Electronically Seamless Domain Wall of Chiral Charge Density Wave in 1T-TiSe2

Domain walls (DWs) have been recognized to play crucial roles in various interesting properties and functionalities in quantum materials. A notable example is DWs in charge density wave (CDW), which are believed to provide metallic electron channels for emerging superconductivity. However, electronic states of DWs and the microscopic mechanism toward superconductivity have been elusive. Here, we clarify the atomic/electronic structure of DWs of the chiral CDW emerging in 1T-TiSe2, using scanning tunneling microscopy and density functional calculations. We reveal unambiguously the microscopic origin of chiral CDW as the C2 distortion in Se layers and its interlayer coupling. We further identify unique DWs connecting CDW domains of opposite chirality. The DWs are endowed with no in-gap state due to the characteristic multibands around the band gap, which defies the widely believed notion for CDW DWs. These results provide an important insight into the role of DWs in emerging superconductivity.

Orbital-selective time-domain signature of nematicity dynamics in the charge-density-wave phase of La1.65Eu0.2Sr0.15CuO4

Understanding the interplay between charge, nematic, and structural ordering tendencies in cuprate superconductors is critical to unraveling their complex phase diagram. Using pump-probe time-resolved resonant X-ray scattering on the (0 0 1) Bragg peak at the Cu [Formula: see text] and O [Formula: see text] resonances, we investigate nonequilibrium dynamics of [Formula: see text] nematic order and its association with both charge density wave (CDW) order and lattice dynamics in La[Formula: see text]Eu[Formula: see text]Sr[Formula: see text]CuO[Formula: see text]. The orbital selectivity of the resonant X-ray scattering cross-section allows nematicity dynamics associated with the planar O 2[Formula: see text] and Cu 3[Formula: see text] states to be distinguished from the response of anisotropic lattice distortions. A direct time-domain comparison of CDW translational-symmetry breaking and nematic rotational-symmetry breaking reveals that these broken symmetries remain closely linked in the photoexcited state, consistent with the stability of CDW topological defects in the investigated pump fluence regime.

Discovery of orbital ordering in Bi2Sr2CaCu2O8+x

The primordial ingredient of cuprate superconductivity is the CuO2 unit cell. Theories usually concentrate on the intra-atom Coulombic interactions dominating the 3d9 and 3d10 configurations of each copper ion. However, if Coulombic interactions also occur between electrons of the 2p6 orbitals of each planar oxygen atom, spontaneous orbital ordering may split their energy levels. This long-predicted intra-unit-cell symmetry breaking should generate an orbitally ordered phase, for which the charge transfer energy ε separating the 2p6 and 3d10 orbitals is distinct for the two oxygen atoms. Here we introduce sublattice-resolved ε(r) imaging to CuO2 studies and discover intra-unit-cell rotational symmetry breaking of ε(r). Spatially, this state is arranged in disordered Ising domains of orthogonally oriented orbital order bounded by dopant ions, and within whose domain walls low-energy electronic quadrupolar two-level systems occur. Overall, these data reveal a Q = 0 orbitally ordered state that splits the oxygen energy levels by ~50 meV, in underdoped CuO2.

Mechanical manipulation for ordered topological defects

Randomly distributed topological defects created during the spontaneous symmetry breaking are the fingerprints to trace the evolution of symmetry, range of interaction, and order parameters in condensed matter systems. However, the effective mean to manipulate topological defects into ordered form is elusive due to the topological protection. Here, we establish a strategy to effectively align the topological domain networks in hexagonal manganites through a mechanical approach. It is found that the nanoindentation strain gives rise to a threefold Magnus-type force distribution, leading to a sixfold symmetric domain pattern by driving the vortex and antivortex in opposite directions. On the basis of this rationale, sizeable mono-chirality topological stripe is readily achieved by expanding the nanoindentation to scratch, directly transferring the randomly distributed topological defects into an ordered form. This discovery provides a mechanical strategy to manipulate topological protected domains not only on ferroelectrics but also on ferromagnets/antiferromagnets and ferroelastics.

Topological soliton molecule in quasi 1D charge density wave

Soliton molecules, bound states of two solitons, can be important for the informatics using solitons and the quest for exotic particles in a wide range of physical systems from unconventional superconductors to nuclear matter and Higgs field, but have been observed only in temporal dimension for classical wave optical systems. Here, we identify a topological soliton molecule formed spatially in an electronic system, a quasi 1D charge density wave of indium atomic wires. This system is composed of two coupled Peierls chains, which are endowed with a Z4 topology and three distinct, right-chiral, left-chiral, and non-chiral, solitons. Our scanning tunneling microscopy measurements identify a bound state of right- and left-chiral solitons with distinct in-gap states and net zero phase shift. Our density functional theory calculations reveal the attractive interaction of these solitons and the hybridization of their electronic states. This result initiates the study of the interaction between solitons in electronic systems, which can provide novel manybody electronic states and extra data-handling capacity beyond the given soliton topology.

