Charge orders, magnetism and pairings in the cuprate superconductors
T. Kloss, X. Montiel, V. S. de Carvalho, H. Freire, C. Pépin
CCharge orders, magnetism and pairings in the cuprate superconductors
T. Kloss , X. Montiel , V. S. de Carvalho , H. Freire , C. Pépin
1. IPhT, L’Orme des Merisiers, CEA-Saclay, 91191 Gif-sur-Yvette, France and2. Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia-GO, Brazil (Dated: December 11, 2015)We review the recent developments in the field of cuprate superconductors with the special focuson the recently observed charge order in the underdoped compounds. We introduce new theoreticaldevelopments following the study of the antiferromagnetic (AF) quantum critical point (QCP) in twodimensions, in which preemptive orders in the charge and superconducting (SC) sectors emerged,that are in turn related by an SU(2) symmetry. We consider the implications of this proliferation oforders in the underdoped region, and provide a study of the type of fluctuations which characterizethe SU(2) symmetry. We identify an intermediate energy scale where the SU(2) pairing fluctuationsare dominant and argue that they are unstable towards the formation of a Resonant Peierls Excitonic(RPE) state at the pseudogap (PG) temperature T ∗ . We discuss the implications of this scenariofor a few key experiments. I. INTRODUCTION
The last decade has seen a strong revival of inter-est in cuprate superconductors, with the observation ofcharge orders in the underdoped regime of these materi-als. Maybe the starting point of this intense period of in-vestigation was the observation by STM of checkerboard-type patterns inside the vortices in Bi-2212 [1, 2]. Subse-quent studies with Fermi surface reconstruction showedthat this feature was generic [3, 4] (also verified in Bi-2201[5, 6]) and that the charge patterns corresponded to twoaxial wave vectors (0 , Q y ) and ( Q x , , incommensuratewith the lattice periodicity, and the magnitude of thewave vectors decreases with oxygen-doping. The chargeexcitation was also found to be non-dispersive in temper-ature, and correlated with the “hot-spots” - the points ofthe Fermi surface where the AF zone boundary is inter-sected. The picture refined itself a bit later, and we nowbelieve the charge order emerges at the tip of the Fermiarcs [7, 8]. The study of charge order in underdopedcuprates stayed in a status quo until the observation ofquantum oscillations (QO) under a strong magnetic fieldin YBCO [9, 10]. This result pointed directly to a re-construction of the Fermi surface induced by magneticfield and received several explanations in terms of stripeand charge patterns until the link was made with thebi-axial charge order observed by STM [11–14]. In par-ticular, models for the reconstruction of the Fermi surfaceinvolved charge ordering with bi-axial wave vectors sim-ilar to those unveiled by STM. A subsequent NuclearMagnetic Resonance (NMR) study finally found somecharge splitting under a magnetic field B ≥ T, whichbrought the final confirmation that charge order under afinite magnetic field is coherent, static and long-ranged[15–17]. The field versus temperature phase diagramwas later completed by ultrasound experiments, whichshowed evidence for a flat transition line at B c = 17 T[18]. For B ≤ B c , YBCO is a d -wave superconductor.The increase of the magnetic field then creates vortices whose cores show the typical charge ordering [16]. For B ≥ B c YBCO shows long range charge order with atypical ordering temperature remarkably similar in mag-nitude with the SC T c . In the PG regime at B = 0 , bothhard x-ray [19, 20] and soft x-rays [21–24] study showedthe presence of a sizable short range excitation at the bi-axial wave vectors. A softening of the phonon spectrumhas been observed in the PG phase, while a softening atthe charge order wave vectors occurs below T c [25, 26].Note that a recent state-of-the-art x-ray experiment at B = 17 T showed that the charge order becomes uni-axialand tri-dimensional at high field [27]. One preliminaryconclusion that one can infer from these experiments isthat the charge and superconducting sectors are of thesame order of magnitude in cuprate superconductors. Wewill use this observation later when we introduces theemergent SU(2) rotations that appear between these twosectors.We turn now to one of the most enduring mysteriesof cuprate superconductors, the pseudogap (PG) regime.The PG phase was observed in 1989 by NMR experi-ments [28, 29], where a gradual drop in the Knight-shiftwas observed at a crossover temperature T ∗ . This gapwas attributed to a loss of density in the electronic car-riers, and it was shown to decrease when the oxygen-doping increases but no obvious symmetry breaking wasassociated with this phase transition. We focus here ona few properties of the PG phase which we will use laterin the SU(2)-interpretation of the experiments. The firstremark that one can make is that the PG is an extremelyrobust feature of the phase diagram. It seems insensitiveto disorder [30, 31] and magnetic field [32] and is closelyassociated to a regime of linear-in- T resistivity on itsright hand side [33–35]. One other very striking observa-tion in the PG regime detected by angle-resolved photoe-mission (ARPES), is the formation of Fermi arcs, insteadof a closed-contour Fermi surface [36–41]. Recently, a mo-mentum scale of similar magnitude as the one observed inthe CDW was associated to the opening of the PG in the a r X i v : . [ c ond - m a t . s up r- c on ] D ec anti-nodal region of the Brillouin zone (BZ) [42–44], andled to an interpretation in terms of a pair-density-wave(PDW) [45, 46]- or a finite momentum superconductingstate Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) [47, 48].Coherent neutron scattering showed a Q = 0 signal [49–54], which was interpreted in terms of intra-unit-cell loopcurrents [55–57]. Although a Q = 0 phase is unable toopen a gap in the electronic density of states, the loop-current line surprisingly follows the T ∗ -line. Note thatNMR [58, 59] and µ SR [60, 61] techniques were not ableto detect such loop current. An explanation could bethe longer time scale of local probes ( ≈ − − − s )compared with the INS time scale ( ≈ − s ). At lowertemperature, a Kerr effect signal has been reported, hint-ing at a breaking of time-reversal (TR) symmetry insidethe PG [62]. This last observation is widely discussed bythe community, but it is necessarily related to the Q = 0 loop currents [63, 64]. The inelastic neutron scattering(INS) is also interesting for revealing collective modes ofthe system. A resonance at meV was found in YBCOthe early days of cuprate superconductivity [65] and atsimilar energies in other compounds [66–69]. It was firstbelieved that this collective excitation existed only in theSC phase, where it has a typical “hour-glass” shape cen-tered around meV at Q = ( π, π ) , as a function ofenergy and wave-vector. It was later shown that the res-onance exists as well in the PG phase above T c , whereit is still centered around meV, but shows a typical“Y”-shape with a long energy-extension at Q = ( π, π ) [70–73]. Many theoretical approaches have been invoked todescribe the resonance below the SC transition [74–77].This observation of the resonance around similar typicalenergies in the SC and PG phases, however, has never re-ceived a theoretical description, and constrains theoriesof the PG to keep some reminiscence of the SC phase.The neutron