Cascade of phase transitions in the vicinity of a quantum critical point
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Cascade of phase transitions in the vicinity of a quantum critical point
H. Meier , , C. P´epin , M. Einenkel , and K. B. Efetov , , ♦ Department of Physics, Yale University, New Haven, Connecticut 06520, USA Institut f¨ur Theoretische Physik III, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany IPhT, CEA-Saclay, L’Orme des Merisiers, 91191 Gif-sur-Yvette, France ♦ Corresponding author, E-mail : [email protected]. (Dated: August 14, 2018)We study the timely issue of charge order checkerboard patterns observed in a variety of cupratesuperconductors. We suggest a minimal model in which strong quantum fluctuations in the vicinityof a single antiferromagnetic quantum critical point generate the complexity seen in the phase dia-gram of cuprates superconductors and, in particular, the evidenced charge order. The Fermi surfaceis found to fractionalize into hotspots and antinodal regions, where physically different gaps areformed. In the phase diagram, this is reflected by three transition temperatures for the formationof pseudogap, charge density wave, and superconductivity (or quadrupole density wave if a suffi-ciently strong magnetic field is applied). The charge density wave is characterized by modulationsalong the bonds of the CuO lattice with wave vectors connecting points of the Fermi surface inthe antinodal regions. These features, previously observed experimentally, are so far unique to thequantum critical point in two spatial dimensions and shed a new light on the interplay betweenstrongly fluctuating critical modes and conduction electrons in high-temperature superconductors.
I. INTRODUCTION
High-temperature (high- T c ) cuprate super-conductors rank among the most complex materialsever discovered. Despite the rich diversity within thecuprate family, all compounds share common featuressuch as the antiferromagnetic Mott insulator phase atzero or small doping. Magnetic fluctuations are ubiqui-tously present in all compounds of the cuprate family.Upon hole-doping of the copper-oxide planes, theybecome superconductors at unusually high transitiontemperatures T c . Ultimately, at intermediate doping,they exhibit the enigmatic pseudo-gap phase character-ized by a gap observed in transport and thermodynamicsup to a temperature T ∗ > T c .In the last years, incommensurate charge modulationshave been reported in many of the families’ compounds.These modulations form a checkerboard pattern and pos-sibly also break nematicity. Complimentary to eachother, these experiments demonstrate that this order isdifferent from stripe spin-charge modulations predictedearlier , observed in La-compounds , and discussedin numerous publications (see, e.g., Refs. 16 and 17), aswell as from the d -wave order proposed in Ref. 18.Among the simplest properties common to all thecuprate compounds is the presence of strong anti-ferromagnetic fluctuations due to the proximity of adoping-driven quantum phase transition between an an-tiferromagnetic and normal metal phase. Approach-ing the complexity of the cuprates from the perspec-tive of this universal singularity, we provide an exten-sive study of a single antiferromagnetic two-dimensionalquantum critical point (QCP). Proximity to quantumphase transitions is generally believed important toexplain the intriguing behavior of high- T c cuprates ,heavy fermions , or doped ferromagnets .Our study unveils that this QCP triggers a cascade of phase transitions with symmetries different from those ofthe parent transition. These phases include d -wave su-perconductivity, a checkerboard structure of quadrupoledensity wave (QDW), a charge density wave (CDW) withanother checkerboard structure turned by 45 ◦ with re-spect to the former, and the “pseudogap state” whichlacks any long range order. The additional charge order(CDW) arises due to interaction of electrons with super-conducting fluctuations in situations when superconduc-tivity itself is destroyed. To the best of our knowledge,formation of CDW due to superconducting fluctuationshas not been considered previously. The complexity ofthe phase diagram is recovered out of a single originalQCP using a low energy effective theory describing in-teraction between low energy fermions and paramagnons,which represent the quantum fluctuations of the antifer-romagnetic order parameter. This unexpected result en-riches the conventional picture of a single QCP andmay provide new insights into the pseudogap phase ofhole-doped cuprates. II. PHYSICAL PICTURE
