Charge ordering in superconducting copper oxides
Alex Frano, Santiago Blanco-Canosa, Bernhard Keimer, Robert J Birgeneau
CCharge Ordering in Superconducting Copper Oxides
Alex Frano, ∗ Santiago Blanco-Canosa,
2, 3, † Bernhard Keimer, ‡ and Robert J. Birgeneau
5, 6, § Department of Physics, University of California, San Diego, California 92093, USA Donostia International Physics Center, DIPC, 20018 Donostia-San Sebastian, Basque Country, Spain IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Basque Country, Spain Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany Department of Physics, University of California, Berkeley, California 94720, USA Department of Materials Science and Engineering,University of California Berkeley, Berkeley, CA 94720, USA.
Charge order has recently been identified as a leading competitor of high-temperature supercon-ductivity in moderately doped cuprates. We provide a survey of universal and materials-specificaspects of this phenomenon, with emphasis on results obtained by scattering methods. In par-ticular, we discuss the structure, periodicity, and stability range of the charge-ordered state, itsresponse to various external perturbations, the influence of disorder, the coexistence and competi-tion with superconductivity, as well as collective charge dynamics. In the context of this journalissue which honors Roger Cowley’s legacy, we also discuss the connection of charge ordering withlattice vibrations and the central-peak phenomenon. We end the review with an outlook on researchopportunities offered by new synthesis methods and experimental platforms, including cuprate thinfilms and superlattices.
A. Introduction.
Quantum materials with strongly correlated electronsare a major platform for the investigation of emer-gent phenomena, which occur when a system’s macro-scopic properties are governed by the collective behav-ior of a particle ensemble rather than by its individualcomponents. The intricate interplay between static anddynamic spin, charge, orbital, and pair correlations ofthe valence electrons and their influence on the macro-scopic phase behavior of quantum materials continuesto intrigue a large community of researchers. Copperoxide superconductors are an archetypical platform forresearch on emergent quantum phenomena .
Despitetheir simple lattice architecture and band structure (withCuO square planes as the only generic electronically ac-tive building block, and a single Cu d -orbital and two O p -orbitals dominating the electronic states near the Fermilevel), they exhibit a set of electronic phases with rad-ically different macroscopic properties as a function ofthe concentration of mobile charge carriers in the CuO planes. Among the multiple phases that have been pro-posed and studied in the cuprates, an antiferromagneticinsulator as well as unconventional metallic and super-conducting phases (Figure 1) have long been accepted as“universal” – that is, they are found in all chemically dis-tinct families of cuprates that can be prepared with thecorresponding doping level.This brief review addresses recent developments in re-search on charge order, which was recently recognized asanother universal feature of the copper oxide phase di-agram. The concept of charge order has a long historyin solid-state research. Already in the 1950s, Peierls rec-ognized that the total energy of a one-dimensional metalcould be lowered through a lattice distortion with a wavevector matching the Fermi wave vector. Evidence forPeierls distortions (also known as “charge density waves”, CDWs) was later found in many binary and ternary com-pounds with weakly correlated quasi-one-dimensional (-1D) or quasi-2D electron systems.
In most quasi-2Dsystems, CDW formation reconstructs the Fermi surface,but does not remove it entirely. Another paradigm ofcharge ordering emerged in research on transition metaloxides comprising metal ions with different stable valencestates, such as manganates, nickelates, and cobaltates.
Depending on the doping level, various superstruc-tures of these valence states have been observed. Theytend to be commensurate with the underlying lattice, andthey are often associated with commensurate magneticorder. Originally predicted in 1989, the experimental dis-covery of charge order in the cuprates dates back to1995, when Tranquada and coworkers performed neu-tron diffraction experiments on a La-based cuprate witha doping level of 1/8 holes per square-lattice unit cell. They found Bragg reflections characteristic of a mag-netic superstructure with a periodicity that is eight timeslarger than the basic crystallographic unit cell, as well asanother set of reflections indicating a four-unit-cell mod-ulation of the crystal lattice (presumably in response toa modulation of the valence electron density). Becauseof the commensurate nature of the charge modulationand the coexistence with magnetic order, the orderedstate was discussed in a real-space picture, which sub-sequently evolved into a new paradigm for charge order– often referred to as the “striped phase”. In this pic-ture, nanoscale regions with weakly perturbed antiferro-magnetic order are separated by nonmagnetic “rivers ofcharge” where the mobile holes can delocalize withoutbreaking magnetic bonds.It was soon realized, however, that the striped phaseonly occurs in La-based superconductors, where it is sta-bilized by soft tilt distortions of the oxygen octahedrasurrounding the Cu ions in this lattice structure. a r X i v : . [ c ond - m a t . s t r- e l ] F e b Neutron diffraction experiments on other cuprates, in-cluding YBa Cu O x , did not detect magnetic super-structures associated with the striped phase. How-ever, the similarities of the magnetic excitations in mod-erately doped cuprates with and without stripeorder revealed by inelastic neutron scattering experi-ments (with important contributions from Roger Cowleyand his coworkers ) stimulated theories of dynami-cal stripes, where quantum fluctuations obliterate staticmagnetic order. Another class of theories proposedbond-centered charge order, where the charge modula-tion predominantly affects the oxygen ions, and spins-1/2on a Cu-O-Cu bonds form non-magnetic singlets that arenot directly visible in neutron diffraction experiments. This scenario was supported by scanning tunneling spec-troscopy (STS) studies of Ca − x Na x CuO Cl and Bi-based superconductors, which found direct evidence ofbond-centered charge modulations. These materials,however, are known for their comparatively high level ofchemical disorder and inhomogeneity. The charge order-ing patterns found by STS are short-range ordered withcorrelation lengths not exceeding several unit cells, rais-ing the question to what extent they might actually becaused by disorder, in analogy to “Friedel oscillations”in elemental metals. Recent numerical simulations ofthe doped Hubbard and t − J models have yieldedfirm evidence of stripe order (both static and fluctuat-ing), with details that depend sensitively on the modelparameters and the doping level. The most recent calcu-lations of the Hubbard and t − J models capture some ofthe interplay between density wave orders and supercon-ductivity. Recently, the application of resonant x-ray scattering(RXS) to the cuprates opened up new perspectives onthis complex issue. Taking advantage of photons tunedto the dipole-active L -absorption edge of Cu, resonantscattering can discriminate between diffraction featuresoriginating from modulations of the valence electron den-sity in the CuO planes, and diffuse scattering generatedby random lattice displacements due to chemical sub-stitution, which dominates non-resonant x-ray scatteringmaps of most doped cuprates (where both valence andcore electrons contribute). RXS experiments found su-perstructure reflections indicative of charge order in mod-erately doped YBa Cu O x , which is regarded asone of the cuprate families least affected by chemical dis-order because the oxygen dopant atoms are arranged inchains of several tens of nanometers length. Around thesame time, nuclear magnetic resonance and non-resonantx-ray diffraction measurements also found evidence ofcharge order in YBa Cu O x . The x-ray data fur-ther showed that the wavevector characterizing charge or-dering in YBa Cu O x is incommensurate with the un-derlying crystal lattice, in qualitative analogy to chargedensity waves in weakly correlated metals. This observa-tion stimulated a large number of weak-coupling theorieson the origin of this phenomenon and its relationship tothe Fermi surface inferred from transport measurements on YBa Cu O x . In particular, the presence of elec-tron pockets revealed by Hall effect and quantum oscilla-tion experiments on moderately hole-doped cuprates wasattributed to a CDW-induced Fermi surface reconstruc-tion and there has been much debate about possi-ble contributions of charge order and charge fluctuationsto the spontaneous lattice-rotational symmetry breaking(“electronic nematicity”) that is widely observed in theunderdoped regime. Following up on the early work on YBa Cu O x , alarge body of experiments has confirmed that charge or-der is a universal feature of moderately doped cuprates,with a set of properties that are shared between all chemi-cally distinct families of cuprates where this phenomenonhas been studied (Table 2). At the same time, pro-nounced materials-specific variations in this phenomenol-ogy were also found, which is not unexpected becausemodulations of the valence-electron density couple tothe materials-specific lattice structure via long-rangedCoulomb interactions. Indeed, crystallographic refine-ments in the charge-ordered state in YBa Cu O x haveshown substantial displacements of nearly all ions in theunit cell, and the correlation lengths of charge orderedstates in different cuprates suggest a high sensitivity tochemical disorder associated with doping. The relevanceof crystallographic disorder was further highlighted bytheoretical work demonstrating that 2D incommensurateelectronic superstructures are highly sensitive to phasondisorder. In this brief review – which expands on previous re-views of the subject – we will summarize the univer-sal properties of charge order in cuprates including itsgeometry, temperature dependence, dimensionality andspatial range. We will also discuss materials-specific vari-ations, highlighting recent discoveries whose universalityis still under investigation. Honoring Roger Cowley andhis legacy of scattering techniques, we will focus on x-rayscattering results and draw parallels to his seminal workon soft-mode structural phase transitions and associated“central peaks” .