resonance was also observed in mono-layertetragonal compounds (Hg-1201), where the long energyextension at Q = ( π, π ) persists below T c [78].Collective modes of a material give useful insights toprobe symmetries of an effective model. One exampleis a resonance observed in the Raman A g channel, thatappears at energies very similar to the ones where a col-lective mode was observed by INS [79–81]. Raman scat-tering typically probes the symmetries of the Fermi sur-face and the presence of “two gaps” in the underdopedregime of the cuprates was observed below T c [82–84].This fact was corroborated in a series of ARPES exper-iments on BSCO from which the gap velocity v ∆ at thenodes was extracted and shown to differ from the Fermivelocity. Three regions in the phase diagram were iden-tified [85]. Starting from the over-doped region and de-creasing the doping, v ∆ is shown to first increase thento reach a plateau in the underdoped region -down todopings of the order of 5 %, and after that it drops atlower dopings when the system gets close to the insulat-ing Mott-transition. The key question associated with the PG phase is whether it is a “strong-coupling” phe-nomenon, emerging as a direct consequence of the Motttransition [86–90], or whether it is a a very unusual collec-tive phenomenon which is sensitive to other peculiaritiesof the physics of the cuprates, like its low dimensional-ity, the antiferromagnetic fluctuations or its fermiology[91–94]. In this work, we argue that the key to explainthe mystery of the PG phase resides in an underlyingemergent SU(2) symmetry, which produces SU(2) pairingfluctuations at intermediate energy scales. These fluctu-ations are in turn unstable toward the formation of a newkind of excitonic state, the (RPE) state, which is respon-sible for gapping out the Fermi surface in the anti-nodalregion of the BZ [94].The paper is organized as follows: In section II, wepresent the basics of the emergent symmetry model withSU(2) symmetry. Section III discusses the competitionbetween the U(1) and SU(2) paring fluctuations in theframework of the non linear σ model. In particular, wepropose to explain the PG state as a new type of chargeorder: the Resonant Peierls Excitonic (RPE) state com-ing from the SU(2) fluctuations. We also demonstratethat the CDW state is a secondary instability producedby U(1) fluctuations mediated by a Leggett mode. In sec-tion IV, we discuss the possible experimental evidence ofthis phase before to conclude in section V. II. THE EMERGENT SU(2) SYMMETRY
The concept of emergent symmetry in the context ofthe cuprate superconductors can be traced back to thework of Yang and later Zhang [95, 96] where a representa-tion with pseudo-spin operators was introduced which ro-tated the d -wave SC state onto a d -wave bi-partite chargeorder. The lowering and raising pseudo-spin operators η + , η − = ( η + ) † , and η z , which follow from the definition η + = (cid:88) k c † k ↑ c †− k + Q ↓ (1a) η z = (cid:88) k (cid:16) c † k ↑ c k ↑ + c † k + Q ↓ c k + Q ↓ − (cid:17) . (1b)The operators (1) form an SU(2) Lie algebra. Noticeably,the η -pairing stat – which is equivalent to finite center ofmass pairing of vector Q = ( π, π ) , or FFLO state – is aneigenstate of the Hubbard Hamiltonian, both for positiveand negative U. The simplest irreducible representationfor the pseudo-spin is the triplet vector ∆ m , with m = {− , , } defined as ∆ = − √ (cid:88) k c † k ↑ c †− k ↓ , (2a) ∆ = 12 (cid:88) k ,σ c † k σ c k + Q σ , (2b) ∆ − = − ∆ † , (2c)which represents the two conjugated s-wave SC states (∆ − , ∆ ) and the charge ordering state ∆ . The SU(2)pseudo-spin operators (1) rotate each component of themultiplet (2) into one another in the standard way ( l isthe rank of the irreducible representation Eq. (2), here l = 1 ) (cid:2) η ± , ∆ m (cid:3) = (cid:112) l ( l + 1) − m ( m ± m ± , (3a) [ η z , ∆ m ] = m ∆ m . (3b)The effective theory describing the pseudo-spin symme-try is the SO (4) [ SO (4) = ( SU (2) × SU (2)) /Z ] non-linear σ -model which excites thermally from the SC stateto the ordered state. This model describes transitionsfrom one state to the other within the generic frameworkof “spin-flop” transitions. In the case above one has apseudo spin-flop from the s-wave SC to the CDW states,whereas the standard spin-flop transition from easy axisto easy plane belongs to the SO(3) group [97]. The con-cept of SU(2)-symmetry was used later on in an effec-tive theory of the PG leading to a rotation from the d -wave superconductor to the d-density wave state [98].Here the generators of the symmetry are simply iη + , iη − and η z and the effective theory is the O (4) non-linear σ -model. Let us mention a similar rotation between thenematic d -wave bond order ∆ nem = (cid:80) k σ d k c † k σ c k σ and d -wave states ∆ + dsc = − √ (cid:80) k d k c † k ↑ c † k ↓ , and ∆ − dsc = √ (cid:80) k d k c k ↓ c k ↑ where d k = cos k x − cos k y [99]. Thepseudo-spin generators in this case take the form L + , L − = ( L + ) † , L with L + = (cid:88) k c † k ↑ c †− k ↓ , L = (cid:88) k σ (cid:16) c † k σ c k σ − (cid:17) . (4)Note that the chemical potential couples to the generator η z (or L ) and thus a finite chemical potential breaks theSU(2) symmetry in favor of the SC state.Another rotation, this time from the SC state towardsthe AF state, was introduced early on and became fa-mous as the SO(5) theory [97, 101, 102] . The SO(5)theory is the one of a non-linear σ -model which oper-ates on a five state “superspin” (cid:0) n , n , n , n , n (cid:1) -two SC states (cid:0) n = ∆ s , n = ∆ † s (cid:1) and three AF vec-tors (cid:0) n = s + , n = s − , n = s z (cid:1) [97]. The superspin n a is a vector representation of the SO(5) algebra. TheSO(5) theory was based on the idea that both the SCand the AF states are key players of the physics of doping T PG T s Fermi liquidstrange metal
TT T T MF T MF T MF T MF T c T c T c T c T s T s T s T p T p T p T p T n T n T n T n (cid:43) c (cid:43) c (cid:43) c (cid:43) c (cid:43) c ’ (cid:43) (cid:43) c ’ (cid:43)(cid:43) (cid:43)
1) 2)3) 4)
FIG. 1. (Color online) Schematic phase diagram of theSO(5) model [100]. Four types of scenarios are discussed inRef. [100]: 1) a direct first order transition with a bi-criticalpoint, 2) two second order transitions with an intermediatecoexistence regime, 3) one single second order transition ter-minating at a QCP at zero temperature and 4) two secondorder transitions with a quantum disordered phase. Althoughthe SO(5) symmetry is broken in scenario 1), 2) and 3) at zerotemperature, thermal fluctuations lead to a restoration belowthe mean-field critical temperature T MF . Adapted from Ref.