Higgs criticality in a two-dimensional metal
HHiggs criticality in a two-dimensional metal
Debanjan Chowdhury and Subir Sachdev
1, 2 Department of Physics, Harvard University,Cambridge Massachusetts-02138, USA. Perimeter Institute of Theoretical Physics, Waterloo Ontario-N2L 2Y5, Canada. (Dated: March 12, 2015)
Abstract
We analyze a candidate theory for the strange metal near optimal hole-doping in the cuprate super-conductors. The theory contains a quantum phase transition between metals with large and small Fermisurfaces of spinless fermions carrying the electromagnetic charge of the electron, but the transition doesnot directly involve any broken global symmetries. The two metals have emergent SU(2) and U(1) gaugefields respectively, and the transition is driven by the condensation of a real Higgs field, carrying a finitelattice momentum and an adjoint SU(2) gauge charge. This Higgs field measures the local antiferromagneticcorrelations in a ‘rotating reference frame’. We propose a global phase diagram around this Higgs transition,and describe its relationship to a variety of recent experiments on the cuprate superconductors. a r X i v : . [ c ond - m a t . s t r- e l ] M a r . INTRODUCTION Several recent experiments have provided strong evidence for a dramatic change in the natureof the low temperature electronic state of the hole-doped cuprate superconductors near optimaldoping ( x = x c ). Moreover, zero field photoemission experiments carried out in the normal statehave seen evidence for a ‘large’ Fermi-surface for x > x c , consistent with the overall Luttingercount , and disconnected Fermi ‘arcs’ near the nodal regions for x < x c . At high fields, quantumoscillations also reveal a ‘large’ Fermi-surface for x > x c , but a closed electron-like Fermi-surfacewith an area that constitutes a small fraction of the entire Brillouin-zone for x < x c . It is thereforequite natural to associate the transition with decreasing x at x = x c with the loss of a ‘large’ Fermi-surface and the simultaneous opening of a pseudogap. There has also been significant experimentalprogress in understanding the structure of the density-wave ordering at lower doping, whichis likely responsible for the reconstructed electron-like Fermi-surface seen in quantum oscillationexperiments .In this paper we will use these advances to motivate and develop a previously proposed model for the physics of the strange metal near optimal doping. We argue that the rich phenomenologyobserved in the underdoped cuprates is primarily driven by a transition between non-Fermi liquidmetals with large and small Fermi surfaces which does not directly involve any broken globalsymmetry. All states with broken symmetry observed at low temperatures and low doping arenot part of the critical field theory , but are derived as low energy instabilities of the parentsmall Fermi surface phase. This diminished role for broken symmetries is consistent with absenceof any observed order with a significant correlation length at higher temperatures. We will alsoconstruct a global phase diagram to describe the many phases and crossovers around the strangemetal.A quantum phase transition which does not involve broken symmetries is necessarily associatedwith a topological change in the character of the ground state wavefunction. Emergent gauge fieldsare a powerful method of describing this topological structure, and they remain applicable alsoto the gapless metallic phases of interest to us here. Given the fundamental connection betweenemergent gauge fields and the size of the Fermi surface, which was established in Ref. 24 usingOshikawa’s method , we are naturally led to a quantum phase transition in which there is a changein the structure of the deconfined gauge excitations. Indeed, this describes a Higgs transition in a We shall ignore the subtleties associated with the presence of quenched disorder, except when it acts as a sourceof momentum decay for DC transport, as discussed later. .The primary new motivation for the model of Ref. 21 arises from our recent work analyzing the d -form factor density waves observed in scanning tunnelling microscopy and X-ray experiments .In this work , we argued that such density waves arise most naturally as an instability of a metallichigher temperature pseudogap state with small Fermi surfaces described as a ‘fractionalizedFermi liquid’ (FL*); other works with related ideas on the pseudogap are Refs. 30–36. Specifically,we used a theory of the FL* involving a background U(1) spin liquid with bosonic spinons :it is therefore convenient to dub this metallic state for the pseudogap as a U(1)-FL*. Theseresults are also easily extended to a Z spin liquid, and we will consider this case in Appendix A.The presence of a small Fermi surface without symmetry breaking requires topological order andemergent gauge fields , and so also a Higgs transition to the large Fermi surfaces at larger doping:here we provide a natural embedding of a FL* theory into such a transition, and we expect similarapproaches are possible for other possible topological orders in the underdoped regime.We now consider the evolution of the U(1)-FL*, and its small electronic Fermi surfaces, to theconventional ‘large’ Fermi surface Fermi liquid state at large doping. There is an existing conven-tional theory of the transformation from small to large Fermi surfaces driven by the disappearanceof antiferromagnetic order. This is a transition between two Fermi liquids, and the vicinity ofthe transition is described by the Hertz-Millis theory and its field-theoretic extensions ,as shown in Fig. 1. Here, we describe a detour from this direct route in which two new non-Fermi liquid phases appear between the conventional phases of Hertz-Millis theory. The detouris described by a SU(2) gauge theory, and the transition from small to large Fermi surfaces isnow a Higgs transition without any local order parameter, in which the emergent gauge structuredescribing the topological order in the ground state changes from U(1) to SU(2). The Higgs fieldof this transition is a measure of the local antiferromagnetic correlations in a rotating referenceframe to be introduced below in Eq. (1). However the Fermi surface excitations in this FL* phase carry the same quantum numbers as the electron, and donot couple minimally to the emergent (deconfined) gauge-fields. /g (A)AFM order with small Fermi pockets of electrons (B) Fermi liquid with large Fermi surface of electrons Increasing SDW order (C) U(1) ACL with small holon pockets (D) SU(2) ACL with large Fermi surface of spinless fermionsConventionalFermi liquidsHertz-Milliscriticalityof AFM order M Non-FermiliquidsHiggscriticalitywith no orderparameter
FIG. 1: Sketch of the metallic phases of our theory. Only phase A has a broken global symmetry, associatedwith the presence of long-range antiferromagnetic (AFM) order. The conventional Fermi liquid phases at thetop have a transition from small to large Fermi surfaces accompanied by the loss of AFM order. The dashedarrow represents a direct route between these phases, which could be a description of the electron-dopedcuprates. The full arrow around the point M is our proposed route with increasing doping in the hole-dopedcuprates. The U(1)-FL* descends from the U(1) ACL, as shown in Fig. 2. Note that the U(1)-FL* has a‘small’ Fermi surface of electrons due to the presence of topological order, while phase A above has a ‘small’Fermi surface of electrons because of translational symmetry breaking.