B. Structure and periodicity.
Over the past few years, all major cuprate families havebeen investigated by RXS, and evidence of charge orderwith ordering vector along the Cu-O-Cu bond directionhas been found at moderate doping levels in almost allsystems (Figure 1 and Table 1). The sole exception isa glassy, rotationally symmetric state that has recentlybeen observed in lightly electron-doped cuprates. Thebond-oriented charge order disagrees with early theoriesof Fermi surface instabilities in spin-fermion theories, but later work showed that instabilities matching the ex-perimental observations can also occur in one-band andthree-band models. Alternatively, the charge-orderingwavevector has also been described in a real-space pic-ture in the spirit of the original “stripe” model, where Electron doping n (%) Hole doping p (%) SC ChargefluctuationsCO AFM T e m pe r a t u r e ( K ) CO FIG. 1.
A contextual summary of CO physics incuprates.
The phase diagram of doped cuprates, show-ing antiferromagnetic (AFM), superconducting (SC), charge-ordered (CO) regions. T ∗ denotes the onset of the pseudogap. it characterizes the periodicity of a nanoscale array ofphase-separated regions. Bi-based cuprates were studied both by RXS and STS,and the ordering wavevectors inferred from both meth-ods were found to be quantitatively consistent.
Thelength of the ordering wavevector decreases with in-creasing doping in all hole-doped systems except in theLa CuO system, and it increases with increasing elec-tron doping, qualitatively consistent with the doping evo-lution of the Fermi surface diameter (Fig. 2). A quanti-tative comparison between RXS and angle-resolved pho-toemission (ARPES) data on single layer Bi Sr CuO δ showed that the length of the charge-ordering wavevectormatches the distance between the tips of the “Fermi arcs”that are universally observed in the pseudogap regime ofmoderately hole-doped cuprates. However, ARPESdata do not indicate the formation of a gap at the Fermi-arc tips in the charge-ordered state, indicating thatsimple modifications of the classical CDW picture do notdescribe the data. Recent Raman spectroscopy exper-iments have revealed similar energy scales of the com-peting phases (charge order, superconductivity, and thepseudogap). In the La CuO system, the low-temperature charge-ordering wavevector increases with increasing doping, op-posite to all other hole-doped cuprates. This behav-ior has been explained as a consequence of the prox-imity of independent spin and charge instabilities withnearly commensurate wave vectors. A lock-in of thetwo wavevectors as a function of temperature observedby recent x-ray and neutron diffraction experiments sup-ports this scenario. In other hole-doped systems includ-ing especially YBa Cu O x , the wavevectors character-izing low-energy spin and charge fluctuations are quitedifferent, so that such a lock-in is less favorable.The ionic displacement patterns associated with chargeorder have been investigated by both resonant and non-resonant x-ray diffraction. Most of the evidence reportedso far indicates that the charge is concentrated in the center of the Cu-O-Cu bonds, with an anti-correlationof the amplitudes along the two orthogonal axes of thesquare lattice. This pattern, which is often re-ferred to as “ d -wave” charge order, is also consistent withSTS observations and with the results of calculationsbased on the 2D single-band Hubbard model . How-ever, different degrees of admixture of other symmetrycomponents have also been reported, which is notsurprising in view of the complex structure of real cupratecompounds and the significant displacement of ions out-side the CuO planes indicated by a full refinement of thex-ray diffraction pattern of underdoped YBa Cu O x . C. Domains and influence of disorder.
The low-temperature correlation lengths of the charge-ordered state vary greatly among the different cupratefamilies, ranging from a few unit cells in electron-dopedand Bi- and Hg-based hole-doped superconductors all theway to several tens of lattice spacings in YBa Cu O x (Table 2). This is generally consistent with the propen-sity for chemical disorder in these systems, which is min-imal in YBa Cu O x because of its regular array ofoxygen dopant ions. The shorter correlation lengths ob-served in other hole- and electron-doped cuprates areonly weakly T -dependent, indicating that disorder lim-its the growth of charge-ordered domains. Short-rangecharge order with T -independent correlation length maybe amenable to a description in terms of defect-induced“Friedel oscillations” akin to the ones that have been ob-served in weakly correlated metals. This description seems inadequate, however, for thecharge-ordered states in YBa Cu O x and La CuO whose correlation lengths increase strongly upon cool-ing, indicating an incipient critical divergence that is cutoff at the onset of superconductivity. This behavior sug-gests the relevance of many-body interactions in drivingcharge ordering. At the same time, non-resonant inelas-tic x-ray scattering (IXS) with very high energy resolu-tion indicates that the diffraction peaks correspondingto quasi-2D charge-order in YBa Cu O x have an en-ergy width of at most ∼ µ eV, which implies that theorder is truly static, albeit short-range in space. Thisfinding is closely related to the “central peak” studied byRoger Cowley in soft-mode-driven structural phase tran-sitions. Cowley proposed a model according to whichlattice defects, which are unavoidable in real materials,pin some of the soft critical vibrations, resulting in aninhomogeneous state close to the transition.
A re-lated situation arises from the confluence of disorder andinteractions in YBa Cu O x .Experiments sensitive to fluctuations with lower ener-gies and larger time scales have confirmed the presenceof static short-range charge order. In particular, nuclearmagnetic resonance experiments revealed a line broaden-ing attributable to charge ordering at a temperature com-parable to the onset seen by x-rays. Even longer time n (%) p (%) q CO (rlu) C h a r g e flu c t u a t i on s FIG. 2.
The momentum versus doping behavior.
The dependence of the charge ordering wavevector q CO as function ofdoping for various families, denoted in the compounds’ respective reciprocal lattice units (rlu) defined along the Cu-O bonddirection: YBa Cu O x ( (cid:63) ),214-LCO ( • ), Bi-2201 ( ),Bi-2212 ( ), Hg-1201 ( • ), and electron-doped materials ( ). The regionin pink denotes the q -range where fluctuations are observed in hole-doped materials. Cuprate System Low-T q -vectorrange (rlu) † High-T fluctuations q -vector range (rlu) Dimensionality CO in over-doped? La CuO (214-LCO) 0.2 – 0.245 ∼ (cid:5) UnknownYBa Cu O y ∈ ( ∼ . , ∼ . (cid:5) and 3D n/aBi Sr CuO δ (Bi-2201) 0.33 – 0.29 ∼ Yes Bi Sr CaCu O δ (Bi-2212) 0.33 – 0.25 ∼ ∗ No HgBa CuO δ (Hg-1201) 0.30 – 0.265 ∈ ( ∼ . , ∼ . UnknownElectron doped 0.16 – 0.28 ∗ UnknownTABLE 1.
Summary of the key properties of charge order in cuprate families. † : range of q -vectors denoted in orderof increasing dopant concentration. (cid:5) : phase-correlated along c -axis, with peaks at half-integer values of L , the out-of-planereciprocal lattice units. ∗ : the 2-dimensionality of these systems is inferred but has not been directly confirmed. scales are probed by coherent x-ray scattering, which wasrecently deployed on La − x Ba x CuO . These measure-ments confirmed the static nature of the charge orderat all temperatures where the corresponding peak canbe detected. Moreover, the same technique unveiled amemory effect of the charge order domains, shown in . The authors observe how charge order domains form-ing at low temperature retain their internal structureeven after cycling the temperature above the onset of thecharge order (but below the high-temperature tetragonal-to-orthorhombic transition at 240 K) and back down.However, when the sample is heated above 240 K – abovewhich the system becomes tetragonal – then cooled down, the charge order domains form in a different spatial con-figuration, and the memory is lost. The authors concludethat the structural features developing at the tetragonal-to-orthorhombic transition at 240 K determine the chargeorder pinning landscape at low temperatures.In principle, charge instabilities with q -vectors alongthe principal axes of the square lattice can either form“single- q ” states with uniaxial order or a “double- q ” statewith checkerboard order. These two states are difficultto distinguish in scattering experiments on cuprates withtetragonal lattice structure, because the superposition ofdomains of two single- q states resembles the diffractionpattern of the double- q state. However, both RXS onYBa Cu O x (where the Cu-O chains generate an or-thorhombic crystal structure) and STS observations onCa CuO Cl and Bi-based superconductors indicatedomains with uniaxial charge order. The recon-structed Fermi surface inferred from quantum oscillationexperiments, on the other hand, is more readily under-stood in models with biaxial charge order.
