Interplay between charge order and superconductivity in cuprate superconductors
Deheng Gao, Yiqun Liu, Huaisong Zhao, Yingping Mou, Shiping Feng
IInterplay between charge order and superconductivity in cuprate superconductors
Deheng Gao and Yiqun Liu
Department of Physics, Beijing Normal University, Beijing 100875, China
Huaisong Zhao
College of Physics, Qingdao University, Qingdao 266071, China
Yingping Mou and Shiping Feng ∗ Department of Physics, Beijing Normal University, Beijing 100875, China
One of the central issues in the recent study of cuprate superconductors is the interplay of chargeorder with superconductivity. Here the interplay of charge order with superconductivity in cupratesuperconductors is studied based on the kinetic-energy-driven superconducting (SC) mechanism bytaking into account the intertwining between the pseudogap and SC gap. It is shown that theappearance of the Fermi pockets is closely associated with the emergence of the pseudogap. How-ever, the distribution of the spectral weight of the SC-state quasiparticle spectrum on the Fermiarc, or equivalently the front side of the Fermi pocket, and back side of Fermi pocket is extremelyanisotropic, where the most part of the spectral weight is located around the tips of the Fermi arcs,which in this case coincide with the hot spots on the electron Fermi surface (EFS). In particular, ascharge order in the normal-state, this EFS instability drives charge order in the SC-state, with thecharge-order wave vector that is well consistent with the wave vector connecting the hot spots onthe straight Fermi arcs. Furthermore, this charge-order state is doping dependent, with the charge-order wave vector that decreases in magnitude with the increase of doping. Although there is acoexistence of charge order and superconductivity, this charge order antagonizes superconductivity.The results from the SC-state dynamical charge structure factor indicate the existence of a quanti-tative connection between the low-energy electronic structure and collective response of the electrondensity. The theory also shows that the pseudogap and charge order have a root in common, theyand superconductivity are a natural consequence of the strong electron correlation.
PACS numbers: 74.72.Kf, 74.25.Jb, 74.20.Mn,71.45.Lr, 71.18.+y
I. INTRODUCTION
The understanding of the mechanism of superconduc-tivity in cuprate superconductors remains one of themost intriguing problems in condensed matter physics.The parent compound of cuprate superconductors is ahalf-filled Mott insulator , which occurs to be due tothe strong electron correlation . Superconductivityis derived from doping this parent Mott insulator ,indicating that superconductivity and the related ex-otic physics in the doped regime are also dominated bythe same strong electron correlation. In conventionalsuperconductors , an energy gap exists in the electronicenergy spectrum only below the superconducing (SC)transition temperature T c , which is corresponding to theenergy for breaking a Cooper pair of the electrons andcreating two excited states. However, in cuprate super-conductors above T c but below a characteristic tempera-ture T ∗ , an energy gap called the pseudogap exists . Inparticular, this pseudogap is most notorious in the un-derdoped regime, where the charge carrier concentrationis too low for the optimal superconductivity .However, the strong electron correlation also inducesthe system to find new way to lower its total energy, of-ten by spontaneous breaking of the native symmetriesof the lattice . This tendency leads to that the pseu-dogap regime harbours diverse manifestations of the or- dered electronic phases, and then a characteristic fea-ture in the complicated phase diagram of cuprate su-perconductors is the interplay between different orderedelectronic states and superconductivity . In partic-ular, by virtue of systematic studies using the scan-ning tunneling microscopy (STM), resonant X-ray scat-tering (RXS), angle-resolved photoemission spectroscopy(ARPES), and many other measurement technique ,it has been found recently that charge order is a univer-sal phenomenon in cuprate superconductors, which ex-ists within the pseudogap phase, appearing below a tem-perature T CO well above T c in the underdoped regime,and coexists with superconductivity below T c . T CO isthe temperature where charge order develops, and is ofthe order of the pseudogap crossover temperature T ∗ .This near coincidence of T ∗ and T CO , as well as the co-existence of charge order and superconductivity below T c , suggests that a crucial role in the pseodogap phaseis played by charge order . These experimental obser-vations also identified that charge order in the pseudo-gap phase of cuprate superconductors emergences con-sistently in surface and bulk, and in momentum and realspace. Furthermore, the combination of the RXS dataand electron Fermi surface (EFS) measured results usingARPES revealed a quantitative link between the charge-order wave vector Q CO and the momentum vector con-necting the tips of the straight Fermi arcs , which a r X i v : . [ c ond - m a t . s up r- c on ] M a r in this case coincide with the hot spots on EFS, indi-cating that the hot spots play an important role in thecharge-order formation. This correspondence also showsthe existence of a quantitative connection between thecollective response of the electron density and the low-energy electronic structure. As a natural consequence ofa doped Mott insulator, the charge-order state is alsodoping dependent, with the magnitude of the charge-order wave vector Q CO that decreases upon the increaseof doping, in the analogy of the unusual behavior of thedoping dependence of the pseudogap . These experi-mental results observed on cuprate superconductors therefore show that charge order more intrinsically inter-twines with superconductivity. In this case, some crucialquestions are raised: (i) does the strong electron correla-tion play a role in the charge-order state and its interplaywith superconductivity? (ii) is charge order also the re-sult of the emergence of the pseudogap? (iii) do chargeorder and the SC order compete?Since the discovery of charge order and its evolutionwith doping and temperature in the pseudogap phase ofcuprate superconductors, the intense efforts at the ex-perimental and theoretical levels have been put forth inorder to understand the physical origin of charge orderand of its interplay with superconductivity . On theone hand, the possible special role played by the tipsof the Fermi arcs has been discussed phenomenologi-cally within the context of a magnetically-driven charge-order instability , where the charge-order wave vec-tors spanning the hot spots are a manifestation of thepseudogap formation due to charge order, rather thanbeing suggestive of pre-existing