Identification of a nematic pair density wave state in Bi2Sr2CaCu2O8+x

Electron-pair density wave (PDW) states are now an intense focus of research in the field of cuprate correlated superconductivity. PDWs exhibit periodically modulating superconductive electron pairing that can be visualized directly using scanned Josephson tunneling microscopy (SJTM). Although from theory, intertwining the d-wave superconducting (DSC) and PDW order parameters allows a plethora of global electron-pair orders to appear, which one actually occurs in the various cuprates is unknown. Here, we use SJTM to visualize the interplay of PDW and DSC states in Bi2Sr2CaCu2O8+x at a carrier density where the charge density wave modulations are virtually nonexistent. Simultaneous visualization of their amplitudes reveals that the intertwined PDW and DSC are mutually attractive states. Then, by separately imaging the electron-pair density modulations of the two orthogonal PDWs, we discover a robust nematic PDW state. Its spatial arrangement entails Ising domains of opposite nematicity, each consisting primarily of unidirectional and lattice commensurate electron-pair density modulations. Further, we demonstrate by direct imaging that the scattering resonances identifying Zn impurity atom sites occur predominantly within boundaries between these domains. This implies that the nematic PDW state is pinned by Zn atoms, as was recently proposed [Lozano et al., Phys. Rev. B 103, L020502 (2021)]. Taken in combination, these data indicate that the PDW in Bi2Sr2CaCu2O8+x is a vestigial nematic pair density wave state [Agterberg et al. Phys. Rev. B 91, 054502 (2015); Wardh and Granath arXiv:2203.08250].

Enhanced charge density wave coherence in a light-quenched, high-temperature superconductor

Superconductivity and charge density waves (CDWs) are competitive, yet coexisting, orders in cuprate superconductors. To understand their microscopic interdependence, a probe capable of discerning their interaction on its natural length and time scale is necessary. We use ultrafast resonant soft x-ray scattering to track the transient evolution of CDW correlations in YBa₂Cu₃O_6+x after the quench of superconductivity by an infrared laser pulse. We observe a nonthermal response of the CDW order characterized by a near doubling of the correlation length within ≈1 picosecond of the superconducting quench. Our results are consistent with a model in which the interaction between superconductivity and CDWs manifests inhomogeneously through disruption of spatial coherence, with superconductivity playing the dominant role in stabilizing CDW topological defects, such as discommensurations.

Disentangling Coexisting Structural Order Through Phase Lock-In Analysis of Atomic-Resolution STEM Data

As a real-space technique, atomic-resolution STEM imaging contains both amplitude and geometric phase information about structural order in materials, with the latter encoding important information about local variations and heterogeneities present in crystalline lattices. Such phase information can be extracted using geometric phase analysis (GPA), a method which has generally focused on spatially mapping elastic strain. Here we demonstrate an alternative phase demodulation technique and its application to reveal complex structural phenomena in correlated quantum materials. As with other methods of image phase analysis, the phase lock-in approach can be implemented to extract detailed information about structural order and disorder, including dislocations and compound defects in crystals. Extending the application of this phase analysis to Fourier components that encode periodic modulations of the crystalline lattice, such as superlattice or secondary frequency peaks, we extract the behavior of multiple distinct order parameters within the same image, yielding insights into not only the crystalline heterogeneity but also subtle emergent order parameters such as antipolar displacements. When applied to atomic-resolution images spanning large (~0.5 × 0.5 μm2) fields of view, this approach enables vivid visualizations of the spatial interplay between various structural orders in novel materials.

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.

Atomic-Resolution Cryogenic Scanning Transmission Electron Microscopy for Quantum Materials