The en-ergy balance between both states is subtle and may beinfluenced by disorder. It is therefore conceivable thata checkerboard array forms under high magnetic fields,where the quantum oscillation experiments are carriedout.An extreme case of interaction-driven charge or-der is the three-dimensional (3D) charge-ordered state,which was recently discovered by x-ray experiments onYBa Cu O x in high magnetic fields and un-der high uniaxial strain. This state is heralded by sharpBragg peaks centered at integer values of the momentumtransfer perpendicular to the CuO layers, in contrast tothe diffraction rods indicating quasi-2D charge order un-der ambient conditions (Fig. 3). Relatedly, resonant softx-ray scattering has recently demonstrated a 3D chargeorder peak in YBa Cu O δ films grown epitaxially onSrTiO in the absence of magnetic fields (Fig. 3(d)).This state is genuinely long-range ordered, with domainsizes of ∼
100 lattice spacings in-plane and ∼
10 spacingsout-of-plane inferred from the width of the Bragg reflec-tions (Table 2). The field- and strain-induced 3D chargeorder is uniaxial, with a propagation vector that is di-rected along the orthorhombic axis parallel to the Cu-Ochains. The length of the ordering wavevector matchesthe one of the 2D charge ordered state in the same ma-terials in the absence of strain and magnetic fields.The universality of 3D charge order and its role inenabling the exceptionally pronounced quantum oscilla-tions in underdoped YBa Cu O x are subjects of cur-rent investigation. This state may be favored by the elec-tronically active Cu-O chains in the YBa Cu O x crys-tal structure, which generate an additional open sheetof the Fermi surface that is not present in most othercuprates. The complex electronic structure generatedby the CuO chains, its modification by uniaxial or epi-taxial strain, and its influence on charge ordering phe-nomena are interesting subjects of future theoretical andexperimental research.The data described above strongly suggests that dis-order is central to the behavior of the charge order. Itindicates possible realizations of both the 2D and 3D ran-dom field XY models.
Note that for the XY model,random fields are much more destructive of the orderin 2D compared with 3D. In both cases it prevents theachievement of true long-range order. This difference inthe effects of structural (i.e. random field) disorder intwo and three dimensions could explain why the quasi-2D charge order is of much shorter range compared tothat in the 3D correlated charge ordered state.
D. Charge order in the phase diagram.
Table 1 and Fig. 1 summarize the charge orderingphenomenology across various cuprate systems. Mostresearch on charge order has thus far focused on the“pseudogap” regime, which sets in below the tempera-ture T ∗ marked in the phase diagram of Fig. 1. Twomajor explanations have been advanced for the origin ofthe pseudogap: (i) a crossover due to electronic corre-lations that also generate antiferromagnetic short-rangeorder, and (ii) a phase transition with a broken sym-metry that is “hidden” to most experimental probes.The former explanation holds for both hole-doped andelectron-doped superconductors, where the onset of thepseudogap is less well defined. Comprehensive x-rayscattering surveys of YBa Cu O x and La − x Sr x CuO ,whose charge-ordered states are driven by many-body in-teractions, indicate a gradual onset of diffuse features as-sociated with charge ordering. Although there is someambiguity in the definition of the corresponding onsettemperature, the x-ray data clearly indicate that T CO is lower than T ∗ and displays a different doping depen-dence. Whereas T ∗ decreases monotonically with in-creasing hole doping, T CO subtends a “dome” in thephase diagram analogous to the superconducting dome,but centered at a lower doping level. This findingclearly rules out models according to which static chargeorder is the origin of the pseudogap.Based on this phenomenology, charge order has beenwidely viewed as an instability secondary to the one driv-ing the formation of the pseudogap. In particular, if theformation of the pseudogap and Fermi arcs can be un-derstood as a consequence of strong correlations, mod-els of charge order can take these features of the elec-tronic structure as a starting point. However, recentexperiments have challenged this viewpoint. The firstchallenge came from a resonant x-ray scattering studyof overdoped (Bi,Pb) . Sr . CuO δ , which uncovereddiffraction features remarkably similar to the ones inthe underdoped regime of the same material, and withwavevectors consistent with a simple extrapolation of thedoping dependent trend established there. The maxi-mum amplitude of these features is observed near theLifshitz point, where the Fermi surface passes througha van Hove singularity and its topology changes fromhole-like to electron-like. The charge-order correlationlength inferred from their momentum-space width de-creases as a function of increasing temperature, akin tofindings in the underdoped regime, but the onset temper-ature is well above room temperature. Since the effectivestrength of the electronic correlations decreases in highlyoverdoped cuprates, a different mechanism has to be in-voked to explain the exceptionally robust charge-orderedstate in overdoped Bi Sr CuO δ . A particular scenariois based on the high density of states at the Fermi levelgenerated by the van Hove singularity, which may be con-ducive to charge density wave formation – in analogy tothe density-of-states enhancement originating from the T e m pe r a t u r e M agne t i cfi e l d U n i a x i a l s t r a i n E p i t a x i a l s t r a i n SC 2D CDW+3D CDW2D CDWSC
Hole doping –0.4 –0.3 –0.2 H in ( H , 0, L ) I n t en s i t y d i ff e r en c e ( a r b . un i t s ) L i n ( H , , L ) Max I n t en s i t y Min K in (0, K , L ) L i n ( , K , L ) .
64 3 .
68 3 .
72 3 .
64 3 .
68 3 .
72 3 .
64 3 .
68 3 .
72 3 .
64 3 .
68 3 . L in (0,0.315, L ) Uniaxial Strain EpitaxyMagnetic field(a) (b)(c) (d)
FIG. 3.
Dimensionality of charge order in YBa Cu O x . (a) A generic phase diagram indicating the crossover from2D to 3D charge order upon the application of magnetic field, uniaxial strain, and epitaxial strain (adapted from Ref. ).(b) With the application of a static and pulsed magnetic field of (cid:38)
20 T, the CO diffraction signal evolves froman elongated rod-like feature at 0 T along the out-of-plane reciprocal lattice direction L (indicating a 2D phenomenon) to asharp peak centered at integer values of L . The in-plane propagation vector (whose coordinates are given in square-latticenotation of YBa Cu O x and labeled H, K ) remains unchanged (adapted from ). (c) L -scans of the CO peak revealinghow the extended rod evolves into a sharp peak at L = 1 upon the application of up to uniaxial strain (adapted from ). (d)A reciprocal space map of a YBa Cu O x film measured by RXS indicating both a rod feature and a sharp peak at L = 1(adapted from ). narrowing of the valence band by many-body interactionsin the underdoped regime. Future work should aim for aconsistent description of the fermiology and the structureof the charge-ordered state in the overdoped regime. Inparticular, charge ordering is expected to induce a recon-struction of the Fermi surface that should be observableby ARPES. It is also important to assess the universal- ity of this state in the cuprates. Initial experiments onhighly overdoped Bi Sr CaCu O δ did not uncover ev-idence of charge ordering. A second challenge for the prevailing view of chargeordering as a secondary instability came from a re-cent report by Arpaia et al. who attributed thenearly temperature-independent, sloping background de-tected by various resonant scattering experiments onYBa Cu O x to a charge fluctuation sig-nal covering a broad region of reciprocal space (Fig-ure 1). The wavevector characterizing this signal is closeto, but distinct from the diffraction features heraldingan incipient long-range charge-ordered state discussedabove. Other distinct characteristics are the nonzero en-ergy width of the broad feature, which indicates fluctu-ating rather than static order, and its persistence up toat least room temperature. If this finding can be gener-alized to other families of cuprates, it may indicate thepresence of dynamical nearest-neighbor bond correlationsover a temperature range comparable to that of the an-tiferromagnetic spin correlations, and possibly reflectingthe same tendency to form singlets. E. Competition and coexistence withsuperconductivity.
A priori one might expect a strong competition be-tween charge order directed along the Cu-O-Cu bondsand d -wave superconductivity which also exhibits maxi-mal strength in this direction. This expectation is indeedconfirmed by several distinct experimental signatures.First, the amplitude and correlation length inferred fromthe charge-ordering reflections in YBa Cu O x andLa − x Sr x CuO exhibit cusp-like maxima at T c , indi-cating that the charge order is weakened by the onsetof superconductivity. Second, x-ray scattering experi-ments under high magnetic fields showed that the reflec-tions sharpen and their amplitude is restored, presum-ably because superconductivity is weakened by orbitaldepairing. Third, the superconducting dome in thephase diagram exhibits a dip at doping levels p ∼ . This effect is most pronounced inLa − x Ba x CuO where T c is suppressed almost entirelyby the striped phase with coexisting charge and spinorder. Finally, the enhancement of superconductivityvia ultrafast phonon pumping has been connectedto a suppression of the charge order peak intensity inboth La − x Ba x CuO and in YBa Cu O x . The strength of these signatures varies greatly amongdifferent materials, reflecting the degree of lattice dis-order that pins the charge order correlations. In Bi-and Hg-based superconductors, the superconductivity-induced anomalies of the charge-ordering reflections areminimal.