Note that the Higgs transition in Fig. 1 is between metallic states which we denote as ‘algebraiccharge liquids’ (ACL). The small and large Fermi surfaces in the ACLs are those of spinless fermionswhich carry the electromagnetic charge of the electron. For the U(1) ACL, a bound state formsbetween the spinless fermions and a spin S = 1 / , leading to small Fermi surfacesof fermionic quasiparticles carrying the same quantum numbers as the electron in the U(1)-FL*:so photoemission will detect a small Fermi surface of electrons in the U(1)-FL*. We anticipatethat similar effects are also present in the SU(2) ACL metal: there is a large density of statesof thermally excited S = 1 / II. OVERVIEW
Let us begin with a simplified picture of the optimal doping strange metal with a large Fermisurface. We consider a model of electrons c iα on the sites i of a square lattice, with α = ↑ , ↓ a SU(2)spin index. We transform the electrons to a rotating reference frame using a SU(2) rotation R i and (spinless-)fermions ψ i,p with p = ± , c i ↑ c i ↓ = R i ψ i, + ψ i, − , (1)where R † i R i = R i R † i = 1. Note that this representation immediately introduces a SU(2) gaugeinvariance (distinct from the global SU(2) spin rotation) ψ i, + ψ i, − → U i ψ i, + ψ i, − , R i → R i U † i , (2)under which the original electronic operators remain invariant, c iα → c iα ; here U i is a SU(2) gauge-transformation acting on the p = ± index. So the ψ p fermions are SU(2) gauge fundamentals,they carry the physical electromagnetic global U(1) charge, but they do not carry the SU(2) spinof the electron. The density of the ψ p is the same as that of the electrons. Such a rotatingreference frame perspective was used in the early work by Shraiman and Siggia on lightly-dopedantiferromagnets , and the importance of its gauge structure was clarified in Ref. 21.The strange metal is obtained by forming a large Fermi surface state of the ψ p fermions, while R i fluctuate isotropically over all SU(2) rotations with a moderate correlation length. This descriptionsuggests a simple trial wavefunction for this strange metal. Begin with a large Fermi surface (LFS) This allows us to describe phases without long-range antiferromagnetic order. ** U(1) FL* FL T * SU(2) ACL AF T U(1) ACL x d -BDW Higgs QCP SC FIG. 2: Our proposed phase diagram for the hole-doped cuprates, building on a theory for Higgs criticalityfor the optimal doping QCP. The green and red lines correspond to those in Fig. 1. The algebraic chargeliquids (ACLs) have Fermi surfaces of spinless ψ fermions which carry the electromagnetic charge: in theSU(2) ACL the Fermi surface is ‘large’ and is coupled to an emergent SU(2) gauge field, while in the U(1)ACL the Fermi surface is ‘small’ and coupled to an emergent U(1) gauge field. The fractionalized Fermiliquid (FL*) descends from the U(1) ACL by the binding of ψ fermions to neutral spinons. The d -BDW isthe d -form factor bond density wave, the SC is the d -wave superconductor, and the FL is the large Fermisurface Fermi liquid. We are not concerned here with the physics of the extremely underdoped region. Also,we expect that the crossovers within the superconducting phase will exhibit a ‘back-bending’ which isnot shown above, and which we do not discuss further here. The dashed lines at T ∗ and T ∗∗ are crossovers,while the Higgs QCP at T = 0 is a sharp phase transition. state of free ψ p fermions: (cid:89) k inside LFS , p = ± ψ † p ( k ) | (cid:105) . (3)Expand this out in position space, insert the inverse of Eq. (1) to write the wavefunction in terms6f R and the physical electrons c α , and finally average over R , to obtain ˆ (cid:89) i dR i W [ { R j } ] (cid:89) k inside LFS , p = ± (cid:34)(cid:88) i e i k · r i R iαp c † iα (cid:35) | (cid:105) , (4)where W is a variational weight-function of the R i , invariant under global spin rotations. For W = 1, we have a zero correlation length for R i , and we obtain a wavefunction for the c α involvingonly empty and doubly-occupied sites. With non-trivial W , the correlation length of R increases,we also build in spin singlet pairs of c α electrons on nearby sites. Comparing to the Gutzwiller-projected trial states commonly used for the underdoped cuprates , this wavefunction includesthe possibility of doubly-occupied sites and assigns different complex weights to the off-site singletpairs.For a more precise and complete description of the strange metal, which accounts for the gaugestructure in Eq. (1), we must turn to a quantum effective action for the ψ p which necessarilyincludes an emergent SU(2) gauge field. In the terminology of Ref. 38, such a theory of spinless,gapless fermions coupled to an emergent gauge field is an ‘algebraic charge liquid’ (ACL), andhence we have labeled the strange metal as SU(2) ACL in Fig. 2. This name implies that theSU(2) gauge symmetry is unbroken ( i.e. not ‘Higgsed’), and in such a situation the ψ p fermionshave a large Fermi surface with a shape similar to that of the electron Fermi surface in Fermi liquidstate at large doping.Now let us consider the transition to the U(1) ACL in Fig. 2. This is described by the con-densation of a real Higgs field H a , where a = 1 , , R (see also Ref. 52 for an illuminatinganalogy). The condensation of the Higgs field breaks the gauge symmetry from SU(2) to U(1) andreconstructs the ψ p Fermi surface from large to small. It is this Higgs transition which describesthe optimal doping QCP in Fig. 2, and analyzing its structure is the main purpose of the presentpaper. In the case where H a is complex, the Higgs phase can break the gauge symmetry downto Z , and we consider this case in Appendix A. The Shraiman-Siggia analyses of doped anti-ferromagnets were effectively within such a Higgs-condensed regime, and this obscured the gaugestructure of their formulation .Let us also note from Fig. 2 that the U(1) ACL is the parent of the U(1)-FL*. This wasdiscussed in Refs. 37,38, and will be reviewed below: the U(1)-FL* arises by the formation ofbound states between the spinons, R , and ψ fermions around the small Fermi surface. We expectthat a similar phenomenon also happens at low T in the SU(2) ACL at lower temperatures, so that7he photoemission largely reflects the structure of the large Fermi surface of the SU(2) ACL.The phase diagram in Fig. 2 is meant to be schematic; determining the exact nature of thevarious crossover and phase-transition lines is beyond the scope of this work. The Higgs transitionis present only at T = 0, and there is only a crossover at T > d − BDWvanish with the same power law as a function of ( x − x c ) at the QCP . Theoretically speaking,other possibilities are also allowed. A. Field theory
We now specify the imaginary time Lagrangian of the optimal doping QCP in Fig. 2, and itsvicinity. For now, the Lagrangian will not include the R bosons: we assume that R fluctuationsare short-ranged, but the associated spin-gap in the SU(2) ACL phase of Fig. 2 is small because ofproximity to the multi-critical point M in Fig. 1; we will include the R contributions in Section III.Then we have, L QCP = L ψ + L H + L Y . (5)The first term describes a large Fermi surface of ψ fermions minimally coupled to a SU(2) gaugefield A aµ = ( A aτ , A a ): L ψ = (cid:88) i ψ † i,p [( ∂ τ − µ ) δ pp (cid:48) + iA aτ σ app (cid:48) ] ψ i,p (cid:48) + (cid:88) i,j t ij ψ † i,p (cid:20) e iσ a A a · ( r i − r j ) (cid:21) pp (cid:48) ψ j,p (cid:48) , (6)where t ij are the fermion hopping parameters, r i are the spatial co-ordinates of the sites, µ is thechemical potential, and σ a are Pauli matrices acting on the SU(2) gauge indices.The Higgs Lagrangian is denoted L H , and it has a form familiar from its particle-physicsincarnations, L H = 12 ( ∂ τ H a − i(cid:15) abc A bτ H c ) + ˜v ∇ H a − i(cid:15) abc A b H c ) + s H a ) + u