ConspectusThe rich physics permeating the phase diagrams of quantum materials have commanded the attention of the solid-state chemistry, materials science, and condensed-matter physics communities, sparking immense research into quantum phase transitions including superconducting, ferroic, and charge-order transitions. Many of these transitions occur at low temperatures and involve electronic, magnetic, or lattice order, which emerges on the atomic to mesoscopic scales. The complex interplay of these states and the heterogeneity that arises due to competition and intertwining of phases, however, is not fully understood and requires probes that capture ordering over multiple length scales down to the local atomic symmetries. Advances in scanning transmission electron microscopy (STEM) have enabled atomic-resolution imaging as well as mapping of functional picometer-scale atomic displacements inside materials. In this Account, we discuss our group's work to expand the reach of atomic-resolution STEM to cryogenic temperatures (cryo-STEM) to study quantum materials with focus on charge-ordered systems.Charge-ordered phases, in which electrons as well as the atomic lattice form periodic patterns that lift the translational symmetries of the crystal, are not only intertwined with superconductivity but also underlie other exotic electronic phenomena such as colossal magnetoresistance and metal-insulator transitions. The periodic lattice distortions (PLDs) modulate the positions of the crystal's nuclei, which can be readily probed by electron microscopy. In a set of examples, we demonstrate cryo-STEM as a powerful technique for probing local order, nanometer-scale heterogeneities, and topological defects in charge-ordered manganites and in transition metal dichalcogenide charge density wave (CDW) systems.With the nearly commensurate-to-commensurate CDW transition upon cooling in 1T-TaS₂, we show that nanoscale lattice textures in CDW phases can be revealed through direct imaging. These early atomic-resolution results, however, also highlighted the need for improvements in cryo-STEM imaging, which led to a push to advance data collection and analysis for direct spatial mapping and quantification of PLDs. By introducing an image registration algorithm developed specifically to accommodate fast, low signal-to-noise image acquisitions of crystalline lattices, we address previous limitations due to sample drift in cryo-STEM experiments. This has enabled subangstrom cryo-STEM imaging with sufficient signal-to-noise to reveal the low temperature structure of 1T'-TaTe₂. Furthermore, it allows mapping and quantification of PLD atomic displacements in the charge-ordered manganites Bi_0.35Sr_0.18Ca_0.47MnO₃ and Nd_0.5Sr_0.5MnO₃ with picometer precision at ∼95 K to resolve not only distinct ordered phases (i.e., site- and bond-centered charge order) but also their nanoscale coexistence within the same sample.Atomic-resolution cryo-STEM opens new opportunities for understanding the microscopic underpinnings of quantum phases. In this Account, we focus on spatial mapping of lattice degrees of freedom in phases that are present at temperatures down to liquid nitrogen. Further advances in instrumentation are needed to expand the temperature range and to also enable atomic-resolution measurements that rely on weaker signals such as electron energy loss spectroscopy (EELS) for probing of electronic structure or 4D-STEM approaches to map electric and magnetic fields.

Imaging the energy gap modulations of the cuprate pair-density-wave state

The defining characteristic^1,2 of Cooper pairs with finite centre-of-mass momentum is a spatially modulating superconducting energy gap Δ(r), where r is a position. Recently, this concept has been generalized to the pair-density-wave (PDW) state predicted to exist in copper oxides (cuprates)^3,4. Although the signature of a cuprate PDW has been detected in Cooper-pair tunnelling⁵, the distinctive signature in single-electron tunnelling of a periodic Δ(r) modulation has not been observed. Here, using a spectroscopic technique based on scanning tunnelling microscopy, we find strong Δ(r) modulations in the canonical cuprate Bi₂Sr₂CaCu₂O_8+δ that have eight-unit-cell periodicity or wavevectors Q ≈ (2π/a₀)(1/8, 0) and Q ≈ (2π/a₀)(0, 1/8) (where a₀ is the distance between neighbouring Cu atoms). Simultaneous imaging of the local density of states N(r, E) (where E is the energy) reveals electronic modulations with wavevectors Q and 2Q, as anticipated when the PDW coexists with superconductivity. Finally, by visualizing the topological defects in these N(r, E) density waves at 2Q, we find them to be concentrated in areas where the PDW spatial phase changes by π, as predicted by the theory of half-vortices in a PDW state^6,7. Overall, this is a compelling demonstration, from multiple single-electron signatures, of a PDW state coexisting with superconductivity in Bi₂Sr₂CaCu₂O_8+δ.

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.

Distinction between pristine and disorder-perturbed charge density waves in ZrTe3

Charge density waves (CDWs) in the cuprate high-temperature superconductors have evoked much interest, yet their typical short-range nature has raised questions regarding the role of disorder. Here we report a resonant X-ray diffraction study of ZrTe[Formula: see text], a model CDW system, with focus on the influence of disorder. Near the CDW transition temperature, we observe two independent signals that arise concomitantly, only to become clearly separated in momentum while developing very different correlation lengths in the well-ordered state that is reached at a distinctly lower temperature. Anomalously slow dynamics of mesoscopic charge domains are further found near the transition temperature, in spite of the expected strong thermal fluctuations. Our observations signify the presence of distinct experimental fingerprints of pristine and disorder-perturbed CDWs. We discuss the latter also in the context of Friedel oscillations, which we argue might promote CDW formation via a self-amplifying process.