[100]. these compounds and are close enough in energy so thatin between their respective phase transition an SO(5)-symmetric state is found where SC and AFM are undis-tinguishable. This phase was naturally associated withthe PG of the cuprates. A typical SO(5) non-linear σ -model was introduced to describe the effective physics ofthe system, and four typical phase diagrams were derivedwhich are depicted in Fig. 1. The mechanism favoringone of the states in the non-linear σ -model can be un-derstood as a spin-flop transition- also called “super spinflop” transition for the SO(5) symmetries. As mentionedabove, one gets a very accurate picture by thinking ofthe spin-flop transition of the antiferromagnetic state in auniform magnetic field B along the easy z -axis [103, 104].The magnetic field creates an easy plane xy , so that ata critical value of the field, the Néel wave vector changesits orientation abruptly from the z -axis to the xy -plane.Hence although in each of the above cases the symme-tries are different, the underlying physics is as simpleas the one on a spin-flop transition. The four typicalphase diagrams show the various phases as a function oftemperature and an external parameter which breaks thesymmetry and are depicted in Fig. 1. They correspondto the cases where: 1) a first order transition betweenthe two states terminates at a bi-critical point; 2) thereis a coexistence phase between the two orders; 3) thetransition between the two orders terminates at T = 0 at a quantum critical point (QCP) or 4) the two ordersare disconnected. In cases 1), 2) and 3), although thesymmetry is broken at zero temperature, thermal fluctu-ations restore the SO(5)- or SU(2)- symmetry, leading toan invariant phase under the mean field transition T MF .In the case of the SO(5)-symmetry, the typical phase di-agram of the cuprates has the shape depicted in case 4),where the two orders are disconnected from one another,and this situation leads to a splitting of the big group tothe SO (3) × U (1) subgroups describing fluctuations asso-ciated to each order separately. Furthermore, the SO(5)symmetry led to the prediction of a giant proximity effectin a SC-AF-SC junction [105, 106]. Unfortunately, thisproximity effect was never verified experimentally [107].One reason invoked here was that the SC state is by na-ture itinerant while the AF state is an insulator in thosecompounds. Hence the typical energy difference betweenthose two states is big, of the order of the Coulomb U.In the present work, we revive the concept of emergentsymmetry, with an the SU(2) symmetry which rotatesfrom a d -wave SC state to an incommensurate d -wavecharge order. The pseudo-spin generators have the sameform as depicted in Eqn. (1), with the exception thatthe wave vector Q = Q is now the charge order wavevector and is not necessarily commensurate with the lat-tice. The l = 1 irreducible representation is given by the d -wave version of Eqn. (2) with namely ∆ = χ CDW , ∆ = ∆ † dsc and ∆ − = − ∆ † , namely ∆ = − √ (cid:88) k d k c † k ↑ c †− k ↓ , (5a) ∆ = 12 (cid:88) k ,σ d k c † k σ c k + Q σ , (5b) ∆ − = 1 √ (cid:88) k d k c k ↓ c − k ↑ . (5c)The CDW ordering wave vector could be the axialCDW wave vector observed through many recent experi-ments ( STM, Quantum oscillations, X-rays, ARPES) orit could be another wave vector carefully chosen so thatthe SU(2) symmetry is fully respected. As it turns out,the Eight Hot Spots (EHS) model depicted in Fig. 2 pro-vides an exact realization of such a symmetry, as was firstmentioned in Ref.[108–110]. This model is a simplifiedversion of the spin-fermion model, which describes thevicinity of an AF QCP within a metallic substrate [111].At this point it is useful to recall that the spin-fermionmodel played an important role at the beginning of thetheoretical investigation on the cuprates [112–114]. For arecent link to strongly correlated systems note also Ref.[115]. Two different views were (and are still) competingfor the understanding of the phase diagram of these com- Q QQ diag FIG. 2. (Color online) Schematic representation of the firstBrillouin zone with the Fermi surface of cuprate supercon-ductors. Hotspots are located at the intersection points ofthe AFM zone boundary with the Fermi surface. The diago-nal coupling vector Q diag between two hotspots is shown inblue and the AFM ordering vector Q in magenta. pounds. Observing that the SC phase is close to a Mottinsulator, a first group of theoreticians consider that thesystem is fundamentally strongly correlated, namely thatthe Coulomb energy U = 1 eV is affecting the qualitativebehavior down to very low temperatures [86, 90, 116].This viewpoint has been extensively developed aroundthe resonating valence bond (RVB) suggestion made byAnderson as early as 1987 [86], and now explored viaextensive numerical calculations which can capture thestrongly interacting behavior [90, 117]. Another part ofthe physics community defends the viewpoint that theMott transition has a strong qualitative influence up to6-7% doping, beyond which the physics of the system ismainly driven by the presence of AF fluctuations [118].Proponents of this viewpoint hence start the theoreti-cal investigation with the spin-fermion (SF) model whichcouples conduction electrons to AF paramagnon modes Φ = (Φ x , Φ y , Φ z ) on the brink of criticality with the prop-agator (cid:68) Φ αω, q Φ β − ω, − q (cid:69) = δ αβ c − ω + ( q − Q ) + ξ − AF , (6)where Q is the AF wave vector and ξ − AF is the effec-tive mass of the paramagnons which defines the distanceto the QCP. Note, however, that in the present theorythe SF model can be considered as an effective theoryfor which the SU(2) symmetry is approximately verified-in the case of hot regions [119], or exactly verified inthe case of the EHS model [108, 110]. Although theproximity to the AF QCP may not be verified in thecuprates, we believe that the concept of emergent SU (2) symmetry is robust and will remain when dust settlesdown. The conduction electrons have the kinetic en-ergy H K = (cid:80) k,σ c † k,σ (cid:15) k c k,σ and interact with the para-magnons through a simple spin-spin interaction term H int = J (cid:80) i Φ i · c † i σc i . In the EHS model, a furthersimplification is implemented with the reduction of theFermi surface to eight “hot spots” which are the pointsat T = 0 where the electrons scatter through the AF Φ -modes. When the electron dispersion (cid:15) k (cid:39) v hs k islinearized at the hot spots, the model possesses an ex-act SU (2) symmetry defined by the operators of Eqn.(1) but with Q = Q diag , being the diagonal wave vectordepicted in Fig. 2. This model was further studied inRef.[110] and an SU (2) precursor of the AF state wasfound, where quadrupolar density wave (QDW) with di-agonal wave vector, which is equivalent to a d -wave CDWwith diagonal wave vector, is degenerate with the d -waveSC state. This new state can be understood as a kindof non-Abelian superconductor with order parameter ˆ b that, instead of having a U(1) phase, has an SU (2) uni-tary matrix fluctuating between the charge and SC sector[110, 119] ˆ b = b (cid:18) ∆ CDW ∆ SC − ∆ ∗ SC ∆ ∗ CDW (cid:19) SU (2) (7)subject to the constraint | ∆ CDW | + | ∆ SC | = 1 . Withinthe framework of the EHS model, and the related O (4) non linear σ -model, several experimental findings weresuccessfully addressed [120–122]. The general picturefollows closely the ideas expressed in the SO (5) the-ory, which are valid for all theories of emergent symme-tries. A small curvature term in the electron dispersionbreaks the symmetry in favor of the SC state. Henceat T = 0 the system is a superconductor. Once thetemperature is raised, thermal fluctuations then excitethe system between the two pseudo-spin states, restoringthe SU (2) invariance below the PG dome. Conversely,an applied magnetic field breaks the SU (2) symmetryin favor of the CDW state and beyond a certain criti-cal field B c , a “pseudo spin-flop” is observed where theground state “flips” from the SC state to CDW order.This “pseudo spin-flop” was precisely observed in exper-iments performed under magnetic field, with a criticalfield B c ∼ T [9, 13, 123]. In particular, the ultra-sound experiment [18] shows that the typical B versusT