24 [( H a ) ] . (7)The Higgs potential is determined by the parameters s and u , and transition across the QCP iscontrolled by the variation in s . As usual, for negative s , the Higgs field condenses, and this breaksthe gauge symmetry from SU(2) to U(1); and for positive s , the Higgs field is gapped, and thenthe SU(2) gauge symmetry remains unbroken. Corresponding to ‘case C’ in Ref. 53. L Y . As in particle-physics, this is a trilinear couplingbetween the Higgs field and the fermions, but now it has a different spatial structure: L Y = − λ H ai e i K · r i ψ † i,p σ app (cid:48) ψ i,p (cid:48) , (8)where K = ( π, π ) is the antiferromagnetic wavevector. This spatial structure indicates that H a transforms non-trivially under lattice translations: H a → H a e i K · a under translation by a ; (9)note that this is permitted because e i K · a = ± a . The transformation inEq. (9) arises from the role of the Higgs field as a measure of the antiferromagnetic correlationsin a rotating reference frame. In the presence of the Higgs condensate, this Yukawa couplingreconstructs the ψ Fermi surface from large to small, and the e i K · r i factor is crucial in the structureof this reconstruction. While in the particle physics context the Higgs condensate gives the fermionsa mass gap, here the fermions acquire a gap only on certain portions of the large Fermi surface,and a small Fermi surface of gapless fermions remains.We note that the effective gauge theory will also acquire a Yang-Mills term for the SU(2) gaugefield A a when high energy degrees of freedom are integrated out. As is well known in theories ofemergent gauge fields, such a term helps stabilize deconfined phases of the type considered here.We do not write this term out explicitly here, but will include its contributions in Section IV A,and specifically in the L A term in Eq. (21). B. DC transport
The body of our paper will describe a field theoretic analysis of the non-Fermi liquid proper-ties of L QCP . This combines recent progress in the theories of Fermi surfaces coupled to orderparameters and gauge fields . Here we mention one notable result on the electrical re-sistivity in the quantum-critical region of the Higgs transition. As in recent work on otherquantum critical points of metals, we consider the situation in which there is a strong momentumbottleneck i.e. there is rapid exchange of momentum between the fermionic and bosonic degreesof freedom, and the resistivity is determined by the rate of loss of momentum. In particular, it ispossible for the resistivity to be dominated by the scattering of neutral bosonic degrees of freedom,rather than that of charged fermionic excitations near the Fermi surface. In our model, we arguethat an important source for momentum decay is the coupling of the Higgs field to disorder L dis = V ( r ) [ H a ( r )] , (10)9here V ( r ) is quenched Gaussian random variable with (cid:104)(cid:104) V ( r ) (cid:105)(cid:105) = 0 ; (cid:104)(cid:104) V ( r ) V ( r (cid:48) ) (cid:105)(cid:105) = V δ ( r − r (cid:48) ) , (11)where the double angular brackets indicate an average over quenched disorder. Comparing withEq. (7), we see that V ( r ) can be viewed as a random local variation in the value of s , the tun-ing parameter which determines the position of the QCP. We will show that the analysis of thecontribution of L dis to the resistivity closely parallels the computation in Ref. 58 for the spin-density-wave quantum critical point. And as in Ref. 58, we find a resistivity for weak disorderwhich is proportional to V , ρ ( T ) ∼ V T − z ) /z , (12)where ∆ = d + z − ν − is the scaling dimension of the ( H a ) operator, ν is the correlation lengthexponent and z is the dynamical exponent. As we will see in Section IV B, this predicts a linear-in- T resistivity for the leading order values of the exponents.The outline for the rest of our paper is as follows. In Section III, we arrive at the above gauge-theoretic description starting from the theory of a metal with fluctuating antiferromagnetism anddiscuss the mean-field phase diagram as a function of the relevant tuning parameters. In SectionIV, we describe the properties of the QCP using a low-energy description of the Fermi-surfacecoupled to a gauge-field and the critical fluctuations of the Higgs’ field. Finally in Section V, wediscuss the relation of our proposed phase-diagram to the actual phase-diagram in the hole-dopedcuprates. Appendix A contains the extension to spiral order and Z gauge theory, while technicaldetails are in Appendix B. III. SU(2) GAUGE THEORY OF ANTIFERROMAGNETIC METALS
We summarize the derivation in Ref. 21 of the SU(2) gauge theory, starting from a model ofelectrons on the square lattice coupled to the fluctuations of collinear antiferromagnetism at thewavevector K = ( π, π ). The case of collinear antiferromagnetism at other wavevectors was alsoconsidered in Ref. 21, and we treat spiral antiferromagnets at incommensurate wavevectors inAppendix A.We begin with a model of electrons coupled to the quantum fluctuations of antiferromagnetism10epresented by the unit vector n i(cid:96) , with (cid:96) = x, y, z and (cid:80) (cid:96) n i(cid:96) = 1. The Lagrangian is given by L = L f + L n + L fn , L f = (cid:88) i c † iα [( ∂ τ − µ ) δ ij − t ij ] c jα , L n = 12 g (cid:20) ( ∂ τ n (cid:96) ) + v ( ∇ n (cid:96) ) (cid:21) , L fn = − λ (cid:88) i e i K · r i n i(cid:96) · c † iα σ (cid:96)αβ c iβ . (13)In the above g measures the strength of quantum fluctuations associated with the orientation of n (cid:96) , λ is an O (1) spin-fermion coupling and v is a characteristic spin-wave velocity.Now we insert the parametrization in Eq. (1) into Eq. (13) and proceed to derive an effectivetheory for ψ p and R . The formulation of the latter theory is aided by the introduction of a SU(2)gauge connection A aµ = ( A aτ , A a ). As is familiar in many discussions of emergent gauge fields incorrelated electron systems, this gauge field arises after decoupling hopping terms via an auxiliaryfield; here we skip these intermediate steps, and simply write down appropriate hopping terms forthe ψ p and R which are made gauge-invariant by suitable insertions of the gauge connection.With the parameterization in Eq. (1) we notice that the coupling L fn in Eq. (13) maps preciselyonto the Yukawa coupling in Eq. (8) with H ai σ app (cid:48) = n i(cid:96) R ∗ iαp σ (cid:96)αβ R iβp (cid:48) , (14)and so we define the Higgs field H ai by H ai ≡ n i(cid:96) Tr[ σ (cid:96) R i σ a R † i ] . (15)This identifies H a as the antiferromagnetic order in the rotating reference frame defined by Eq. (1).An important property of this definition is that the field H a is invariant under a global SU(2) spinrotation V , which rotates the direction of the physical electron spin and of the antiferromagneticorder, c i ↑ c i ↓ → V c i ↑ c i ↓ , R i → V R i . (16)Note that the SU(2) spin rotation is a left multiplication of R above, while the SU(2) gaugetransformation in Eq. (2) is a right multiplication of R . With these properties, Eq. (15) impliesthat H a transforms as a vector under the SU(2) gauge transformation in Eq. (2).11 U(2) gauge
