Unusual magnetic order in the pseudogap region of the superconductor HgBa2CuO4+delta

Superconductivity due to fluctuating loop currents

Orbital magnetism and the loop currents (LCs) that accompany it have been proposed to emerge in many systems, including cuprates, iridates, and kagome superconductors. In the case of cuprates, LCs have been put forward as the driving force behind the pseudogap, strange-metal behavior, and dx2-y2-wave superconductivity. Here, we investigate whether fluctuating intra-unit-cell LCs can cause unconventional superconductivity. For odd-parity LCs, we find that they are repulsive in all pairing channels near the underlying quantum-critical point (QCP). For even-parity LCs, their fluctuations give rise to unconventional pairing, which is not amplified in the vicinity of the QCP, in sharp contrast to pairing mediated by spin-magnetic, nematic, or ferroelectric fluctuations. Applying our formalism to the cuprates, we conclude that fluctuating intra-unit-cell LCs are unlikely to yield dx2-y2-wave superconductivity. If LCs are to be relevant for the cuprates, they must break translation symmetry.

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.

Topological spin texture in the pseudogap phase of a high-Tc superconductor

An outstanding challenge in condensed-matter-physics research over the past three decades has been to understand the pseudogap (PG) phenomenon of the high-transition-temperature (high-T_c) copper oxides. A variety of experiments have indicated a symmetry-broken state below the characteristic temperature T* (refs. ^1-8). Among them, although the optical study⁵ indicated the mesoscopic domains to be small, all these experiments lack nanometre-scale spatial resolution, and the microscopic order parameter has so far remained elusive. Here we report, to our knowledge, the first direct observation of topological spin texture in an underdoped cuprate, YBa₂Cu₃O_6.5, in the PG state, using Lorentz transmission electron microscopy (LTEM). The spin texture features vortex-like magnetization density in the CuO₂ sheets, with a relatively large length scale of about 100 nm. We identify the phase-diagram region in which the topological spin texture exists and demonstrate the ortho-II oxygen order and suitable sample thickness to be crucial for its observation by our technique. We also discuss an intriguing interplay observed among the topological spin texture, PG state, charge order and superconductivity.

All topological bands of all nonmagnetic stoichiometric materials

Topological quantum chemistry and symmetry-based indicators have facilitated large-scale searches for materials with topological properties at the Fermi energy (E_F). We report the implementation of a publicly accessible catalog of stable and fragile topology in all of the bands both at and away from E_F in the 96,196 processable entries in the Inorganic Crystal Structure Database. Our calculations, which represent the completion of the symmetry-indicated band topology of known nonmagnetic materials, have enabled the discovery of repeat-topological and supertopological materials, including rhombohedral bismuth and Bi₂Mg₃. We find that 52.65% of all materials are topological at E_F, roughly two-thirds of bands across all materials exhibit symmetry-indicated stable topology, and 87.99% of all materials contain at least one stable or fragile topological band.

Comparing Thickness and Doping-Induced Effects on the Normal States of Infinite-Layer Electron-Doped Cuprates: Is There Anything to Learn?

We grew Sr_1-xLa_xCuO₂ thin films and SrCuO₂/Sr_0.9La_0.1CuO₂/SrCuO₂ trilayers by reflection high-energy diffraction-calibrated layer-by-layer molecular beam epitaxy, to study their electrical transport properties as a function of the doping and thickness of the central Sr_0.9La_0.1CuO₂ layer. For the trilayer samples, as already observed in underdoped SLCO films, the electrical resistivity versus temperature curves as a function of the central layer thickness show, for thicknesses thinner than 20 unit cells, sudden upturns in the low temperature range with the possibility for identifying, in the normal state, the T* and a T** temperatures, respectively, separating high-temperature linear behavior and low-temperature quadratic dependence. By plotting the T* and T** values as a function of T_C^onset for both the thin films and the trilayers, the data fall on the same curves. This result suggests that, for the investigated trilayers, the superconducting critical temperature is the important parameter able to describe the normal state properties and that, in the limit of very thin central layers, such properties are mainly influenced by the modification of the energy band structure and not by interface-related disorder.