Before delving into details of the microscopic deriva-tion, let us first develop the physical picture and phe-nomenology. In Sec. III we provide a microscopic studyto back up the physical picture, and finally, in Sec. IVwe address the question how our results may help tounderstand physical phenomena observed in the high- T c cuprates. A. Spin-fermion model and pseudogap state
We adopt the two-dimensional spin-fermion model for the antiferromagnetic QCP as the “minimal model” FIG. 1. (a) Brillouin zone and Fermi surface. Quantum crit-ical paramagnons single out eight hotspots that we organizein two quartets (L = 1 and L = 2). (b) Extended modelof hotspot (red) and antinodal states (blue). Non-singularparamagnons with wave vectors K , . . . , K mediate the in-teraction between hotspot and antinodal states. (c) Cooperpair generation at antinodes A and B . in which we seek to understand the diversity of the non-magnetic phases. As has been known for a while , thismodel features a superconducting instability of the nor-mal metal state. More recently, linearizing the quasipar-ticle spectrum near so-called “hotspots”, Metlitski andSachdev pointed out an SU(2) particle-hole symmetryof the effective Lagrangian that might lead to another in-stability toward a “bond order” state. About two yearslater, it has been noticed that, in fact, a state with acomplex order parameter comprising both superconduc-tivity and an unusual charge order forms below a certain T ∗ . These phenomena significantly expand the earliereffective picture of free but Landau-damped param-agnons.In the model considered, spin- fermion quasiparti-cles ψ = ( ψ ↑ , ψ ↓ ), which occupy states close to the Fermisurface shown in Fig. 1(a), couple to paramagnons φ =( φ x , φ y , φ z ) with propagator (cid:10) φ αω, q φ β − ω, − q (cid:11) = δ αβ c − ω + ( q − Q ) + ξ − , (1)where c is the velocity of paramagnon excitations. Atthe QCP, the length ξ AF diverges so that the param-agnon propagator becomes singular at the antiferromag-netic ordering wave vector Q = ( ± π/a, ± π/a ), where a is the lattice constant of the Cu layer. Quasiparticlesemitting or absorbing such singular paramagnons existonly in the vicinity of eight hotspots, see Fig. 1. In afirst approximation, we thus focus on these hotspots.The energy scale Γ ∼ λ , where λ is fermion–paramagnon coupling constant, determines a tempera-ture T ∗ ∼ . d -wave superconducting subordersshows up and completely changes major properties ofthe system. The order parameter in this regime can be represented in the form b u , where b ∼ Γ is an amplitudeand u an unitary matrix in particle-hole space, u = (cid:18) ∆ QDW ∆ SC − ∆ ∗ SC ∆ ∗ QDW (cid:19) . (2)In this matrix, ∆ QDW and ∆ SC are complex ampli-tudes for charge order and superconductivity, respec-tively. Unitarity imposes | ∆ QDW | + | ∆ SC | = 1. In fact,there are two independent order parameters of the formof Eq. (2), one for each of the two quartets of hotspots,Fig. 1(a), which in the “hotspot-only” approximation areeffectively decoupled.The charge order competing with superconductivity ischaracterized by a quadrupole moment spatially modu-lated with wave vectors Q and Q , see Fig. 2. Thesewave vectors connect hotspots opposite to each otherwith respect to the center of the Brillouin zone but areequivalently represented as in the inset of Fig. 2. The re-sulting checkerboard structure of this quadrupole-densitywave (QDW) is shown in Fig. 2(a). The QDW (or, equiv-alently, “bond order”) instability for wave vectors Q , has been recently confirmed in an unrestricted Hartree-Fock study. The matrix order parameter u , Eq. (2), obtained frommean-field equations at temperatures T < T ∗ is highlydegenerate. At low enough temperatures, this degener-acy is lifted by curvature and magnetic field effects, theformer favoring superconductivity, the latter QDW. Athigh enough temperatures (but still below T ∗ ) thermalfluctuations restore the degeneracy and thus establisha pseudogap phase without a specific long-range order.The effective O(4) non-linear σ -model for fluctuations of u (derived in Ref. 27) as well as a more recent O(6)-model show in many aspects a good agreement withexperiments. B. Antinodal states