SU(2) spin
U(1) e . m . charge c n ψ H R S are labelled by their dimension of 2 S + 1. The U(1) column contains the chargeunder the U(1) gauge field. We have now assembled all the steps taken after substituting Eq. (1) into Eq. (13). TheLagrangian of the resulting gauge theory is then obtained as L SU(2) = L QCP + L R , (17)where L QCP was described below Eq. (5) in Section II A, and L R is the Lagrangian for R . Thestructure of the latter is determined by the transformations of R in Eqs. (2) and Eq. (16). So wehave L R = 12 g Tr (cid:20) ( ∂ τ R − iA aτ Rσ a )( ∂ τ R † + iA aτ σ a R † ) + v ( ∇ R − i A a Rσ a )( ∇ R † + i A a σ a R † ) (cid:21) . (18)This completes our derivation of the SU(2) gauge theory.It is useful here to collect the transformations of the fields under the SU(2) gauge transformation,the global SU(2) spin rotation, and electromagnetic U(1) charge, as summarized in table I.Finally, we can make contact with other approaches by expressing R as R i = z i ↑ − z ∗ i ↓ z i ↓ z ∗ i ↑ , (19)with | z i ↑ | + | z i ↓ | = 1, but this parameterization will not be useful to us. Consider the situation inthe Higgs phase, where the field H a is condensed. Then we are free to choose a gauge in which theHiggs condensate is H a = (0 , , n i(cid:96) = 12 H ai Tr[ σ (cid:96) R i σ a R † i ]= z ∗ iα σ (cid:96)αβ z iβ for H a = (0 , , . (20)The last relationship is the familiar connection between the O(3) and CP variables, but note thatit holds here only within the phase where the Higgs field is condensed i.e. in the U(1) ACL.12 . Mean field phase diagram We now describe the phases of L SU(2) obtained in a simple mean field theory in which weallow condensates of the bosonic field R and H a . These phases are obtained by varying the tuningparameters s and g , and were shown in Fig. 1; in Fig. 3, we label the phases by their condensates.The phases are: s /g (A) AFM order with small Fermi pockets (B) Fermi liquid with large Fermi surface(C) U(1) ACL with small holon pockets (D) SU(2) ACL with large Fermi surface M h R i = 0 , h H a i = 0 h R i 6 = 0 , h H a i = 0 h R i 6 = 0 , h H a i 6 = 0 h R i = 0 , h H a i 6 = 0 FIG. 3: The phase diagram for the theory in Eq. (17) as a function of s and 1 /g (also shown in Fig. 1). Thecolor-coding of the phases corresponds to that in Fig. 2. The multicritical point, M, corresponds to g = g c and s = 0. This paper is concerned with the critical properties associated with the transition (C) ↔ (D). • The Higgs phase, labelled as (A) in Figs. 1,3, where both SU(2) spin and SU(2) gauge arebroken, leading to (cid:104) R (cid:105) (cid:54) = 0 , (cid:104) H a (cid:105) (cid:54) = 0. The gauge-excitations, ( A τ , A ), are gapped here.This phase describes the AFM-metal where the large Fermi-surface gets reconstructed intohole (and electron) pockets due to condensation of H a ∼ n , the N´eel order parameter. • The SU(2) confining phase, labelled as (B) in Figs. 1,3. Note that the SU(2) spin here re-mains unbroken. We have (cid:104) R (cid:105) (cid:54) = 0 , (cid:104) H a (cid:105) = 0, which is necessary to preserve spin-rotationinvariance since n = 0 from Eq. (15). This is the usual Fermi liquid phase, with a largeFermi-surface. • The Higgs phase, labelled as (C) in Figs. 1,3, where the SU(2) gauge is broken, but theSU(2) spin remains unbroken, leading to (cid:104) R (cid:105) = 0 , (cid:104) H a (cid:105) (cid:54) = 0. By recalling the physicalinterpretation of the fields, this amounts to a locally well developed amplitude of the AFM,without any long-range orientational order. We can choose H a ∼ (0 , ,
1) by carrying out13 gauge-transformation, which immediately implies that a U(1) subgroup of the SU(2) gauge remains unbroken, so that the A z photon remains gapless. Thus this phase describes a U(1)algebraic charge liquid , or, the holon-metal . However, due to the locally well developedAFM order, the Fermi-surface is reconstructed into ψ p holon pockets that are minimallycoupled to a U(1) gauge-field.As a function of temperature, there could be a continuous crossover from a U(1) ACL toa U(1) FL ∗ (or a “holon-hole” metal), where some of the holons ( ψ ± ) start forming boundstates with the gapped spinons ( z α ) . • The final phase (D) in Figs. 1,3 has the full symmetry, with none of the fields condensed: (cid:104) R (cid:105) = (cid:104) H a (cid:105) = 0. Instead of the above U(1) ACL, where only A z was gapless, in this phasethere are a triplet of gapless SU(2) photons coupled to a large Fermi-surface. This phasecan be described as a SU(2) algebraic charge liquid . Formally, this phase a spin gap, but weassume that T is greater than the gap in the metallic regions of Fig. 2 because of proximityto the point M in Fig. 1. At low enough T , this phase is unstable to superconductivity .We should emphasize that the above mean-field analysis has been rudimentary; e.g. we cannotrule out the possibility that higher order couplings could induce first-order transitions, that couldeven eliminate an intermediate phase.The next section shall present the theory for the interplay between the fluctuations of the gaugeand Higgs’ fields, within a low-energy field-theoretic formulation. IV. LOW-ENERGY FIELD THEORY
We are interested in studying the properties of the QCP between the SU(2) ACL and the U(1)ACL. At the QCP, s = 0, the entire Fermi-surface is coupled to the transverse fluctuations ofa SU(2) gauge field. There have been studies in the particle physics literature of Fermi surfacescoupled to non-Abelian gauge fields ; however these have been restricted to spatial dimension d = 3, where a RPA analysis gives almost the complete answer. In spatial dimension d = 2 ofinterest to us here, we shall follow the approach taken for Abelian gauge theories which uses apatch decomposition of the Fermi surface. The same approach transfers easily to the non-Abeliancase; indeed because of the Landau damping of the gauge bosons, there is little difference betweenthe Abelian and non-Abelian cases , as will also be clear from our analysis in Section IV A.Apart from their coupling to a SU(2) gauge field, the fermionic ψ p particles are also coupled14o a quantum critical Higgs field. This coupling is strongest at 8 ‘hot spots’ around the Fermisurface, and in Section IV B we shall be able to use the methods developed from the case of aspin-density-wave transition of Fermi liquids Some of the details of the computations appear in Appendix B. m =1 m =2 m =3 m =4 yx yx + FIG. 4: The shaded grey regions represent the occupied states. The transverse gauge-field fluctuationscouple strongly to the flavor current arising from the ψ ± patches and destroy the ψ quasiparticles all aroundthe Fermi-surface. Across the Higgs transition, the fluctuations of the H a field couple most strongly to thefour-pairs ( m = 1 , ..,
4) of “hot-spots”, shown as the filled circles.