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.

Superconductivity and the Jahn–Teller Polaron

In this article, we review the essential properties of high-temperature superconducting cuprates, which are unconventional isotope effects, heterogeneity, and lattice responses. Since their discovery was based on ideas stemming from Jahn–Teller polarons, their special role, together with the Jahn–Teller effect itself, is discussed in greater detail. We conclude that the underlying physics of cuprates cannot stem from purely electronic mechanisms, but that the intricate interaction between lattice and charge is at its origin.

Unveiling unconventional magnetism at the surface of Sr2RuO4

Materials with strongly correlated electrons often exhibit interesting physical properties. An example of these materials is the layered oxide perovskite Sr₂RuO₄, which has been intensively investigated due to its unusual properties. Whilst the debate on the symmetry of the superconducting state in Sr₂RuO₄ is still ongoing, a deeper understanding of the Sr₂RuO₄ normal state appears crucial as this is the background in which electron pairing occurs. Here, by using low-energy muon spin spectroscopy we discover the existence of surface magnetism in Sr₂RuO₄ in its normal state. We detect static weak dipolar fields yet manifesting at an onset temperature higher than 50 K. We ascribe this unconventional magnetism to orbital loop currents forming at the reconstructed Sr₂RuO₄ surface. Our observations set a reference for the discovery of the same magnetic phase in other materials and unveil an electronic ordering mechanism that can influence electron pairing with broken time reversal symmetry.

Strange Metals as Ersatz Fermi Liquids

A long-standing mystery of fundamental importance in correlated electron physics is to understand strange non-Fermi liquid metals that are seen in diverse quantum materials. A striking experimental feature of these metals is a resistivity that is linear in temperature (T). In this Letter we ask what it takes to obtain such non-Fermi liquid physics down to zero temperature in a translation invariant metal. If in addition the full frequency (ω) dependent conductivity satisfies ω/T scaling, we argue that the T-linear resistivity must come from the intrinsic physics of the low energy fixed point. Combining with earlier arguments that compressible translation invariant metals are "ersatz Fermi liquids" with an infinite number of emergent conserved quantities, we obtain powerful and practical conclusions. We show that there is necessarily a diverging susceptibility for an operator that is odd under inversion and time reversal symmetries, and has zero crystal momentum. We discuss a few other experimental consequences of our arguments, as well as potential loopholes, which necessarily imply other exotic phenomena.

Chiral flux phase in the Kagome superconductor AV3Sb5

We argue that the topological charge density wave phase in the quasi-2D Kagome superconductor AV₃Sb₅ is a chiral flux phase. Considering the symmetry of the Kagome lattice, we show that the chiral flux phase has the lowest energy among those states which exhibit 2×2 charge orders observed experimentally. This state breaks the time-reversal symmetry and displays anomalous Hall effect. The explicit pattern of the density of state in real space is calculated. These results are supported by recent experiments and suggest that these materials are new platforms to investigate the interplay between topology, superconductivity and electron-electron correlations.

From SrTiO3 to Cuprates and Back to SrTiO3: A Way Along Alex Müller’s Scientific Career

K.A. Müller took a long route in science leaving many traces and imprints, which have been and are still today initiations for further research activities. We “walk” along this outstanding path but are certainly not able to provide a complete picture of it, since the way was not always straight, often marked by unintended detours, which had novel impact on the international research society.

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.

Extent of Fermi-surface reconstruction in the high-temperature superconductor HgBa2CuO4+δ

High magnetic fields have revealed a surprisingly small Fermi surface in underdoped cuprates, possibly resulting from Fermi-surface reconstruction due to an order parameter that breaks translational symmetry of the crystal lattice. A crucial issue concerns the doping extent of such a state and its relationship to the principal pseudogap and superconducting phases. We employ pulsed magnetic-field measurements on the cuprate [Formula: see text]Cu[Formula: see text] to identify signatures of Fermi-surface reconstruction from a sign change of the Hall effect and a peak in the temperature-dependent planar resistivity. We trace the termination of Fermi-surface reconstruction to two hole concentrations where the superconducting upper critical fields are found to be enhanced. One of these points is associated with the pseudogap endpoint near optimal doping. These results connect the Fermi-surface reconstruction to both superconductivity and the pseudogap phenomena.