The nontrivial order parameter (2) has been derivedtaking into account only interactions mediated by thecritical modes with momenta ∼ Q corresponding to thestrongest antiferromagnetic fluctuations. Fermi surfaceregions beyond the hotspots have not yet been touchedby the theoretical treatment and remained gapless in the“hotspot-only” approximation.For the superconducting suborder, however, it is clearthat the gap should cover the entire Fermi surface , withthe exception of the nodes of the d -wave gap functionthat are situated at the intercept points of the Fermisurface and the diagonals of the Brillouin zone. In par-ticular, we expect a significant superconducting gap alsoat the so-called antinodes situated at the zone edges, seeFig. 1(b). The main result of the present study is that su-perconductivity is not the only possible order close to theantinodes, and we are going to show that another chargeorder (CDW) can appear in this region and challengesuperconductivity there. Favorably for CDW, oppositeantinodes are effectively nested for a singular interaction.Flatness of the antinodal Fermi surface is not requisitebut may enhance this effect.To be specific, we extend the study of the spin-fermionmodel by considering both hotspots and antinodes, seeFig. 1(b). In the leading approximation, fermion quasi-particles located close to the antinodes interact withhotspot fermions by exchanging non-singular param-agnons with propagator h φ j φ j i ≃ [(∆ K ) + ξ − ] − where∆ K = | K − Q | is the distance between hotspots andnearest antinodes, cf. Fig. 1. This interaction is clearlyweaker than the interaction between hotspots connectedby Q . On the other hand, it allows quantum criticality tospread into the so far untouched antinodal regions. Thesmallness of the non-singular propagators justifies a per-turbative treatment, whereas singular paramagnons haveto be fully accounted for.We now discuss the effects due to the paramagnon-mediated interaction between hotspot and antinodalquasiparticles. We begin with the case of establishedsuperconductivity at the hotspots and then, more inter-estingly, for the case of hotspots gapped by QDW orpseudogapped hotspots.Below T c , hotspot fermions form Cooper pairs. In thiscase, we may neglect fluctuations and replace pairs ofhotspot fermion fields by their mean-field average ∆ SC ∼ b h ψ †↑ , ψ †↓ , i . Let us consider two antinodal quasiparticlessituated, e.g., at antinodes A and B , see Fig. 1(c). Vir-tually exchanging a paramagnon with wave vector K ,they are scattered to hotspots and . There, they areaffected by the established superconducting order ∆ SC and thus form Cooper pairs themselves. Interestingly, asimilar virtual process is impossible for the particle-holesuborder (QDW) since in this case both particle and holewould have to emit a paramagnon of the same wave vec-tor.As a result, the explicit mean-field analysis, cf.Eq. (26), yields a superconducting gap at the antinodes,which by a factor of α ∼ Γ v (cid:2) (∆ K ) + ξ − ] (3)is smaller than the hotspot gap. Notably, cf. againEq. (26), the antinodal gap has d -wave symmetry. More-over, by continuity the antinodal superconductivity fixesthe relative phase of the so far decoupled superconduct-ing suborders of the two hotspot quartets in Fig. 1(a), en-suring overall d -wave symmetry of the superconductingorder parameter. We note that this mean-field result ac-tually does not require separating the Fermi surface intohotspots and antinodal regions and has been obtainedwith full momentum resolution. C. Charge density wave
When hotspot superconductivity is destroyed by ei-ther thermal fluctuations or a strong magnetic field, the superconducting gap at the hotspots has zero mean, h ∆ SC i = 0, implying absence of antinodal superconduc-tivity as well. However, antinodal quasiparticles still cou-ple to non-zero superconducting fluctuations ∆ SC ( r , τ )induced at the antinodes by the same mechanism thatproduced the antinodal superconducting gap in the pre-ceding section.In this situation, the superconducting fluctuations me-diate an effective interaction between antinodal fermions.Close to the transition, the mass ξ − of superconductingfluctuations is small and the effective interaction becomescritical. This also leads to effective nesting of oppositeantinodes. As a result, this situation is remarkably simi-lar to the initial situation of hotspot fermions interactingvia critical paramagnons. While quantum-critical para-magnons reorganize the ground state of hotspot quasipar-ticles into the pseudogap state, the critical superconduct-ing fluctuations play a very similar role at the antinodesand trigger in analogy a transition to another phase. Thisrepeated triggering of orders thus constitutes a cascade of phase transitions.The order parameter formed at the antinodes is pureparticle-hole pairing. It cannot be a form of superconduc-tivity because it has to be “orthogonal” to the supercon-ducting fluctuations