ̃(𝑞0)Σ̃(𝑞0)2 + 𝑓 (𝑘 𝑓 + ®𝑞)2+( ∫𝑞𝑜𝑢𝑡1𝑚2𝑏+ (| ®𝑞 | −𝑄Λ)2 + (𝑞0 + 𝑘0)2 + 𝐵(𝑞0 + 𝑘0, | ®𝑞 |,Λ 𝑓 ) + 𝑅𝑏 ( | ®𝑞 |)) 𝑣 𝑓(1 + Λ 𝑓𝑘 𝑓)Σ̃(𝑞0)(Σ̃(𝑞0)2 + 𝑣2𝑓Λ 𝑓2 (1 + Λ 𝑓𝑘 𝑓)2)2−( ∫𝑞𝑖𝑛1𝑚2𝑏+ (| ®𝑞 | −𝑄Λ)2 + (𝑞0 + 𝑘0)2 + 𝐵(𝑞0 + 𝑘0, | ®𝑞 |,Λ 𝑓 ) + 𝑅𝑏 ( | ®𝑞 |)) 𝑣 𝑓(1 − Λ 𝑓𝑘 𝑓)Σ̃(𝑞0)(Σ̃(𝑞0)2 + 𝑣2𝑓Λ 𝑓2 (1 − Λ 𝑓𝑘 𝑓)2)2 ).(D3)We used the fact that Im(Σ(−𝑞0)) = − Im(Σ(𝑞0)). After the angular integration and expansion of the 𝐵(𝑄,Λ 𝑓 ):𝜕Λ 𝑓 Im(Σ(𝑘0)) = − 𝑔22𝜋3∫ ∞0d𝑞0{ ∫ Λ 𝑓0d𝑞𝜋𝑞(𝜕Λ 𝑓Λ)(Λ + 𝜕Λ𝑄Λ (𝑞 −𝑄Λ))(𝑚2𝑏+ 3𝑄2Λ+ 𝑞20 + A𝜋 |𝑞0 |𝑣 𝑓𝑄Λ)2 (𝐿1 (𝑞0 − 𝑘0, 𝑣 𝑓Λ 𝑓 ) − 𝐿1 (𝑞0 + 𝑘0, 𝑣 𝑓Λ 𝑓 ))++∫ Λ+𝑄ΛΛ 𝑓d𝑞𝑞(𝜙3 − 𝜙1) (𝜕Λ 𝑓Λ)(Λ + 𝜕Λ𝑄Λ (𝑞 −𝑄Λ))(𝑚2𝑏+ 3𝑄2Λ+ 𝑞20 + A𝜋 |𝑞0 |𝑣 𝑓𝑄Λ)2 (𝐿1 (𝑞0 − 𝑘0, 𝑣 𝑓Λ 𝑓 ) − 𝐿1 (𝑞0 + 𝑘0, 𝑣 𝑓Λ 𝑓 ))++ 2∫ Λ+𝑄ΛΛ 𝑓d𝑞(𝜕Λ 𝑓Λ)(Λ + 𝜕Λ𝑄Λ (𝑞 −𝑄Λ))𝑣 𝑓(𝑚2𝑏+ 3𝑄2Λ+ 𝑞20 + A𝜋 |𝑞0 |𝑣 𝑓𝑄Λ)2 (𝐼0( 𝑞0 − 𝑘0𝑣 𝑓 𝑞, 𝜙1)− 𝐼0( 𝑞0 + 𝑘0𝑣 𝑓 𝑞, 𝜙1))+ (D4)+∫ Λ 𝑓0d𝑞𝜋𝑣2𝑓𝑞Λ 𝑓𝑚2𝑏+ 3𝑄2Λ+ 𝑞20 + A𝜋 |𝑞0 |𝑣 𝑓𝑄Λ(𝐿2 (𝑞0 − 𝑘0, 𝑣 𝑓Λ 𝑓 ) − 𝐿2 (𝑞0 + 𝑘0, 𝑣 𝑓Λ 𝑓 ))++∫ Λ+𝑄𝑙Λ 𝑓d𝑞(𝜙3 − 𝜙1)𝑣2𝑓 𝑞Λ 𝑓𝑚2𝑏+ 3𝑄2Λ+ 𝑞20 + A𝜋 |𝑞0 |𝑣 𝑓𝑄Λ(𝐿2 (𝑞0 − 𝑘0, 𝑣 𝑓Λ 𝑓 ) − 𝐿2 (𝑞0 + 𝑘0, 𝑣 𝑓Λ 𝑓 ))}.We focus on the large 𝑘 𝑓 limit, where 𝜋 − 𝜙3 → 𝜙1. The definitions of the function 𝐿𝑛 and 𝐼0 are presented below:𝜙1 := arccos(Λ 𝑓𝑞+Λ 𝑓22𝑘 𝑓 𝑞− 𝑞2𝑘 𝑓)𝑘 𝑓≫Λ,Λ 𝑓−−−−−−−−→ arccos(Λ 𝑓𝑞)(D5a)𝜙3 := arccos(−Λ 𝑓𝑞+Λ 𝑓22𝑘 𝑓 𝑞− 𝑞2𝑘 𝑓)𝑘 𝑓≫Λ,Λ 𝑓−−−−−−−−→ arccos(−Λ 𝑓𝑞)(D5b)𝐿𝑛 (𝑎, 𝑏) :=𝑎(𝑎2 + 𝑏2)𝑛 (D5c)𝐼0 (𝑎, 𝜙) :=∫ 𝜙0d𝜙′𝑎𝑎2 + cos2 (𝜙′) =arctan(𝑎 tan(𝜙)√1+𝑎2)√1 + 𝑎2. (D5d)Appendix E: Numerical detailsNumerical results of the flow of fermionic self-energy presented in Sec. III.B. involves two components: solving aset of nonlinear differential equations and evaluation of 2-dimensional internal integrals changing