Recent experiments on La CuO systemsdid not show an enhancement of the charge order re-flections in magnetic fields, likely due to pinningby lattice distortions. Even in YBa Cu O x , the re-flections indicative of quasi-2D short-range order underambient conditions are not entirely suppressed below T c ,indicating inhomogeneous coexistence of superconductiv-ity and charge order. It is interesting to note the anal- ogy between lattice defects, which pin incommensuratecharge-order fluctuations, and spinless Zn impurities,which are known to pin low-energy incommensurate spinfluctuations in the cuprates. RXS experiments on Zn-substituted YBa Cu O x revealed that these impuritiesalso suppress charge order, evidencing a three-phasecompetition between superconductivity, incommensuratemagnetic order, and incommensurate charge order in thissystem.The only instance in which the competition betweencharge order and superconductivity seems complete,without inhomogeneous coexistence from pinning to lat-tice defects, is the 3D charge-ordered state generated inYBa Cu O x under uniaxial strain. Non-resonant x-ray experiments showed that the reflections from thisstate are completely obliterated in the superconductingstate, underscoring its nature as a genuine thermody-namic phase. This observation points out new perspec-tives to study the interplay between interaction-drivenquantum phases with minimal influence of disorder.The near-degeneracy of the charge-ordered and super-conducting phases in underdoped cuprates has stimu-lated theories that describe the fluctuations in the pseu-dogap states in terms of a composite order parameterthat comprises both superconducting and charge-ordercomponents.
While these models reproduce boththe gradual onset of the quasi-2D charge-order reflectionsand the cusp of their intensity at T c , deviations from thecorresponding predictions are apparent both at low andat high temperatures, presumably due to the influenceof disorder and/or electron phonon interactions. We alsonote predictions of a charge modulation with a wavevec-tor twice as long as the charge-ordering vector from pair-density-wave correlations, which have received somesupport from recent scanning tunneling microscopy ex-periments. Observations of such reflections in x-rayexperiments have thus far not been reported.
F. Collective charge dynamics and phononanomalies.
As in the case of spin excitations, measurements ofthe dynamical structure factor associated with collectivecharge fluctuations have the potential to uncover detailedinformation about the interactions that drive the chargeordering instability. Such excitations have recently beendirectly observed via resonant inelastic x-ray scattering(RIXS) at energies up to 60 meV, and were shown tohybridize strongly with phonons via electron-phonon in-teractions so that they are difficult to separate fromthe phonon spectrum. Magnetic RIXS experiments onelectron-doped Nd − x Ce x CuO have uncovered anoma-lies in the paramagnon modes at the wavevector char-acterizing charge order, indicating a dynamical couplingbetween collective spin and charge excitations at energiesup to about 300 meV. This observation is consistentwith the high temperature scale of the broad charge fluc-
State In-plane ξ (˚A) Out-of-plane ξ at L = n (˚A) Out-of-plane ξ at L = n (˚A) Unperturbed 60-70; n/aMagnetic field 100 at H <
20 T, 180 at
H >
28 T at L = 1; 100at H = 16 T at L = n ; H = 16 T at L =1 ∼
34 at H = 20 T, 50 at H =28 T;
47 at H = 16 T Uniaxial strain 310 at 1% ∼
94 at 1% Epitaxial strain 80-100 ∼ TABLE 2.
Correlation lengths of 2D and 3D charge orders in YBa Cu O x . A comparison of the correlation lengths ξ of the charge order in YBa Cu O x under different experimental conditions. tuation signal from quasielastic RIXS experiments dis-cussed above.Recent RIXS experiments on La − x Ba x CuO have been conducted with the aim of obtaining a unifiedpicture of the doping-dependent q -vector across families(Fig. 4). The q -dependent RIXS signal as a function oftemperature (Figure 4(a)) shows a shift from the com-mensurate value of ∼ q -vector value approaches that ofthe other families, charge and spin correlations decoupleabove 55 K. Since the scattered intensity vanishes rapidlyabove 80 K, it is not clear whether the upwards trend inthe q -vector reaches 0.3 rlu at higher temperatures.However, the trend of q -vector versus temperatureof the charge fluctuations is also material-dependent(Fig. 4). A recent RIXS report on double-layerBi Sr CaCu O δ showed an opposite trend in the q -vector as function of temperature. At low tempera-tures, the q -vector of the elastic peak is roughly 0.3 rlubut shifts to lower values upon heating, approaching0.25 rlu. Detailed theoretical work is required to assessthe origin of the temperature dependence of the charac-teristic q -vector for the charge response and its relation-ship to the electronic structure and lattice dynamics ofspecific compounds.Imprints of collective charge fluctuations on thephonon spectrum have also been observed by inelas-tic neutron scattering and high-resolution non-resonantinelastic x-ray scattering, which are sensitive to theionic positions via interactions with nuclei and core elec-trons, respectively (Figure 5). Neutron scattering ex-periments on La − x Sr x CuO and YBa Cu O x foundpronounced anomalies at the charge ordering wave vec-tor of the Cu-O-Cu bond-stretching and bond-bendingphonons that are common to all cuprates. High-resolution inelastic x-ray scattering experiments have de-tected sharp anomalies also in acoustic and low-energyoptical phonon branches of these materials. In un-derdoped YBa Cu O x , these anomalies involve pro-nounced phonon broadening (up to ∼
40% of the phononenergy) and sharp Kohn anomalies in the dispersionabove and below T c , respectively. The complete softening of a phonon was seen at the 3D charge orderingtransition of YBa Cu O x under uniaxial strain. The connection between the lattice dynamics and thecharge order has also been explored by means of IXS incuprate families beyond YBa Cu O x (Figure 5(c)). Inthe La CuO family, a broadening of an acoustic phononbranch at 10-12 meV was recently observed. The rangeof q -values over which the broadening occurs is tem-perature dependent. At low temperatures the broaden-ing is centered narrowly around q CO ∼ .
23 rlu. Thelinewidth of the phonon mode decreases when coolingthrough the charge order onset temperature. However,at 300 K the reciprocal space region where the broad-ening occurs spans a much wider range between [0.2-0.3] rlu. This result, consistent with the RIXS report wediscussed previously, indicates a commonality be-tween the La CuO system and the other cuprate fami-lies.Recent experiments on Bi Sr CaCu O δ foundphonon anomalies at the charge-ordering wavevector thatpersist above the onset of static charge-order, possi-bly signalling a soft-phonon precursor. However, thetemperature-independent broadening was also observedin the overdoped regime, far away from the charge orderas is currently established, and alternative explanationsbased on conventional lattice dynamics have also beenproposed.
Another study using inelastic neutron scat-tering confirmed the broadening in the single-layer com-pound Bi Sr CuO δ at optimal doping (where staticcharge order has also not been reported). The nature ofthis phonon anomaly is still an open question which couldbe resolved by modern RIXS experiments with higherspectral resolution that are becoming capable of directlydetermining the electron-phonon coupling. Taken together, these experiments imply a significantinfluence of electron-phonon interactions on the energet-ics of charge ordering, in addition to the electronic en-ergies that have been considered in most of the theoret-ical work on this issue. Some of the materials-specificvariations of charge order in the cuprates may well re-flect variations of the electron-phonon coupling in differ-ent lattice structures. We note that the recent observa-tions on classical CDW materials have also been inter- (a) (b) E ne r g y l o ss ( e V ) -0.3 -0.2 -0.1 0.0 H (rlu), K=0 E ne r g y l o ss ( e V ) -0.3 -0.2 -0.1 0.0 H (rlu), K=0 A T = 54 K B T = 59 K C T = 90 K x10
T= 54 KT= 90 K (r l u ) q C O Locked/unlock CO E ne r g y l o ss ( e V ) I n t. ( a . u . )
240 K0.150.100.050.150.100.05 Q ua s i - e l a s t i c i n t en s i t y ( a . u . ) P honon i n t en s i t y ( a . u . ) Q || (rlu) Q || (rlu) Q || (rlu) q CO FIG. 4.