that mediate the effective interac-tion. Furthermore, particle-hole pairing at antinodes A and B , see Fig. 1(b), is independent from particle-holepairing at C and D . This can be seen as, e.g., wavevectors K and K mediate interactions at antinodes A and B but have no meaning for C and D , where involvedparamagnons carry wave vectors K and K . Invarianceunder rotations of 90 ◦ then inevitably leads to a bidi-rectional charge density wave (CDW) order at the antin-odes.The explicit analysis (see Sec. III) follows the samesteps as the mean-field scheme of Ref. 27 for the pseudo-gap state. This leads us to a similar universal mean-fieldequation, see Eq. (35), with all relevant energies mea-sured in units of the energyΓ CDW ∼ α Γ (4)with α ≪ T < T
CDW ∼ . CDW . In realis-tic cuprate systems, we may expect T c < T CDW < T ∗ aswell as comparable energy scales, Γ CDW ∼ Γ. The calcu-lation of charge density ρ ( r ) in the CDW phase leads toa spatial modulation of the form ρ CDW ( r ) ∼ e Γ v (cid:8) cos( Q x r + ϕ x ) + cos( Q y r + ϕ y ) (cid:9) . (5)The wave vectors Q x and Q y (see Fig. 2) connect op-posite antinodes and correspond to a modulation alongthe bonds of the Cu lattice. The resulting pattern is acheckerboard as shown in Fig. 2(b), similarly to the pat-tern of QDW shown in Fig. 2(a). Notably, the CDW andQDW patterns are turned by 45 ◦ with respect to each FIG. 2. Checkerboard charge order for the pseudogap sub-order of (a) QDW and (b) antinodal CDW. Modulation vec-tors Q i giving the periods R i = 2 π Q i / | Q i | are shown in theinset. (c) Qualitative dependence of the superconducting andcharge order gaps on the position on the Fermi surface (HS= hotspots, AN = antinodes). other. Variables ϕ x,y denote offset phases. Figure 2(c)summerizes the results of our study by providing a sketchof the emergent orders as a function of the position onthe Fermi surface. III. MICROSCOPIC ANALYSISA. Effective Lagrangian
We begin our microscopic analysis by developing a con-venient and compact notation for the subsequent calcu-lations. We are mainly interested in the low-lying excita-tions close to the hotspots and antinodes, which we nu-merate according to Fig. 1(b) with numbers j = 1 , . . . , J = A , . . . , D, respectively. In thisspirit, we represent a general quasiparticle field ψ ( r ) as ψ ( r ) = X j =1 e i p j r ψ j ( r ) + D X J =A e i p J r χ J ( r ) , (6)where p j and p J denote the positions of hotspots j andantinodes J , respectively, in the Brillouin zone. Thefields for hotspot quasiparticles ψ j and for antinodalones χ J fluctuate only slowly in space on scales muchlarger than the lattice constant a .Following Ref. 27, we introduce three pseudospin sec- tors L ⊗ Λ ⊗ Σ to organize the hotspot states, ψ = (cid:18) ψ ψ (cid:19) Σ (cid:18) ψ ψ (cid:19) Σ Λ (cid:18) ψ ψ (cid:19) Σ (cid:18) ψ ψ (cid:19) Σ Λ L . (7)Inspecting the structure defined in Eq. (7), we see thatthe sector L organizes the hotspots in the two quartetsalong the diagonals of the Brillouin zone, cf. Fig. 1(a).Sector Λ distinguishes inside each of the quartets the twopairs of hotspots connected by the antiferromagnetic or-dering wave vector Q . Finally, the pseudospin Σ cor-responds to the two hotspots within each of such pairs.The antinodal fields are similarly combined into χ = (cid:18) χ A χ B (cid:19) Υ (cid:18) χ C χ D (cid:19) Υ Ξ , (8)where Ξ and Υ are two more pseudospins for the fourantinodes in Fig. 1(b). Operators acting on these vari-ous pseudospin spaces are conveniently expanded in Paulimatrices denoted by, e.g., Υ for the first Pauli matrixin Υ space. Each of the field components ψ j and χ J in Eqs. (7) and (8) is itself a spinor for the physicalspin, for which we use as usual the Pauli matrix nota-tion σ = ( σ , σ , σ ).In the approximation of linearized Fermi surfaces closeto hotspots and antinodes, the non-interacting part L of the Lagrangian reads L = χ † (cid:0) ∂ τ − iˆ v ∇ (cid:1) χ + ψ † (cid:0) ∂ τ − i ˆ V ∇ (cid:1) ψ , (9)where the velocity operator for the antinodal states readsˆ v = − v (cid:2) Υ (1 + Ξ ) e x + Υ (1 − Ξ ) e y (cid:3) . (10)Herein, e x and e y are unit vectors in the directions ofCu bonds and v is the (antinodal) Fermi velocity. Thehotspot velocity operator ˆ V is a little more complicated.Since we do not use this operator in the present studydirectly, we refer the reader to Ref. 27.During the analysis, it will be convenient to studycharge and superconducting correlations on equal foot-ing. Therefore, we introduce another pseudospin τ dis-tinguishing particle and hole states,Ψ = 1 √ (cid:18) ψ i σ ψ ∗ (cid:19) τ , X = 1 √ (cid:18) χ i σ χ ∗ (cid:19) τ . (11)The matrix C = − τ σ allows for a definition of charge-conjugation ¯Ψ = Ψ t C , ¯X = X t C . (12)In particle-hole space notation, the Lagrangian (9) be-comes L = − ¯X (cid:0) ∂ τ − iˆ v ∇ (cid:1) X − ¯Ψ (cid:0) ∂ τ − i ˆ V ∇ (cid:1) Ψ , (13)which concludes the non-interacting part of the effectivetheory.In order to incorporate the interaction mediated byparamagnons φ into the model, we again single outthe relevant modes. These are those harmonics ofthe field φ with wave vector close to Q for hotspot–hotspot interaction and wave vectors at K , . . . , K forhotspot–antinode interactions, see Fig. 1(b). We assumethat K − K is not an inverse lattice vector, which for ageneral curved Fermi surface is the correct assumption.In the compact notation, the Lagrangian for interac-tion at wave vectors ∼ Q is written as L int , Q = λ ¯ΨΣ ( φ σ )Ψ . (14)Here λ is the coupling constant for the paramagnon–fermion interaction. The general correlation function for φ , Eq. (1), translates to the correlation (cid:10) φ α ,ω, q φ β , − ω, − q (cid:11) = δ αβ c − ω + q + ξ − (15)for the field φ entering Eq. (14).For the interaction at wave vectors K , . . . , K , we in-troduce fields φ ± k that are related to field φ of Eq. (1)as φ ± k, q ,ω = φ ± K k + q ,ω (16)with correlations (cid:10) φ αk,ω, q φ β − k, − ω, − q (cid:11) ≃ δ αβ (∆ K ) + ξ − . (17)While hotspot–hotspot paramagnons φ become criti-cal at the antiferromagnetic QCP ( ξ AF → ∞ ), param-agnons φ ± k are effectively static as ∆ K = | K − Q | ≫| q | , c − ω at low energies. The corresponding Lagrangianreads L int , K = 2 λ X k =1 (cid:16) ¯Ψ T k ( φ k σ )X + ¯X T t k ( φ k σ )Ψ (cid:17) , (18)where matrices T k describe the various scattering pro-cesses between hotspots and antinodes. They are givenby T = (cid:18) t B3 t A1 (cid:19) τ , T = (cid:18) t A6 t B8 (cid:19) τ ,T = (cid:18) t D7 t C5 (cid:19) τ , T = (cid:18) t C4 t D2 (cid:19) τ , (19)where the 8 × t Jj are defined by ψ † t Jj χ = ψ † j χ J . While non-trivial effects due to the hotspot La-grangian L int , Q , Eq. (14), have been extensively studiedin Refs. 26 and 27, we are now in a position to extendthe physical picture by effects emerging in the antinodalregion, which couple nontrivially to the hotspots via La-grangian L int , K , Eq. (18). B. Emerging orders
1. Pseudogap state
Coupling between hotspot fermions and quantum-critical paramagnons φ , Eq. (15), has been studied fora long time. In Ref. 27, it was shown that close to theQCP ( ξ AF → ∞ ) below a temperature T ∗ ∼ Γ ∼ λ ,an unusual order parameter composed of two compet-ing suborders appears. These are superconductivity withcomplex amplitudes ∆ and ∆ and a charge order ofa spatially modulated quadrupole moment (quadrupoledensity wave, QDW) with amplitudes ∆ and ∆ ,cf. Eq. (2). Upper indices refer to the two decoupledquartets of hotspots given by L = 1 and L = 2 states, re-spectively, cf. Fig. 1(a). This order, hereafter referred toas “pseudogap”, constitutes a stable saddle-point mani-fold in the theory L + L int , Q . We incorporate it in termsof a mean-field term that replaces L int , Q , Eq. (14), in themodel. This term is given by L PG = ¯Ψ b (i ∂ τ ) O PG Ψ , (20)where b ( ε ) is a function of fermionic Matsubara frequen-cies ε and O PG is a matrix in the pseudospin spaces thatreflects the symmetry of the order parameter. It reads O PG = iΣ (cid:18) u − u † (cid:19) Λ (cid:18) u − u † (cid:19) Λ L . (21)Here, u and u are SU(2) matrices in particle-hole spacefor each of the two quartets of hotspots.Let us expand the u j in particle-hole space Pauli ma-trices τ i , u j = ∆ j + i (cid:0) ∆ j τ + ∆ j τ + ∆ j τ (cid:1) , (22)so that ∆ j QDW = ∆ j + i∆ j and ∆ j SC = ∆ j + i∆ j . Num-bers ∆ jn are real and satisfy the constraint P n =0 [∆ jn ] =1 imposed by unitarity. At low energies, we mayapproximate the function b ( ε ) as a (positive) constant, b ( ε ) ≃ b .Study of fluctuations of the pseudogap b ( ε ) O showsthat below a temperature T c < T ∗ , one of the suborders—QDW or superconductivity— is suppressed, providedsymmetry-breaking effects such as curvature of the Fermisurface are included in the consideration. In the absenceof the magnetic field, finite curvature makes the com-posite order parameter prefer superconductivity as theground state, whereas a sufficiently strong magnetic fieldcan make a charge modulated state (QDW) energeticallymore favourable. Between T c and T ∗ , neither are ca-pable of forming a long-range order and the system is ina regime of strong thermal fluctuations between the twosuborders.