phase diagram in Fig. 3 is very similar to Fig.1-2).Within the EHS model, this experiment was addressedin Ref. [121]. Note that a co-existence phase is presentin this phase diagram, which accentuates the similaritywith the phase diagram 2) in Fig. 1 of the SO (5) the-ory. Notice as well that the CDW and SC temperaturesare of the same order of magnitude, which was never thecase for the AF and SC states. It is another indicationthat the SU (2) symmetry is more likely verified in theunderdoped cuprates than the SO (5) symmetry.Of course, a question can be raised at this point,which is that the exact realization of the SU (2) sym-metry within the EHS model gives a charge wave vectoron the diagonal, while only axial charge order was exper-imentally observed [1, 2, 6, 20, 21, 24, 42, 43, 124]. It is FIG. 3. (Color online) Experimental B - T phase diagramfrom sound velocity measurements in YBCO from Ref. [18].The “pseudo spin-flop” is visible from the SC to CDW tran-sition beyond a critical field B c (cid:39) T. an important question in the SU (2) theory and we willaddress it in details in the next section. For the momentlet us notice that similar rotations as in Eqn.(1) can begenerated for the axial wave vector Q = { Q x , Q y } ob-served experimentally, which rotates similar multiplets asin Eqn.(5) but for the axial wave vector. This idea of a ro-tation between the d -wave SC state and the axial chargeorder [122] was used to explain that the CDW signal ispeaked at T c [20, 21]. It was also used in explaining the A g mode observed in Raman scattering as a collectivemode associated to this specific rotation [125–127].The notion an emergent symmetry is more general thanany of its specific representations. It is indeed very nice tohave a model, although very simplified, where the SU (2) symmetry is exactly realized (at all energy scales), butthe main concern is whether this symmetry is approxi-mately realized at finite temperatures in the underdopedregion of the phase diagram. That is the interest of theconcept of emergent symmetry: although it can be ex-actly realized in only a few effective models, if the split-ting between the two pseudo-spin states is smaller thanthe typical energy of each state, it can also be approxi-mately realized at low energies in the more realistic 2D t − t (cid:48) Hubbard model (this was verified explicitly in Refs.[128, 129] using two-loop RG techniques).Another remark that can be made at this stage, is thatanother type of SU (2) symmetry was identified early on,which consists of performing a particle-hole transforma-tion on each site c † iσ → c i − σ , which translates in thereciprocal space as c † kσ → c k − σ for all k vector. This sym-metry is interesting for the phase diagram of the cupratesbecause it is exact at half-filling and will be graduallybroken with doping [130, 131]. The operators for thissymmetry group take the form η + ph = (cid:88) k c † k ↑ c † k ↓ (8a) η z = (cid:88) k σ (cid:16) c † k σ c k σ − (cid:17) , (8b)while one irreducible l = 1 representation associated toit can be taken as [132] ∆ = − √ (cid:88) k d k c † k ↑ c †− k ↓ , (9a) ∆ = 12 (cid:88) k ,σ d k c † k σ c − k σ , (9b) ∆ − = 1 √ (cid:88) k d k c k ↓ c − k ↑ . (9c)Interestingly, within the EHS model, the operators (8)and (9) are, respectively, the same as (1) and (5) with adiagonal wave vector, since there the summation over k is reduced to the eight hot spots. The EHS model is alsoan exact realization of the SU (2) symmetry associatedwith particle-hole transformation.Using (8) one can also ask oneself what is the SU (2) partner of the observed axial CDW. To fix the ideas letus take a uni-axial CDW order with ordering wave vector Q x relating two hot spots. One can then construct the l = 1 irreducible representation using the particle- holetransformation. This gives ∆ = − √ (cid:88) k d k c † k ↑ c †− k + Q y ↓ , (10a) ∆ = 12 (cid:88) k ,σ d k c † k σ c k + Q x σ , (10b) ∆ − = 1 √ (cid:88) k d k c − k + Q y ↓ c k ↑ , (10c)which means that the SU (2) partner of the Q x CDW isthe pair density wave (PDW), namely a non zero centerof mass SC state, with Q y wave vector. This notion ofPDW state was introduced recently to explain the veryunusual ARPES data tracing the formation of the PGin Bi-2201[45, 133]. In this theory, the formation of thePDW is suggested as the primary mechanism for the for-mation of the PG state, which means that the observedCDW is a secondary order. As such, it should be ob-served at a wave vector twice as big as the PDW wavevector Q CDW = 2 Q P DW . In contrast, if the mecha-nism governing the underdoped region is a hidden pseu-dospin SU (2) symmetry, then the partner of a bi-axialCDW is a bi-axial PDW with the same wave vectors Q CDW = Q P DW (see the Refs. [46, 129, 134, 135]). Thelatter scenario has recently been verified experimentally[136].
III. NON LINEAR σ -MODEL, AND SU (2) VS. U (1) PAIRING FLUCTUATIONS
The idea of emergent symmetries received a recent cri-tique, that when the group of symmetry is large enough,the symmetric phase is unstable to smaller subgroups[137]. For example, the symmetric phase associated tothe SO (5) symmetry which was intended to describe thePG shall decompose into the SU (2) × U (1) group de-scribing fluctuations around the AF and SC phases re-spectively. Similarly, the SU (2) symmetry which rotatesbetween the CDW and SC channel shall decompose intothe U (1) × U (1) groups. In this section, we consider se-riously the criticism that the symmetric phase of largenon-abelian groups is unstable, but wonder more partic-ularly about the fate of SC fluctuations.The role of SC fluctuations in the physics of cupratesis indeed very mysterious. We know that they are a feworders of magnitude more intense that in standard met-als like Al , or Cu [138], but experiments detecting purethe Josephson effect were observed only a few tens of de-grees above T c [30, 139, 140]. In the deeply underdopedphase, U (1) SC fluctuations form a dome shape that wewill discuss further in this section [140–144] . Direct ob-servation of pre-formed pairs in the PG phase was alwaysnegative, but a giant proximity effect was observed in theLanthanum-compounds induced in the PG phase when itis surrounded by optimal SC phases [145–147]. The veryeasy injection of pairs from optimally doped into the PGphase suggests that the PG phase is related to the SCphase through a hidden symmetry. Such proximity ef-fects were predicted in the case of the SO (5) symmetryand never observed [105], but specific predictions in thecase of the SU (2) symmetry were never discussed in de-tail. A. The SU (2) SC fluctuations