A. Fermi-surface coupled to gauge-field
Here we describe the low energy theory of the SU(2) ACL, away from the Higgs condensationat the QCP. We need only consider a SU(2) gauge field coupled to the large Fermi surface of the ψ p fermions. As in the U(1) case , we can make a patch decomposition of the Fermi surface,and treat antipodal pairs of patches separately. For a single pair of antipodal patches, we havethe fermions ψ ± p (see Fig. 4), with ± the patch index, and p the usual SU(2) gauge index. This iscoupled to the transverse components of the SU(2) gauge field, A a .15 = L f + L A + L int ,L f = ψ † + p ( ∂ τ − i∂ x − ∂ y ) ψ + p + ψ †− p ( ∂ τ + i∂ x − ∂ y ) ψ − p ,L A = 12 e ( ∂ y A ax ) ,L int = A ax ( ψ † + p σ app (cid:48) ψ + p (cid:48) − ψ †− p σ app (cid:48) ψ − p (cid:48) ) (21)Let us review the one-loop renormalization of the gauge and fermionic matter fields. We startby looking at the self-energy of the gauge-field due to the particle-hole bubble (Fig. 5a). We have,Π A ( q ) = 2 (cid:88) s ˆ d(cid:96) τ d (cid:96) (2 π ) G s ( (cid:96) ) G s ( (cid:96) + q ) , (22)where (cid:96) = ( (cid:96) τ , (cid:96) ) and the bare fermionic propagator is given by, G s ( (cid:96) ) = 1 − i(cid:96) τ + s(cid:96) x + (cid:96) y . (23)The final result is of the form ,Π A ( q ) = c b | q τ || q y | , where c b = 12 π . (24)The computations are summarized in Appendix B 1. FIG. 5: One loop contributions to the (a) gauge-field, and, (b) Fermion self-energies. Curly lines representthe A propagators, D ( (cid:96) ), while solid lines represent the ψ propagators, G ( (cid:96) ). Computing the fermionic self-energy due to the bosonic-propagator dressed with the RPA levelpolarization bubble (Fig. 5b) leads to,Σ s,pp (cid:48) ( k ) = − σ apα σ aαp (cid:48) ˆ d(cid:96) τ d (cid:96) (2 π ) D ( (cid:96) ) G s ( k − (cid:96) ) , (25)= − δ pp (cid:48) ˆ d(cid:96) τ d (cid:96) (2 π ) D ( (cid:96) ) G s ( k − (cid:96) ) , (26) We note that since the fermions are strictly in two-dimensions, the non-universal factor of Λ, the UV cutoff, dropsout. The factor that appears in general is of the form Λ d − , where d is the number of space-dimensions. D ( (cid:96) ) is the gauge-field propagator, D − ( (cid:96) ) = (cid:18) c b | (cid:96) τ || (cid:96) y | + 1 e (cid:96) y (cid:19) . (27)We then obtain, Σ s ( k ) = − i ˆ d(cid:96) τ d(cid:96) y (2 π ) sgn( k τ − (cid:96) τ ) c b | (cid:96) τ | / | (cid:96) y | + (cid:96) y /e , (28)Σ s ( k ) = − ic f sgn( k τ ) | k τ | / , where c f = 2 √ (cid:18) e π (cid:19) / . (29)This self-energy contribution is larger than the bare ∂ τ term at low energies. Therefore, uponincluding the RPA contribution into the fermionic propagator, we have, G s ( (cid:96) ) = 1 − ic f sgn( (cid:96) τ ) | (cid:96) τ | / + s(cid:96) x + (cid:96) y , (30)which is the well known result for the quasiparticles being damped all along the Fermi-surface. B. Higgs criticality at the QCP
Now we consider the QCP at which the Higgs boson condensed from the non-Fermi liquid SU(2)ACL state described in the previous subsection. Across this Higgs transition from the SU(2) ACLto the U(1) ACL, the Fermi-surface gets reconstructed—this is controlled by the real Higgs field, H a , which carries lattice momentum, K = ( π, π ). By the same arguments used for the onset ofspin-density-wave order in a Fermi liquid , the low energy physics of the QCP is dominated bythe vicinity of the hot-spots: these are points on the Fermi surface which are connected by K (seeFig. 4). The computation for the present non-Fermi liquid Fermi surface proceeds just as for theFermi liquid case, by linearizing the bare dispersion for the fermions around the hot spots: L = L hs + L H + L fH ,L hs = ψ † m p ( ∂ τ − i v m · ∇ ) ψ m p + ψ † m p ( ∂ τ − i v m · ∇ ) ψ m p ,L fH = 1 (cid:112) N f H a · ( ψ † m p σ app (cid:48) ψ m p (cid:48) + ψ † m p σ app (cid:48) ψ m p (cid:48) ) , (31)where L H already appeared in Eq. (7); m is the hot-spot pair index (Fig. 4).Let us first look at the one-loop self energy of the H a field (Fig. 6a). This is given by,Π H ( q ) = 2 (cid:88) m ˆ d(cid:96) τ d (cid:96) (2 π ) (cid:20) G m ( (cid:96) + q ) G m ( (cid:96) ) + G m ( (cid:96) + q ) G m ( (cid:96) ) (cid:21) , (32)17 IG. 6: One loop contributions to the (a) Higgs-field, and, (b) Fermion self-energies. The dashed linesrepresent the Higgs’ field propagator, χ . where we now use the non-Fermi liquid fermion Green’s function renormalized by the gauge fieldfluctuations, as discussed in Section IV A: G mα ( (cid:96) ) = 1 − ic f sgn( (cid:96) τ ) | (cid:96) τ | / − v α · (cid:96) . (33)Note the z = 3 / ∼ ∂ τ ).Upon including contributions from all pairs of hot-spots, we obtain (see Appendix B 2),Π H ( q ) = Π H ( q = 0) + γ | q τ | , where γ = n πv x v y , (34)where n = 4 is the number of pairs of hot spots. Note that the c f dependence has completelydropped out and the above result is precisely the expression that we would have obtained if we hadstarted with the bare fermion Green’s functions (or, any anomalous power ∼ | (cid:96) τ | β ). This result isnot surprising—it just reproduces the “Landau-damped” form of the propagator for H a . As weknow, the only requirement for the appearance of Landau-damping is the existence of particle-holeexcitations around the Fermi-surface in the limit of ω →