RIXS studies of trends in charge order q -vector. (a) RIXS measurements from the La CuO based systemreveal a shift in the in-plane q -vector (with coordinates H, K in square-lattice notation) of the low-temperature CO peak atzero energy loss (top left) with respect to that of high-temperature fluctuations (bottom left). The right panel shows thedecoupling of the incommensurability of the CO and spin density wave in the high-temperature phase (yellow region). Uponwarming, the shift is from ∼ ∼ (adapted from ). (b) The reverse trend is seen by RIXS in Bi Sr CaCu O δ . The top panel shows a RIXS intensity maptaken at 240 K revealing a broad feature at 60 meV around ∼ q -vector of the CO. A shift from ∼ ∼ Q || along the Cu-O bond direction (adapted from ). preted as evidence of an important role of momentum-dependent electron-phonon interactions on the formationof the charge density wave. G. Outlook
Over the past few years, a multitude of spectro-scopic methods including resonant x-ray scattering, high-resolution non-resonant inelastic x-ray scattering, andscanning tunneling spectroscopy has yielded a new “uni-versal” phase diagram of the cuprates that now includesa prominent charge-ordering dome centered at moder-ate doping levels. Recent experiments have begun toadd multiple new dimensions to this diagram throughtunable control parameters such as high magnetic fields,uniaxial and hydrostatic pressure, and intense THz lightfields. These experiments are still at a stage of rapid de-velopment and will likely yield fresh insight in the nearfuture. In particular, in situ combinations of spectro-scopic, transport, and thermodynamic measurements un-der the influence of external parameters have the poten-tial to elucidate the influence of microscopic charge cor-relations on the macroscopic properties in a much moresystematic fashion than prior work monitoring narrowtwo-dimensional segments of this multidimensional phase diagram.Another development that has only just begun takesadvantage of recent progress in the synthesis of thinfilms and heterostructures to systematically manipulatethe electron system in the cuprates. In particular, thehigh sensitivity of RXS allows measurements on filmsthat are only a few unit cells thick and thus adopt thelattice structure of the underlying substrate. For in-stance, RXS experiments on thin YBa Cu O x filmswith a tetragonal structure imposed by the SrTiO sub-strate uncovered a robust 3D charge-ordered state thatis not present in bulk YBa Cu O x under ambientconditions (Fig. 3(d)). In contrast to the state re-alized under high uniaxial strain, however, the inten-sity of the Bragg reflections characteristic of 3D chargeorder depends monotonically on temperature, withoutevidence of competition with superconductivity. Thisindicates a profound influence of the substrate on theelectronic ground state of the cuprate film. Exper-iments on cuprate-manganate superlattices revealed amassive rearrangement of the orbital occupation andelectronic structure at the hetero-interfaces and anunexpectedly long-range influence of the interfaces onthe electron-phonon interaction and charge-orderingphenomena. Very recent experiments on relatedsystems have led to the discovery of a highly unusual0 (a) (b) E ne r g y ( m e V ) K in (0, K,
50 100 150 200 250
50 100 150 200 250 T = T c Q = (0, 0.31, 6.5)0.0 0.1 0.2 0.3 0.4 0.5 Temperature (K) q CO Temperature (K) Q = (0, 0.25, 6.5) F W H M ( m e V ) E ne r g y ( m e V ) K=1.6
Energy (meV) Energy (meV)
K=1.6
K=1.65 K=1.65
K=1.672 K=1.672
K=1.7
T=55 K N o r m a li z ed I n t en s i t y ( c oun t s / s ) K=1.7
T=155 K E ne r g y ( m e V ) K in (0, K , 14.5) M M T = 12 K: T = 45 K: T = 70 K: T = 130 K: T = 200 K: T = 300 K q F W H M ( m e V ) F W H M ( m e V )
12 K130 K K in (0, K , 14.5)0.2 0.3 CO q CO q CO q CO Underdoped(with static CO) Overdoped(without static CO) H in ( H , 0, 0) H in ( H , 0, 0) E ne r g y ( m e V ) E ne r g y ( m e V ) (c) (d) FIG. 5.
Phonon anomalies revealed by IXS. (a) In YBa Cu O x , two low-energy phonon branches reveal dips in theirdispersion curves at ∼ q -vector (denoted by dotted black line) at low temperatures (left panel). The phononssoften (top right) and broaden (bottom right) at the onset of superconductivity (adapted from ). (b) Another group observedthe same broadening but not the softening (adapted from ). (c) In La − x Ba x CuO , similar phonon branches soften withdecreasing temperature at ∼ q -vector (denoted by dotted black lines). In addition, the full width at halfmaximum (FWHM) of the higher energy phonon broadens in the region of the charge order q -vector at both low (top right)and high (bottom right) temperatures, potentially as a precursor of the CO (adapted from ). (d) A similar broadening wasobserved in Bi Sr CaCu O δ at all temperatures and for underdoped (with static CO) and overdoped (without static CO)samples. Green dotted lines are guide to the eye. The white arrow marker indicates the CO q -vector (adapted from ). Themomentum coordinates in all panels are given in square-lattice notation and labeled ( H, K, L ). magnetic-field-induced insulator-superconductor transi-tion, which may reflect an imprint of the field-sensitivecharge-ordered state in the manganate onto the CuO layers. These results indicate the vast potential ofmetal-oxide heterostructures as a platform for system-atic manipulations of the electronic properties of thecuprates.To fully realize this potential, these experimental de-velopments must go hand in hand with advances in the- oretical research, which have recently led to increasinglyrealistic descriptions of correlation effects in complexsolids. Since the combination of disorder and correlationscontinues to present a formidable challenge, recent resultson bulk crystals and superlattices with minimal disorderand full translational periodicity present new opportuni-ties for ab-initio modeling. In particular, the discoveryof 3D charge order in oxygen-ordered YBa Cu O x hasestablished a model system in which both the host lattice1and the electronic superstructure exhibit complete long-range order. This situation should be highly favorablefor realistic modeling of the electron-phonon interactionand its influence on the stability of the charge-orderedstate.The ultimate aim of research on the cuprates is a mi-croscopic theory of high-temperature superconductivity.In this context, the results collected over the past fewyears unambiguously demonstrate that static charge or-der of any kind (with and without coexisting spin order)competes against superconductivity. Important openquestions are the role of soft collective charge fluctua-tions in proximity to the charge-ordered state, and therelevance of long-range entanglement generically associ-ated with quantum criticality. Transport experi-ments on YBa Cu O x in high magnetic fields indi-cated two separate superconducting domes, which areassociated with a divergence of the electronic mass. The observation that these domes are centered near theend points of the stability range of quasi-2D charge or-der in YBa Cu O x suggests a possible support-ing role of quantum-critical charge fluctuations for su-perconductivity, on top of the spin fluctuations that areknown to favor d -wave superconductivity. However, acausal relationship could not be firmly established dueto the influence of disorder, which leads to a gradual on-set of quasi-2D charge order as a function of both tem-perature and doping and to inhomogeneous coexistencewith superconductivity at low temperatures. Moreover,scattering methods probing the microscopic charge or-der and dynamics could thus far not be performed underthe high-field conditions required for quantum transportexperiments.Finally, we note that a wide variety of compoundswith less correlated electron systems also exhibit phasediagrams with proximate superconducting and charge-ordered phases, ranging from organic charge-transfersalts all the way to structural homologues of theiron-pnictide high-temperature superconductors. In each of these cases, the influence of collective chargefluctuations on the mechanism of superconductivity andtheir interplay with spin correlations and lattice vibra-tions remains unresolved. The recent advances in materi- als synthesis and experimental methodology we have dis-cussed provide exciting perspectives for decisive progresson this central issue.
H. Acknowledgements
First of all, we would like to thank and honor Roger A.Cowley for his five decades of inspiring experimental andtheoretical research in solid state physics. There is al-most no subject in our field where Roger has not writtenone of the foundational papers, be it on structural phasetransitions, lattice dynamics, magnetic excitons, low di-mensional, quantum magnetism, disordered systems, su-perconductivity and superfluidity. His papers are modelsof clarity and precision. Roger was an ideal collaborator,modest and self-deprecating but at the same time bothcreative and brilliant. His passing was a great loss to ourfield but his legacy will last.Work at Lawrence Berkeley National Laboratory wasfunded by the U.S. Department of Energy, Office ofScience, Office of Basic Energy Sciences, Materials Sci-ences and Engineering Division under Contract No. DE-AC02-05-CH11231 within the Quantum Materials Pro-gram (KC2202). Work at DIPC is supported by IKER-BASQUE and the MINECO of Spain through the projectPGC2018-101334-A-C22.We would like to thank Peter Abbamonte, ElizabethBlackburn, Martin Bluschke, Lucio Braicovich, JohanChang, Xiang Chen, Riccardo Comin, Andrea Damas-celli, Mark Dean, Tom Devereaux, Giacomo Ghiringelli,Martin Greven, David Hawthorn, Stephen Hayden,Feizhou He, Yu He, Vladimir Hinkov, Markus H¨ucker,Steve Kivelson, Wei-Sheng Lee, Yuan Li, Toshinao Loew,Claudio Mazzoli, Matteo Minola, Marco Moretti, Ed-uardo da Silva Neto, Juan Porras, Christian Sch¨ußler-Langeheine, Wojciech Tabis, Matthieu Le Tacon, JohnTranquada, George Sawatzky, Enrico Schierle, SuchitraSebastian, Padraic Shafer, Z.X. Shen, Ronny Sutarto,Eugen Weschke, and Ming Yi for fruitful discussions andcollaborations. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] Keimer, B.; Moore, J. E.
Nature Physics , , 1045–1055. Keimer, B.; Kivelson, S. A.; Norman, M. R.; Uchida, S.;Zaanen, J.
Nature , , 179–186. Norman, M. R.; Pines, D.; Kallinc, C.
Advances in Physics , , 715 – 733. Taillefer, L.
Annual Review of Condensed Matter Physics , , 51–70. Peierls, R.
Quantum Theory of Solids ; Oxford ClassicTexts in the Physical Sciences; Oxford University Press:Oxford, New York, 2001. Monceau, P.