In this section we assume that at an intermediate en-ergy scale SC fluctuations are present, protected by an SU (2) symmetry between the CDW and SC channel.The microscopic derivation of the non linear σ -model de-scribing the SU (2) fluctuations can be found, in the con-text of the EHS model in Ref.[110], and in the contextwhere regions of the Brillouin zone instead of points are“hot” -or hot anti-nodal regions-, in Ref. [94]. The mass-less O (4) effective free energy has the following form F SU (2) = T (cid:88) ε,ω ˆ k , q trδ ˆ u †− k,q (cid:2) J ,k ω + J ,k q (cid:3) δ ˆ u k,q (11)where ˆ u k , q = (cid:18) ∆ CDW ∆ SC − ∆ ∗ SC ∆ ∗ CDW (cid:19) is the SU (2) matrixassociated with the condition | ∆ CDW | + | ∆ SC | = 1 , a) Π Π b) k x (cid:72) Ξ ks (cid:76) FIG. 4. (Color online) a) Visualization of the SU(2) symme-try breaking contribution ( ξ s k ) in the positive region of thefirst Brillouin zone. It is small in the blue region and vanishesfor the two black lines crossing the hotspot, but grows in thenodal line to the upper edge. b) Variation of ( ξ s k ) along theFermi surface (shown as a gray line in panel a)) parametrizedas a function of k x . While ( ξ s k ) vanishes at the hotspot andstays small at the antinodes at k x = π , it grows at the leftside when approaching the nodal position. and the tr runs on the SU (2) structure. The coeffi-cients write J ,k = | M k | / (cid:12)(cid:12) G − (cid:12)(cid:12) , and J ,k = J ,k v k ,where M k is the magnitude of the mean-field SU (2) or-der parameter ˆ M k , q = (cid:18) ˆ m k ,q ˆ m † k ,q (cid:19) Λ , with ˆ m k , q = M k ˆ u k , q , which has a × structure in the τ × Λ - SU (2) spaces where τ is the particle-hole transformation and Λ is the Q -translation. The Green’s function writes ˆ G − = ˆ G − ,k + ˆ M k, , with ˆ G − ,k = iω − ( τ ξ s k − ξ a k ) Λ , with ξ s,a k = ( (cid:15) k ± (cid:15) − k − Q ) / , and (cid:15) k being the electron disper-sion. Note that no information was given on the value ofthe Q -wave vector for the CDW sector. It corresponds inall generality to the SU (2) operators (1) and (2). The ex-act SU (2) symmetry is verified when ξ s k = 0 which effec-tively kills the τ -term in the equation for ˆ G − ,k . ξ s k hencemodels the symmetry breaking term associated with thisspecific wave vector and contributing to the free energyas F SB = T (cid:88) ε,ω ˆ k , q J ,k tr (cid:104) δ ˆ u †− k,q τ δ ˆ u k,q τ (cid:105) , (12)with J ,k = 14 | m ,k | ( ξ s k ) | G − | . (13)The shape of the symmetry-breaking term Eqn.(13) isvisualized in Fig. 4. One can observe the anisotropy ofthe mass in various directions in the Brillouin Zone : themass is much bigger in the nodal direction than in theanti-nodal one. B. Resonant Peierls Excitonic (RPE) state
We now integrate the SU (2) SC fluctuations out ofthe partition function, and evaluate the consequences of a) b)FIG. 5. (Color online) left) Density of the charge orderparameter | χ k, − k | in the first Brillouin zone from the RPEstate. The charge density follows the Fermi surface, but dueto a SU(2) dependent mass contribution, does not stabilize inthe nodal region. right) Charge order parameter around thehotspot position for a constant p F ordering vector. FromRef. [94].FIG. 6. (Color online) Set of degenerate p F couplingsbetween electrons on opposed Fermi surfaces in the antinodalregion due to the RPE state. From Ref. [94]. them in the charge channel. We get the following effectiveaction S eff [ c ] = (cid:88) kk (cid:48) q π k,k (cid:48) ,q c †↑ k c ↑ k (cid:48) c †↓− k + q c ↓− k (cid:48) + q , (14)with π k,k (cid:48) ,q = (cid:104) ¯∆ k,q ∆ k (cid:48) ,q (cid:105) = π ( δ k, − k (cid:48) + δ k,k (cid:48) )( J ω n + J ( v · q ) + a ,k ) , (15)where the form of the SC fluctuations comes fromEqns.(11,12). The self-consistent Dyson equation ( or“gap equation”) writes χ k,k (cid:48) = (cid:88) q π k,k (cid:48) ,q [ ˆ G ( q − k, q − k (cid:48) )] , (16)with χ k,k (cid:48) = (cid:80) q π k,k (cid:48) ,q (cid:104) c † σ k c σ k (cid:48) (cid:105) and with [ ˆ G k,k (cid:48) ] = −(cid:104) c † σ ( k ) c σ ( k (cid:48) ) (cid:105) = − χ k,k (cid:48) ( i(cid:15) n − ξ k )( i(cid:15) (cid:48) n − ξ k (cid:48) ) − χ k,k (cid:48) . (17)To get some idea about the nonlocal nature the orderparameter χ k , k (cid:48) , it is illustrative to consider one of thelabels having a constant shift k (cid:48) = k + P . The order pa-rameter can then be decomposed as χ P , k = χ P F P [94],where χ P is a plain wave and F k a formfactor for theelectron-hole pair with a finite extent, see Fig. 5b). Thetotal order χ k , k (cid:48) is thus a superposition of all local χ P , k orders, where P is running over the whole set of degen-erate p F couplings as visualized in Fig. 6. The gap inthe antinodal region is ∆ P G k = (cid:80) { P } χ P , k . Note that inEqns. (16,17) the external wave vectors k , k (cid:48) are a pri-ori not defined, but are let free to find self-consistentlythe most favorable solution. We studied numerically thepossible excitonic solutions of the gap equations and theresult is depicted in Fig. 5. We obtained an excitonicstate in which a large number of wave vectors are degen-erate with a typically k − k (cid:48) = 2 k F which are spread outin the anti-nodal region of the Brillouin zone producing adepletion of the density of states in this region (see Fig.6). Due to the angular dependence of the fluctuationmass ( a ,node (cid:29) a ,anti − node ), we obtain a preferentialgapping out of the anti-nodal region, which is charac-teristic of the SU (2) SC fluctuations compared to theoriginal U (1) . C. Long range charge order
At this stage we have proposed a theory for the PGphase of cuprates superconductors. In the following wewill give some more arguments that this theory is apromising candidate to describe high T c superconduc-tors. We know from a body of experimental evidence,though, that the PG phase is distinct from the observeduni-axial CDW. At zero field, the CDW dome decreaseswhen the oxygen doping decreases, which is at variancewith the PG T ∗ -line, which increases with the chemi-cal potential (or oxygen doping) [21, 24, 124, 148–150].Moreover, recent studies of the Fermi surface reconstruc-tion under a magnetic field of T infer that the PG isformed before the Fermi surface is reconstructed by CDWorder -recall that the CDW becomes long range and threedimensional beyond B = 17 T. There are many proposalsfor the PG phase [45, 46, 90–92, 117, 122, 137, 151–154],but since ours consists of a special type of excitonic liquid,it is important to shed light on the relationship betweenthe excitonic RPE phase and the observed axial CDWwhich is stabilized under magnetic field.Within the EHS model, or within all sorts of simpleweak coupling RPA evaluation, we find that the axialcharge order is a secondary instability, weaker than CDWwith a wave vector on the diagonal. The question thatis then raised, is why nothing at all is observed on thediagonal, whether it is by STM [2, 5, 6, 8] or by X-rays[20, 21, 24, 148] The simplest explanation is that thepseudogap is forming, gapping out the anti-nodal regionof the Brillouin Zone, and wiping out the CDW instabil-ity on the diagonal. When the mechanism of formation of the pseudogap instability has operated, then the sec-ondary instability can be visible, at the tip of the Fermiarcs. Many suggestions have been made for the forma-tion of axial CDW order. The fact that this wave vectoris present as a secondary instability in any weak couplingtheory, and stabilized for example, in the presence of ad-ditional effect like Coulomb interactions [134, 155, 156],within both one-loop and two-loop RG [128, 129, 157–159] or starting from a three band model [152], or invok-ing the proximity to the van Hove singularity [160] hasbeen outlined in many works, including ours. All thesestudies are based on the observation that axial CDW isdistinct from the formation of the PG state and starts toget formed at the tip of the arcs. In all mentioned sce-narios though, it is quit unclear why the CDW dome isincreasing with doping, in contrast with the PG line T ∗ .The SU (2) -paradigm which related the axial CDW to SCfluctuations that gives a good explanation for the maxi-mum of amplitude of the CDW order at T c [122] seemsto have been lost in the attempts to rotate the leadingwave