0. In the general case, this always leads to ∼ | q τ | / | q y | for a bosonic order-parameter coupled to a fermion-bilinear. When the order parameteritself carries a finite momentum K , as is the case here, then the denominator in the damping termgets cut off and leads to ∼ | q τ | .Equipped with the above expression, let us now compute the self-energy of the fermions in thevicinity of the hot-spots (Fig. 6b),Σ ,pp (cid:48) ( p ) = σ apα σ aαp (cid:48) ˆ d(cid:96) τ d (cid:96) (2 π ) G ( p − (cid:96) ) χ ( (cid:96) ) , (35)= 3 δ pp (cid:48) ˆ d(cid:96) τ d (cid:96) (2 π ) G ( p − (cid:96) ) χ ( (cid:96) ) , (36)where the propagator tuned to the critical point ( s = 0) is given by : χ − ( (cid:96) ) = ( γ | (cid:96) τ | + (cid:96) ).18e are interested in the singular power law frequency dependence of the self-energy at the hot-spots. For future use, it is useful to express the Green’s function in Eq. (33) in the more generalform G mα ( (cid:96) ) = 1 − iζ f sgn( (cid:96) τ ) | (cid:96) τ | β − v α · (cid:96) , (37)where the exponent β = 2 / p = 0) becomes (see AppendixB 3), Σ ( p τ ) = 3 i ˆ d(cid:96) τ π tan − (cid:18) (cid:113) γv | (cid:96) τ | − ζ f | p τ − (cid:96) τ | β ζ f | p τ − (cid:96) τ | β (cid:19) sgn( (cid:96) τ − p τ ) (cid:113) v γ | (cid:96) τ | − ζ f | p τ − (cid:96) τ | β , (38)which correctly reproduces Σ ( p τ = 0) = 0. Furthermore, note that if ζ f = 0, i.e. if the anomalousself-energy contribution were to be absent, then the above reduces to the well known form Σ ( p τ ) = 3 i ˆ d(cid:96) τ π sgn( (cid:96) τ − p τ ) (cid:112) v γ | (cid:96) τ | = − i π (cid:112) γv sgn( p τ ) (cid:112) | p τ | , (39)reproducing the z = 2 result. Let us now proceed to evaluate the expression in the presence of afinite ζ f . Rescaling (cid:96) τ = xp τ leads to,Σ ( p τ ) = 3 i π (cid:112) γv sgn( p τ ) | p τ | − β × ˆ dx π tan − (cid:18) (cid:112) | p τ | − β | x | − c | − x | β √ c | − x | β (cid:19) sgn( x − (cid:112) | p τ | − β | x | − c | − x | β , (40)where the dimensionless parameter, c = ζ f /γv . An asymptotic analysis of Eq. (40) shows thatΣ ( | p τ | → ∼ − i sgn( p τ ) (cid:112) | p τ | , for β ≥ /
2. (41)So the low energy singularity of the self-energy is independent of the non-Fermi liquid exponent β in the fermion Green’s function, and has the same value as in the spin density wave case withoutthe gauge field. This is the key observation of the present subsection. To estimate the co-efficient,we can use a self-consistent approach in which we use a self energy in Eq. (37) with β = 1 / β = 2 /
3. Assembling all the constraints, the final expressiontakes the following z = 2 formΣ ( p τ ) = 3 i π (cid:112) γv I ( c ) sgn( p τ ) (cid:112) | p τ | , (42)19here the function I ( c ) is defined in Appendix B 4 and I ( c →
0) = − z = 2, while the singularities arising from the SU(2) gauge field coupling around theFermi surface have z = 3 /
2. At a given length scale, the contributions with the larger z dominatebecause they have a lower energy. Hence the Higgs criticality of a non-Fermi liquid maps onto thespin density wave criticality of a Fermi liquid.With this conclusion in hand, we can now directly apply the results of Ref. 58 on the DCresistivity to the Higgs QCP. The approach of Ref. 58 requires that there is quasiparticle breakdownaround the entire Fermi surface, and the fermionic excitations rapidly equilibriate with all thebosonic modes. While this was only marginally true for the spin-density-wave quantum criticalpoint considered in Ref. 58, it is easily satisfied for the Higgs QCP being considered here: the SU(2)gauge field makes the entire Fermi surface “hot”, while the Higgs field fluctuations induce additionalfermion damping at the hot spots on the Fermi surface. As in the previous case, it is possible fordisorder to couple to the square of the Higgs field because such an operator is gauge-invariant, aswe noted in Eq. (10). And the corresponding contribution to the resistivity is in Eq. (12). For theexponents d = 2, z = 2, and ν = 1 / ρ ( T ) ∼ V T . V. DISCUSSION
The primary goal of this paper has been to propose a candidate theory for the quantum phasetransition near optimal doping in the cuprates. We analyzed the QCP between metals with ‘large’and ‘small’ Fermi-surfaces, which did not involve any broken global symmetries, but instead in-volved a Higgs’ transition between metals with emergent SU(2) and U(1) gauge fields. The Higgsfield acts as a measure of the local antiferromagnetic order in the rotating reference frame definedby Eq. (1). As we discussed in Sections I and II, the symmetry broken phases observed in theunderdoped cuprates arise as low temperature instabilities of the ‘small’ Fermi-surface metal.The underlying QCP we studied was between two metals (the U(1) ACL and the SU(2) ACL)in which the Fermi surface excitations are coupled to emergent gauge fields, and so there are noLandau quasiparticles. However, electron-like quasiparticles do re-emerge around a small Fermiin the U(1)-FL*, and we will discuss similar features around the large Fermi surface in the SU(2)20CL below. The reconstruction to the ‘small’ Fermi-surface in the ACL phases is driven by thecondensation of the Higgs field, and the Higgs critical point has additional singular structure in thevicinity of the “hot-spots”. The Higgs criticality has associated with it an interplay of both z = 3 / z = 2 physics in the vicinity of the hot-spots. We showedthat near the Higgs QCP the z = 2 physics dominates, and hence many critical properties maponto the previously studied problem of the onset of spin density wave order in a Fermi liquid .Let us now conclude with a discussion of the relationship of our proposed phase diagram inFig. 