Symmetry of charge order in cuprates

Universal Stripe Symmetry of Short-Range Charge Density Waves in Cuprate Superconductors

The omnipresence of charge density waves (CDWs) across almost all cuprate families underpins a common organizing principle. However, a longstanding debate of whether its spatial symmetry is stripe or checkerboard remains unresolved. While CDWs in lanthanum- and yttrium-based cuprates possess a stripe symmetry, distinguishing these two scenarios is challenging for the short-range CDW in bismuth-based cuprates. Here, high-resolution resonant inelastic x-ray scattering is employed to uncover the spatial symmetry of the CDW in Bi2 Sr2 - x Lax CuO6 + δ . Across a wide range of doping and temperature, anisotropic CDW peaks with elliptical shapes are found in reciprocal space. Based on Fourier transform analysis of real-space models, the results are interpreted as evidence of unidirectional charge stripes, hosted by mutually 90°-rotated anisotropic domains. This work paves the way for a unified symmetry and microscopic description of CDW order in cuprates.

Critical nematic correlations throughout the superconducting doping range in Bi2-zPbzSr2-yLayCuO6+x

Charge modulations have been widely observed in cuprates, suggesting their centrality for understanding the high-T_c superconductivity in these materials. However, the dimensionality of these modulations remains controversial, including whether their wavevector is unidirectional or bidirectional, and also whether they extend seamlessly from the surface of the material into the bulk. Material disorder presents severe challenges to understanding the charge modulations through bulk scattering techniques. We use a local technique, scanning tunneling microscopy, to image the static charge modulations on Bi_2-zPb_zSr_2-yLa_yCuO_6+x. The ratio of the phase correlation length ξ_CDW to the orientation correlation length ξ_orient points to unidirectional charge modulations. By computing new critical exponents at free surfaces including that of the pair connectivity correlation function, we show that these locally 1D charge modulations are actually a bulk effect resulting from classical 3D criticality of the random field Ising model throughout the entire superconducting doping range.

Monte Carlo study of cuprate superconductors in a four-bandd-pmodel: role of orbital degrees of freedom

Understanding the various competing phases in cuprate superconductors is a long-standing challenging problem. Recent studies have shown that orbital degrees of freedom, both Cue_gorbitals and Oporbitals, are a key ingredient for a unified understanding of cuprate superconductors, including the material dependence. Here we investigate a four-bandd-pmodel derived from the first-principles calculations with the variational Monte Carlo method, which allows us to elucidate competing phases on an equal footing. The obtained results can consistently explain the doping dependence of superconductivity, antiferromagnetic and stripe phases, phase separation in the underdoped region, and also novel magnetism in the heavily-overdoped region. The presence ofporbitals is critical to the charge-stripe features, which induce two types of stripe phases withs)-wave andd-wave bond stripe. On the other hand, the presence ofdz2orbital is indispensable to material dependence of the superconducting transition temperature (Tc), and enhances local magnetic moment as a source of novel magnetism in the heavily-overdoped region as well. These findings beyond one-band description could provide a major step toward a full explanation of unconventional normal state and highTcin cuprate supercondutors.

Controllable Emergent Spatial Spin Modulation in Sr_{2}IrO_{4} by In Situ Shear Strain

Symmetric anisotropic interaction can be ferromagnetic and antiferromagnetic at the same time but for different crystallographic axes. We show that the competition of anisotropic interactions of orthogonal irreducible representations can be a general route to obtain new exotic magnetic states. We demonstrate it here by observing the emergence of a continuously tunable 12-layer spatial spin modulation when distorting the square-lattice planes in the quasi-two-dimensional antiferromagnetic Sr_{2}IrO_{4} under in situ shear strain. This translation-symmetry-breaking phase is a result of an unusual strain-activated anisotropic interaction which is at the fourth order and competing with the inherent quadratic anisotropic interaction. Such a mechanism of competing anisotropy is distinct from that among the ferromagnetic, antiferromagnetic, and/or the Dzyaloshinskii-Moriya interactions, and it could be widely applicable and highly controllable in low-dimensional magnets.

High-T c Cuprates: a Story of Two Electronic Subsystems

A review of the phenomenology and microscopy of cuprate superconductors is presented, with particular attention to universal conductance features, which reveal the existence of two electronic subsystems. The overall electronic system consists of 1 + p charges, where p is the doping. At low dopings, exactly one hole is localized per planar copper-oxygen unit, while upon increasing doping and temperature, the hole is gradually delocalized and becomes itinerant. Remarkably, the itinerant holes exhibit identical Fermi liquid character across the cuprate phase diagram. This universality enables a simple count of carrier density and yields comprehensive understanding of the key features in the normal and superconducting state. A possible superconducting mechanism is presented, compatible with the key experimental facts. The base of this mechanism is the interaction of fast Fermi liquid carriers with localized holes. A change in the microscopic nature of chemical bonding in the copper oxide planes, from ionic to covalent, is invoked to explain the phase diagram of these fascinating compounds.

B_{1g}-Phonon Anomaly Driven by Fermi Surface Instability at Intermediate Temperature in YBa_{2}Cu_{3}O_{7-δ}

We performed temperature- and doping-dependent high-resolution Raman spectroscopy experiments on YBa_{2}Cu_{3}O_{7-δ} to study B_{1g} phonons. The temperature dependence of the real part of the phonon self-energy shows a distinct kink at T=T_{B1g} above T_{c} due to softening, in addition to the one due to the onset of the superconductivity. T_{B1g} is clearly different from the pseudogap temperature with a maximum in the underdoped region and resembles charge density wave onset temperature, T_{CDW}. We attribute the B_{1g}-phonon softening to an energy gap on the Fermi surface induced by a charge density wave order, which is consistent with the results of a recent electronic Raman scattering study. Our work demonstrates a way to investigate Fermi surface instabilities above T_{c} via phonon Raman studies.