Atomic manipulation of the gap in Bi2Sr2CaCu2O8+x

Single-atom manipulation within doped correlated electron systems could help disentangle the influence of dopants, structural defects, and crystallographic characteristics on local electronic states. Unfortunately, the high diffusion barrier in these materials prevents conventional manipulation techniques. Here, we demonstrate the possibility to reversibly manipulate select sites in the optimally doped high-temperature superconductor Bi₂Sr₂CaCu₂O_8+x using the local electric field of the tip of a scanning tunneling microscope. We show that upon shifting individual Bi atoms at the surface, the spectral gap associated with superconductivity is seen to reversibly change by as much as 15 milli-electron volts (on average ~5% of the total gap size). Our toy model, which captures all observed characteristics, suggests that the electric field induces lateral movement of local pairing potentials in the CuO₂ plane.

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.

The Ancient Romans’ Route to Charge Density Waves in Cuprates

An account is given of the main steps that led the research group in Rome, to which the author belongs, to the formulation of the charge-density-wave scenario for high- T c superconducting cuprates. The early finding of the generic tendency of strongly correlated electron systems with short range interactions to undergo electron phase separation was subsequently contrasted with the homogenizing effect of the long-range Coulomb interaction. The two effects can find a compromise in the formation of incommensurate charge density waves. These charge density waves are inherently dynamical and are overdamped as a consequence of the possibility to decay in electron-hole pairs, yet tend to maintain a (quantum) critical character, which is mirrored in their marked momentum and frequency dependence and in their strong variation with temperature and doping. These dynamical incommensurate charge density waves act as mediators of pairing lading to high- T c superconductivity, and provide the scattering mechanism that produces the observed violation of the Fermi-liquid paradigm in the metallic phase.

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.

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.

Nature and evolution of incommensurate charge order in manganites visualized with cryogenic scanning transmission electron microscopy

Charge order is a modulation of the electron density and is associated with unconventional phenomena, including colossal magnetoresistance and metal–insulator transitions. Determining how the lattice responds provides insights into the nature and symmetry of the ordered state. Scanning transmission electron microscopy can measure lattice displacements with picometer precision, but its use has been limited to room-temperature phases only. Here, we demonstrate high-resolution imaging at cryogenic temperature and map the nature and evolution of charge order in a manganite. We uncover picometer-scale displacive modulations whose periodicity is strongly locked to the lattice and visualize temperature-dependent phase inhomogeneity in the modulations. These results pave the way to understanding the underlying structure of charge-ordered states and other complex phenomena.

Incommensurate charge order in hole-doped oxides is intertwined with exotic phenomena such as colossal magnetoresistance, high-temperature superconductivity, and electronic nematicity. Here, we map, at atomic resolution, the nature of incommensurate charge–lattice order in a manganite using scanning transmission electron microscopy at room temperature and cryogenic temperature (∼93 K). In diffraction, the ordering wave vector changes upon cooling, a behavior typically associated with incommensurate order. However, using real space measurements, we discover that the ordered state forms lattice-locked regions over a few wavelengths interspersed with phase defects and changing periodicity. The cations undergo picometer-scale (∼6 pm to 11 pm) transverse displacements, suggesting that charge–lattice coupling is strong. We further unearth phase inhomogeneity in the periodic lattice displacements at room temperature, and emergent phase coherence at 93 K. Such local phase variations govern the long-range correlations of the charge-ordered state and locally change the periodicity of the modulations, resulting in wave vector shifts in reciprocal space. These atomically resolved observations underscore the importance of lattice coupling and phase inhomogeneity, and provide a microscopic explanation for putative “incommensurate” order in hole-doped oxides.

The study of electronic nematicity in an overdoped (Bi, Pb)2Sr2CuO6+δ superconductor using scanning tunneling spectroscopy

The pseudogap (PG) state and its related intra-unit-cell symmetry breaking remain the focus in the research of cuprate superconductors. Although the nematicity has been studied in Bi₂Sr₂CaCu₂O_8+δ, especially underdoped samples, its behavior in other cuprates and different doping regions is still unclear. Here we apply a scanning tunneling microscope to explore an overdoped (Bi, Pb)₂Sr₂CuO_6+δ with a large Fermi surface (FS). The establishment of a nematic order and its real-space distribution is visualized as the energy scale approaches the PG.

Evidence for a Nematic Phase in La_{1.75}Sr_{0.25}NiO_{4}

Determining the nature of electronic states in doped Mott insulators remains a challenging task. In the case of tetragonal La_{2-x}Sr_{x}NiO_{4}, the occurrence of diagonal charge and spin stripe order in the ground state is now well established. In contrast, the nature of the high-temperature "disordered" state from which the stripe order develops has long been a subject of controversy, with considerable speculation regarding a polaronic liquid. Following the recent detection of dynamic charge stripes, we use neutron scattering measurements on an x=0.25 crystal to demonstrate that the dispersion of the charge-stripe excitations is anisotropic. This observation provides compelling evidence for the presence of electronic nematic order.