Advances in Physics , , 325–581. Wilson, J.; Salvo, F. D.; Mahajan, S.
Advances in Physics , , 117–201. Gibbs, D.; Mohanty, K. M.; Bohr, J.
Phys. Rev. B , , 562–564. Helgesen, G.; Hill, J. P.; Thurston, T. R.; Gibbs, D.;Kwo, J.; Hong, M.
Phys. Rev. B , , 2990–3004. Staub, U.; Meijer, G. I.; Fauth, F.; Allenspach, R.; Bed-norz, J. G.; Karpinski, J.; Kazakov, S. M.; Paolasini, L.; d’Acapito, F. Phys. Rev. Lett. , , 126402. Lu, Y.; Frano, A.; Bluschke, M.; Hepting, M.; Macke, S.;Strempfer, J.; Wochner, P.; Cristiani, G.; Logvenov, G.;Habermeier, H.-U.; Haverkort, M. W.; Keimer, B.;Benckiser, E.
Phys. Rev. B , , 165121. Mori, S.; Chen, C. H.; Cheong, S.-W.
Nature , ,473–476. Tranquada, J.
Journal of Physics and Chemistry of Solids , , 2150 – 2154. Babkevich, P.; Freeman, P. G.; Enderle, M.; Prab-hakaran, D.; Boothroyd, A. T.
Nature Communications , , 11632. Khomskii, D. I.
Transition Metal Compounds ; CambridgeUniversity Press, 2014. Zaanen, J.; Gunnarsson, O.
Phys. Rev. B , , 7391–7394. Machida, K.
Physica C: Superconductivity , , 192– 196. Tranquada, J. M.; B. J. Sternlieb, J. D. A.; Nakamura, Y.;Uchida, S.
Nature , , 561. Abbamonte, P.; Rusydi, A.; Smadici, S.; Gu, G. D.;Sawatzky, G. A.; Feng, D. L.
Nature Physics , ,155–158. Fink, J.; Schierle, E.; Weschke, E.; Geck, J.;Hawthorn, D.; Soltwisch, V.; Wadati, H.; Wu, H.-H.;D¨urr, H. A.; Wizent, N.; B¨uchner, B.; Sawatzky, G. A.
Phys. Rev. B , , 100502. Fink, J.; Soltwisch, V.; Geck, J.; Schierle, E.; Weschke, E.;B¨uchner, B.
Phys. Rev. B , , 092503. H¨ucker, M.; v. Zimmermann, M.; Gu, G. D.; Xu, Z. J.;Wen, J. S.; Xu, G.; Kang, H. J.; Zheludev, A.; Tran-quada, J. M.
Phys. Rev. B , , 104506. Wilkins, S. B.; Dean, M. P. M.; Fink, J.; H¨ucker, M.;Geck, J.; Soltwisch, V.; Schierle, E.; Weschke, E.; Gu, G.;Uchida, S.; Ichikawa, N.; Tranquada, J. M.; Hill, J. P.
Phys. Rev. B , , 195101. Wu, H.-H.; Buchholz, M.; Trabant, C.; Chang, C. F.; Ko-marek, A. C.; Heigl, F.; Zimmermann, M. v.; Cwik, M.;Nakamura, F.; Braden, M.; Sch¨ußler-Langeheine, C.
Na-ture Communications , , 1–5. Jacobsen, H.; Zaliznyak, I. A.; Savici, A. T.; Winn, B. L.;Chang, S.; H¨ucker, M.; Gu, G. D.; Tranquada, J. M.
Phys.Rev. B , , 174525. Fabbris, G.; H¨ucker, M.; Gu, G. D.; Tranquada, J. M.;Haskel, D.
Phys. Rev. B , , 060507. Tranquada, J. M.; Woo, H.; Perring, T. G.; Goka, H.;Gu, G. D.; Xu, G.; Fujita, M.; Yamada, K.
Nature , , 534–538. Bourges, P.; Sidis, Y.; Fong, H. F.; Regnault, L. P.;Bossy, J.; Ivanov, A.; Keimer, B.
Science , ,1234–1237. Hayden, S. M.; Mook, H. A.; Dai, P.; Perring, T. G.;Do˘gan, F.
Nature , , 531–534. Stock, C.; Buyers, W. J. L.; Cowley, R. A.; Clegg, P. S.;Coldea, R.; Frost, C. D.; Liang, R.; Peets, D.; Bonn, D.;Hardy, W. N.; Birgeneau, R. J.
Phys. Rev. B , ,024522. Stock, C.; Cowley, R. A.; Buyers, W. J. L.; Frost, C. D.;Taylor, J. W.; Peets, D.; Liang, R.; Bonn, D.;Hardy, W. N.
Phys. Rev. B , , 174505. Kivelson, S. A.; Bindloss, I. P.; Fradkin, E.;Oganesyan, V.; Tranquada, J. M.; Kapitulnik, A.;Howald, C.
Reviews of Modern Physics , ,1201–1241. Vojta, M.
Advances in Physics , , 699–820. Hoffman, J. E.; Hudson, E. W.; Lang, K. M.; Madha-van, V.; Eisaki, H.; Uchida, S.; Davis, J. C.
Science , , 466–469. Hanaguri, T.; Lupien, C.; Kohsaka, Y.; Lee, D.-H.;Azuma, M.; Takano, M.; Takagi, H.; Davis, J. C.
Nature , , 1001–1005. Kohsaka, Y.; Taylor, C.; Fujita, K.; Schmidt, A.;Lupien, C.; Hanaguri, T.; Azuma, M.; Takano, M.;Eisaki, H.; Takagi, H.; Uchida, S.; Davis, J. C.
Science , , 1380–1385. Wise, W. D.; Boyer, M. C.; Chatterjee, K.; Kondo, T.;Takeuchi, T.; Ikuta, H.; Wang, Y.; Hudson, E. W.
NaturePhysics , , 696–699. Parker, C. V.; Aynajian, P.; Neto, E. H. d. S.; Pushp, A.;Ono, S.; Wen, J.; Xu, Z.; Gu, G.; Yazdani, A.
Nature , , 677–680. Dalla Torre, E. G.; Benjamin, D.; He, Y.; Dentelski, D.;Demler, E.
Phys. Rev. B , , 205117. Huang, E. W.; Mendl, C. B.; Liu, S.; Johnston, S.;Jiang, H.-C.; Moritz, B.; Devereaux, T. P.
Science , , 1161–1164. Zheng, B.-X.; Chung, C.-M.; Corboz, P.; Ehlers, G.;Qin, M.-P.; Noack, R. M.; Shi, H.; White, S. R.; Zhang, S.;Chan, G. K.-L.
Science , , 1155–1160. Choubey, P.; Tu, W.-L.; Lee, T.-K.; Hirschfeld, P. J.
NewJournal of Physics , , 013028. Jiang, H.-C.; Devereaux, T. P.
Science , , 1424–1428. Jiang, H.-C.; Weng, Z.-Y.; Kivelson, S. A.
Physical ReviewB , , 140505. Ghiringhelli, G. et al.
Science , , 821–825. Achkar, A. J. et al.
Physical Review Letters , ,167001. Bobroff, J.; Alloul, H.; Ouazi, S.; Mendels, P.; Ma-hajan, A.; Blanchard, N.; Collin, G.; Guillen, V.;Marucco, J.-F.
Phys. Rev. Lett. , , 157002. Chang, J.; Blackburn, E.; Holmes, A. T.; Chris-tensen, N. B.; Larsen, J.; Mesot, J.; Liang, R.;Bonn, D. A.; Hardy, W. N.; Watenphul, A.; Zimmer-mann, M. v.; , E. M.; Hayden, S. M.
Nature Physics , , 871. Wu, T.; Mayaffre, H.; Kr¨amer, S.; Horvatic, M.;Berthier, C.; Hardy, W. N.; Liang, R.; Bonn, D. A.;Julien, M.-H.
Nature , , 191. Wu, T.; Mayaffre, H.; Kr¨amer, S.; Horvatic, M.;Berthier, C.; Kuhns, P. L.; Reyes, A. P.; Liang, R.;Hardy, W. N.; Bonn, D. A.; Julien, M.-H.
Nature Com-munications , , 2113. Wu, T.; Mayaffre, H.; Kr¨amer, S.; Horvatic, M.;Berthier, C.; Hardy, W. N.; Liang, R.; Bonn, D. A.;Julien, M.-H.
Nature Communications , , 6438. Metlitski, M. A.; Sachdev, S.
Phys. Rev. B , ,075128. Wang, Y.; Chubukov, A.
Physical Review B , ,035149. Doiron-Leyraud, N.; Proust, C.; LeBoeuf, D.; Leval-lois, J.; Bonnemaison, J.-B.; Liang, R.; Bonn, D. A.;Hardy, W. N.; Taillefer, L.
Nature , , 565. Lalibert´e, F. et al.