2. Antinodal superconductivity
Averaging the Lagrangian (18) over the paramagnonfluctuations φ k , Eq. (17), yields an effective 4-point in-teraction vertex L int = − λ (∆ K ) X k =1 ¯X τ T t k τ σ Ψ ¯Ψ σ τ T k τ X . (23)The model L + L PG + L int , Eqs. (13), (20), and (23), isthe effective model our subsequent study on the physicsat the antinodes is based on.In a mean-field scheme to decouple the interaction L int ,Eq. (23), we replace the Ψ ¯Ψ operator by its mean-fieldcorrelation function, which by Eqs (13) and (20) is givenby h Ψ ¯Ψ i m . f . = J ( T )4 π O . (24)The function J ( T ) is defined as J ( T ) = Ω Tv X ε b ( ε ) p ε + b ( ε ) (25)and Ω ∼ λ /v is the volume of the hotspot, cf. Ref. 27.Inside the pseudogap regime, the function J ( T ) ∼ λ /v is in a good approximation independent of the tempera-ture T , while it turns to zero when T approaches T ∗ .Inserting Eq. (24) into Eq. (23) yields L int ≃ L m . f . with the mean-field Lagrangian given by L m . f . = 3 λ J ( T ) π (∆ K ) (cid:2) ¯X Ξ Υ (cid:8) ∆ τ + ∆ τ (cid:9) X (cid:3) . (26)Herein, ∆ = (∆ − ∆ ) / = (∆ − ∆ ) / SC = ∆ + i∆ of the hotspotsuperconductivity. Importantly, in this mean-field treat-ment, only the superconducting suborder of the hotspotpseudogap gives a contribution, while the QDW does noteffectively couple to the fields ¯X and X so that it does notplay a direct role at the antinodes. Equation (26) thusdemonstrates that the hotspot superconductivity inducesa superconducting order parameter at the antinodes bythe same mechanism sketched in Fig. 1(c) and discussedin Sec. II. The presence of Ξ reflects the d -wave symme-try of the superconducting order. The order parameterof antinodal superconductivity is maximal if∆ , = − ∆ , , (27)which should be energetically the favoured configura-tion. Note that the matching condition (27) reduces theO(4) × O(4) symmetry of the hotspot order to a con-strained O(6) model, cf. Ref. 31.Let us estimate the strength of the superconductinggap induced at the antinodes. According to Eq. (25), weestimate J ( T ) inside the pseudogap as J ( T ) ∼ λ /v ,which is smaller than the high energy scale given by the momentum distance ∆ K between hotspots and antin-odes. Thus, while the hotspot pseudogap is of order Γ ∼ λ , the induced antinodal superconducting gap is of or-der λ [ λ / ( v ∆ K ) ] ∼ α Γ ≪ Γ, cf. Eq. (3). We empha-size once more that the antinodal superconductivity isinduced only if the hotspot system is in the supercon-ducting state.
3. Antinodal charge-density wave order
Let us now address the case when hotspot super-conductivity is destroyed by either thermal fluctuationsabove T c (pseudogap state) or by a strong enough mag-netic field at arbitrary temperature. In the latter case,we obtain QDW at T < T c or the pseudogap stateat T > T c instead of the superconductor. Then, themean-field decoupling in Eq. (26) does not induce a fi-nite gap at the antinodes as ∆ = ∆ = 0. How-ever, superconducting fluctuations are still present evenif h ∆ ( r , τ ) i = h ∆ ( r , τ ) i = 0. These fluctuations havebeen studied with the help of a non-linear σ -model inRef. 27.At not too high temperatures above the superconduct-ing critical temperature T c at zero field or below T c in asufficiently strong magnetic field destroying the super-conductivity, the superconducting fluctuations ∆ SC ( r , τ )are small and the σ -model yields the effective Lagrangian L fluct ≃ gλ (cid:0) | ∂ µ ∆ SC | + ξ − | ∆ SC | (cid:1) (28)with ∂ µ = ( u − ∂ τ , ∇ ), g ∼ u ∼ v the velocity of the fluctuation modes. For T > T c it is not easy to carry out explicit calculations in the pseu-dogap state. However, it is well-known that there is nophase transition in the two-dimensional fully isotropicO(4)-symmetric σ -model as all excitations have a gap. Inour situation this means that correlation functions of su-perconducting fluctuations can still formally be obtainedfrom Eq. (28) but the constants entering this equationshave now to be considered as effective parameters whosevalues can hardly be calculated analytically. In the sub-sequent analysis, we assume that the length ξ SC divergeson the critical line separating the superconducting regionfrom QDW or pseudogap phase.In the Gaussian approximation of Eq. (28), we imme-diately integrate the fluctuation modes out of the La-grangian (26) (, where ∆ SC = ∆ + i∆ is now assumedto fluctuate both in space and time). Then, we obtainthe effective interaction between the antinodal fermions, L int , fluct = − λ J ( T ) π g (∆ K ) X j =1 (cid:0) ¯X( r , τ ) Ξ Υ τ j X( r , τ ) (cid:1) × Φ( r − r ′ , τ − τ ′ ) (cid:0) ¯X( r ′ , τ ′ ) Ξ Υ τ j X( r ′ , τ ′ ) (cid:1) , (29)where Φ q ,ω = 1 u − ω + q + ξ − (30)is the propagator of superconducting fluctuations. Atthe transition, ξ SC → ∞ and this propagator is singularin the infrared limit, which makes the antinodal pointseffectively hot. Moreover, opposite antinodes are effec-tively nested. We emphasize, though, that, in analogywith the hotspot fermions interacting via critical para-magnons, this effective nesting is due to the singularform of the propagator of superconducting fluctuationsin the vicinity of the superconductor transition where thelength ξ SC diverges. This does not necessarily require ageometrically flat Fermi surface at the antinodes. The interaction (29) generates an instability toward charge-density wave (CDW) order.