Direct observation of electronic-liquid-crystal phase transitions and their microscopic origin in La1/3Ca2/3MnO3

The ground-state electronic order in doped manganites is frequently associated with a lattice modulation, contributing to their many interesting properties. However, measuring the thermal evolution of the lattice superstructure with reciprocal-space probes alone can lead to ambiguous results with competing interpretations. Here we provide direct observations of the evolution of the superstructure in La1/3Ca2/3MnO3 in real space, as well as reciprocal space, using transmission electron microscopic (TEM) techniques. We show that the transitions are the consequence of a proliferation of dislocations plus electronic phase separation. The resulting states are well described by the symmetries associated with electronic-liquid-crystal (ELC) phases. Moreover, our results resolve the long-standing controversy over the origin of the incommensurate superstructure and suggest a new structural model that is consistent with recent theoretical calculations.

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.

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.

Inhomogeneity of charge-density-wave order and quenched disorder in a high-Tc superconductor

It has recently been established that the high-transition-temperature (high-Tc) superconducting state coexists with short-range charge-density-wave order and quenched disorder arising from dopants and strain. This complex, multiscale phase separation invites the development of theories of high-temperature superconductivity that include complexity. The nature of the spatial interplay between charge and dopant order that provides a basis for nanoscale phase separation remains a key open question, because experiments have yet to probe the unknown spatial distribution at both the nanoscale and mesoscale (between atomic and macroscopic scale). Here we report micro X-ray diffraction imaging of the spatial distribution of both short-range charge-density-wave 'puddles' (domains with only a few wavelengths) and quenched disorder in HgBa2CuO4 + y, the single-layer cuprate with the highest Tc, 95 kelvin (refs 26-28). We found that the charge-density-wave puddles, like the steam bubbles in boiling water, have a fat-tailed size distribution that is typical of self-organization near a critical point. However, the quenched disorder, which arises from oxygen interstitials, has a distribution that is contrary to the usually assumed random, uncorrelated distribution. The interstitial-oxygen-rich domains are spatially anticorrelated with the charge-density-wave domains, because higher doping does not favour the stripy charge-density-wave puddles, leading to a complex emergent geometry of the spatial landscape for superconductivity.

Incipient charge order observed by NMR in the normal state of YBa2Cu3Oy

The pseudogap regime of high-temperature cuprates harbours diverse manifestations of electronic ordering whose exact nature and universality remain debated. Here, we show that the short-ranged charge order recently reported in the normal state of YBa2Cu3Oy corresponds to a truly static modulation of the charge density. We also show that this modulation impacts on most electronic properties, that it appears jointly with intra-unit-cell nematic, but not magnetic, order, and that it exhibits differences with the charge density wave observed at lower temperatures in high magnetic fields. These observations prove mostly universal, they place new constraints on the origin of the charge density wave and they reveal that the charge modulation is pinned by native defects. Similarities with results in layered metals such as NbSe2, in which defects nucleate halos of incipient charge density wave at temperatures above the ordering transition, raise the possibility that order-parameter fluctuations, but no static order, would be observed in the normal state of most cuprates if disorder were absent.

Pushing the limits: an instrument for hard X-ray imaging below 20 nm

Hard X-ray microscopy is a prominent tool suitable for nanoscale-resolution non-destructive imaging of various materials used in different areas of science and technology. With an ongoing effort to push the 2D/3D imaging resolution down to 10 nm in the hard X-ray regime, both the fabrication of nano-focusing optics and the stability of the microscope using those optics become extremely challenging. In this work a microscopy system designed and constructed to accommodate multilayer Laue lenses as nanofocusing optics is presented. The developed apparatus has been thoroughly characterized in terms of resolution and stability followed by imaging experiments at a synchrotron facility. Drift rates of ∼2 nm h(-1) accompanied by 13 nm × 33 nm imaging resolution at 11.8 keV are reported.

Direct phase-sensitive identification of a d-form factor density wave in underdoped cuprates

The identity of the fundamental broken symmetry (if any) in the underdoped cuprates is unresolved. However, evidence has been accumulating that this state may be an unconventional density wave. Here we carry out site-specific measurements within each CuO2 unit cell, segregating the results into three separate electronic structure images containing only the Cu sites [Cu(r)] and only the x/y axis O sites [Ox(r) and O(y)(r)]. Phase-resolved Fourier analysis reveals directly that the modulations in the O(x)(r) and O(y)(r) sublattice images consistently exhibit a relative phase of π. We confirm this discovery on two highly distinct cuprate compounds, ruling out tunnel matrix-element and materials-specific systematics. These observations demonstrate by direct sublattice phase-resolved visualization that the density wave found in underdoped cuprates consists of modulations of the intraunit-cell states that exhibit a predominantly d-symmetry form factor.