Nature Communications , , 432. Tabis, W. et al.
Nature Communications , , 5875. Doiron-Leyraud, N. et al.
Nature Communications , , 1–7. Fernandes, R. M.; Orth, P. P.; Schmalian, J.
Annual Re- view of Condensed Matter Physics , , 133–154. Forgan, E. M.; Blackburn, E.; Holmes, A. T.; Briffa, A.K. R.; Chang, J.; Bouchenoire, L.; Brown, S. D.;Liang, R.; Bonn, D.; Hardy, W. N.; Christensen, N. B.;Zimmermann, M. V.; H¨ucker, M.; Hayden, S. M.
NatureCommunications , , 10064. Nie, L.; Maharaj, A. V.; Fradkin, E.; Kivelson, S. A.
Phys-ical Review B , , 085142. Tranquada, J. M.
AIP Conference Proceedings , , 114–187. Comin, R.; Damascelli, A.
Annual Review of CondensedMatter Physics , , 369–405. Cowley, R. A.; Salje, E. K. H.
Philosophical Transactionsof the Royal Society of London. Series A: Mathematical,Physical and Engineering Sciences , , 2799–2814. A. Cowley, R.; M. Shapiro, S.
Journal of the PhysicalSociety of Japan , , 111001. Kang, M. et al.
Nature Physics , , 335–340. Thomson, A.; Sachdev, S.
Phys. Rev. B , , 115142. Capati, M.; Caprara, S.; Castro, C. D.; Grilli, M.; Sei-bold, G.; Lorenzana, J.
Nature Communications , ,7691. Miao, H.; Lorenzana, J.; Seibold, G.; Peng, Y. Y.;Amorese, A.; Yakhou-Harris, F.; Kummer, K.;Brookes, N. B.; Konik, R. M.; Thampy, V.; Gu, G. D.;Ghiringhelli, G.; Braicovich, L.; Dean, M. P. M.
Pro-ceedings of the National Academy of Sciences , ,12430–12435. Miao, H.; Ishikawa, D.; Heid, R.; Le Tacon, M.; Fab-bris, G.; Meyers, D.; Gu, G. D.; Baron, A.; Dean, M.
Physical Review X , , 011008. Blanco-Canosa, S.; Frano, A.; Loew, T.; Lu, Y.; Por-ras, J.; Ghiringhelli, G.; Minola, M.; Mazzoli, C.;Braicovich, L.; Schierle, E.; Weschke, E.; Le Tacon, M.;Keimer, B.
Physical Review Letters , , 187001. Blackburn, E.; Chang, J.; H¨ucker, M.; Holmes, A. T.;Christensen, N. B.; Liang, R.; Bonn, D. A.; Hardy, W. N.;R¨utt, U.; Gutowski, O.; Zimmermann, M. v.; , E. M.;Hayden, S. M.
Phys. Rev. Lett. , , 137004. Bluschke, M.; Yaari, M.; Schierle, E.; Bazalitsky, G.;Werner, J.; Weschke, E.; Keren, A.
Phys. Rev. B , , 035129. Arpaia, R.; Caprara, S.; Fumagalli, R.; De Vec-chi, G.; Peng, Y. Y.; Andersson, E.; Betto, D.;De Luca, G. M.; Brookes, N. B.; Lombardi, F.; Sal-luzzo, M.; Braicovich, L.; Di Castro, C.; Grilli, M.; Ghir-inghelli, G.
Science , , 906–910. Gerber, S. et al.
Science , , 949–952. Chang, J.; Blackburn, E.; Ivashko, O.; Holmes, A. T.;Christensen, N. B.; H¨ucker, M.; Liang, R.; Bonn, D. A.;Hardy, W. N.; R¨utt, U.; Zimmermann, M. v.; For-gan, E. M.; Hayden, S. M.
Nature Communications , , 11494. Bluschke, M.; Frano, A.; Schierle, E.; Putzky, D.;Ghorbani, F.; Ortiz, R.; Suzuki, H.; Christiani, G.;Logvenov, G.; Weschke, E.; Birgeneau, R. J.;da Silva Neto, E. H.; Minola, M.; Blanco-Canosa, S.;Keimer, B.
Nature Communications , , 2978. Kim, H.-H. et al.
Science , , 1040–1044. Comin, R. et al.
Science , , 390–392. Peng, Y. Y. et al.
Nature Materials , , 697–702. da Silva Neto, E.; Aynajian, P.; Frano, A.; Comin, R.;Schierle, E.; Weschke, E.; Gyenis, A.; Wen, J.; Schnee-loch, J.; Xu, Z.; Ono, S.; Gu, G.; Le Tacon, M.; Yaz- dani, A. Science , , 393–396. Chaix, L. et al.
Nature Physics , , 952–956. He, Y. et al.
Phys. Rev. B , , 035102. Tabis, W. et al.
Phys. Rev. B , , 134510. da Silva Neto, E. H.; Comin, R.; He, F.; Sutarto, R.;Jiang, Y.; Greene, R. L.; Sawatzky, G. A.; Damascelli, A. Science , , 282–285. da Silva Neto, E. H. et al. Physical Review B , ,161114. Jang, H.; Asano, S.; Fujita, M.; Hashimoto, M.; Lu, D. H.;Burns, C. A.; Kao, C.-C.; Lee, J.-S.
Phys. Rev. X , , 041066. Reber, T. J.; Plumb, N. C.; Sun, Z.; Cao, Y.; Wang, Q.;McElroy, K.; Iwasawa, H.; Arita, M.; Wen, J. S.; Xu, Z. J.;Gu, G.; Yoshida, Y.; Eisaki, H.; Aiura, Y.; Dessau, D. S.
Nature Physics , , 606–610. Loret, B.; Auvray, N.; Gallais, Y.; Cazayous, M.; For-get, A.; Colson, D.; Julien, M.-H.; Paul, I.; Civelli, M.;Sacuto, A.
Nature Physics , , 771–775. Comin, R. et al.
Nature Materials , , 796. Comin, R.; Sutarto, R.; da Silva Neto, E. H.; Chau-viere, L.; Liang, R.; Hardy, W. N.; Bonn, D. A.; He, F.;Sawatzky, G. A.; Damascelli, A.
Science , , 1335–1339. Fujita, K.; Hamidian, M. H.; Edkins, S. D.; Kim, C. K.;Kohsaka, Y.; Azuma, M.; Takano, M.; Takagi, H.;Eisaki, H.; Uchida, S.-i.; Allais, A.; Lawler, M. J.;Kim, E.-A.; Sachdev, S.; Davis, J. C. S.
Proceedings of theNational Academy of Sciences , , E3026–E3032. Hamidian, M. H.; Edkins, S. D.; Kim, C. K.; Davis, J. C.;Mackenzie, A. P.; Eisaki, H.; Uchida, S.; Lawler, M. J.;Kim, E.-A.; Sachdev, S.; Fujita, K.
Nature Physics , , 150–156. Sachdev, S.; La Placa, R.
Phys. Rev. Lett. , ,027202. Achkar, A. J.; He, F.; Sutarto, R.; McMahon, C.;Zwiebler, M.; H¨ucker, M.; Gu, G. D.; Liang, R.;Bonn, D. A.; Hardy, W. N.; Geck, J.; Hawthorn, D. G.
Nature Materials , , 616–620. McMahon, C.; Achkar, A. J.; da Silva Neto, E. H.;Djianto, I.; Menard, J.; He, F.; Sutarto, R.; Comin, R.;Liang, R.; Bonn, D. A.; Hardy, W. N.; Damascelli, A.;Hawthorn, D. G. arXiv:1904.12929 , Le Tacon, M.; Bosak, A.; Souliou, S. M.; Dellea, G.;Loew, T.; Heid, R.; Bohnen, K.-P.; Ghiringhelli, G.;Krisch, M.; Keimer, B.
Nature Physics , , 52–58. Chen, X.; Thampy, V.; Mazzoli, C.; Barbour, A.;Miao, H.; Gu, G.; Cao, Y.; Tranquada, J.; Dean, M.;Wilkins, S.
Physical Review Letters , , 167001. Chen, X. M.; Mazzoli, C.; Cao, Y.; Thampy, V.; Bar-bour, A. M.; Hu, W.; Lu, M.; Assefa, T. A.; Miao, H.;Fabbris, G.; Gu, G. D.; Tranquada, J. M.; Dean, M. P. M.;Wilkins, S. B.; Robinson, I. K.
Nature Communications , , 1435. Sebastian, S. E.; Harrison, N.; Lonzarich, G. G.
Reportson Progress in Physics , , 102501. Chan, M. K.; Harrison, N.; McDonald, R. D.;Ramshaw, B. J.; Modic, K. A.; Bariˇsi´c, N.; Greven, M.
Nature Communications , , 1–9. Jang, H. et al.
Proceedings of the National Academy ofSciences , , 14645–14650. Jang, H. et al.