Fermi arcs that are un-stable to charge order. However, a different proposal at-tributes the pesudogap phase to the pair-density-wavestate , while charge order only appears in the pesu-dogap phase as a subsidiary order parameter, and thetips of the Fermi arcs themselves result from an EFSinstability around the antinodal region that is distinctfrom charge order. In particular, the physical origin ofcharge order has been studied based on the t - J modelby taking into account the pseudogap effect , where thecharge-order state is driven by the pseudogap-inducedEFS instability, with the charge-order wave vector corre-sponding to the straight hot spots on EFS. This study also indicates that charge order is intimately related topseudogap, and they are a natural consequence of thestrong electron correlation in cuprate superconductors.On the other hand, it has been argued that the emer-gence of charge order in the SC-state is consistent withthe picture of the anticorrelation between charge orderand superconductivity , i.e., these two order param-eters are related, as opposed to simply coexisting andcompeting. Moreover, a possible common origin of themain instabilities in cuprate superconductors has beensuggested, namely, the possibility that the sequence ofordering tendencies ( Q = 0 order precedes charge or-der, which in turn precedes the SC order) and the phasediagram as a whole are driven by the strong electron correlation . However, up to now, the finial consen-sus on the physical origin of charge order and of its in-terplay with superconductivity has not reached. In thispaper, we study the physical origin of charge order andof its interplay with superconductivity in cuprate super-conductors within the framework of the kinetic-energy-driven SC mechanism, where the SC-state quasiparti-cle excitation spectrum is obtained explicitly by takinginto account the intertwining between the SC gap andpseudogap. Based on this SC-state quasiparticle exci-tation spectrum, the main features of charge order inthe SC-state of cuprate superconductors are qualitativelyreproduced , including the doping dependence of thecharge-order wave vector. In particular, we show that ascharge order in the normal-state , charge order in theSC-state is also driven by the pseudogap-induced EFSinstability, with the charge-order wave vector that is wellconsistent with the wave vector connecting the straighthot spots on EFS. Although there is a coexistence ofcharge order and superconductivity below T c , this chargeorder antagonizes superconductivity.This paper is organized as follows. In Sec. II, we brieflyintroduces the general formalism of the SC-state quasi-particle spectral function of the t - J model in the charge-spin separation fermion-spin representation obtained interms of the full charge-spin recombination scheme. Thequantitative characteristics of the interplay of charge or-der with superconductivity are discussed in Sec. III,where we show that the physical origin of charge ordercan be interpreted in terms of the formation of the pseu-dogap by which it means a reconstruction of EFS to formthe Fermi pockets, while the intimate interplay betweencharge order and superconductivity is similar to the in-trinsical intertwining between the SC gap and pseuogap.In other words, the pseudogap and charge order have aroot in common, they and superconductivity are a nat-ural consequence of the strong electron correlation. Fi-nally, we give a summary and discussions in Sec. IV. II. FORMALISM
Superconductivity in cuprate superconductors is a phe-nomenon in which an assembly of electrons goes into theelectron pair-condensed phase as a consequence of thedominance of the interaction between electrons by the ex-change of a collective-mode . This exchanged collective-mode acts like a bosonic glue to hold the electron pairstogether, and is closely related to the SC-state quasipar-ticle excitations determined by the low-energy electronicstructure . On the other hand, the charge-orderstate is defined as a broken-symmetry state occurringwhen electrons self-organize into the periodic structures .Therefore charge order and its interplay with supercon-ductivity should be reflected in the low-energy electronicstructure. The electronic structure of cuprate supercon-ductors in the SC-state is manifested itself by the energyand momentum dependence of the SC-state quasiparti-cle excitation spectrum I ( k , ω ), which is closely relatedto the SC-state quasiparticle spectral function as , I ( k , ω ) = | M ( k , ω ) | n F ( ω ) A ( k , ω ) , (1)where M ( k , ω ) is a matrix element between the initialand final electronic states, and therefore depends on theelectron momentum, on the energy and polarization ofthe income photon. However, following the commonpractice, the magnitude of M ( k , ω ) has been rescaled tothe unit in this paper. n F ( ω ) is the fermion distribu-tion, while A ( k , ω ) is the SC-state quasiparticle spectralfunction, and is related directly with the imaginary partof the single-electron diagonal Green’s function G ( k , ω )as A ( k , ω ) = − G ( k , ω ). This SC-state quasiparti-cle excitation spectrum in Eq. (1) is measurable via theARPES technique and can provide the crucial informa-tion on EFS, the quasiparticle dispersions, and even themomentum-resolved magnitude of the SC gap .The quasiparticle excitation spectrum I ( k , ω ) in Eq.(1) also shows that the microscopic understanding of thephysical origin of charge order and of its interplay withsuperconductivity regains a central role in the contextof the essential physics of cuprate superconductors, sincethe calculation of I ( k , ω ) must be performed within themicroscopic mechanism of superconductivity. After in-tensive investigations over more than three decades, nowit is widely believed that the t - J model on a square lat-tice contains the essential ingredients to describe super-conductivity and the related exotic physics in cupratesuperconductors . Its Hamiltonian is given by, H = − (cid:88) (cid:104) l ˆ a (cid:105) σ t l ˆ a C † lσ C l +ˆ aσ + µ (cid:88) lσ C † lσ C lσ + J (cid:88) (cid:104) l ˆ η (cid:105) S l · S l +ˆ η , (2)supplemented by the local constraint (cid:80) σ C † lσ C lσ ≤ C † lσ ( C lσ ) is creation(annihilation) operator for electrons with spin orienta-tion σ = ↑ , ↓ on lattice site l , S l = ( S x l , S y l , S z l ) is spinoperator, µ is the chemical potential, and J is the ex-change interaction between the nearest-neighbor (NN)sites ˆ η . In this paper, we restrict the hopping of elec-trons t l ˆ a to the NN sites ˆ η and next NN sites ˆ τ withthe amplitudes t l ˆ η = t and t l ˆ τ = − t (cid:48) , respectively, while (cid:104) l ˆ a (cid:105) means that l runs over all sites, and for each l , overits NN sites ˆ a = ˆ η or next NN sites ˆ a = ˆ τ . Hereafter,the parameters are chosen as t/J = 2 . t (cid:48) /t = 0 . . The magnitude of J and the lattice constant of the square lattice are the en-ergy and length units, respectively. This t - J model (2)therefore is characterised by a competition between thekinetic energy, which makes electrons itinerant, and themagnetic energy, which makes electrons localized. Thestrong electron correlation manifests itself by the localconstraint of no double electron occupancy, and thereforethe crucial requirement is to impose this local constraintproperly . In order to satisfy this local constraint,we employ the fermion-spin formalism , in which theelectron operators C l ↑ and C l ↓ in the t - J model (2) are replaced by C l ↑ = h † l ↑ S − l and C l ↓ = h † l ↓ S + l , where thespinful fermion operator h lσ = e − i Φ lσ h l keeps track of thecharge degree of freedom of the constrained electron to-gether with some effects of spin configuration rearrange-ments due to the presence of the doped hole itself (chargecarrier), while the spin operator S l represents the spindegree of freedom of the constrained electron, and thenthe local constraint of no double occupancy is satisfied ateach site. In this fermion-spin representation, the origi-nal t - J model (2) can be rewritten as, H = (cid:88) (cid:104) l ˆ a (cid:105) t l ˆ a ( h † l +ˆ a ↑ h l ↑ S + l S − l +ˆ a + h † l +ˆ a ↓ h l ↓ S − l S + l +ˆ a ) − µ (cid:88) lσ h † lσ h lσ + J eff (cid:88) (cid:104) l ˆ η (cid:105) S l · S l +ˆ η , (3)where S − l = S x l − iS y l and S + l = S x l + iS y l are the spin-lowering and spin-raising operators for the spin S = 1 / J eff = (1 − δ ) J , and δ = (cid:104) h † lσ h lσ (cid:105) = (cid:104) h † l h l (cid:105) is the charge-carrier doping concentration. As an im-portant consequence, the kinetic-energy terms in the t - J model (2) have been transferred as the interaction be-tween charge carriers and spins, and therefore dominatesthe essential physics of cuprate superconductors.For a microscopic description of the SC-state of cupratesuperconductors, the kinetic-energy-driven SC mecha-nism has been established based on the t - J model (3)in the fermion-spin representation , where the inter-action between charge carriers and spins directly fromthe kinetic energy by the exchange of spin excitationsgenerates the SC-state in the particle-particle channeland pseudogap state in the particle-hole channel, andtherefore there is a coexistence of the SC gap and pseu-dogap below T c . However, for the discussions of theelectronic state properties, the exact knowledge of thesingle-electron Green’s function (then the quasiparticlespectral function) is of crucial importance . Withinthe framework of the charge-spin separation , thissingle-electron Green’s function can be evaluated interms of the charge-spin recombination. In the conven-tional charge-spin recombination scheme , the single-electron Green’s function in space-time is a product ofthe charge-carrier and spin Green’s functions, and theresulting Fourier transform is a convolution of the charge-carrier and spin Green’s functions. However, in the earlydays of superconductivity, it has been formally demon-strated that a microscopic theory based on the charge-spin separation can not give a consistent description ofEFS and the related quasiparticle excitations in termsof the conventional charge-spin recombination . Inthis case, within the kinetic-energy-driven SC mecha-nism, we have developed recently a full charge-spinrecombination scheme to fully recombine a charge car-rier and a localized spin into an electron, where it hasbeen realized that the coupling form between the elec-tron quasiparticle and spin excitation is the same as thatbetween the charge-carrier quasiparticle and spin excita-tion. Based on this full charge-spin recombination, theobtained single-electron Green’s function in the normal-state can produce a large EFS with the area that fulfillsLuttinger’s theorem . Following these discussions, the single-electron diagonal and off-diagonal Green’s func-tions G ( k , ω ) and (cid:61) † ( k , ω ) of the t - J model (3) in thefermion-spin representation have been obtained as , G ( k , ω ) = 1 ω − ε k − Σ ( k , ω ) − [Σ ( k , ω )] / [ ω + ε k + Σ ( k , − ω )] , (4a) (cid:61) † ( k , ω ) = − Σ ( k , ω )[ ω − ε k − Σ ( k , ω )][ ω + ε k + Σ ( k , − ω )] − [Σ ( k , ω )] , (4b)where the bare electron excitation spectrum ε k = − Ztγ k + Zt (cid:48) γ (cid:48) k + µ , with γ k = (cos k x + cos k y ) / γ (cid:48) k =cos k x cos k y , and Z is the number of the NN or next NNsites on a square lattice, while the electron self-energiesΣ ( k , ω ) in the particle-hole channel and Σ ( k , ω ) in theparticle-particle channel have been evaluated in termsof the full charge-spin recombination, and are given ex-plicitly in Ref. 47. In particular, both the electronself-energies Σ ( k , ω ) in the particle-hole channel andΣ ( k , ω ) in the particle-particle channel are generated bythe same interaction of electrons with spin excitations.Since the electron self-energy Σ ( k , ω ) in the particle-particle channel is a coupling of the energy and momen-tum dependence of the electron pair interaction strengthand electron pair order parameter, it is defined as theenergy and momentum dependence of the SC gap ,¯∆ s ( k , ω ) = Σ ( k , ω ), where following the common prac-tice, the imaginary part of Σ ( k , ω ) has been ignored.On the other hand, the electron self-energy Σ ( k , ω ) inthe particle-hole channel can be divided into two parts:Σ ( k , ω ) = ReΣ ( k , ω ) + i ImΣ ( k , ω ), where ReΣ ( k , ω )and ImΣ ( k , ω ) are the real and imaginary parts, respec-tively. With the above single-electron diagonal Green’sfunction (4a), the SC-state quasiparticle spectral func-tion A ( k , ω ) now can be obtained explicitly as, A ( k , ω ) = 2Γ( k , ω )[ ω − ε k − Re ¯Σ( k , ω )] + Γ ( k , ω ) , (5)and then the dispersion of the quasiparticle state as afunction of momentum and therefore EFS itself can beprobed by ARPES measurements, where the SC-statequasiparticle scattering rate Γ( k , ω ) and the real part ofthe modified electron self-energy Re ¯Σ( k , ω ) are given by,Γ( k , ω ) = | ImΣ ( k , ω ) − ¯∆ ( k , ω )ImΣ ( k , − ω )[ ω + ε k + ReΣ ( k , − ω )] + [ImΣ ( k , − ω )] (cid:12)(cid:12)(cid:12)(cid:12) , (6a)Re ¯Σ( k , ω ) = ReΣ ( k , ω )+ ¯∆ ( k , ω )[ ω + ε k + ReΣ ( k , − ω )][ ω + ε k + ReΣ ( k , − ω )] + [ImΣ ( k , − ω )] . (6b) Substituting this SC-state quasiparticle spectral function A ( k , ω ) in Eq. (5) into Eq. (1), we therefore can obtainedthe SC-state quasiparticle excitation spectrum I ( k , ω ).In comparison with the normal-state quasiparticle scat-tering rate Γ N ( k , ω ), it is thus shown that there is aadditional suppression of the spectral weight below T c due to the SC gap opening. III. INTERPLAY BETWEEN CHARGE ORDERAND SUPERCONDUCTIVITY