Photo-enhanced antinodal conductivity in the pseudogap state of high-Tc cuprates

A major challenge in understanding the cuprate superconductors is to clarify the nature of the fundamental electronic correlations that lead to the pseudogap phenomenon. Here we use ultrashort light pulses to prepare a non-thermal distribution of excitations and capture novel properties that are hidden at equilibrium. Using a broadband (0.5–2 eV) probe, we are able to track the dynamics of the dielectric function and unveil an anomalous decrease in the scattering rate of the charge carriers in a pseudogap-like region of the temperature (T) and hole-doping (p) phase diagram. In this region, delimited by a well-defined T*_neq(p) line, the photoexcitation process triggers the evolution of antinodal excitations from gapped (localized) to delocalized quasiparticles characterized by a longer lifetime. The novel concept of photo-enhanced antinodal conductivity is naturally explained within the single-band Hubbard model, in which the short-range Coulomb repulsion leads to a k-space differentiation between nodal quasiparticles and antinodal excitations.

The pseudogap phase exhibited by the cuprates is almost as enigmatic as superconductivity in these materials itself. A time-resolved study performed by Cilento et al. suggests that this state can be photoexcited into a transient non-equilibrium state that is more conductive than the equilibrium state.

Direct observation of multiferroic vortex domains in YMnO3

Topological vortices with swirling ferroelectric, magnetic and structural anti-phase relationship in hexagonal RMnO₃ (R = Ho to Lu, Y, and Sc) have attracted much attention because of their intriguing behaviors. Herein, we report the structure of multiferroic vortex domains in YMnO₃ at atomic scale using state-of-the-art aberration-corrected scanning transmission electron microscopy (STEM). Two types of displacements were identified among six domain walls (DWs); six translation-ferroelectric domains denoted by α+, γ−, β+, α−, γ+ and β−, respectively, were recognized, demonstrating the interlocking nature of the anti-vortex domain. We found that the anti-vortex core is about four unit cells wide. In addition, we reconstructed the vortex model with three swirling pairs of DWs along the [001] direction. These results are very critical for the understanding of topological behaviors and unusual properties of the multiferroic vortex.

Simultaneous transitions in cuprate momentum-space topology and electronic symmetry breaking

The existence of electronic symmetry breaking in the underdoped cuprates and its disappearance with increased hole density p are now widely reported. However, the relation between this transition and the momentum-space (k-space) electronic structure underpinning the superconductivity has not yet been established. Here, we visualize the Q = 0 (intra-unit-cell) and Q ≠ 0 (density-wave) broken-symmetry states, simultaneously with the coherent k-space topology, for Bi₂Sr₂CaCu₂O(8+δ) samples spanning the phase diagram 0.06 ≤ p ≤ 0.23. We show that the electronic symmetry-breaking tendencies weaken with increasing p and disappear close to a critical doping p(c) = 0.19. Concomitantly, the coherent k-space topology undergoes an abrupt transition, from arcs to closed contours, at the same p(c). These data reveal that the k-space topology transformation in cuprates is linked intimately with the disappearance of the electronic symmetry breaking at a concealed critical point.

Quenched disorder and vestigial nematicity in the pseudogap regime of the cuprates

The cuprate high-temperature superconductors have been the focus of unprecedentedly intense and sustained study not only because of their high superconducting transition temperatures, but also because they represent the most exquisitely investigated examples of highly correlated electronic materials. In particular, the pseudogap regime of the phase diagram exhibits a variety of mysterious emergent behaviors. In the last few years, evidence from NMR and scanning tunneling microscopy (STM) studies, as well as from a new generation of X-ray scattering experiments, has accumulated, indicating that a general tendency to short-range-correlated incommensurate charge density wave (CDW) order is "intertwined" with the superconductivity in this regime. Additionally, transport, STM, neutron-scattering, and optical experiments have produced evidence--not yet entirely understood--of the existence of an associated pattern of long-range-ordered point-group symmetry breaking with an electron-nematic character. We have carried out a theoretical analysis of the Landau-Ginzburg-Wilson effective field theory of a classical incommensurate CDW in the presence of weak quenched disorder. Although the possibilities of a sharp phase transition and long-range CDW order are precluded in such systems, we show that any discrete symmetry-breaking aspect of the charge order--nematicity in the case of the unidirectional (stripe) CDW we consider explicitly--generically survives up to a nonzero critical disorder strength. Such "vestigial order," which is subject to unambiguous macroscopic detection, can serve as an avatar of what would be CDW order in the ideal, zero disorder limit. Various recent experiments in the pseudogap regime of the hole-doped cuprates are readily interpreted in light of these results.