2 to the experimentally obtained phase diagram in the non-La-based cuprates. The d − SCand d − BDW both arise as instabilities of the U(1) FL*, as has been discussed in Refs. 27,62 (theSU(2) ACL is also unstable to superconductivity ). The high temperature pseudogap phase for T ∗∗ < T < T ∗ is a U(1) ACL or more appropriately a holon-hole metal . There is a crossover tothe U(1) FL* phase at T = T ∗∗ , where all the holons have formed bound-states with the spinons.An important feature of the U(1) FL* phase is that its transport and photoemission signatures aremostly identical to those of a Fermi liquid. The primary difference from a Landau Fermi liquid isthat the volume enclosed by the Fermi surface is proportional to the density of holes, x , and notto the Luttinger density 1 + x . The U(1) FL* phase also has an emergent U(1) gauge field, asrequired by the topological arguments in Ref. 24, but the Fermi surface quasiparticles are gaugeneutral. The recent remarkable observation of Fermi liquid transport properties in the pseudogapphase of Hg1201 below T ∗∗ , with some possibly non-Fermi liquid behavior between T ∗∗ and T ∗ , can therefore be viewed as strong support for the existence of a FL* derived out of a parentACL. In particular, the alternative ‘fluctuating order’ picture of the pseudogap does not naturallylead to such temperature dependent crossovers from non-Fermi liquid to Fermi liquid regimes.For the La-based cuprates, there is a larger doping regime with magnetic order, overlappingwith the regime of charge order. This can be accommodated in our phase diagram by movingthe full red arrow in Fig. 1 just to the other side of the point M, and allowing for incommensurateorder as in Appendix A.An important challenge for future experiments is to detect direct experimental signatures ofthe complete small Fermi surface of the proposed FL* phase. We presume that it is the smallquasiparticle residue on the ‘back side’ of the small Fermi surface which is responsible for thearc-like features in the photoemission spectrum . Therefore, we need a probe which does notinvolve adding or removing an electron from the sample, and so is not sensitive to the quasiparticleresidue. Possibilities are Friedel oscillations, the Kohn anomaly, or ultrasonic attenuation.Within our proposed phase diagram in Fig. 2, the strange metal phase is to be viewed as a21U(2) ACL at the Higgs critical point, and proximate to the multicritical point M to ensure thespin gap is smaller than T . The DC transport properties of this phase are controlled by thecoupling of the gauge-invariant square of the Higgs field to long wavelength disorder, followingan analysis of Ref. 58 for the spin density wave critical point. Such a coupling leads to a linear-in-temperature resistivity. Also, as emphasized in Ref. 58, the residual resistivity is proportionalto short wavelength disorder which can scatter fermions across the Fermi surface. So there isno direct relationship between the residual resistivity and the slope of the linear resistivity. Itwould be interesting in future work to explore the role of intrinsic umklapp scattering events inthe transport properties of such strange metals in the strong coupling regime.The electron spectral function in the SU(2) ACL is a convolution of the spectra of the ψ fermionsand the R bosons. As in the computation in Ref. 37, we assume the R spectrum is thermallyoverdamped (because of the proximity to M), and the electron spectral function primarily reflectsthe ψ spectrum; we also expect precursors of the bound state formation between the ψ and the R to enhance the ψ features in the electron spectrum , just as in the U(1)-FL*. Then the electronspectral functions should have an anisotropic structure around the Fermi surface, with the weakergauge field-induced damping in the nodal region, and the stronger Higgs field-induced damping inthe anti-nodal region. Also note that while the Higgs field coupling does show up in the resistivityas discussed above, the gauge fields coupling has a weaker effect on transport. This is becausegauge-invariance prevents a non-derivative coupling between the gauge field and perturbationsthat violate momentum conversation. An important open question is whether this rich theoreticalstructure can be made consistent with the complex experimental features of the conductivity andmagnetotransport in the strange metal .Our linear- T resistivity is proportional to disorder, as in the previous model in Ref. 58. However,because the disorder couples to the Higgs field, the relevant disorder is long-wavelength. This isin contrast to short wavelength disorder, which can lead to efficient large momentum scattering offermions around the Fermi surface. Modifying the coefficient of the resistivity therefore requiresmodifying long-wavelength disorder, and this may be difficult to do because of the intrinsic disorderfrom the dopant ions. Inducing short-wavelength disorder, by including e.g. Zn impurities, maynot be effective in modifying the co-efficient of the linear- T resistivity. These features can act astests of our proposed mechanism for the resistivity of the strange metal .Finally, we note from Figs. 2 and 1,3 that the SU(2) ACL survives for an extended region beyondthe Higgs QCP. This implies strange metal behavior over a finite range of doping as T →
0, andnot only at a single QCP. Transport measurements in magnetic fields which have suppressed22uperconductivity appear to be consistent with such a non-Fermi liquid phase. Acknowledgments
We thank Seamus Davis and Martin Greven for useful discussions. D.C. is supported by theHarvard-GSAS Merit fellowship. This research was supported by the NSF under Grant DMR-1360789, the Templeton foundation, and MURI grant W911NF-14-1-0003 from ARO. Research atPerimeter Institute is supported by the Government of Canada through Industry Canada and bythe Province of Ontario through the Ministry of Research and Innovation.