Superconductivity in cuprate superconductors is an in-stability of the normal-state, and this normal-state fromwhich it emerges over much of the phase diagram is thepseudogap phase . In this section, we show that thecharge-order formation is a natural result of the emer-gence of the pseudogap, and then in analogy to the in-terplay between the pseudogap and SC gap, the charge-order correlation is more intimately entangled with su-perconductivity in cuprate superconductors . A. Fermi pockets induced by a reconstruction ofelectron Fermi surface
EFS of cuprate superconductors can be measured viathe ARPES technique and its shape can have deepconsequences for the anomalous properties . The natureand topology of EFS in the pseudogap phase of cupratesuperconductors has been debated for many years. Theearly ARPES experimental studies found a large EFSconsistent with the band structure calculations .Later, the ARPES measurements with the enhancementof the resolution revealed that the large EFS in the pseu-dogap phase does not remain intact, but breaks up intothe disconnected Fermi arcs . Recently, the great im-provements in the resolution of the ARPES experimentalmeasurements allowed to resolve additional features inthe ARPES spectrum. Among these new achievementsis the observation of the Fermi pockets in the pseudogapphase of cuprate superconductors , with the area ofthe Fermi pockets that is strongly dependent on the dop-ing concentration . In particular, these Fermi pockets (b)(a) FIG. 1: (Color online) (a) The map of the quasiparticle excitation spectral intensity I ( k ,
0) and (b) the quasiparticle excitationspectrum I ( k ,
0) in the [ k x , k y ] plane at δ = 0 .
12 with T = 0 . J for t/J = 2 . t (cid:48) /t = 0 .
3. The pairing of electrons andholes at k and k + Q HS [red lines in (a)] drives the charge-order formation, whereas the electron pairing at k and − k states[yellow lines in (a)] is responsible for superconductivity. can persist into the SC-state . On the other hand, thecharge-order state in cuprate superconductors is closelyrelated to the EFS reconstruction . This is why thedetermination of the shape of EFS and the related dis-tribution of the SC-state quasiparticle excitations in thepseudogap phase of cuprate superconductors is believedto be key issue for the understanding of the physical ori-gin of charge order and of its intimate interplay withsuperconductivity.The intensity of the SC-state quasiparticle excitationspectrum I ( k , ω ) in Eq. (1) at zero energy is used tomap out the underlying EFS, i.e., the locations of EFSin momentum space is determined directly by , ε k + Re ¯Σ ( k ,
0) = 0 , (7)and then the lifetime of the SC-state quasiparticles atEFS is dominated by the inverse of the SC-state quasi-particle scattering rate Γ( k ,
0) in Eq. (6a). For a su-perconductor, EFS is defined just above T c . However, astraightforward calculation shows that ε k +Re ¯Σ ( k ,
0) =0 in Eq. (7) is also equivalent to, ε k + ReΣ ( k ,
0) = 0 , (8)which shows that the locations of EFS in momentumspace in the SC-state is almost the same as that in thenormal-state. This is why we can define operationally EFS of cuprate superconductors in the SC-state as thecontours in momentum space determined from the low-energy spectral weight . However, the SC quasipar-ticle scattering rate (then the lifetime of the SC quasi-particle) at EFS has been modified by the SC gap as,Γ( k ,
0) = (cid:12)(cid:12)(cid:12)(cid:12)
ImΣ ( k , − ¯∆ ( k )ImΣ ( k , (cid:12)(cid:12)(cid:12)(cid:12) . (9) In Fig. 1, we plot (a) the map of the SC-state quasi-particle excitation spectral intensity I ( k ,
0) in Eq. (1)and (b) the SC-state quasiparticle excitation spectrum I ( k ,
0) in the [ k x , k y ] plane at doping δ = 0 .
12 with tem-perature T = 0 . J . In the d-wave type SC-state, ifthe single-particle coherence from the electron self-energyΣ ( k ,
0) in the particle-hole channel is neglected, EFS asthe single contour in momentum space is gapped, lead-ing to the four isolated gapless points at the nodes in themomentum space . However, when the single-particlecoherence from Σ ( k ,
0) is included as shown in Fig. 1,some unconventional features emerge: (i) the originalsingle-contour EFS in momentum space is split by theelectron self-energy Σ ( k ,
0) into two contours k F and k BS , respectively, where the redistribution of the low-energy spectral weight of the SC-state quasiparticle ex-citation spectrum leads to a reconstruction of EFS; (ii)the low-energy spectral weight at the contours k F and k BS are suppressed by the SC-state quasiparticle scatter-ing rate Γ( k , k F and k BS break up into the disconnected segmentsaround the nodal region; (iii) the contour k F intersectsthe contour k BS at the tips of these disconnected seg-ments to form a Fermi pocket, where following the com-mon practice , the disconnected segment around thenodal region at the contour k F is referred to the Fermiarc, and is also defined as the front side of the Fermipocket, while the other at the contour k BS around thenodal region is associated with the back side of the Fermipocket. Moreover, this Fermi pocket is not symmetricallylocated in the Brillouin zone (BZ), i.e., it is not centeredaround [ π/ , π/ ,it is thus shown that the Fermi pockets in cuprate su-perconductors appeared in the SC-state can persist intothe normal-state, and the location, shape and area of theFermi pockets in the SC-state are almost the same asthat in the normal-state. These results are in qualita-tive agreement with the experimental data obtained bymeans of the ARPES experimental measurements and magnetoresistance quantum oscillation , wherethe definitive Fermi pockets in both the SC- and normal-states have been observed. B. Coexistence of charge order andsuperconductivity
In Fig. 1, it is also shown clearly that the part of thespectral weight of the SC-state quasiparticle excitationspectrum at the Fermi arc has been transferred to theback side of the Fermi pocket by the self-energy Σ ( k , ,the most part of the spectral weight is located aroundthe tips of the Fermi arcs, which in this case coincidewith the hot spots on EFS, where the spectral weighthas a largest value (see Fig. 1b), reflecting a fact thatthe most of the SC quasiparticles occupies region aroundthe eight isolated hot spots on EFS, and then these SCquasiparticles around the hot spots contribute effectivelyto the SC-state quasiparticle scattering process . How-ever, charge order in cuprate superconductors is char-acterized by the charge-order wave vector, which is justdetermined by the wave vector connecting the straighthot spots on EFS . In the present study based onthe kinetic-energy-driven SC mechanism, we find that thetheoretical result of the SC quasiparticle scattering wavevector between the hot spots on the straight Fermi arcsshown in Fig. 1a at the doping δ = 0 .