vector from the diagonal to the axes.In this paper, we would like to offer an alternativescenario which to our knowledge has not been proposedyet. It is based on the observation that the CDW domefollows closely the U (1) SC fluctuation dome that canbe detected through many probes, like resistivity study[30, 31], Nernst effect [141, 143, 144], Josephson tunnel-ing. The U (1) standard SC fluctuations are usually veryweak in SC states because Coulomb interactions pushthem above the plasma frequency[161]. In the case ofthe optimally doped cuprates, due to low dimensionality,phase fluctuations are observed in a window of roughly15 K above T c [139]. For more underdoped compounds,the approach of the Mott insulating states produces adrastic increase of phase fluctuations due to the phase-particle number Heisenberg uncertainty relations [162].We would like to exploit this idea here and suggest thatthe axial CDW order is an effect of the U (1) SC fluctu-ations. We use a recent remark of Ref. [163] where itis noted that after the formation of the PG, the Fermisurface has split into four sections-or arcs, which are nowindependent from each other. In these conditions, onecan get a Leggett mode between the tips of the arcs,which in turn can induce a charge order with the correctwave vectors, see Fig. 7. The gap equation for the axialCDW mediated by the Leggett mode writes χ k σ, k + Q σ = T (cid:88) ω n q π Q , q [ G − k + q , − k − Q + q ] , (18)with Q = Q x / y being the axial wave vector, G givenby Eqn.(17) with the replacement ξ k → ξ k +∆ P G k to takethe gapping of the FS into account, whereas π Q , q is the U (1) correlation of the phase fluctuations at the tip of La Sr x CuO T ( K ) Sr content x onset T T *T c S u p e r c o n d u c t i n g T c C D W O n s e t
Temperature (K)
H o l e d o p i n g p
FIG. 7. (Color online) upper panel) Temperature-holedoping phase diagram deduced from Nernst effect experi-ments (from Ref.[144]). The gray area corresponds to CDWphase where Nernst coefficient is not zero. lower panel)Temperature-hole doping phase diagram with the supercon-ducting critical temperature (black dots) and the onset tem-perature of the CDW axial order (red dots) deduced fromRXS experiments in YBa Cu O x (extrated from Ref. [24]). QQ diag Q x Q y FIG. 8. (Color online) Leggett mode (highlighted in red),where charge order is induced by U (1) phase fluctuations atthe tip of the arcs with the axial wave vector Q x/y . the arcs, as represented in Fig. 8 π Q , q = (cid:68) T ∆ k , − k + q ∆ † k + Q , − k − Q + q (cid:69) U (1) = π J (cid:48) ω n + J (cid:48) ( v · q ) + m , (19)with k being the wave vector at the tip of the one Fermi arc and k + Q being the wave vector at the tip of theadjacent Fermi arc. We use a generic form of the prop-agator π Q , q , where π , J (cid:48) , J (cid:48) and m are mon-universalparameters the dependence on Q is neglected. We haveperformed a numerical study of Eqn.(18) which confirmsthat U (1) SC fluctuations mediated by a Leggett modeproduce axial CDW with the desired wave vector. Thisproposal has the merit to consistently link both the for-mation of the PG and the observed axial CDW to SCfluctuations, the former being described by the SU (2) non linear σ -model while the latter are the standard U (1) phase fluctuations. IV. DISCUSSION
In the phase diagram of high temperature cuprates afew key players can be identified [131, 164]. There isat half-filling the Mott insulating transition with typicalenergy of 1 eV associated to it. Antiferromagnetism isubiquitous in the whole phase diagram, with an orderedphase of typically T Neel ≈ K at half-filling, very closeto the Mott transition, and strong, but short range AFfluctuations in the underdoped regime. In the proposal ofthis paper, the mysterious PG phase of high temperaturecuprates is attributed to a new kind of excitonic state, theRPE, which can be understood as a new type of “liquid”of excitons, with a superposition of degenerate wave vec-tors. This state is a consequence of integrating out the SCfluctuation, protected by an emergent SU (2) symmetrybetween the SC and charge channel. In the discussion ofthis proposal, the first thing to recall is that although an-tiferromagnetism is not directly responsible for the PG, itis nevertheless the underlying force driving the emergenceof precursor orders. In the early version of this theory,the EHS model has been studied as a reference modelwhere the SU (2) symmetry is verified [110, 165]. In thismodel the eight hot spots are singled out of the Fermisurface, and long range AFM fluctuations stabilize thecomposite SU (2) - order parameter, composed by a diag-onal quadrupolar density wave and SC. In more genericversions of this theory, the model is extended to “hotregions” of the Brillouin zone - the anti-nodal regions,where AF acts predominantly and the SU (2) symmetryis most strongly verified [119]. Antiferromagnetism didnot disappear from the phase diagram, but rather hasa very special relation to the PG by defining the widthof the “hot regions”, thus limiting the domain of actionof the RPE state, and also being the driving force bothbehind SC pairing and the SU (2) symmetry.The concept of emergent symmetry though, is morerobust and general than even the idea of Quantum Crit-icality and it is under such a generic paradigm that wewant to cast out the underdoped region of cuprate super-conductivity. The main idea is that charge orders are thenatural partner and competitors of SC pairing in the un-0derdoped region of the cuprates, and typical pseudo “spinflops” between the two orders are to be expected, and webelieve already observed under magnetic field [18].The experimental consequences of a phase diagramcontrolled by an SU (2) emergent pseudospin symmetryare numerous, and it is very likely that our proposal forthe RPE state may be confirmed or in-firmed within thenext few years.
1. Spectroscopic signatures
One can first ask about the spectroscopic signature ofsuch an excitonic state. What can be seen in STM orX-rays. Our claim here is that we can reproduce thevery recent findings on Bi-2212 [7, 166], that the pure d -wave component of the axial CDW extends up to thePG temperature, see Fig. 9. In the RPE state, indeed,the excitons form not only around many degenerate wavevectors, but with a finite width around each wave vector.The real space picture is that the particle-hole pairing a)b)FIG. 9. (Color online) Upper panel a) Tunneling conduc-tance measurements from Ref. [8] of underdoped cuprates.Two characteristic energies, a lower one for Bogoliubov quasi-particles and a higher one corresponding to the pseudogap areobserved. Lower panel b) Energy dependence of the s - and d - wave form factors, indicating that the higher gap-energyscale corresponds to the d -wave form. From Ref. [8] is non local in space, and modulated by many wave vec-tors. When the induced charge on the oxygen is evaluated and Fourier transformed, one finds that it is 90% d -wave(100 % for the diagonal wave vector and a bit less forthe others), and at the same time, the axial wave vec-tors are more favored compared to the diagonal due toits nesting properties in the anti-nodal region [167]. Theconsequence is that the charge on the Oxygen shows apreponderant spectrum with axial wave vectors Q x and Q y . At this stage our conclusion is that the RPE stateis already observed by STM and X-rays, which have cap-tured its preponderant contribution on the axial wavevectors.