Coexisting commensurate and incommensurate charge ordered phases in CoO

The subtle interplay of strong electronic correlations in a distorted crystal lattice often leads to the evolution of novel emergent functionalities in the strongly correlated materials (SCM). Here, we unravel such unprecedented commensurate (COM) and incommensurate (ICOM) charge ordered (CO) phases at room temperature in a simple transition-metal mono-oxide, namely CoO. The electron diffraction pattern unveils a COM ([Formula: see text]=[Formula: see text] and ICOM ([Formula: see text]) periodic lattice distortion. Transmission electron microscopy (TEM) captures unidirectional and bidirectional stripe patterns of charge density modulations. The widespread phase singularities in the phase-field of the order parameter (OP) affirms the abundant topological disorder. Using, density functional theory (DFT) calculations, we demystify the underlying electronic mechanism. The DFT study shows that a cation disordering ([Formula: see text]) stabilizes Jahn-Teller (JT) distortion and localized aliovalent [Formula: see text] states in CoO. Therefore, the lattice distortion accompanied with mixed valence states ([Formula: see text]) states introduces CO in CoO. Our findings offer an electronic paradigm to engineer CO to exploit the associated electronic functionalities in widely available transition-metal mono-oxides.

Charge order textures induced by non-linear couplings in a half-doped manganite

The self-organization of strongly interacting electrons into superlattice structures underlies the properties of many quantum materials. How these electrons arrange within the superlattice dictates what symmetries are broken and what ground states are stabilized. Here we show that cryogenic scanning transmission electron microscopy (cryo-STEM) enables direct mapping of local symmetries and order at the intra-unit-cell level in the model charge-ordered system Nd_1/2Sr_1/2MnO₃. In addition to imaging the prototypical site-centered charge order, we discover the nanoscale coexistence of an exotic intermediate state which mixes site and bond order and breaks inversion symmetry. We further show that nonlinear coupling of distinct lattice modes controls the selection between competing ground states. The results demonstrate the importance of lattice coupling for understanding and manipulating the character of electronic self-organization and that cryo-STEM can reveal local order in strongly correlated systems at the atomic scale.

In this paper, the authors demonstrate that cryogenic scanning transmission electron microscopy allows for the direct mapping of the local arrangements and symmetries of electronic order, providing a useful method for studying strongly correlated systems. They show this using the example of Nd1/2Sr1/2MnO3, a model charge ordered material.

Locally commensurate charge-density wave with three-unit-cell periodicity in YBa2Cu3Oy

In order to identify the mechanism responsible for the formation of charge-density waves (CDW) in cuprate superconductors, it is important to understand which aspects of the CDW's microscopic structure are generic and which are material-dependent. Here, we show that, at the local scale probed by NMR, long-range CDW order in YBa2Cu3Oy is unidirectional with a commensurate period of three unit cells (λ = 3b), implying that the incommensurability found in X-ray scattering is ensured by phase slips (discommensurations). Furthermore, NMR spectra reveal a predominant oxygen character of the CDW with an out-of-phase relationship between certain lattice sites but no specific signature of a secondary CDW with λ = 6b associated with a putative pair-density wave. These results shed light on universal aspects of the cuprate CDW. In particular, its spatial profile appears to generically result from the interplay between an incommensurate tendency at long length scales, possibly related to properties of the Fermi surface, and local commensuration effects, due to electron-electron interactions or lock-in to the lattice.

Orbital symmetries of charge density wave order in YBa2Cu3O6+x

Charge density wave (CDW) order has been shown to compete and coexist with superconductivity in underdoped cuprates. Theoretical proposals for the CDW order include an unconventional d-symmetry form factor CDW, evidence for which has emerged from measurements, including resonant soft x-ray scattering (RSXS) in YBa₂Cu₃O_6+x (YBCO). Here, we revisit RSXS measurements of the CDW symmetry in YBCO, using a variation in the measurement geometry to provide enhanced sensitivity to orbital symmetry. We show that the (0 0.31 L) CDW peak measured at the Cu L edge is dominated by an s form factor rather than a d form factor as was reported previously. In addition, by measuring both (0.31 0 L) and (0 0.31 L) peaks, we identify a pronounced difference in the orbital symmetry of the CDW order along the a and b axes, with the CDW along the a axis exhibiting orbital order in addition to charge order.

Comprehensive Electrical Control of Metamagnetic Transition of a Quasi-2D Antiferromagnet by In Situ Anisotropic Strain

Effective nonmagnetic control of the spin structure is at the forefront of the study for functional quantum materials. This study demonstrates that, by applying an anisotropic strain up to only 0.05%, the metamagnetic transition field of spin-orbit-coupled Mott insulator Sr₂ IrO₄ can be in situ modulated by almost 300%. Simultaneous measurements of resonant X-ray scattering and transport reveal that this drastic response originates from the complete strain-tuning of the transition between the spin-flop and spin-flip limits, and is always accompanied by large elastoconductance and magnetoconductance. This enables electrically controllable and electronically detectable metamagnetic switching, despite the antiferromagnetic insulating state. The obtained strain-magnetic field phase diagram reveals that C₄ -symmetry-breaking anisotropy is introduced by strain via pseudospin-lattice coupling, directly demonstrating the pseudo-Jahn-Teller effect of spin-orbit-coupled complex oxides. The extracted coupling strength is much weaker than the superexchange interactions, yet crucial for the spontaneous symmetry-breaking, affording the remarkably efficient strain-control.

Multiorbital charge-density wave excitations and concomitant phonon anomalies in Bi2Sr2LaCuO6+δ

Charge-density waves (CDWs) are a ubiquitous form of electron density modulation in cuprate superconductors. Unveiling the nature of quasistatic CDWs and their dynamical excitations is crucial for understanding their origin––similar to the study of antiferromagnetism in cuprates. However, dynamical CDW excitations remain largely unexplored due to the limited availability of suitable experimental probes. Here, using resonant inelastic X-ray scattering, we observe dynamical CDW excitations in Bi₂Sr₂LaCuO_6+δ (Bi2201) superconductors through its interference with the lattice. The distinct anomalies of the bond-buckling and the bond-stretching phonons allow us to draw a clear picture of funnel-shaped dynamical CDW excitations in Bi2201. Our results of the interplay between CDWs and the phonon anomalies shed light on the nature of CDWs in cuprates.