12 is Q HS = 0 . of thecharge-order wave vector Q CO ≈ .
265 oberved in theunderdoped cuprate superconductors, indicating that ascharge order in the normal-state , charge order in theSC-state is also driven by the EFS instability. More-over, these results also show that (i) the charge-order-induced reconstruction of EFS into the Fermi pockets iscaused by a finite charge-order wave vector Q HS61–65 ; (ii)the charge-order correlation is developed in the normal-state , and can persists into the SC-state, leading to acoexistence of charge order and superconductivity below T c . These results are also well consistent with the exper-imental observations . Furthermore, we have shownthat the magnitude of the charge-order wave vector Q HS is also related to the next NN hopping t (cid:48) of electrons,i.e., it increases with the increase of t (cid:48) , and therefore theexperimentally observed differences of the magnitudes of the charge-order wave vector Q CO among the differentfamilies of cuprate superconductors at the same dopingconcentration can be attributed to the different values of t (cid:48) .In the conventional superconductors , the SC quasi-particles are the phase-coherent linear superpositions ofelectrons and holes, while the SC condensate is made upof electron pairs, which are bound-states of two electronswith opposite momenta and spins. However, the charge-order quasiparticles are superpositions of electrons (orholes), and then the charge-order state is likewise a paircondensate, but of electrons and holes, whose net mo-mentum determines the wave-length of charge order .Within the framework of the kinetic-energy-driven SCmechanism, the structure of the coexistence of supercon-ductivity and charge order in cuprate superconductorsis also shown in Fig. 1a, where the pairing of electronsand holes at k and k + Q HS (red lines) separated by thecharge-order wave vector Q HS drives the charge-order for-mation, whereas the electron pairing at k and − k (yel-low lines) states induces superconductivity. In this SC-state with coexisting charge order, the quasiparticles be-come superpositions of four, rather just two quasiparticleeigenstates . C. Doping dependence of charge-order wave vector
FIG. 2: (Color online) The map of the quasiparticle excitationspectral intensity I ( k ,
0) at δ = 0 .
15 with T = 0 . J for t/J = 2 . t (cid:48) /t = 0 . The above result in Fig. 1 indicates that the charge-order wave vector is just corresponding to the straighthot spots on EFS, however, the position of the hotspots is doping dependent. In this case, the charge-order state in cuprate superconductors is characterizednot only by the charge-order wave vector, but also byits doping dependence . To address the evolution of thecharge-order wave vector with doping, we plot the mapof the SC-state quasiparticle excitation spectral inten-sity I ( k ,
0) at the doping δ = 0 .
15 with T = 0 . J in Fig. 2. Comparing it with Fig. 1a for the sameset of parameters except for the doping δ = 0 .
15, it isthus shown that with the increase of doping, the po-sition of the hot spots shifts towards to the antinodes,which leads to that the magnitude of the charge-orderwave vector decreases with the increase of doping. To seethis doping dependence of the charge-order wave vectormore clearly, we have performed a series of calculationsfor the SC-state quasiparticle excitation spectrum I ( k , Q HS (blue line) as afunction of doping with T = 0 . J is plotted in Fig.3. For comparison, the result of T c (black line)as a function of doping is also replotted in Fig. 3. It isshown clearly that T c has a distinct dome-shaped dopingdependence, i.e., it increases with the increase of dopingin the underdoped regime, and reaches a maximum inthe optimal doping, then decreases with the increase ofdoping in the overdoped regime. However, in contrast tothe case of T c in the underdoped regime, the magnitudeof Q HS smoothly decreases with the increase of doping inthe underdoped regime, in qualitative agreement with theexperimental data . Furthermore, in comparison withthe corresponding results of the doping dependence of thepseudogap , it is thus shown that the behavior ofthe doping dependence of the charge-order wave vector isvery similar to that of the doping dependence of the pseu-dogap, indicating that the appearance of charge order isclosely related to the emergence of the pseudogap. FIG. 3: (Color online) The charge-order wave vector (blueline) in T = 0 . J and superconducting transition tempera-ture T c (black line) for t/J = 2 . t (cid:48) /t = 0 . In spite of a coexistence of charge order and super-conductivity below T c , above obtained results also showa change in the EFS topology from a single contour in momentum space, where the SC-state quasiparticle ex-citation spectrum is gapless at the nodes and thereforethe SC quasiparticle lifetime on the nodes is infinitelylong, to the Fermi pockets, where the SC quasiparticleenergies have been heavily renormalized by the electronself-energy Σ ( k , ω ) in the particle-hole channel and thenthey acquire a finite lifetime τ ( k ,