2. Proximity effect
A second remark is that emergent symmetries rotatingthe SC phase to another type of order predict proximityeffects when the PG phase is sandwiched between twooptimally doped superconductors. The intensity of theinduced current in the junction persists for thickness ofthe gap material much greater than the superconductingcorrelation length. This ”Giant” Proximity Effect (GPE)is not explicable by the standard theory of the proximityeffect between two SC junction, but can be understoodin the situation where the SC state is ”quasi-degenerate”to another phase of matter and Cooper pair can thusbe easily injected from the SC state to the other state.The situation is thus very promising for emergent sym-metries, and has been extensively studied in the case ofthe SO (5) -symmetry [102, 106], where specific predic-tions for the current as a function of the phase differenceacross the junction can be made as well for the SU (2) symmetry, see Fig. 10. Note that a giant proximity effecthas already been observed in various compounds but hasnot been observed for the specific setup of the SO (5) -group. One straightforward application of our theory isto check whether SU (2) fluctuations, where the rotationis between the SC state and the CDW state can accountfor the experimental data [147, 168–173]. SCSC PG
FIG. 10. (Color online) Proposition of a SC-PG-SC junc-tion to study the giant proximity effect within the SU(2) the-ory. At a given temperature T that is homogeneous over thejunction, the two outer SC layers are superconducing and atoptimal doping such that T < T c . The inner PG layer is anunderdoped SC in the pseudogap phase ( T ∗ > T > T c ). Fromthe giant proximity effect we expect the PG phase to becomesuperconducting by lowering the thickness of the inner layer.
3. Magnetic field phase diagram
The phase diagram found as a function of magneticfield and temperature, derived with a variety of exper-iments [9, 13, 16–18] is typical for a super “spin-flop”between two states related by a symmetry (see Fig.11).Note that three dimensional CDW has been recently ob-served by X-ray scattering above B = 17 T [27]. The
FIG. 11. (Color online) B - T phase diagram obtained fromthe spin-fermion model considering order parameter fluctua-tions around the mean-field value with a nonlinear σ modelfrom Ref. [120]. CDW and SC orders have the same order of magnitude inthis diagram, and the transition between the two is verysudden, like in a generic spin-flop XY model [120, 121].Moreover, an SU (2) partner of the axial CDW has re-cently been reported, i.e. the PDW with the same wavevector [136]. Although it is not a direct proof of the un-derlying symmetry, it seems to rule out other scenariosfor the PG state where the PDW is primary while theCDW orders are secondary, and hence occur at twice thesame wave vector as the PDW.
4. Collective modes
Emergent symmetries also have signatures in terms ofcollective modes. In a recent work we argued that the A g -mode observed in Raman scattering very close inenergy to the neutron mode is such a signature of theSC-CDW SU (2) symmetry [125], see Fig. 12. The col-lective mode used in this work was associated with the η -operator of Eqn.(1) with axial wave vector, thus associ-ated to the triplet representation Eqn.(10). The presenceof the two orders in conjunctions was needed in order forthe Raman scattering vertex not to vanish. The modelcould account for the absence of observation of this orderin the B g and B g channels.Inelastic neutron scattering has reported since the veryearly days the presence of a collective mode in the un-derdoped regime, centered around the AFM wave vector Q = ( π, π ) , and at a finite energy around E = 41 meVfor the compound YBCO [65, 71]. Many theories, based ( a ) ( b ) ( c ) w a v e n u m b e r w ( c m - 1 ) Im c Raman( w ) a.u. Im c Raman( w ) a.u. B t h e o B e x p A t h e o A e x p A t h e o * Im c S( w ) a.u. h = 1 0 m e VJ = 1 0 7 . 5 m e VV = 5 4 . 5 m e V p = 0 . 1 6 FIG. 12. (Color online) Raman scattering response froma collective mode. a) shows the calculated Neutron suscep-tibilities at the momentum Q = ( π, π ) as a function of thefrequency. In b) (and c) ), the experimental (solid line) andcalculated (dashed line) Raman response in the A g ( B g )Raman channels. The Raman resonance arises at the samefrequency than the Neutron resonance at Q (Fig (a) and (b))below the superconducting coherent peak energy observablein the B g symmetry (Fig (b) and (c)). This collective moderesonance appears in the A g symmetry since the Raman re-sponse is screened by long range Coulomb interaction in thissymmetry. From Ref. [125]. on an RPA treatment of a magnetic spinon mode belowthe SC gap have been produced in order to explain thisvery characteristic feature of the cuprates [77, 102, 174].The SO (5) theory was originally devoted to the studyof this mode [77]. The RPA theories, reproduce suc-cessfully the position of incommensurate signal around Q = ( π, π ) , having the typical “hour-glass” shape in theenergy-momentum space. The present theories have dif-ficulties to account for the fact that this signal remainsinside the PG phase, changing form from the “hour-glass”to a “Y” shape, namely acquiring some extra low energyspectral weight at Q = ( π, π ) . The proposed RPE stateis an excitonic state with excitations around a bunch of k F - wave vectors in the anti-nodal region. Thus it be-haves a little bit as a “charge superconductor”, that inthe simplest models, will gap out the electronic degreesof freedom precisely as a superconductor would do. Webelieve the RPE state can also account for the extra spec-tral weight at Q = ( π, π ) , which will be presented in afuture work [167].2
5. ARPES
We turn now to angle-resolved photoemission spec-troscopy (ARPES), which have been very influential inour understanding of the PG phase of these materials,especially with the seminal observation of Fermi arcs inthis phase . A recent ARPES experiment on Bi-2201 hasbeen very important in our understanding of the forma-tion of the PG [41, 43]. Dispersion cuts close to the ZoneEdge show that the PG opens at a typical momentumlarger than the momentum relating the two Fermi points k F . Moreover when the dispersion cuts get closer to thenodes, the PG closes from below rather than from above.It has been argued that this set of peculiar features canonly be explained by a PDW state (a finite momentumSC state), since only the particle-hole reversal specificto the pairing state can account for the closing of thegap from below [45, 175]. We argue that the RPE state FIG. 13. (Color online) Electronic dispersion measured byARPES in the superconducting (blue and green) and normalstate (red). Cuts taken at constant k x : close to the zone edge(O)( k x = π ), close to the nodal region (T, k x = 0 . π ) and inan intermediate case (R, k x = 0 . π ). The closing of the gapcan be observed from the antinodal to the nodal zone. FromRef. [43]. provides another explanation for this fascinating set ofdata. Besides the multiplicity of the wave vectors, thekey ingredient is the non locality of the excitons. In par-ticular, in the reciprocal space, they form within a finitewindow in the anti-nodal region, which can account forthe natural closing in energy of the gap, both from aboveand below (ARPES does not see the positive energies),so that we have only to account for the negative part ofthe spectrum [167].
6. Loop currents
The observation of a Q = 0 signal in neutron scat-tering at a temperature line following T ∗ [49] is one ofthe mysteries of the PG phase, which has been inter-preted in terms of the formation of intra-unit-cell Θ II -loop currents [90]. Although it is commonly understoodthat a Q = 0 phase transition does not open a gap in theelectronic spectrum, and thus the Θ II -loop-current phasealone cannot be responsible for the origin of the PG, any proposal for the PG phase has to account for the signalobserved in the neutron scattering experiment. We have FIG. 14. (Color online) Coexistence between Θ II -loop cur-rent order parameter and bidirectional d -wave CDW as a func-tion of the interaction strength λ . From Ref. [135]. produced two studies within the EHS model regardingthe possibility of coexistence of charge orders and loopcurrents [135, 159]. In Ref. [159], we have shown, withina saddle-point approximation, that the Θ II -loop-currentorder cannot coexist with a d-wave CDW with diagonalwave vectors. As a result, we have offered this scenario asthe possible reason, which explains why a d-wave chargeorder was never observed along the diagonal direction inthe cuprates. In a subsequent work [135], we have demon-strated that a similar behavior is displayed by the d-waveCDW along axial directions described by uni-directionalwave vectors (i.e. of the stripe-type), since the Θ II -loop-current order is also detrimental to the latter order. Bycontrast, we have shown that bi-directional (i.e. checker-board) d-wave CDW and PDW along axial momenta,which are in turn related by the emergent SU (2) pseu-dospin symmetry pointed out previously, are compatiblewith the Θ II -loop-current order, since all these orders cancoexist with one another in the phase diagram (see, e.g.,Fig. 14). These theoretical predictions agree, most spec-tacularly, with recent STM results [136] and also withx-ray experiments [24].