Charge-density waves (CDWs) are ubiquitous in underdoped cuprate superconductors. As a modulation of the valence electron density, CDWs in hole-doped cuprates possess both Cu-3d and O-2p orbital character owing to the strong hybridization of these orbitals near the Fermi level. Here, we investigate underdoped Bi₂Sr_1.4La_0.6CuO_6+δ using resonant inelastic X-ray scattering (RIXS) and find that a short-range CDW exists at both Cu and O sublattices in the copper-oxide (CuO₂) planes with a comparable periodicity and correlation length. Furthermore, we uncover bond-stretching and bond-buckling phonon anomalies concomitant to the CDWs. Comparing to slightly overdoped Bi₂Sr_1.8La_0.2CuO_6+δ, where neither CDWs nor phonon anomalies appear, we highlight that a sharp intensity anomaly is induced in the proximity of the CDW wavevector (Q_CDW) for the bond-buckling phonon, in concert with the diffused intensity enhancement of the bond-stretching phonon at wavevectors much greater than Q_CDW. Our results provide a comprehensive picture of the quasistatic CDWs, their dispersive excitations, and associated electron-phonon anomalies, which are key for understanding the competing electronic instabilities in cuprates.

Charge ordering in superconducting copper oxides

Charge order has recently been identified as a leading competitor of high-temperature superconductivity in moderately doped cuprates. We provide a survey of universal and materials-specific aspects of this phenomenon, with emphasis on results obtained by scattering methods. In particular, we discuss the structure, periodicity, and stability range of the charge-ordered state, its response to various external perturbations, the influence of disorder, the coexistence and competition with superconductivity, as well as collective charge dynamics. In the context of this journal issue which honors Roger Cowley's legacy, we also discuss the connection of charge ordering with lattice vibrations and the central-peak phenomenon. We end the review with an outlook on research opportunities offered by new synthesis methods and experimental platforms, including cuprate thin films and superlattices.

Z3-vestigial nematic order due to superconducting fluctuations in the doped topological insulators NbxBi2Se3 and CuxBi2Se3

A state of matter with a multi-component order parameter can give rise to vestigial order. In the vestigial phase, the primary order is only partially melted, leaving a remaining symmetry breaking behind, an effect driven by strong classical or quantum fluctuations. Vestigial states due to primary spin and charge-density-wave order have been discussed in iron-based and cuprate materials. Here we present the observation of a partially melted superconductivity in which pairing fluctuations condense at a separate phase transition and form a nematic state with broken Z3, i.e., three-state Potts-model symmetry. Thermal expansion, specific heat and magnetization measurements of the doped topological insulators NbxBi2Se3 and CuxBi2Se3 reveal that this symmetry breaking occurs at [Formula: see text] above [Formula: see text], along with an onset of superconducting fluctuations. Thus, before Cooper pairs establish long-range coherence at Tc, they fluctuate in a way that breaks the rotational invariance at Tnem and induces a crystalline distortion.

Time-reversal asymmetry without local moments via directional scalar spin chirality

Invariably, time-reversal symmetry (TRS) violation in a state of matter is identified with static magnetism in it. Here, a directional scalar spin chiral order (DSSCO) phase is introduced that disobeys this basic principle: it breaks TRS but has no density of static moments. It can be obtained by melting the spin moments in a magnetically ordered phase but retaining residual broken TRS. Orbital moments are then precluded by the spatial symmetries of the spin rotation symmetric state. It is allowed in one, two and three dimensions under different conditions of temperature and disorder. Recently, polar Kerr effect experiments in the mysterious pseudogap phase of the underdoped cuprates hinted at a strange form of broken TRS below a temperature T _K, that exhibits a hysteretic 'memory effect' above T _K and begs reconciliation with nuclear magnetic resonance (which sees no moments), x-ray diffraction (which finds charge ordering tendencies) and the Nernst effect (which detects nematicity). Remarkably, the DSSCO provides a phenomenological route for reconciling all these observations, and it is conceivable that it onsets at the pseudogap temperature ∼T*. A six-spin interaction mediated by enhanced fluctuations of velocity asymmetry between left- and right-movers above the onset of charge ordering in the cuprates is proposed as the driving force behind DSSCO formation. A testable prediction of the existence of the DSSCO in the cuprates is a Kerr signal above T _K triggered and trainable by a current driven along one of the in-plane axes, but not by a current along the other.

Competing stripe and magnetic phases in the cuprates from first principles

Realistic description of competing phases in complex quantum materials has proven extremely challenging. For example, much of the existing density-functional-theory-based first-principles framework fails in the cuprate superconductors. Various many-body approaches involve generic model Hamiltonians and do not account for the interplay between the spin, charge, and lattice degrees of freedom. Here, by deploying the recently constructed strongly constrained and appropriately normed (SCAN) density functional, we show how the landscape of competing stripe and magnetic phases can be addressed on a first-principles basis both in the parent insulator YBa2Cu3O6 and the near-optimally doped YBa2Cu3O7 as archetype cuprate compounds. In YBa2Cu3O7, we find many stripe phases that are nearly degenerate with the ground state and may give rise to the pseudogap state from which the high-temperature superconducting state emerges. We invoke no free parameters such as the Hubbard U, which has been the basis of much of the existing cuprate literature. Lattice degrees of freedom are found to be crucially important in stabilizing the various phases.

Charge ordering in high-temperature superconductors visualized by scanning tunneling microscopy

Since the discovery of stripe order in La_1.6-x Nd_0.4Sr _x CuO₄ superconductors in 1995, charge ordering in cuprate superconductors has been intensively studied by various experimental techniques. Among these studies, scanning tunneling microscope (STM) plays an irreplaceable role in determining the real space structures of charge ordering. STM imaging of different families of cuprates over a wide range of doping levels reveal similar checkerboard-like patterns, indicating that such a charge ordered state is likely a ubiquitous and intrinsic characteristic of cuprate superconductors, which may shed light on understanding the mechanism of high-temperature superconductivity. In another class of high-temperature superconductors, iron-based superconductors, STM studies reveal several charge ordered states as well, but their real-space patterns and the interplay with superconductivity are markedly different among different materials. In this paper, we present a brief review on STM studies of charge ordering in these two classes of high-temperature superconductors. Possible origins of charge ordering and its interplay with superconductivity will be discussed.