0) = Γ − ( k ,
0) on theFermi arc and back side of the Fermi pocket, thereby in-dicating a reconstruction of EFS caused by the onset ofcharge order. In this case, the essential physics of the in-timate interplay of charge order with superconductivityis closely related to the coexistence and competition be-tween the SC gap and pseudogap below T c . This followsa fact that within the framework of the kinetic-energy-driven SC mechanism, the pseudogap state in theparticle-hole channel is generated by the same electroninteraction that also generates SC-state in the article-particle channel. In particular, as we have shown inthe previous discussions , the electron self-energyΣ ( k , ω ) in the particle-hole channel in Eq. (4) can bealso rewritten as,Σ ( k , ω ) ≈ [ ¯∆ PG ( k )] ω + ε k , (10)with the corresponding energy spectrum ε k andthe momentum dependence of the pseudogap ¯∆ PG ( k )can be obtained directly from the electron self-energy Σ ( k , ω ) and its antisymmetric part Σ ( k , ω )as ε k = − Σ ( k , / Σ ( k ,
0) and ¯∆ PG ( k ) =Σ ( k , / (cid:112) − Σ ( k , PG ( k ) is identified as being a role of the single-particlecoherence by which it means a reconstruction of EFS toform the Fermi pockets. In particular, the correspondingimaginary part of Σ ( k , ω ) can be also expressed explic-itly in terms of the pseudogap ¯∆ PG ( k ) as,ImΣ ( k , ω ) ≈ π [ ¯∆ PG ( k )] δ ( ω + ε k ) , (11)which therefore reflects an intimate relation between theSC-state quasiparticle scattering rate in Eq. (6a) andpseudogap ¯∆ PG ( k ).Substituting this electron self-energy Σ ( k , ω ) in Eq.(10) into Eq. (4), the single-electron diagonal and off-diagonal Green’s functions in Eq. (4) can be rewrittenexplicitly as, G ( k , ω ) = (cid:18) U k ω − E k + V k ω + E k (cid:19) + (cid:18) U k ω − E k + V k ω + E k (cid:19) , (12a) (cid:61) † ( k , ω ) = − a k ¯∆ s ( k )2 E k (cid:18) ω − E k − ω + E k (cid:19) + a k ¯∆ s ( k )2 E k (cid:18) ω − E k − ω + E k (cid:19) , (12b)where a k = ( E k − ε k ) / ( E k − E k ), a k = ( E k − ε k ) / ( E k − E k ), and as a natural consequence of thecoexistence of the pseudogap and SC gap, the quasi-particles become superpositions of four eigenstates withthe corresponding energy eigenvalues E k , − E k , E k ,and − E k , respectively, where E k = (cid:112) [ K k + K k ] / E k = (cid:112) [ K k − K k ] /
2, and the kernel functions, K k = ε k + ε k + 2 ¯∆ ( k ) + ¯∆ ( k ) , (13a) K k = (cid:113) ( ε k − ε k ) b k + 4 ¯∆ ( k ) b k + ¯∆ ( k ) , (13b)with b k = ε k − ε k + 2 ¯∆ ( k ), and b k = ( ε k − ε k ) +¯∆ ( k ), while the coherence factors of the SC-state withthe coexisting pseudogap state are given by, U k = 12 (cid:20) a k (cid:18) ε k E k (cid:19) − a k (cid:18) ε k E k (cid:19)(cid:21) , (14a) V k = 12 (cid:20) a k (cid:18) − ε k E k (cid:19) − a k (cid:18) − ε k E k (cid:19)(cid:21) , (14b) U k = − (cid:20) a k (cid:18) ε k E k (cid:19) − a k (cid:18) ε k E k (cid:19)(cid:21) , (14c) V k = − (cid:20) a k (cid:18) − ε k E k (cid:19) − a k (cid:18) − ε k E k (cid:19)(cid:21) , (14d)and satisfy the sum rule for any wave vector k : U k + V k + U k + V k = 1, where a k = [ ¯∆ PG ( k )] / ( E k − E k ). In analogy to the normal-state case , this energyband splitting induced by the pseudogap therefore leadsto form two contours k F and k BS in momentum space asshown in Fig. 1. FIG. 4: (Color online) The map of the quasiparticle scatteringrate at δ = 0 .
12 with T = 0 . J for t/J = 2 . t (cid:48) /t = 0 . On the other hand, the contribution to the SC-statequasiparticle excitation spectrum comes from two typicalexcitations: the electron-hole and the electron pair exci-tations. In this case, the result in Eq. (9) also shows thatthe contribution to the SC-state quasiparticle scattering rate Γ( k ,
0) from the first term ImΣ ( k ,
0) of the right-hand side in Eq. (9) mainly comes from the electron-holeexcitations, and is intimately related to the emergence ofthe pseudogap. This process can therefore persist intothe normal-state pseudogap phase, and leads to a num-ber of the anomalous properties . In particular, thesharp peak structure of the energy and momentum de-pendence of ImΣ ( k , ω ) is directly responsible for theremarkable peak-dip-hump structure in the quasiparti-cle excitation spectrum of cuprate superconductors .However, the additional contribution to the SC-statequasiparticle scattering rate Γ( k ,
0) from the second term¯∆ ( k ) / ImΣ ( k ,
0) of the right-hand side in Eq. (9) origi-nates from the electron pair excitations. This additionalprocess is caused by the SC gap opening, and vanishes inthe normal-state. In Fig. 4, we plot the map of the inten-sity of the SC-state quasiparticle scattering rate Γ( k , δ = 0 .
12 with T = 0 . J . Apparently,Γ( k ,
0) is strong dependence of momentum, reflecting afact that both the pseudogap ¯∆ PG ( k ) and SC gap ¯∆ s ( k )are extremely anisotropic in momentum space. To seethis fact more clearly, we plot the angular dependenceof (a) | ImΣ ( k F , | and (b) ¯∆ s ( k F ,
0) along EFS fromthe antinode to the node at the doping δ = 0 .