with the flow.26kf2ΛfqoutqinΛFIG. B.3. Part of the regularized Fermi surface with marked regions of internal integration yielding nonvanisahing contributions.Here ®𝑞𝑖𝑛 denotes all vectors for which |𝑘 𝑓 𝑒𝑥 + ®𝑞 | < 𝑘 𝑓 and ®𝑞𝑜𝑢𝑡 for which |𝑘 𝑓 𝑒𝑥 + ®𝑞 | > 𝑘 𝑓 .To tackle the first task we used the embedded Runge-Kutta method with adaptive step size to minimize the numberof RHS evaluations. The absolute and relative tolerances which dictate the variation of step size were set to 10−12 and10−13 respectively. Disregarding the fermionic self-energy in the flow equation for bosonic correlation function allowsto calculate the flow of bosonic quantities, tune to the criticality with desired precision and then calculate the flow ofthe fermionic self-energy taking into account previously obtained flow of the bosonic parameters. We implemented anadaptive step size algorithm for the fermionic sector in a way that allows, if necessary, to use additional steps betweenthe time points implemented for the bosonic flow. Therefore the fermionic step size was always smaller or equal tothe bosonic step size.Evaluating the integrals required a detailed preceding analysis, especially concerning the Λ-dependence of theintegrands. In Eq. (D4) the integrands viewed as functions of internal frequency 𝑞0 exhibit peaks located at externalfrequency 𝑘0 of the width of order Λ2 or Λ3 depending on the external frequency. The frequency integral can beschematically written as𝐼𝑞0 =∫ ∞0d𝑞0 ( 𝑓 (𝑞0 + 𝑘0) − 𝑓 (𝑞0 − 𝑘0)) 𝑔(𝑞0) , (E1)where 𝑓 (𝑞0) is even with respect to its internal argument while 𝑔(𝑞0) is odd. Now we change the variables in such away