Depth-Resolved Modulation of Metal-Oxygen Hybridization and Orbital Polarization across Correlated Oxide Interfaces

Interface-induced modifications of the electronic, magnetic, and lattice degrees of freedom drive an array of novel physical properties in oxide heterostructures. Here, large changes in metal-oxygen band hybridization, as measured in the oxygen ligand hole density, are induced as a result of interfacing two isovalent correlated oxides. Using resonant X-ray reflectivity, a superlattice of SrFeO₃ and CaFeO₃ is shown to exhibit an electronic character that spatially evolves from strongly O-like in SrFeO₃ to strongly Fe-like in CaFeO₃ . This alternating degree of Fe electronic character is correlated with a modulation of an Fe 3d orbital polarization, giving rise to an orbital superstructure. At the SrFeO₃ /CaFeO₃ interfaces, the ligand hole density and orbital polarization reconstruct in a single unit cell of CaFeO₃ , demonstrating how the mismatch in these electronic parameters is accommodated at the interface. These results provide new insight into how the orbital character of electrons is altered by correlated oxide interfaces and lays out a broadly applicable approach for depth-resolving band hybridization.

Strong long-range electron-phonon interaction as possible driving force for charge ordering in cuprates

A model resulting in charge ordering (CO) similar to that observed in cuprate superconductors is under study. It includes strong long-range electron-phonon interaction (EPI) and high density of correlated carriers. Coexistence of large bipolarons and delocalized carriers is a feature of such system. We develop generalized variation method to calculate the bipolaron size (CO period) in the ground normal state of such system at various doping. The approach allows the revealing of a possible physical reason of strongly different doping behavior of the CO wave vector in different cuprates. Obtained doping dependences of the CO period and temperature of the CO decay demonstrate quantitative agreement with those observed in cuprates. Predicted in the suggested approach ratio of the CO wave vector to the wave vector of the high-energy anomaly (HEA) in ARPES spectrum is in consent with that in cuprates. Calculated resonant x-rays scattering on the CO emerging in the model is in good agreement with experiments on cuprates including the asymmetry of the CO peaks' cross-section. A gap arises in the spectrum of delocalized carriers near antinodal direction due to their scattering on the periodic potential created by autolocalized carriers, analogously to photon crystal effect.

Evidence for a vestigial nematic state in the cuprate pseudogap phase

The CuO₂ antiferromagnetic insulator is transformed by hole-doping into an exotic quantum fluid usually referred to as the pseudogap (PG) phase. Its defining characteristic is a strong suppression of the electronic density-of-states D(E) for energies |E| < [Formula: see text], where [Formula: see text] is the PG energy. Unanticipated broken-symmetry phases have been detected by a wide variety of techniques in the PG regime, most significantly a finite-Q density-wave (DW) state and a Q = 0 nematic (NE) state. Sublattice-phase-resolved imaging of electronic structure allows the doping and energy dependence of these distinct broken-symmetry states to be visualized simultaneously. Using this approach, we show that even though their reported ordering temperatures T _DW and T _NE are unrelated to each other, both the DW and NE states always exhibit their maximum spectral intensity at the same energy, and using independent measurements that this is the PG energy [Formula: see text] Moreover, no new energy-gap opening coincides with the appearance of the DW state (which should theoretically open an energy gap on the Fermi surface), while the observed PG opening coincides with the appearance of the NE state (which should theoretically be incapable of opening a Fermi-surface gap). We demonstrate how this perplexing phenomenology of thermal transitions and energy-gap opening at the breaking of two highly distinct symmetries may be understood as the natural consequence of a vestigial nematic state within the pseudogap phase of Bi₂Sr₂CaCu₂O₈.

Charge-Lattice Coupling in Hole-Doped LuFe_{2}O_{4+δ}: The Origin of Second-Order Modulation

Understanding singularities in ordered structures, such as dislocations in lattice modulation and solitons in charge ordering, offers great opportunities to disentangle the interactions between the electronic degrees of freedom and the lattice. Specifically, a modulated structure has traditionally been expressed in the form of a discrete Fourier series with a constant phase and amplitude for each component. Here, we report atomic scale observation and analysis of a new modulation wave in hole-doped LuFe_{2}O_{4+δ} that requires significant modifications to the conventional modeling of ordered structures. This new modulation with an unusual quasiperiodic singularity can be accurately described only by introducing a well-defined secondary modulation vector in both the phase and amplitude parameter spaces. Correlated with density-functional-theory (DFT) calculations, our results reveal that those singularities originate from the discontinuity of lattice displacement induced by interstitial oxygen in the system. The approach of our work is applicable to a wide range of ordered systems, advancing our understanding of the nature of singularity and modulation.

Evolution of Pairing Orders between Pseudogap and Superconducting Phases of Cuprate Superconductors

One of the most puzzling problems of high temperature cuprate superconductor is the pseudogap phase (PG) at temperatures above the superconducting transition temperature in the underdoped regime. The PG phase is found by the angle-resolved photoemission spectra (ARPES) to have a gap at some regions in momentum space and a fraction of Fermi surface remained, known as Fermi arcs. The arc turns into a d-wave SC gap with a node below the SC transition temperature. Here, by studying a strongly correlated model at low temperatures, we obtained a phase characterized by two kinds of pairing order parameters with the total momentum of the Cooper pair to be zero and finite. The finite momentum pairing is accompanied with a spatial modulation of pairing order, i.e. a pair density wave (PDW). These PDW phases are intertwined with modulations of charge density and intra-unit cell form factors. The coexistence of the two different pairing orders provides the unique two-gaps like spectra observed by ARPES for superconducting cuprates. As temperature raises, the zero-momentum pairing order vanishes while the finite momentum pairing orders are kept, thus Fermi arcs are realized. The calculated quasiparticle spectra have the similar doping and temperature dependence as reported by ARPES and scanning tunneling spectroscopy (STS). The consequence of breaking symmetry between x and y due to the unidirectional PDW and the possibility to probe such a PDW state in the PG phase is discussed.

Unusual behavior of cuprates explained by heterogeneous charge localization

The discovery of high-temperature superconductivity in cuprates ranks among the major scientific milestones of the past half century, yet pivotal questions regarding the complex phase diagram of these materials remain unanswered. Generally thought of as doped charge-transfer insulators, these complex oxides exhibit pseudogap, strange-metal, superconducting, and Fermi liquid behavior with increasing hole-dopant concentration. Motivated by recent experimental observations, here we introduce a phenomenological model wherein exactly one hole per planar copper-oxygen unit is delocalized with increasing doping and temperature. The model is percolative in nature, with parameters that are highly consistent with experiments. It comprehensively captures key unconventional experimental results, including the temperature and the doping dependence of the pseudogap phenomenon, the strange-metal linear temperature dependence of the planar resistivity, and the doping dependence of the superfluid density. The success and simplicity of the model greatly demystify the cuprate phase diagram and point to a local superconducting pairing mechanism.