Cu(Ir₁ - xCrx)₂S₄: a model system for studying nanoscale phase coexistence at the metal-insulator transition

Increasingly, nanoscale phase coexistence and hidden broken symmetry states are being found in the vicinity of metal-insulator transitions (MIT), for example, in high temperature superconductors, heavy fermion and colossal magnetoresistive materials, but their importance and possible role in the MIT and related emergent behaviors is not understood. Despite their ubiquity, they are hard to study because they produce weak diffuse signals in most measurements. Here we propose Cu(Ir_{1 − x}Cr_x)₂S₄ as a model system, where robust local structural signals lead to key new insights. We demonstrate a hitherto unobserved coexistence of an Ir⁴⁺ charge-localized dimer phase and Cr-ferromagnetism. The resulting phase diagram that takes into account the short range dimer order is highly reminiscent of a generic MIT phase diagram similar to the cuprates. We suggest that the presence of quenched strain from dopant ions acts as an arbiter deciding between the competing ground states.

Formation of gapless Fermi arcs and fingerprints of order in the pseudogap state of cuprate superconductors

We use angle-resolved photoemission spectroscopy and a new quantitative approach based on the partial density of states to study properties of seemingly disconnected portions of the Fermi surface (FS) that are present in the pseudogap state of cuprates called Fermi arcs. We find that the normal state FS collapses very abruptly into Fermi arcs at the pseudogap temperature (T*). Surprisingly, the length of the Fermi arcs remains constant over an extended temperature range between T* and T(pair), consistent with the presence of an ordered state below T*. These arcs collapse again at the temperature below which pair formation occurs (T(pair)) either to a point or a very short arc, whose length is limited by our experimental resolution. The tips of the arcs span between points defining a set of wave vectors in momentum space, which are the fingerprints of the ordered state that causes the pseudogap.

The k-space origins of scattering in Bi2Sr2CaCu2O8+x

We demonstrate a general, computer automated procedure that inverts the reciprocal space scattering data (q-space) that are measured by spectroscopic imaging scanning tunnelling microscopy (SI-STM) in order to determine the momentum space (k-space) scattering structure. This allows a detailed examination of the k-space origins of the quasiparticle interference (QPI) pattern in Bi2Sr2CaCu2O8+x within the theoretical constraints of the joint density of states (JDOS). Our new method allows measurement of the differences between the positive and negative energy dispersions, the gap structure and an energy dependent scattering length scale. Furthermore, it resolves the transition between the dispersive QPI and the checkerboard ([Formula: see text] excitation). We have measured the k-space scattering structure over a wide range of doping (p ∼ 0.22-0.08), including regions where the octet model is not applicable. Our technique allows the complete mapping of the k-space scattering origins of the spatial excitations in Bi2Sr2CaCu2O8+x, which allows for better comparisons between SI-STM and other experimental probes of the band structure. By applying our new technique to such a heavily studied compound, we can validate our new general approach for determining the k-space scattering origins from SI-STM data.

Bond order in two-dimensional metals with antiferromagnetic exchange interactions

We present an unrestricted Hartree-Fock computation of charge-ordering instabilities of two-dimensional metals with antiferromagnetic exchange interactions, allowing for arbitrary ordering wave vectors and internal wave functions of the particle-hole pair condensate. We find that the ordering has a dominant d symmetry of rotations about lattice points for a range of ordering wave vectors, including those observed in recent experiments at low temperatures on YBa2Cu3O(y). This d symmetry implies the charge ordering is primarily on the bonds of the Cu lattice, and we propose incommensurate bond order parameters for the underdoped cuprates. The field theory for the onset of Néel order in a metal has an emergent pseudospin symmetry which "rotates" d-wave Cooper pairs to particle-hole pairs [M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128 (2010)]; our results show that this symmetry has consequences even when the spin correlations are short ranged and incommensurate.

Surface-enhanced charge-density-wave instability in underdoped Bi2Sr(2-x)La(x)CuO(6+δ)

Neutron and X-ray scattering experiments have provided mounting evidence for spin and charge ordering phenomena in underdoped cuprates. These range from early work on stripe correlations in Nd-LSCO to the latest discovery of charge-density-waves in YBa2Cu3O(6+x). Both phenomena are characterized by a pronounced dependence on doping, temperature and an externally applied magnetic field. Here, we show that these electron-lattice instabilities exhibit also a previously unrecognized bulk-surface dichotomy. Surface-sensitive electronic and structural probes uncover a temperature-dependent evolution of the CuO2 plane band dispersion and apparent Fermi pockets in underdoped Bi2 Sr(2-x) La(x) CuO(6+δ) (Bi2201), which is directly associated with an hitherto-undetected strong temperature dependence of the incommensurate superstructure periodicity below 130 K. In stark contrast, the structural modulation revealed by bulk-sensitive probes is temperature-independent. These findings point to a surface-enhanced incipient charge-density-wave instability, driven by Fermi surface nesting. This discovery is of critical importance in the interpretation of single-particle spectroscopy data, and establishes the surface of cuprates and other complex oxides as a rich playground for the study of electronically soft phases.