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.

Low temperature hidden Fermi-liquid charge transport in under doped La x Sr1-x CuO2 infinite layer electron-doped thin films

We have studied the low temperature electrical transport properties of La _x Sr_1-x CuO₂ thin films grown by oxide molecular beam epitaxy on (1 1 0) GdScO₃ and TbScO₃ substrates. The transmission electron microscopy measurements and the x-ray diffraction analysis confirmed the epitaxy of the obtained films and the study of their normal state transport properties, removing the ambiguity regarding the truly conducting layer, allowed to highlight the presence of a robust hidden Fermi liquid charge transport in the low temperature properties of infinite layer electron doped cuprate superconductors. These results are in agreement with recent observations performed in other p  and n doped cuprate materials and point toward a general description of the superconducting and normal state properties in these compounds.

Diagonal nematicity in the pseudogap phase of HgBa2CuO4+δ

The pseudogap phenomenon in the cuprates is arguably the most mysterious puzzle in the field of high-temperature superconductivity. The tetragonal cuprate HgBa₂CuO_4+δ, with only one CuO₂ layer per primitive cell, is an ideal system to tackle this puzzle. Here, we measure the magnetic susceptibility anisotropy within the CuO₂ plane with exceptionally high-precision magnetic torque experiments. Our key finding is that a distinct two-fold in-plane anisotropy sets in below the pseudogap temperature T^*, which provides thermodynamic evidence for a nematic phase transition with broken four-fold symmetry. Surprisingly, the nematic director orients along the diagonal direction of the CuO₂ square lattice, in sharp contrast to the bond nematicity along the Cu-O-Cu direction. Another remarkable feature is that the enhancement of the diagonal nematicity with decreasing temperature is suppressed around the temperature at which short-range charge-density-wave formation occurs. Our result suggests a competing relationship between diagonal nematic and charge-density-wave order in HgBa₂CuO_4+δ.

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.

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.

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.

A tale of two metals: contrasting criticalities in the pnictides and hole-doped cuprates

The iron-based high temperature superconductors share a number of similarities with their copper-based counterparts, such as reduced dimensionality, proximity to states of competing order, and a critical role for 3d electron orbitals. Their respective temperature-doping phase diagrams also contain certain commonalities that have led to claims that the metallic and superconducting (SC) properties of both families are governed by their proximity to a quantum critical point (QCP) located inside the SC dome. In this review, we critically examine these claims and highlight significant differences in the bulk physical properties of both systems. While there is now a large body of evidence supporting the presence of a (magnetic) QCP in the iron pnictides, the situation in the cuprates is much less apparent, at least for the end point of the pseudogap phase. We argue that the opening of the normal state pseudogap in cuprates, so often tied to a putative QCP, arises from a momentum-dependent breakdown of quasiparticle coherence that sets in at much higher doping levels but which is driven by the proximity to the Mott insulating state at half filling. Finally, we present a new scenario for the cuprates in which this loss of quasiparticle integrity and its evolution with momentum, temperature and doping plays a key role in shaping the resultant phase diagram.

Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa2Cu3O y

The origin of the pseudogap region below a temperature T* is at the heart of the mysteries of cuprate high-temperature superconductors. Unusual properties of the pseudogap phase, such as broken time-reversal and inversion symmetry are observed in several symmetry-sensitive experiments: polarized neutron diffraction, optical birefringence, dichroic angle-resolved photoemission spectroscopy, second harmonic generation, and polar Kerr effect. These properties suggest that the pseudogap region is a genuine thermodynamic phase and are predicted by theories invoking ordered loop currents or other forms of intra-unit-cell (IUC) magnetic order. However, muon spin rotation (μSR) and nuclear magnetic resonance (NMR) experiments do not see the static local fields expected for magnetic order, leaving room for skepticism. The magnetic resonance probes have much longer time scales, however, over which local fields could be averaged by fluctuations. The observable effect of the fluctuations in magnetic resonance is then dynamic relaxation. We have measured dynamic muon spin relaxation rates in single crystals of YBa₂Cu₃O _y (6.72 < y < 6.95) and have discovered "slow" fluctuating magnetic fields with magnitudes and fluctuation rates of the expected orders of magnitude that set in consistently at temperatures T_mag ≈ T*. The absence of any static field (to which μSR would be linearly sensitive) is consistent with the finite correlation length from neutron diffraction. Equally important, these fluctuations exhibit the critical slowing down at T_mag expected near a time-reversal symmetry breaking transition. Our results explain the absence of static magnetism and provide support for the existence of IUC magnetic order in the pseudogap phase.