Re-entrant charge order in overdoped (Bi,Pb)2.12Sr1.88CuO6+δ outside the pseudogap regime

In the underdoped regime, the cuprate high-temperature superconductors exhibit a host of unusual collective phenomena, including unconventional spin and charge density modulations, Fermi surface reconstructions, and a pseudogap in various physical observables. Conversely, overdoped cuprates are generally regarded as conventional Fermi liquids possessing no collective electronic order. In partial contradiction to this widely held picture, we report resonant X-ray scattering measurements revealing incommensurate charge order reflections for overdoped (Bi,Pb)_2.12Sr_1.88CuO_6+δ (Bi2201), with correlation lengths of 40-60 lattice units, that persist up to temperatures of at least 250 K. The value of the charge order wavevector decreases with doping, in line with the extrapolation of the trend previously observed in underdoped Bi2201. In overdoped materials, however, charge order coexists with a single, unreconstructed Fermi surface without nesting or pseudogap features. The discovery of re-entrant charge order in Bi2201 thus calls for investigations in other cuprate families and for a reconsideration of theories that posit an essential relationship between these phenomena.

Pseudo-spin skyrmions in the phase diagram of cuprate superconductors

Topological states of matter are at the root of some of the most fascinating phenomena in condensed matter physics. Here we argue that skyrmions in the pseudo-spin space related to an emerging SU(2) symmetry enlighten many mysterious properties of the pseudogap phase in under-doped cuprates. We detail the role of the SU(2) symmetry in controlling the phase diagram of the cuprates, in particular how a cascade of phase transitions explains the arising of the pseudogap, superconducting and charge modulation phases seen at low temperature. We specify the structure of the charge modulations inside the vortex core below T _c, as well as in a wide temperature region above T _c, which is a signature of the skyrmion topological structure. We argue that the underlying SU(2) symmetry is the main structure controlling the emergent complexity of excitations at the pseudogap scale T ^*. The theory yields a gapping of a large part of the anti-nodal region of the Brillouin zone, along with q  =  0 phase transitions, of both nematic and loop currents characters.

Photoemission perspective on pseudogap, superconducting fluctuations, and charge order in cuprates: a review of recent progress

In the course of seeking the microscopic mechanism of superconductivity in cuprate high temperature superconductors, the pseudogap phase- the very abnormal 'normal' state on the hole-doped side- has proven to be as big of a quandary as superconductivity itself. Angle-resolved photoemission spectroscopy (ARPES) is a powerful tool for assessing the momentum-dependent phenomenology of the pseudogap, and recent technological developments have permitted a more detailed understanding. This report reviews recent progress in understanding the relationship between superconductivity and the pseudogap, the Fermi arc phenomena, and the relationship between charge order and pseudogap from the perspective of ARPES measurements.

Quasiparticle conductance-voltage characteristics for break junctions involving d-wave superconductors: charge-density-wave effects

Quasiparticle tunnel conductance-voltage characteristics (CVCs), [Formula: see text], were calculated for break junctions (BJs) made up of layered d-wave superconductors partially gapped by charge-density waves (CDWs). The current is assumed to flow in the ab-plane of electrodes. The influence of CDWs is analyzed by comparing the resulting CVCs with CVCs calculated for BJs made up of pure d-wave superconductors with relevant parameters. The main CDW-effects were found to be the appearance of new CVC peculiarities and the loss of CVC symmetry with respect to the V-sign. Tunnel directionality was shown to be one of the key factors in the formation of [Formula: see text] dependences. In particular, the orientation of electrodes with respect to the current channel becomes very important. As a result, [Formula: see text] can acquire a large variety of forms similar to those for tunnel junctions between superconductors with s-wave, d-wave, and mixed symmetry of their order parameters. The diversity of peculiarities is especially striking at finite temperatures. In the case of BJs made up of pure d-wave superconductors, the resulting CVC can include a two-peak gap-driven structure. The results were compared with the experimental BJ data for a number of high-T _c oxides. It was shown that the large variety of the observed current-voltage characteristics can be interpreted in the framework of our approach. Thus, quasiparticle tunnel currents in the ab-plane can be used as an additional mean to detect CDWs competing with superconductivity in cuprates or other layered superconductors.

Charge-density-wave order takes over antiferromagnetism in Bi2Sr2-x La x CuO6 superconductors

Superconductivity appears in the cuprates when a spin order is destroyed, while the role of charge is less known. Recently, charge density wave (CDW) was found below the superconducting dome in YBa₂Cu₃O_y when a high magnetic field is applied perpendicular to the CuO₂ plane, which was suggested to arise from incipient CDW in the vortex cores that becomes overlapped. Here by ⁶³Cu-nuclear magnetic resonance, we report the discovery of CDW induced by an in-plane field, setting in above the dome in single-layered Bi₂Sr_2−xLa_xCuO₆. The onset temperature T_CDW takes over the antiferromagnetic order temperature T_N beyond a critical doping level at which superconductivity starts to emerge, and scales with the pseudogap temperature T*. These results provide important insights into the relationship between spin order, CDW and the pseudogap, and their connections to high-temperature superconductivity.

Whilst superconductivity usually appears when magnetic order is suppressed, the role of charge is less known. Here, Kawasaki et al. report a charge density wave (CDW) above the superconducting transition induced by an in-plane magnetic field in Bi₂Sr_2-xLa_xCuO₆, with the CDW onset temperature scaling with the pseudogap temperature.

Dimensional Crossover of Charge-Density Wave Correlations in the Cuprates

Short-range charge-density wave correlations are ubiquitous in underdoped cuprates. They are largely confined to the copper-oxygen planes and typically oscillate out of phase from one unit cell to the next in the c direction. Recently, it was found that a considerably longer-range charge-density wave order develops in YBa_{2}Cu_{3}O_{6+x} above a sharply defined crossover magnetic field. This order is more three-dimensional and is in-phase along the c axis. Here, we show that such behavior is a consequence of the conflicting ordering tendencies induced by the disorder potential and the Coulomb interaction, where the magnetic field acts to tip the scales from the former to the latter. We base our conclusion on analytic large-N analysis and Monte Carlo simulations of a nonlinear sigma model of competing superconducting and charge-density wave orders. Our results are in agreement with the observed phenomenology in the cuprates, and we discuss their implications to other members of this family, which have not been measured yet at high magnetic fields.