Performance and characterization of the prototype nm-scale spatial resolution scanning multilayer Laue lenses microscope

Synchrotron based x-ray microscopy established itself as a prominent tool for noninvasive investigations in many areas of science and technology. Many facilities around the world routinely achieve sub-micrometer resolution with a few instruments capable of imaging with the spatial resolution better than 100 nm. With an ongoing effort to push the 2D/3D resolution down to 10 nm in the hard x-ray regime both fabrication of the nano-focusing optics and stability of a microscope become extremely challenging. In this work we present our approach to overcome technical challenges on the path towards high spatial resolution hard x-ray microscopy and demonstrate the performance of a scanning fluorescence microscope equipped with the multilayer Laue lenses focusing optics.

A cryogenic scattering-type scanning near-field optical microscope

Scattering-type scanning near-field optical microscopy (s-SNOM) provides few nanometer optical spatial resolution and is compatible with nearly any form of linear and nonlinear optical spectroscopy. We have developed a versatile s-SNOM instrument operating under cryogenic and variable temperature (∼20-500 K) and compatible with high magnetic fields (up to 7 T). The instrument features independent tip and sample scanning and free-space light delivery with an integrated off-axis parabolic mirror for tip-illumination and signal collection with a numerical aperture of N.A. = 0.45. The optics operate from the UV to THz range allowing for continuous wave, broadband, and ultrafast s-SNOM spectroscopy, including different variants of tip-enhanced spectroscopy. We discuss the instrument design, implementation, and demonstrate its performance with mid-infrared Drude response s-SNOM probing of the domain formation associated with the metal-insulator transitions of VO2 (TMIT ≃ 340 K) and V2O3 (TMIT ≃ 150 K). This instrument enables the study of mesoscopic order and domains of competing quantum phases in correlated electron materials over a wide range of controlled electric and magnetic fields, strain, current, and temperature.

Collective magnetism at multiferroic vortex domain walls

Cross-coupled phenomena of multiferroic domains and domain walls are of fundamental scientific and technological interest. Using cryogenic magnetic force microscopy, we find alternating net magnetic moments at ferroelectric domain walls around vortex cores in multiferroic hexagonal ErMnO(3), which correlate with each other throughout the entire vortex network. This collective nature of domain wall magnetism originates from the uncompensated Er(3+) moments at domain walls and the self-organization of the vortex network. Our results demonstrate that the collective domain wall magnetism can be controlled by external magnetic fields and represent a major advancement in the manipulation of local magnetic moments by harnessing cross-coupled domain walls.

Universality of modulation length and time exponents

We study systems with a crossover parameter λ, such as the temperature T, which has a threshold value λ(*) across which the correlation function changes from exhibiting fixed wavelength (or time period) modulations to continuously varying modulation lengths (or times). We introduce a hitherto unknown exponent ν(L) characterizing the universal nature of this crossover and compute its value in general instances. This exponent, similar to standard correlation length exponents, is obtained from motion of the poles of the momentum (or frequency) space correlation functions in the complex k-plane (or ω-plane) as the parameter λ is varied. Near the crossover (i.e., for λ→λ(*)), the characteristic modulation wave vector K(R) in the variable modulation length "phase" is related to that in the fixed modulation length "phase" q via |K(R)-q|[proportionality]|T-T(*)|(νL). We find, in general, that ν(L)=1/2. In some special instances, ν(L) may attain other rational values. We extend this result to general problems in which the eigenvalue of an operator or a pole characterizing general response functions may attain a constant real (or imaginary) part beyond a particular threshold value λ(*). We discuss extensions of this result to multiple other arenas. These include the axial next-nearest-neighbor Ising (ANNNI) model. By extending our considerations, we comment on relations pertaining not only to the modulation lengths (or times), but also to the standard correlation lengths (or times). We introduce the notion of a Josephson time scale. We comment on the presence of aperiodic "chaotic" modulations in "soft-spin" and other systems. These relate to glass-type features. We discuss applications to Fermi systems, with particular application to metal to band insulator transitions, change of Fermi surface topology, divergent effective masses, Dirac systems, and topological insulators. Both regular periodic and glassy (and spatially chaotic behavior) may be found in strongly correlated electronic systems.