that the peak of the integrand is located at the origin:𝐼 =∫ ∞0d𝑞0 𝑓(𝑞0 + 𝑘0) (𝑔(𝑞0)− 𝑔(𝑞0 + 2𝑘0) )− 𝑘0∫ 10d𝑡𝑞0 𝑓(𝑡𝑞0𝑘0)𝑔(𝑘0 (𝑡𝑞0 + 1))(E2)− 𝑘02∫ 10d𝑡𝑞0 𝑓( 𝑘02 𝑡𝑞0)𝑔( 𝑘02 (2 − 𝑡𝑞0))− 𝑘02∫ 10d𝑡𝑞0 𝑓( 𝑘02 (2 − 𝑡𝑞0))𝑔( 𝑘02 𝑡𝑞0). (E3)We split the integration domain into high frequency region with 𝑞0 > 5 and low frequency region 𝑞0 < 5. The highfrequency tail we evaluate after performing a substitution:𝑥 = tan(0.5( 𝜋2− arctan(5)) (𝑞0 + 1)). (E4)The low frequency region we divide into smaller subregions on logarithic scale. For each of the regions we implementGauss-Legendre quadrature with 60-200 nodes.27Appendix F: Tadpole auxiliary functionThe function 𝑓 (𝑡) we define as [see Eq. (67)]:𝑓(𝜆𝑞):= −𝑖𝑍2𝑓 ,𝜆𝜋2𝑞𝑣 𝑓{2∫𝐾𝑘0 𝑆 𝑓 ,𝜆 (𝐾)𝐺 𝑓 (𝐾, 𝜆)[𝐺 𝑓 (𝐾+𝑄, 𝜆)+𝐺 𝑓 (𝐾−𝑄, 𝜆)]+∫𝐾𝑘0𝐺2𝑓 (𝐾, 𝜆)[𝑆 𝑓 (𝐾+𝑄, 𝜆)+𝑆 𝑓 (𝐾−𝑄, 𝜆)]}(F1)We split the 𝑓 (𝑡) function into two parts:𝑓1 (𝑡) ≔ −𝑖𝑍2𝑓 ,𝜆𝜋2𝑞𝑣 𝑓2∫𝐾𝑘0 𝑆 𝑓 (𝐾, 𝜆)𝐺 𝑓 (𝐾, 𝜆)[𝐺 𝑓 (𝐾 +𝑄, 𝜆) + 𝐺 𝑓 (𝐾 −𝑄, 𝜆)](F2)𝑓2 (𝑡) ≔ −𝑖𝑍2𝑓 ,𝜆𝜋2𝑞𝑣 𝑓∫𝐾𝑘0𝐺2𝑓 (𝐾, 𝜆)[𝑆 𝑓 (𝐾 +𝑄, 𝜆) + 𝑆 𝑓 (𝐾 −𝑄, 𝜆)]. (F3)The final form is shown as:𝑓1 (𝑡) =14𝑡2, for 𝑡 ≥ 114𝑡2(1 − 2√1 − 𝑡2 + 2𝑡 arccos(𝑡))+ 3 arccos(𝑡)4𝑡−2(− 1 + 𝑡2 + 2(−2 + 𝑡2) log(𝑡))4(1 − 𝑡2)3/2, for 𝑡 ∈ ( 12 , 1)−14(2√1 − 4𝑡2 − 2√1 − 𝑡2+ −1 −√1 − 4𝑡2 + 2√1 − 𝑡2 + 2𝑡 arccos(2𝑡) − 2𝑡 arccos(𝑡)𝑡2)+−3(arcsin(𝑡) − arcsin(2𝑡))4𝑡+ 4 arctanh(√︂−1 + 21 + 2𝑡)+ (2 − 𝑡2) log(𝑡)(1 − 𝑡2)3/2, for 𝑡 ≤ 12(F4)and𝑓2 (𝑡) =0 , for 𝑡 ≥ 1arccos(𝑡)4𝑡+ 1 − 𝑡2 + 2 ln(𝑡)2(1 − 𝑡2)3/2, for 𝑡 ∈ ( 12 , 1)12(arcsin(2𝑡) − arcsin(𝑡)2𝑡+ 1√1 − 𝑡2−√1 − 4𝑡2 + 2 ln(𝑡)(1 − 𝑡2)3/2+ 4 arctanh(√︂1 − 2𝑡1 + 2𝑡)), for 𝑡 ≤ 12. (F5)FIG. B.4. Plot of the scaling function important for the analysis in Sec. IV.C.