Phys. Rev. B , , 224513. Imry, Y.; Ma, S.-k.
Phys. Rev. Lett. , , 1399–1401. Aharony, A.; Pytte, E.
Phys. Rev. B , , 5872–5874. Horio, M.; Fujimori, A.
Journal of Physics: CondensedMatter , , 503001. H¨ucker, M.; Christensen, N. B.; Holmes, A. T.; Black-burn, E.; , E. M.; Liang, R.; Bonn, D. A.; Hardy, W. N.;Gutowski, O.; Zimmermann, M. v.; Hayden, S. M.;Chang, J.
Phys. Rev. B , , 054514. Blanco-Canosa, S.; Frano, A.; Schierle, E.; Porras, J.;Loew, T.; Minola, M.; Bluschke, M.; Weschke, E.;Keimer, B.; Le Tacon, M.
Physical Review B , ,054513. Frano, A. et al.
Nature Materials , , 831–834. Fausti, D.; Tobey, R. I.; Dean, N.; Kaiser, S.; Dienst, A.;Hoffmann, M. C.; Pyon, S.; Takayama, T.; Takagi, H.;Cavalleri, A.
Science , , 189–191. Kaiser, S.; Hunt, C. R.; Nicoletti, D.; Hu, W.; Gierz, I.;Liu, H. Y.; Le Tacon, M.; Loew, T.; Haug, D.; Keimer, B.;Cavalleri, A.
Phys. Rev. B , , 184516. Nicoletti, D.; Casandruc, E.; Laplace, Y.; Khanna, V.;Hunt, C. R.; Kaiser, S.; Dhesi, S. S.; Gu, G. D.; Hill, J. P.;Cavalleri, A.
Phys. Rev. B , , 100503. F¨oerst, M. et al.
Phys. Rev. Lett. , , 157002. F¨orst, M.; Frano, A.; Kaiser, S.; Mankowsky, R.;Hunt, C. R.; Turner, J. J.; Dakovski, G. L.; Minitti, M. P.;Robinson, J.; Loew, T.; Le Tacon, M.; Keimer, B.;Hill, J. P.; Cavalleri, A.; Dhesi, S. S.
Phys. Rev. B , , 184514. H¨ucker, M.; v. Zimmermann, M.; Xu, Z. J.; Wen, J. S.;Gu, G. D.; Tranquada, J. M.
Phys. Rev. B , ,014501. Zwiebler, M.; Schierle, E.; Weschke, E.; B¨uchner, B.;Revcolevschi, A.; Ribeiro, P.; Geck, J.; Fink, J.
Phys.Rev. B , , 165157. Blanco-Canosa, S.; Schierle, E.; Li, Z. W.; Guo, H.;Adachi, T.; Koike, Y.; Sobolev, O.; Weschke, E.; Ko-marek, A. C.; Sch¨ußler-Langeheine, C.
Phys. Rev. B , , 195130. Suchaneck, A.; Hinkov, V.; Haug, D.; Schulz, L.; Bern-hard, C.; Ivanov, A.; Hradil, K.; Lin, C. T.; Bourges, P.;Keimer, B.; Sidis, Y.
Physical Review Letters , ,037207. P´epin, C.; de Carvalho, V. S.; Kloss, T.; Montiel, X.
Phys.Rev. B , , 195207. Hayward, L. E.; Hawthorn, D. G.; Melko, R. G.;Sachdev, S.
Science , , 1336–1339. Lee, P. A.
Phys. Rev. X , , 031017. Edkins, S. D.; Kostin, A.; Fujita, K.; Mackenzie, A. P.;Eisaki, H.; Uchida, S.; Sachdev, S.; Lawler, M. J.;Kim, E.-A.; S´eamus Davis, J. C.; Hamidian, M. H.
Science , , 976–980. da Silva Neto, E. H. et al. Science Advances , ,e1600782. Miao, H.; Fumagalli, R.; Rossi, M.; Lorenzana, J.; Sei-bold, G.; Yakhou-Harris, F.; Kummer, K.; Brookes, N. B.;Gu, G. D.; Braicovich, L.; Ghiringhelli, G.; Dean, M.P. M.
Phys. Rev. X , , 031042. Reznik, D.; Sangiovanni, G.; Gunnarsson, O.; Dev-ereaux, T. P.
Nature , , E6–E7. Raichle, M.; Reznik, D.; Lamago, D.; Heid, R.; Li, Y.;Bakr, M.; Ulrich, C.; Hinkov, V.; Hradil, K.; Lin, C. T.;Keimer, B.
Phys. Rev. Lett. , , 177004. Park, S. R.; Fukuda, T.; Hamann, A.; Lamago, D.;Pintschovius, L.; Fujita, M.; Yamada, K.; Reznik, D.
Phys. Rev. B , , 020506. Blackburn, E.; Chang, J.; Said, A. H.; Leu, B. M.;Liang, R.; Bonn, D. A.; Hardy, W. N.; Forgan, E. M.;Hayden, S. M.
Physical Review B , , 054506. Souliou, S. M.; Gretarsson, H.; Garbarino, G.; Bosak, A.;Porras, J.; Loew, T.; Keimer, B.; Le Tacon, M.
PhysicalReview B , , 020503. Merritt, A. M.; Castellan, J.-P.; Keller, T.; Park, S. R.;Fernandez-Baca, J. A.; Gu, G. D.; Reznik, D.
Phys. Rev.B , , 144502. Bonnoit, C. J.; Gardner, D. R.; Chisnell, R.; Said, A. H.;Okada, Y.; Kondo, T.; Takeuchi, T.; Ikuta, H.; Monc-ton, D. E.; Lee, Y. S. arXiv:1202.4994v1 [cond-mat] , Rossi, M.; Arpaia, R.; Fumagalli, R.; Sala, M. M.;Betto, D.; De Luca, G. M.; Kummer, K.; Brink, J.v. d.; Salluzzo, M.; Brookes, N. B.; Braicovich, L.; Ghir-inghelli, G. arXiv:1902.09163 [cond-mat] , Weber, F.; Rosenkranz, S.; Castellan, J.-P.; Osborn, R.;Hott, R.; Heid, R.; Bohnen, K.-P.; Egami, T.; Said, A. H.;Reznik, D.
Physical Review Letters , , 107403. Chakhalian, J.; Freeland, J. W.; Habermeier, H.-U.; Cris-tiani, G.; Khaliullin, G.; Veenendaal, M. v.; Keimer, B.
Science , , 1114–1117. Driza, N.; Blanco-Canosa, S.; Bakr, M.; Soltan, S.;Khalid, M.; Mustafa, L.; Kawashima, K.; Chris-tiani, G.; Habermeier, H.-U.; Khaliullin, G.; Ulrich, C.;Tacon, M. L.; Keimer, B.
Nature Materials , ,675–681. He, J.; Shafer, P.; Mion, T. R.; Tra, V. T.; He, Q.;Kong, J.; Chuang, Y.-D.; Yang, W. L.; Graf, M. J.;Lin, J.-Y.; Chu, Y.-H.; Arenholz, E.; He, R.-H.
NatureCommunications , , 10852. Khmaladze, J.; Sarkar, S.; Soulier, M.; Lyzwa, F.; de An-dres Prada, R.; Perret, E.; Mallett, B. P. P.; Minola, M.;Keimer, B.; Bernhard, C.
Physical Review Materials , , 084801. Wang, Y.; Chubukov, A. V.
Physical Review B , ,125108. Lederer, S.; Schattner, Y.; Berg, E.; Kivelson, S. A.
Pro-ceedings of the National Academy of Sciences , ,4905–4910. Ramshaw, B. J.; Sebastian, S. E.; McDonald, R. D.;Day, J.; Tan, B. S.; Zhu, Z.; Betts, J. B.; Liang, R.;Bonn, D. A.; Hardy, W. N.; Harrison, N.
Science , , 317–320. Clay, R. T.; Mazumdar, S.
Physics Reports , ,1–89. Allred, J. M.; Taddei, K. M.; Bugaris, D. E.;Krogstad, M. J.; Lapidus, S. H.; Chung, D. Y.; Claus, H.;Kanatzidis, M. G.; Brown, D. E.; Kang, J.; Fernan-des, R. M.; Eremin, I.; Rosenkranz, S.; Chmaissem, O.;Osborn, R.
Nature Physics , , 493–498. Waßer, F.; Schneidewind, A.; Sidis, Y.; Wurmehl, S.;Aswartham, S.; B¨uchner, B.; Braden, M.
Phys. Rev. B , , 060505. Lee, S.; de la Pe˜na, G.; Sun, S. X.-L.; Mitrano, M.;Fang, Y.; Jang, H.; Lee, J.-S.; Eckberg, C.; Campbell, D.;Collini, J.; Paglione, J.; de Groot, F. M. F.; Abba-monte, P.
Phys. Rev. Lett. , , 147601. Yi, M. et al.
Phys. Rev. Lett. ,121