Local particle-hole pair excitations by SU(2) symmetry fluctuations

Understanding the pseudo-gap phase which opens in the under-doped regime of cuprate superconductors is one of the most enduring challenges of the physics of these compounds. A depletion in the electronic density of states is observed, which is gapping out part of the Fermi surface, leading to the formation of mysterious lines of massless excitations- the Fermi arcs. Here we give a new theoretical account of the physics of the pseudo-gap phase in terms of the emergence of local patches of particle-hole pairs generated by SU(2) symmetry fluctuations. The proliferation of these local patches accounts naturally for the robustness of the pseudo-gap phase to disturbances like disorder or magnetic field and is shown to gap out part of the Fermi surface, leading to the formation of the Fermi arcs. Most noticeably, we show that these patches induce a modulated charge distribution on the Oxygen atoms, in remarkable agreement with recent X-ray and STM observations.

Quasiparticle Scattering off Defects and Possible Bound States in Charge-Ordered YBa_{2}Cu_{3}O_{y}

We report the NMR observation of a skewed distribution of ^{17}O Knight shifts when a magnetic field quenches superconductivity and induces long-range charge-density-wave (CDW) order in YBa_{2}Cu_{3}O_{y}. This distribution is explained by an inhomogeneous pattern of the local density of states N(E_{F}) arising from quasiparticle scattering off, yet unidentified, defects in the CDW state. We argue that the effect is most likely related to the formation of quasiparticle bound states, as is known to occur, under specific circumstances, in some metals and superconductors (but not in the CDW state, in general, except for very few cases in 1D materials). These observations should provide insight into the microscopic nature of the CDW, especially regarding the reconstructed band structure and the sensitivity to disorder.

Influence of the spatially inhomogeneous gap distribution on the quasiparticle current in c-axis junctions involving d-wave superconductors with charge density waves

The quasiparticle tunnel current J(V) between the superconducting ab-planes along the c-axis and the corresponding conductance [Formula: see text] were calculated for symmetric junctions composed of disordered d-wave layered superconductors partially gapped by charge density waves (CDWs). Here, V is the voltage. Both the checkerboard and unidirectional CDWs were considered. It was shown that the spatial spread of the CDW-pairing strength substantially smears the peculiarities of G(V) appropriate to uniform superconductors. The resulting curves G(V) become very similar to those observed for a number of cuprates in intrinsic junctions, e.g. mesas. In particular, the influence of CDWs may explain the peak-dip-hump structures frequently found for high-T c oxides.

Fermi Surface Reconstruction and Drop in the Hall Number due to Spiral Antiferromagnetism in High-T_{c} Cuprates

We show that a Fermi surface reconstruction due to spiral antiferromagnetic order may explain the rapid change in the Hall number as recently observed near optimal doping in cuprate superconductors [Badoux et al., Nature (London) 531, 210 (2016)]. The single-particle spectral function in the spiral state exhibits hole pockets which look like Fermi arcs due to a strong momentum dependence of the spectral weight. Adding charge-density wave order further reduces the Fermi surface to a single electron pocket. We propose quantum oscillation measurements to distinguish between commensurate and spiral antiferromagnetic order. Similar results apply to certain metals in which topological order replaces antiferromagnetic order.

Commensurate 4a0-period charge density modulations throughout the Bi2Sr2CaCu2O8+x pseudogap regime

Theories based upon strong real space (r-space) electron-electron interactions have long predicted that unidirectional charge density modulations (CDMs) with four-unit-cell (4a₀) periodicity should occur in the hole-doped cuprate Mott insulator (MI). Experimentally, however, increasing the hole density p is reported to cause the conventionally defined wavevector Q_A of the CDM to evolve continuously as if driven primarily by momentum-space (k-space) effects. Here we introduce phase-resolved electronic structure visualization for determination of the cuprate CDM wavevector. Remarkably, this technique reveals a virtually doping-independent locking of the local CDM wavevector at [Formula: see text] throughout the underdoped phase diagram of the canonical cuprate Bi₂Sr₂CaCu₂O₈ These observations have significant fundamental consequences because they are orthogonal to a k-space (Fermi-surface)-based picture of the cuprate CDMs but are consistent with strong-coupling r-space-based theories. Our findings imply that it is the latter that provides the intrinsic organizational principle for the cuprate CDM state.

The amplitudes and the structure of the charge density wave in YBCO

We find unknown s- and d-wave amplitudes of the recently discovered charge density wave (CDW) in underdoped cuprates. To do so we perform a combined analysis of experimental data for ortho-II YBa₂Cu₃O_y. The analysis includes data on nuclear magnetic resonance, resonant inelastic X-ray scattering, and hard X-ray diffraction. The amplitude of doping modulation found in our analysis is 3.5 · 10⁻³ in a low magnetic field and T = 60 K, the amplitude is 6.5 · 10⁻³ in a magnetic field of 30T and T = 1.3 K. The values are in units of elementary charge per unit cell of a CuO₂ plane. We show that the data rule out a checkerboard pattern, and we also show that the data might rule out mechanisms of the CDW which do not include phonons.

Quasi-particle interference of heavy fermions in resonant x-ray scattering

Resonant x-ray scattering (RXS) has recently become an increasingly important tool for the study of ordering phenomena in correlated electron systems. Yet, the interpretation of RXS experiments remains theoretically challenging because of the complexity of the RXS cross section. Central to this debate is the recent proposal that impurity-induced Friedel oscillations, akin to quasi-particle interference signals observed with a scanning tunneling microscope (STM), can lead to scattering peaks in RXS experiments. The possibility that quasi-particle properties can be probed in RXS measurements opens up a new avenue to study the bulk band structure of materials with the orbital and element selectivity provided by RXS. We test these ideas by combining RXS and STM measurements of the heavy fermion compound CeMIn₅ (M = Co, Rh). Temperature- and doping-dependent RXS measurements at the Ce-M₄ edge show a broad scattering enhancement that correlates with the appearance of heavy f-electron bands in these compounds. The scattering enhancement is consistent with the measured quasi-particle interference signal in the STM measurements, indicating that the quasi-particle interference can be probed through the momentum distribution of RXS signals. Overall, our experiments demonstrate new opportunities for studies of correlated electronic systems using the RXS technique.