12 with T = 0 . J in Fig. 5. For comparison, the correspondingresult (dash-dotted line) of the monotonic d-wave form(cos k x − cos k y ) / | ImΣ ( k F , | [then the pseudogap ¯∆ PG ( k F )] and ¯∆ s ( k F ) have a strongangular dependence. On the one hand, | ImΣ ( k F , | ex-hibits the largest value around the antinode k AN . How-ever, the actual minimum of | ImΣ ( k F , | does not ap-pear around the node k N , but locates exactly at the hotspot k HS , where | ImΣ ( k HS , | ∼
0. In particular, themagnitude of | ImΣ ( k F , | around the antinode is largerthan that around the node. On the other hand, althoughthe SC gap ¯∆ s ( k F ) has a dominated d-wave symmetrywith the actual maximum at the antinode and the actualminimum [ ¯∆ s ( k N ) = 0] at the node, it exhibits an un-usual momentum dependence around the hot spot regionwith the anomalously small value, which leads to that theangular variation of ¯∆ s ( k F ) on EFS can not be describedby a monotonic d-wave SC gap form (cos k x − cos k y ) / k x − cos k y ) / as shownin Fig. 5b.The combination of both the results of | ImΣ ( k F , | and ¯∆ s ( k F ) shown in Fig. 5, the angular dependenceof the SC-state quasiparticle scattering rate Γ( k F ,
0) onEFS can therefore be obtained, and the result is plottedin Fig. 6. Moreover, we have also calculated the angu-lar dependence of the SC-state quasiparticle scatteringrate Γ( k BS ,
0) along the back side of the Fermi pocketfrom the antinode to the node, and found that its be-havior is very similar to that of Γ( k F , k F ,
0) is mainly domi-nated by the pseudogap, i.e., the SC-state quasiparticle (a) (b)
FIG. 5: The angular dependence of (a) the imaginary part of the electron self-energy in the particle-hole channel and (b)superconducting gap along k F from the antinode to the node at δ = 0 .
12 with T = 0 . J for t/J = 2 . t (cid:48) /t = 0 .
3. Theposition of the hot spot is indicated by the dashed vertical line. The dash-dotted line in (b) is obtained from a numerical fit(cos k x − cos k y ) / k F from the antinode to the node at δ = 0 . T = 0 . J for t/J = 2 . t (cid:48) /t = 0 . scattering rate Γ( k F ,
0) has almost the same momentumdependence as | ImΣ ( k F , | . In particular, Γ( k F ,
0) ex-hibits the strongest scattering at the antinodes, which isconsistent with the experimental results , wherethe strongest quasiparticle scattering appeared at theantinodes has been also widely observed in cuprate su-perconductors. However, the weakest quasiparticle scat-tering does not take place at the nodes, but occurs ex-actly at the hot spots k HS , where the energy spectra ε k HS ≈ − ε k HS , ¯∆ PG ( k HS ), and ¯∆ s ( k HS ) have anoma-lously small values, and then all the energy bands forthe SC-state quasiparticle excitation spectra on the con-tours k F and k BS converge on the hot spots of EFS. Thisextremely anisotropic momentum dependence of the SC- state quasiparticle scattering rates (then the pseudogap)on k F and k BS therefore suppresses heavily the spectralweight of the SC-state quasiparticle excitation spectrumon the contours k F and k BS in the antinodal region, andreduces modestly the spectral weight around the nodalregion. This is also why the tips of these disconnectedsegments on k F and k BS assemble on the hot spots toform a Fermi pocket in the SC-state around the nodalregion. At the same time, this EFS instability there-fore drives charge order with the charge-order wave vec-tor connecting the parallel hot spots on EFS, as chargeorder driven by the EFS instability in the normal-statecase . In other words, the formation of the Fermi pock-ets (then the charge-order state) is a natural consequenceof the extremely anisotropic momentum dependence ofthe pseudogap (then the quasiparticle scattering rates)originated from the electron self-energy due to the in-teraction between electrons by the exchange of spin ex-citations. Furthermore, the position of the hot spots isdoping dependent, and shifts towards to the antinodeswhen doping is increased, which therefore leads to thatthe charge-order wave vector decreases with the increaseof doping . In the normal-state [ ¯∆ s ( k , ω ) = 0], the SC-state quasiparticle scattering rate Γ( k , ω ) in Eq. (6a) isreduced as the normal-state quasiparticle scattering rateΓ NS ( k , ω ) = | ImΣ ( k , ω ) | . Since the essential propertiesof the angular dependence of Γ( k F ,
0) is dominated bythe pseudogap ¯∆ PG ( k ), these Fermi pockets persist intothe normal-state , and then charge order driven bythe EFS instability emerges in the normal-state .In our previous studies , we have shown that(i) there is a coexistence of the SC gap and pseudogapbelow T c ; (ii) the pseudogap is directly related to thesingle-particle coherent weight, and then the SC-statein the kinetic-energy-driven SC mechanism is controlledby both the SC gap and single-particle coherence; (iii)0however, this single-particle coherence competes stronglywith the electron pairing state, which leads T c in cupratesuperconductors to be reduced to lower temperatures, in-dicating that the pseudogap has a competitive role in en-gendering superconductivity. With these previous stud-ies of the intertwining between the pseudogap and SCgap, our present results of the connection of charge orderand pseudogap therefore show that (i) the reconstructionof EFS and the related charge-order state in the pseu-dogap phase can be attributed to the emergence of thepseudogap; (ii) as the role played by the pseudogap, al-though charge order coexists with superconductivity be-low T c , this charge order displays the analogous com-petition with superconductivity; (iii) the Fermi pocket,charge order, and pseudogap are intimately related, i.e.,there is a common origin for the Fermi pocket, chargeorder, and pseudogap, they and superconductivity are anatural consequence of the strong electron correlation incuprate superconductors. D. Quantitative connection between electronicstructure and collective response of electron density
Now we turn our attention to the quantitative con-nection between the collective response of the electron density and low-energy electronic structure. The collec-tive response of the electron density, as manifested bythe SC-state dynamical charge structure factor C ( k , ω ),is closely related to the imaginary part of the SC-statequasiparticle density-density correlation function, C ( k , ω ) = − ω Im ˜Π c ( k , ω ) , (15)with the SC-state quasiparticle density-density correla-tion function ˜Π c ( k , ω ) that is defined as,˜Π c ( R l − R l (cid:48) , t − t (cid:48) ) = (cid:104)(cid:104) T ρ ( R l , t ) ρ ( R l (cid:48) , t (cid:48) ) (cid:105)(cid:105) , (16)where the ρ ( R l , t ) is the density operator, and can beexpressed explicitly as, ρ ( R l ) = e (cid:88) σ C † lσ C lσ . (17)Substituting this density operator (17) into Eqs. (16)and (15), the SC-state dynamical charge structure factor C ( k , ω ) thus can be obtained explicitly in terms of theSC-state quasiparticle spectral functions as, C ( k , ω ) = 2 e N (cid:88) q (cid:90) ∞−∞ dω (cid:48) π [ A ( q + k , ω (cid:48) + ω ) A ( q , ω (cid:48) ) − A (cid:61) ( q + k , ω (cid:48) + ω ) A (cid:61) ( q , ω (cid:48) )] n F ( ω (cid:48) + ω ) − n F ( ω (cid:48) ) ω , (18)where A (cid:61) ( k , ω ) = − (cid:61) † ( k , ω ). It should be notedthat in the calculation of the above SC-state dynamicalcharge structure factor, the vertex correction has beendropped, since it has been shown that the vertex correc-tion is negligibly small in the calculation of the density-density correlation of cuprate superconductors .To explore the global feature of charge order in the SC-state, we have mapped the SC-state dynamical chargestructure factor (18) in the [ k x , k y ] plane, and find thatthere are four resonance peaks, where the charge-orderpropagates only along the parallel directions [ ± Q HS , , ± Q HS ] of BZ, in good agreement with the exper-imental observations . To see the parallel resonancepeaks around the wave vector Q HS clearly, we plot theresult of C ( k , ω ) along the k = [0 ,
0] to k = [ π,
0] di-rection of BZ at the energy ω = 6 . J and the doping δ = 0 .