Indeed, the La-grangian (29) for the interaction of antinodal fermionshas effectively the same form as the effective interactioninduced by paramagnons that is responsible for the for-mation of the pseudogap. We thus introduce a CDWorder parameter in the Lagrangian, L CDW = ¯X b CDW (i ∂ τ ) O CDW X , (31)and obtain in analogy with Ref. 27 the mean-field equa-tion b CDW ( ε ) O CDW = − λ J ( T ) π g (∆ K ) X j =1 T X ε ′ , k ′ Φ k ′ ,ε − ε ′ Ξ Υ τ j b CDW ( ε ′ ) O CDW ε ′ + ( v k ′ ) + b ( ε ′ ) Ξ Υ τ j . (32)Deriving Eq. (32) has required that O CDW anticom-mutes with the velocity operator ˆ v , Eq. (10), which im-plies {O CDW , Υ } = 0 and [ O CDW , Ξ ] = 0. In addi-tion, we assume the normalization O = 1. Fur-thermore, in order to compensate for the minus sign inEq. (32), we need to impose that {O CDW , Υ τ } = 0 and {O CDW , Υ τ } = 0. Summarizing all these constraints,the antinodal order parameter becomes O CDW = (cid:18) ∆ ′ x Υ τ + ∆ ′′ x Υ
00 ∆ ′ y Υ τ + ∆ ′′ y Υ (cid:19) Ξ . (33)Parameters ∆ ′ x and ∆ ′′ x play the roles of real and imagi-nary parts for the order parameter of CDW in x -directionwhile ∆ ′ y and ∆ ′′ y do so for the y -direction. They sat-isfy the nonlinear constraints [∆ ′ x ] + [∆ ′′ x ] = 1 and[∆ ′ y ] + [∆ ′′ y ] = 1.Measuring all quantities of dimension of energy in unitsof Γ CDW = 18 uλ J π gv (∆ K ) , (34)we derive from Eq. (32) a universal self-consistency equa-tion for the CDW amplitude b CDW ( ε ),¯ b CDW ( ε ) = ¯ T X ¯ ε ′ | ¯ ε − ¯ ε ′ | ¯ b CDW (¯ ε ′ ) q ¯ ε ′ + ¯ b (¯ ε ′ ) . (35)In this equation, all quantities z of dimension energy en-ter in the form ¯ z = z/ Γ CDW , The energy scale Γ
CDW ∼ α Γ, cf. Eq. (3), is smaller than both the pseudo-gap energy scale ∼ Γ and the antinodal superconduct-ing gap ∼ α Γ, which appears when the pseudogap hasordered into the superconducting suborder. Numeri-cal investigation of Eq. (35) indicates non-zero solutions
FIG. 3. Dimensionless charge-density gap ¯ b CDW = b CDW / Γ CDW as a function of dimensionless temperature ¯ T interpolated to the frequency ε = 0. A CDW order appearsbelow the temperature T CDW ≈ . CDW . for b CDW ( T, ε ) below a temperature T CDW ≈ . CDW .Figure 3 shows the (interpolated) amplitude b CDW ( T, T .Calculating the charge density in the presence of theorder parameter O CDW , Eq. (33), we obtain formula (5)for the bidirectional CDW modulation, ρ CDW ( r ) ∼ e Γ v (cid:8) cos( Q x r + ϕ x ) + cos( Q y r + ϕ y ) (cid:9) , (36)where ϕ x and ϕ y denote the phases of the CDW order in x and y directions, respectively. Thus, the charge densityis modulated with the wave vectors Q x and Q y connect-ing two opposite antinodal points. This contrasts themodulations of the quadrupole-density D xx generated at the hotspots in the presence of QDW, D xx ( r ) ∼ e (cid:8) | ∆ | cos( Q r + ϕ )+ | ∆ | cos( Q r + ϕ ) (cid:9) . (37)QDW wave vectors Q and Q are turned by 45 ◦ andlonger than the CDW wave vectors by a factor roughly FIG. 4. Qualitative phase diagram summarizing the resultsof Ref. 27 and the present work for zero magnetic field. Closeto the antiferromagnetic (AF) QCP, ξ − = 0, and upon low-ering the temperature, the systems develops first at T ∗ theinstability toward the fluctuating pseudogap state (PG) char-acterized by the order parameter of Eq. (2). At lower tem-peratures T < T
CDW < T ∗ , strong superconducting fluctua-tions induce a transition toward charge density wave (CDW)formed at the antinodes. Finally, below T c , the particle-particle suborder of the pseudogap prevails due to curvatureeffects and establishes d -wave superconductivity. given by √
2. Both orders form checkerboards as illus-trated in Fig. 2. Figure 2(c) shows the type of particle-hole order, i.e. whether QDW or CDW, as a qualitativefunction of the position on the Fermi surface. Whereaswithin our model hotspot and antinodal regions are sep-arated, we expect in realistic systems regions of smalloverlap of the two orders in between.