Superconductor to Mott insulator transition in YBa2Cu3O7/LaCaMnO3 heterostructures

The superconductor-to-insulator transition (SIT) induced by means such as external magnetic fields, disorder or spatial confinement is a vivid illustration of a quantum phase transition dramatically affecting the superconducting order parameter. In pursuit of a new realization of the SIT by interfacial charge transfer, we developed extremely thin superlattices composed of high Tc superconductor YBa2Cu3O7 (YBCO) and colossal magnetoresistance ferromagnet La0.67Ca0.33MnO3 (LCMO). By using linearly polarized resonant X-ray absorption spectroscopy and magnetic circular dichroism, combined with hard X-ray photoelectron spectroscopy, we derived a complete picture of the interfacial carrier doping in cuprate and manganite atomic layers, leading to the transition from superconducting to an unusual Mott insulating state emerging with the increase of LCMO layer thickness. In addition, contrary to the common perception that only transition metal ions may respond to the charge transfer process, we found that charge is also actively compensated by rare-earth and alkaline-earth metal ions of the interface. Such deterministic control of Tc by pure electronic doping without any hindering effects of chemical substitution is another promising route to disentangle the role of disorder on the pseudo-gap and charge density wave phases of underdoped cuprates.

Dynamical Effects in Resonant X-Ray Diffraction

Using resonant magnetic diffraction at the Ni L_{2,3} edge in a LaNiO_{3} superlattice, we show that dynamical effects beyond the standard kinematic approximation can drastically modify the resonant scattering cross section. In particular, the combination of extinction and refraction convert maxima to minima in the azimuthal-angle dependence of the diffracted intensity, which is commonly used to determine orbital and magnetic structures by resonant x-ray diffraction. We provide a comprehensive theoretical description of these effects by numerically solving Maxwell's equations in three dimensions. The understanding and description of dynamical diffraction enhances the capabilities of resonant x-ray scattering as a probe of electronic ordering phenomena in solids.

Emergent gauge fields and the high-temperature superconductors

The quantum entanglement of many states of matter can be represented by electric and magnetic fields, much like those found in Maxwell's theory. These fields 'emerge' from the quantum structure of the many-electron state, rather than being fundamental degrees of freedom of the vacuum. I review basic aspects of the theory of emergent gauge fields in insulators in an intuitive manner. In metals, Fermi liquid (FL) theory relies on adiabatic continuity from the free electron state, and its central consequence is the existence of long-lived electron-like quasi-particles around a Fermi surface enclosing a volume determined by the total density of electrons, via the Luttinger theorem. However, long-range entanglement and emergent gauge fields can also be present in metals. I focus on the 'fractionalized Fermi liquid' (FL*) state, which also has long-lived electron-like quasi-particles around a Fermi surface; however, the Luttinger theorem on the Fermi volume is violated, and this requires the presence of emergent gauge fields, and the associated loss of adiabatic continuity with the free electron state. Finally, I present a brief survey of some recent experiments in the hole-doped cuprate superconductors, and interpret the properties of the pseudogap regime in the framework of the FL* theory.This article is part of the themed issue 'Unifying physics and technology in light of Maxwell's equations'.

Orbital symmetry of charge-density-wave order in La1.875Ba0.125CuO4 and YBa2Cu3O6.67

Recent theories of charge-density-wave (CDW) order in high-temperature superconductors have predicted a primarily d CDW orbital symmetry. Here, we report on the orbital symmetry of CDW order in the canonical cuprate superconductors La1.875Ba0.125CuO4 (LBCO) and YBa2Cu3O6.67 (YBCO), using resonant soft X-ray scattering and a model mapped to the CDW orbital symmetry. From measurements sensitive to the O sublattice, we conclude that LBCO has predominantly s' CDW orbital symmetry, in contrast to the d orbital symmetry recently reported in other cuprates. Furthermore, we show for YBCO that the CDW orbital symmetry differs along the a and b crystal axes and that these both differ from LBCO. This work highlights CDW orbital symmetry as an additional key property that distinguishes the different cuprate families. We discuss how the CDW symmetry may be related to the '1/8-anomaly' and to static spin ordering.

Possibility of charge density wave transition in a SrPt2Sb2 superconductor

The first-order transition at T(0) =  270 K for the platinum-based SrPt2Sb2 superconductor was investigated using x-ray diffraction and magnetic susceptibility measurements. When polycrystalline SrPt2Sb2 was cooled down through T(0), the structure was transformed from monoclinic to a modulated orthorhombic structure, and no magnetic order was formed, which illustrates the possibility of a charge density wave (CDW) transition at T(0). SrPt2Sb2 can thus be a new example to examine the interplay of CDW and superconductivity in addition to SrPt2As2, BaPt2As2, and LaPt2Si2. It is unique that the average structure of the low-temperature phase has higher symmetry than that of the high-temperature phase.

Detection of a Cooper-pair density wave in Bi2Sr2CaCu2O8+x

The quantum condensate of Cooper pairs forming a superconductor was originally conceived as being translationally invariant. In theory, however, pairs can exist with finite momentum Q, thus generating a state with a spatially modulated Cooper-pair density. Such a state has been created in ultracold (6)Li gas but never observed directly in any superconductor. It is now widely hypothesized that the pseudogap phase of the copper oxide superconductors contains such a 'pair density wave' state. Here we report the use of nanometre-resolution scanned Josephson tunnelling microscopy to image Cooper pair tunnelling from a d-wave superconducting microscope tip to the condensate of the superconductor Bi2Sr2CaCu2O8+x. We demonstrate condensate visualization capabilities directly by using the Cooper-pair density variations surrounding zinc impurity atoms and at the Bi2Sr2CaCu2O8+x crystal supermodulation. Then, by using Fourier analysis of scanned Josephson tunnelling images, we discover the direct signature of a Cooper-pair density modulation at wavevectors QP ≈ (0.25, 0)2π/a0 and (0, 0.25)2π/a0 in Bi2Sr2CaCu2O8+x. The amplitude of these modulations is about five per cent of the background condensate density and their form factor exhibits primarily s or s' symmetry. This phenomenology is consistent with Ginzburg-Landau theory when a charge density wave with d-symmetry form factor and wavevector QC = QP coexists with a d-symmetry superconductor; it is also predicted by several contemporary microscopic theories for the pseudogap phase.