12 with T = 0 . J in Fig. 7, where a resonancepeak emerges in the wave vector Q CO ≈ .
27. In partic-ular, this characteristic wave vector Q CO ≈ .
27 is theexactly same with the charge-order wave Q HS = 0 . .To further verify this resonance peak that can be identi-fied as the presence of charge ordering, we plot C ( Q CO , ω )as a function of energy in the wave vector Q CO = 0 . δ = 0 .
12 with T = 0 . J in Fig. 8, wherethe energy is tuned away from the resonance energy, thesharp peak in the wave vector Q CO = 0 .
27 is suppressedheavily, and then eventually disappears, also in qualita-tive agreement with the experimental observation fromthe RXS measurements . This suppression of the res-onance peak by tuning energy away from the resonanceenergy therefore verifies the presence of charge order asthe collective response of the electron density in the SC-state of cuprate superconductors.The above obtained results therefore confirm a quan-titative connection between the low-energy electronicstructure and the collective response of electron densityin cuprate superconductors . The SC-state dynami-cal charge structure factor C ( k , ω ) in Eq. (18) is obtainedin terms of the SC-state quasiparticle spectral functions,1 -0.4 -0.3 -0.2 C ( k , )( a r b . un i t s ) [k x ,0]/2 FIG. 7: Dynamical charge structure factor along the k = [0 , k = [ π,
0] direction of the Brillouin zone at ω = 6 . J and δ = 0 .
12 with T = 0 . J for t/J = 2 . t (cid:48) /t = 0 . C ( k , )( a r b . un i t s ) /J FIG. 8: Dynamical charge structure factor as a function ofenergy in the charge-order wave vector Q CD = 0 .
27 at δ =0 .
12 with T = 0 . J for t/J = 2 . t (cid:48) /t = 0 . in other words, the essential behavior of the collective re-sponse of the electron density in the SC-state of cupratesuperconductors is mainly determined by the SC-statequasiparticle spectral functions [then the single-electrondiagonal and off-diagonal Green’s functions in Eq. (12)and the related pesudogap in Eq. (10)]. However, thesingle-electron diagonal and off-diagonal Green’s func- tions (12) and the related dispersions of the SC-statequasiparticle excitation energies E k and E k can be alsoreproduced qualitatively by a phenomenological Hamil-tonian, H CO = (cid:88) k σ ε k C † k σ C k σ − (cid:88) k ¯∆ s ( k )( C † k ↑ C †− k ↓ + C − k ↓ C k ↑ ) − (cid:88) k σ ε k C † k + Q HS σ C k + Q HS σ + (cid:88) k σ ¯∆ PG ( k )( C † k + Q HS σ C k σ + C † k σ C k + Q HS σ ) , (19)while such type Hamiltonian has been usually employedto phenomenologically discuss the physical origin ofcharge order and of its interplay with superconductiv-ity in cuprate superconductors , where charge or-der induces a reconstruction of EFS to form the Fermipockets, in qualitative agreement with the experimen-tal data. This is why our present study based on thekinetic-energy-driven SC mechanism can give a consis-tent description of the interplay between charge orderand superconductivity in cuprate superconductors. IV. CONCLUSIONS
In conclusion, within the framework of the kinetic-energy-driven SC mechanism, we have discussed thephysical origin of charge order and of its interplay withsuperconductivity in cuprate superconductors by takinginto account the intertwining between the pseudogap andSC gap. In particular, we show that the pseudogap-induced EFS reconstruction generates the formation ofthe Fermi pockets around the nodal region. However, thedistribution of the spectral weight of the SC-state quasi-particle excitation spectrum on the Fermi arc and backside of the Fermi pocket is extremely anisotropic, wherethe most of the SC quasiparticles occupies region aroundthe hot spots on EFS. As charge order in the normal-state, this EFS instability drives the charge-order corre-lation in the SC-state, with the charge-order wave vectorconnecting the straight hot spots on EFS. As a naturalconsequence of a doped Mott insulator, this charge-orderstate is doping dependent, with the magnitude of thecharge-order wave vector that decreases with the increaseof doping. Although charge order appears to be a phe-nomenon that coexists with superconductivity below T c ,this charge order strongly competes with superconduc-tivity. The results from the SC-state dynamical chargestructure factor indicate the existence of a quantitativeconnection between the low-energy electronic structureand collective response of the electron density. Com-bined with the previous results in the normal-state ,the theory also shows that the pseudogap and charge or-der are both consequences of the same physics, they andsuperconductivity are a natural result of the strong elec-tron correlation in cuprate superconductors.2 Acknowledgements
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