Appendix A: Spiral order and Z gauge theory Here we generalize the theory in Eq. (13) to the case of spiral spin order at an incommensuratewavevector K . In this case the antiferromagnetic order is not characterized by a single unit vector n (cid:96) , but by two orthogonal unit vectors n (cid:96) and n (cid:96) which obey n (cid:96) = n (cid:96) = 1 , n (cid:96) n (cid:96) = 0 . (A1)The spin-fermion coupling to the electrons in Eq. (13) is replaced by L fn = − λ (cid:88) i [ n i (cid:96) cos ( K · r i ) + n i (cid:96) sin ( K · r i )] · c † iα σ (cid:96)αβ c iβ , (A2)After the change of variables in Eq. (1), this leads to the Yukawa coupling L Y = − λ (cid:0) H a ∗ i e i K · r i + H ai e − i K · r i (cid:1) ψ † i,p σ app (cid:48) ψ i,p (cid:48) , (A3)where, in contrast to Eq. (8), the Higgs field H a is now complex and is defined by H a ≡
12 ( n (cid:96) + in (cid:96) ) Tr[ σ (cid:96) R i σ a R † i ] , (A4)generalizing Eq. (15). It is now also clear from Eq. (A3) that under translation by a distance a ,the Higgs field transforms as in Eq. (9), where e i K · a can now be complex.The structure of SU(2) gauge theory with a complex Higgs field remains essentially the same asfor the real Higgs discussed in the body of the paper, with one important distinction. The quarticterm in Eq. (7) is replaced by two terms u [ H a ∗ H a ] + u [ H a ] [ H b ∗ ] , (A5)23nd the presence of spiral order requires that u >
0. Then in the Higgs phase, the minimumenergy condensate can always be oriented so that H a = (1 , i, . (A6)Such a Higgs condensate breaks the SU(2) gauge symmetry all the way down to Z . And usingEq. (19), the analog of the relationship in Eq. (20) for the orientation of the spiral order is n (cid:96) + in (cid:96) = 12 H a Tr[ σ (cid:96) Rσ a R † ]= − ε αγ z γ σ (cid:96)αβ z β for H a = (1 , i, . (A7)This co-incides with the conventional representation of the spiral orientation in terms of spinons z α , and it can verified that the values in Eq. (A7) obey Eq. (A1).For the case with u < H a = e iθ (1 , , , (A8)where θ is an arbitrary phase. This corresponds to incommensurate collinear spin order . Appendix B: Feynman diagram computations1. Self-energy: Gauge-field
Since we are interested in the singular structure of Π A ( q ) in Eq. (22), we shall evaluate theintegral over (cid:96) x first, followed by (cid:96) τ , (cid:96) y . Therefore,Π A ( q ) = 2 ˆ d(cid:96) τ d(cid:96) y (2 π ) i [ θ ( (cid:96) τ ) − θ ( (cid:96) τ + q τ )] − iηq τ + q x + q y + 2 (cid:96) y q y + q → − q , (B1)= q τ π ˆ d(cid:96) y (2 π ) − i − iηq τ + q x + q y + 2 (cid:96) y q y + q → − q , (B2)= | q τ | π | q y | . (B3)This leads to the expression for Π A ( q ) in Eq. (24).
2. Self-energy: Higgs’ field
Focusing on just the m = 1 contribution, Eq. (32) becomes,Π Hm =1 ( q ) = 2 ˆ d(cid:96) τ d (cid:96) (2 π ) (cid:20) − ic f sgn( (cid:96) τ + q τ ) | (cid:96) τ + q τ | / − v · ( (cid:96) + q ) 1 − ic f sgn( (cid:96) τ ) | (cid:96) τ | / − v · (cid:96) + q → − q (cid:21) . (B4)24et us define (cid:96) = v · ( (cid:96) + q ) and (cid:96) = v · (cid:96) , so thatΠ Hm =1 ( q ) = 1 v x v y ˆ d(cid:96) τ d(cid:96) d(cid:96) (2 π ) (cid:20) − ic f sgn( (cid:96) τ + q τ ) | (cid:96) τ + q τ | / − (cid:96) − ic f sgn( (cid:96) τ ) | (cid:96) τ | / − (cid:96) + q → − q (cid:21) . (B5)It is not hard to see that the only non-zero contribution comes from the imaginary parts of boththe terms. Then,Π Hm =1 ( q ) = 1 v x v y ˆ d(cid:96) τ d(cid:96) d(cid:96) (2 π ) (cid:20) ic f sgn( (cid:96) τ + q τ ) | (cid:96) τ + q τ | / c f | (cid:96) τ + q τ | / + (cid:96) ic f sgn( (cid:96) τ ) | (cid:96) τ | / c f | (cid:96) τ | / + (cid:96) + q → − q (cid:21) . (B6)Upon carrying out the (cid:96) , (cid:96) − integrals, this becomes,Π Hm =1 ( q ) = − v x v y ˆ d(cid:96) τ π [sgn( (cid:96) τ + q τ ) sgn( (cid:96) τ ) + q → − q ] . (B7)This directly leads to the expression for Π H ( q ) in Eq. (34).
3. Fermion self-energy at the hot-spot
The Fermionic self-energy due to the Higgs’ field fluctuations (Eq. (36)) becomes,Σ ( p ) = 3 ˆ d(cid:96) τ d (cid:96) (2 π ) − iζ f sgn( p τ − (cid:96) τ ) | p τ − (cid:96) τ | β − v · ( p − (cid:96) ) 1 γ | (cid:96) τ | + (cid:96) . (B8)Let us now change coordinates such that (cid:96) ⊥ = ˆ v · (cid:96) and (cid:96) (cid:107) is the component along the Fermi-surfaceof ψ . Then,Σ ( p ) = − ˆ d(cid:96) τ d(cid:96) ⊥ d(cid:96) (cid:107) (2 π ) − iζ f sgn( p τ − (cid:96) τ ) | p τ − (cid:96) τ | β − v · p + v (cid:96) ⊥ γ | (cid:96) τ | + (cid:96) ⊥ + (cid:96) (cid:107) . (B9)It is straightforward to carry out the integral over (cid:96) (cid:107) , which gives,Σ ( p ) = − ˆ d(cid:96) τ d(cid:96) ⊥ (2 π ) − iζ f sgn( p τ − (cid:96) τ ) | p τ − (cid:96) τ | β − v · p + v (cid:96) ⊥ (cid:113) γ | (cid:96) τ | + (cid:96) ⊥ . (B10)Let us now study the form of the self-energy at the hot-spot, p = 0, and extract the p τ dependence.We can symmetrize the above form then to give,Σ ( p τ ) = 3 i ˆ d(cid:96) τ π ˆ ∞ d(cid:96) ⊥ π ζ f sgn( (cid:96) τ − p τ ) | p τ − (cid:96) τ | β ζ f | p τ − (cid:96) τ | β + v (cid:96) ⊥ (cid:113) γ | (cid:96) τ | + (cid:96) ⊥ . (B11)Carrying out the integral over (cid:96) ⊥ (see Appendix B 4) leads to the expression in Eq. (38).25 ( c ) c FIG. 7: I ( c ) evaluated numerically as a function of c . Note that I ( c →
0) = −
1, which reproduces Eq.39.
4. Integrals
We use the integral (for a > , b > ˆ ∞ dx x + a √ x + b = 1 a √ b − a tan − (cid:18) √ b − a a (cid:19) . (B12)The above is valid irrespective of whether a > b or a < b .The integral in Eq. (40) can be evaluated as a function of the dimensionless parameter, c = ζ f /γv , when β = 1 /
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