Glue function of optimally and overdoped cuprates from inversion of the Raman spectra

We address the issue of identifying the mediators of effective interactions in cuprates superconductors. Specifically, we use inversion theory to analyze Raman spectra of optimally and over-doped La2-x Sr x CuO4 samples. This allows us to extract the so-called glue function without making any a priori assumption based on any specific model. We use instead two different techniques, namely the singular value decomposition and a multi-rectangle decomposition. With both techniques we find consistent results showing that: (i) two distinct excitations are responsible for the glue function, which have completely different doping dependence. One excitation becomes weak above optimal doping, where on the contrary the other keeps (or even slightly increases) its strength; (ii) there is a marked temperature dependence on the weight and spectral distribution of these excitations, which therefore must have a somewhat critical character. It is quite natural to identify and characterize these two distinct excitations as damped antiferromagnetic spin waves and damped charge density waves, respectively. This sets the stage for a scenario in which superconductivity is concomitant and competing with a charge ordering instability.

Uncertainty principle for experimental measurements: Fast versus slow probes

The result of a physical measurement depends on the time scale of the experimental probe. In solid-state systems, this simple quantum mechanical principle has far-reaching consequences: the interplay of several degrees of freedom close to charge, spin or orbital instabilities combined with the disparity of the time scales associated to their fluctuations can lead to seemingly contradictory experimental findings. A particularly striking example is provided by systems of adatoms adsorbed on semiconductor surfaces where different experiments--angle-resolved photoemission, scanning tunneling microscopy and core-level spectroscopy--suggest different ordering phenomena. Using most recent first principles many-body techniques, we resolve this puzzle by invoking the time scales of fluctuations when approaching the different instabilities. These findings suggest a re-interpretation of ordering phenomena and their fluctuations in a wide class of solid-state systems ranging from organic materials to high-temperature superconducting cuprates.

Short range smectic order driving long range nematic order: example of cuprates

We present a model for describing the combined presence of nematic and 'smectic' or stripe-like orders seen in recent scanning tunneling microscopy (STM) experiments on cuprates. The smectic order is treated as an electronic charge density wave with an associated Peierls distortion or a 'Pomeranchuk wave'. This primary order is restricted to nanoscale domains by disorder effects, while the secondary coupling to strain generates the nematic order with a considerably longer range. A variety of experimental results are shown to be consistent with our theoretical predictions.

Genesis of charge orders in high temperature superconductors

One of the most puzzling facts about cuprate high-temperature superconductors in the lightly doped regime is the coexistence of uniform superconductivity and/or antiferromagnetism with many low-energy charge-ordered states in a unidirectional charge density wave or a bidirectional checkerboard structure. Recent experiments have discovered that these charge density waves exhibit different symmetries in their intra-unit-cell form factors for different cuprate families. Using a renormalized mean-field theory for a well-known, strongly correlated model of cuprates, we obtain a number of charge-ordered states with nearly degenerate energies without invoking special features of the Fermi surface. All of these self-consistent solutions have a pair density wave intertwined with a charge density wave and sometimes a spin density wave. Most of these states vanish in the underdoped regime, except for one with a large d-form factor that vanishes at approximately 19% doping of the holes, as reported by experiments. Furthermore, these states could be modified to have a global superconducting order, with a nodal-like density of states at low energy.

The microscopic structure of charge density waves in underdoped YBa2Cu3O6.54 revealed by X-ray diffraction

Charge density wave (CDW) order appears throughout the underdoped high-temperature cuprate superconductors, but the underlying symmetry breaking and the origin of the CDW remain unclear. We use X-ray diffraction to determine the microscopic structure of the CDWs in an archetypical cuprate YBa₂Cu₃O_6.54 at its superconducting transition temperature ∼60 K. We find that the CDWs in this material break the mirror symmetry of the CuO₂ bilayers. The ionic displacements in the CDWs have two components, which are perpendicular and parallel to the CuO₂ planes, and are out of phase with each other. The planar oxygen atoms have the largest displacements, perpendicular to the CuO₂ planes. Our results allow many electronic properties of the underdoped cuprates to be understood. For instance, the CDWs will lead to local variations in the electronic structure, giving an explicit explanation of density-wave states with broken symmetry observed in scanning tunnelling microscopy and soft X-ray measurements.

Near to the superconducting state, cuprates display spatially-periodic charge density variations. Here, the authors use x-ray diffraction to determine the microscopic structure, showing how charge density waves in underdoped YBa₂Cu₃O_6.54 break the symmetry of the superconducting layers.

High pressure effects revisited for the cuprate superconductor family with highest critical temperature

How to enhance the superconducting critical temperature (T_c) has been a primary issue since the discovery of superconductivity. The highest T_c reported so far is 166 K in HgBa₂Ca₂Cu₃O_8+δ (Hg1223) at high pressure of 23 GPa, as determined with the reduction onset, but not zero, of resistivity. To clarify the possible condition of the real maximum T_c, it is worth revisiting the effects of pressure on T_c in the highest T_c family. Here we report a systematic study of the pressure dependence of T_c in HgBa₂CaCu₂O_6+δ (Hg1212) and Hg1223 with the doping level from underdoped to overdoped. The T_c with zero resistivity is probed with a cubic-anvil-type apparatus that can produce hydrostatic pressures. Variation, not only increase but also decrease, of T_c in Hg1212 and Hg1223 with pressure strongly depends on the initial doping levels. In particular, we confirm a maximum T_c of 153 K at 22 GPa in slightly underdoped Hg1223.

Superconductivity, zero-resistance electrical transport, exists up to temperatures above 150 K in mercury-based cuprates under pressure. Here, the authors revisit the pressure dependence of the superconducting transition temperature in these materials and find a strong dependence on initial doping.

Andreev-Bragg Reflection from an Amperian Superconductor

We show how an electrical measurement can detect the pairing of electrons on the same side of the Fermi surface (Amperian pairing), recently proposed by Patrick Lee for the pseudogap phase of high-Tc cuprate superconductors. Bragg scattering from the pair-density wave introduces odd multiples of 2k(F) momentum shifts when an electron incident from a normal metal is Andreev reflected as a hole. These Andreev-Bragg reflections can be detected in a three-terminal device, containing a ballistic Y junction between normal leads (1, 2) and the superconductor. The cross-conductance dI1/dV2 has the opposite sign for Amperian pairing than it has either in the normal state or for the usual BCS pairing.