IV. CUPRATE PHYSICS
We now address the phase diagram of cuprates in theproximity of the antiferromagnetic QCP. We emphasizethat our theory applies only to the “metallic” side of theantiferromagnet–normal metal phase transition. The re-gions of too low doping are thus excluded in the followingdiscussion. In the region of intermediate doping, suppres-sion of carrier density below a crossover temperature T ∗ observed in NMR measurements was the first evi-dence for the existence of a “pseudogap” in the electronspectrum. In contrast, d -wave superconductivity appearsonly below a considerably lower temperature T c . In ourtheory, T ∗ is associated with the crossover to the stronglyfluctuating O(4)-symmetric composite order (supercon-ductivity and QDW) close to the hotspots. The phasediagram, see Fig. 4, is further enriched by the forma-tion of CDW order with wave vectors Q x,y (Fig. 2) atthe edge of the Brillouin zone. Also the emergence ofthe CDW order is ultimately due to the proximity to theQCP. The additional phase transition is expected to oc-cur at a temperature T CDW inside the pseudogap phase, T c < T CDW < T ∗ . The charge modulation observed in various recentexperiments has been attributed to the existenceof QDW (or “bond order”) correlations. This pictureis, in principle, in agreement with NMR results andsound propagation measurements . However, STMstudies of BSCCO and experiments with hard andresonant soft X-ray scattering on YBCO have re-vealed a charge modulation along the bonds of the Culattice with modulation vectors close to Q x,y , which arethe CDW wave vectors. Moreover, QDW has a vanish-ing Fourier transform near even Bragg peaks. Therefore,STM and hard X-ray experiments can hardly be expectedto detect the QDW modulation.The seeming contradiction is resolved when we includethe CDW, Eq. (5), in the Cu lattice. Then, this explainsthe experimental results . CDW appears below a crit-ical temperature T CDW that can be considerably lowerthan T ∗ , in line with the results of the hard X-ray exper-iment of Ref. 9. In addition, Hall effect measurements indicate a reconstruction of the Fermi surface that is at-tributed to the formation of CDW. The transition tem-peratures T CDW of these two experiments agree with eachother. Evidence for a transition below T ∗ and related toCDW has also been found recently in a Raman scatteringstudy. The dual effect of the two modulations (QDWand CDW) on the two species of atoms in the CuO planeis a characteristic of our theory and might be tested viaresonant soft X-ray scattering.Very recent STM and resonant elastic X-rayexperiments on BSCCO confirm the CDW wave vec-tors’ orientation along the bonds but indicate that theyconnect hotspots rather than antinodes. In our model,we expect CDW to set in at wave vectors as soon as theQDW gap is small. In realistic systems, this may indeedhappen already not very far from the hotspots, possiblyenhanced by reconstruction of the Fermi surface. Detailsbehind this physics are clearly beyond the range of our“minimal model” and left for a separate study.The emergence of various gaps in k -space around theFermi surface has been reported in Raman scatteringon Bi-2212 and Hg-1201 compounds. It was demon-strated that in overdoped samples the superconductinggap spreads all over the Fermi surface. In contrast, in un-derdoped samples the coherent Cooper pairs are observedmostly near the nodes, whereas the gap at the antinodesis mainly of a non-superconducting origin. This effectcan naturally be explained within our picture becausethe hotspots move to nodes with decreasing the dopingand the superconducting gap at the antinodes should de-crease. At the same time, the CDW gap grows at theantinodes thus “pushing away” the Cooper pairs.We note that after our work has been completed anddistributed as a preprint on arXiv, a work discussingthe issue of the rotation of the charge order wave vectorby 45 ◦ has appeared. A solution of mean-field equa-tions for a new CDW suggested in the latter work, al-though very interesting, is not stable against formationof SC/QDW order of Ref. 27 below its transition temper-ature T ∗ . As a result, new preemptive states predictedin Ref. 42 may be possible only in the vicinity of T ∗ . V. CONCLUSION
Extending the analysis of the spin-fermion model forthe two-dimensional antiferromagnetic QCP to the antin-odal regions, we find below the pseudogap tempera-ture T ∗ another transition to a bidirectional CDW in-duced at the zone edge by superconducting fluctuations.The physics behind this transition is determined by pseu-dogap physics emerging at the hotspots. Our theory thus shows how a complexity of offspring phases arises out ofthe single QCP. The results enable us to address recentlyobserved charge order features in the phase diagram ofthe high- T c cuprates. ACKNOWLEDGMENTS
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