7. Pump probe experiment
A recent pump probe experiment also gives some ev-idence of the presence of strong SC fluctuations at anintermediate energy scale [176] in underdoped cuprates.In the first series of pump probe experiments [177, 178],the cuprate was excited up to . eV and relaxation at thepico-second scale - observed in the optical THz regime,destroyed the Cooper pairs and showed two typical en-ergy scales, one related to the PG regime and one as-sociated with the formation of the coherence SC phase.Those two scales are typically the ones observed, for ex-ample, in the dI/dV response of STM microscope. Butin a recent experiment, the excitation was much weaker,in the mid Infra red regime [176] which enabled to scan3 FIG. 15. (Color online) Schematic phase diagram for YBCOproposed in Ref. [176]. Under out-of-equilibrium conditionsrealized by optical pump-probes, a high mobility phase in theblue shaded area can be realized that extends much above thecritical temperature of equilibrium SC. From Ref. [176]. the properties of the PG phase without destroying theCooper pairs. What was found resembles to a pico-second photograph of SC fluctuations with the superfluiddensity ρ s ∼ ω ∆ σ shooting up in the PG phase, up totemperatures of 300 K , see Fig. 15. This pico-second“photograph” of the superfluid density was shown to fol-low the PG temperature as a function of doping.
8. General phase diagram
A general look at the phase diagram of the cupratessingles out many enduring mysteries, and one of themost enduring one is the linear-in- T resistivity aroundoptimal doping. This phase was identified in a semi-nal work as a Marginal Fermi Liquid (MFL) [179, 180],and it is still a challenge for theories to account for thisphenomenon. Recent in-depth experiments show a morecomplex behavior of the resistivity with temperature,with a part linear in T and a residual part going like T when the strange metal regime is approached fromthe right hand side of the phase diagram [33, 34]. Twoschools of ideas have been advanced to explain this veryunusual phenomenon. In the first school of ideas, it isbelieved that the proximity to the Mott transition cre-ates a very strongly correlated electronic medium wherethe electron mean free path is so weak that we are abovethe Ioffe-Regel limit for the MFL regime [87]. The sec-ond school of ideas advances that the resistivity slopeas a function of temperature is very steep, so that thesecond MFL regime extends far beyond the Ioffe-Regellimit at low enough temperatures. Within this secondviewpoint, the challenge is to suggest a QCP beyondthe dome which could produce a very resistive scatter-ing behavior for the conduction electrons. It is preciselywhat the quantum critical version of the RPE state does.Electronic scattering through quantum critical excitonic doping T AFM pseudogapSCstrange metalfermi liquid T AFM PG doping T AFM PG SC strange metalFermi liquid T c T * Fermi liquidstrange metalQCP AFM
QCP
RPE
FIG. 16. (Color online) Schematic phase diagram of cupratesuperconductors with the RPE state, as proposed in [94]. Thequasi one-dimensional scattering in the vicinity of QCP
RPE produces the resistivity and the electronic self-energy anomalyobserved in the strange metal phase. From Ref. [94]. modes shows a quasi-one dimensional behavior, each elec-tron scattering preferentially through its most favorable k F wave vector [94], and produces a resistivity that be-haves as ρ ∼ T / log T within a Boltzmann semiclassi-cal calculation and the electronic self-energy that reads Σ ( i(cid:15) n ) ∼ i(cid:15) n / log | (cid:15) n | in the “Strange Metal” (SM) regime(see Fig. 16). On the same line of thought, maybe one ofthe most difficult feature of the PG to account for in anytheory is that it is insensitive to a large amount of Zn -doping or irradiation by electrons [31, 140, 181], whichlocate on Cu sites and produce strong disorder which ex-clude the doped Cu -site in the unitary limit [182]. The T ∗ -line is not affected and also the slope of the resistivityin the SM regime does not change [31]. It is difficult forany state of matter to have the sufficient robustness toshow no sensitivity to such a strong perturbation. Oneway the RPE state could resist is through the non-localityof the excitons (i.e. the particles-hole pair), which cantypically being created at site r and removed at site r (cid:48) with the typical correlation (cid:10) c † r c r (cid:48) (cid:11) [167]. V. CONCLUSION
To conclude, within the two large views of the cuprateswhere, on one hand, the physics of the PG is solely de-termined by strong correlations and the proximity to theMott transition, and the other view where the qualita-tive features of the physics of the PG are governed bysome hidden symmetry, the present work makes a cleardiscrimination in favor of the latter. It is claimed herethat the physics of the PG and the SM phase are con-trolled by an emergent SU (2) symmetry. Many proper-ties of the underdoped cuprates can be captured withinthe pseudo-spin theory, the non-linear σ -model associ-ated to this symmetry and the pseudo spin-flop physicsbetween the SC and charge channel. We also claim that SU (2) superconducting fluctuations proliferate at inter-mediate energy scales in the physics of these compounds,and are are the key ingredient to understand the PG4phase. At lower energy, they lead to the formation ofthe RPE state, which we believe has a lot of promis-ing features to be the solution for the PG. At even lowertemperatures, the U (1) phase fluctuations enter the gameand produce coherent axial CDW mediated by a Leggett-mode. The SU (2) symmetry is present in the backgroundof the whole underdoped region, and it is expected thatpseudo-spin partners of the various orders (such as thePDW partner of the CDW order) have recently beenobserved experimentally [136]. Lastly, what is the in-fluence of the Mott transition on the phase diagram ofthe cuprates ? We believe it will qualitatively affect thephysics up to roughly 6% of doping. Below 6% of dop-ing, techniques adequate to describe the very stronglycoupled regime will capture the physics [82, 90, 117] .Beyond 6% doping, the physics is qualitatively protectedby the SU (2) symmetry. A very revealing experimentis the variation of the nodal velocity v ∆ as a functionof doping extracted from ARPES data (Ref. [85]). Atri-sected dome is observed with three distinct regimes,1) below 6% doping, 2) between 6% and 19% of dopingand 3) above 19 % of doping. Within the SU (2) theory,as with all theories controlled by an emergent symmetry,the critical value of 6% of doping is precisely the pointwhere the Mott physics becomes dominant. Within thestrongly correlated viewpoint, the typical doping of 6%is difficult to interpret. A B C E ne r g y ( m e V ) v Δ (laser)v Δ (synch.) Δ node v A FIG. 17. (Color online) ARPES experiments on BSCO per-formed in Ref. [85]. The doping dependence of the gap veloc-ity v ∆ reveals three distinct regime: two regions at low andhigh doping where v ∆ drops and a third regime in-between,where v ∆ reaches a plateau. From Ref. [85]. We are grateful to H. Alloul, Y. Sidis, P. Bourges andA. Ferraz for stimulating and helpful discussions. 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