Intertwined Order and Holography: The Case of Parity Breaking Pair Density Waves

We present a minimal bottom-up extension of the Chern-Simons bulk action for holographic translational symmetry breaking that naturally gives rise to pair density waves. We construct stationary inhomogeneous black hole solutions in which both the U(1) symmetry and spatially translational symmetry are spontaneously broken at a finite temperature and charge density. This novel solution provides a dual description of a superconducting phase intertwined with charge, current, and parity orders.

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.

Time-reversal symmetry breaking hidden order in Sr2(Ir,Rh)O4

Layered 5d transition iridium oxides, Sr₂(Ir,Rh)O₄, are described as unconventional Mott insulators with strong spin-orbit coupling. The undoped compound, Sr₂IrO₄, is a nearly ideal two-dimensional pseudospin-1/2 Heisenberg antiferromagnet, similarly to the insulating parent compound of high-temperature superconducting copper oxides. Using polarized neutron diffraction, we here report a hidden magnetic order in pure and doped Sr₂(Ir,Rh)O₄, distinct from the usual antiferromagnetic pseudospin ordering. We find that time-reversal symmetry is broken while the lattice translation invariance is preserved in the hidden order phase. The onset temperature matches that of the odd-parity hidden order recently highlighted using optical second-harmonic generation experiments. The novel magnetic order and broken symmetries can be explained by the loop-current model, previously predicted for the copper oxide superconductors.

a-b Anisotropy of the Intra-Unit-Cell Magnetic Order in YBa_{2}Cu_{3}O_{6.6}

Within the complex phase diagram of the hole-doped cuprates, seizing the nature of the mysterious pseudogap phase is essential for unraveling the microscopic origin of high-temperature superconductivity. Below the pseudogap temperature T^{⋆}, evidence for intra-unit-cell orders breaking the fourfold rotation symmetry have been provided by neutron diffraction and scanning tunneling spectroscopy. Using polarized neutron diffraction on a detwinned YBa_{2}Cu_{3}O_{6.6} sample, we here report a distinct a-b anisotropy of the intra-unit-cell magnetic structure factor below T^{⋆}, highlighting that intra-unit-cell order in this material breaks the mirror symmetry of the CuO_{2} bilayers. This is likely to originate from a crisscrossed arrangement of loop currents within the CuO_{2} bilayer, resulting in a bilayer mean toroidal axis along the b direction.

Hourglass Dispersion and Resonance of Magnetic Excitations in the Superconducting State of the Single-Layer Cuprate HgBa_{2}CuO_{4+δ} Near Optimal Doping

We use neutron scattering to study magnetic excitations near the antiferromagnetic wave vector in the underdoped single-layer cuprate HgBa_{2}CuO_{4+δ} (superconducting transition temperature T_{c}≈88  K, pseudogap temperature T^{*}≈220  K). The response is distinctly enhanced below T^{*} and exhibits a Y-shaped dispersion in the pseudogap state, whereas the superconducting state features an X-shaped (hourglass) dispersion and a further resonancelike enhancement. A large spin gap of about 40 meV is observed in both states. This phenomenology is reminiscent of that exhibited by bilayer cuprates. The resonance spectral weight, irrespective of doping and compound, scales linearly with the putative binding energy of a spin exciton described by an itinerant-spin formalism.

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.

Charge orders, magnetism and pairings in the cuprate superconductors

We review the recent developments in the field of cuprate superconductors with special focus on the recently observed charge order in the underdoped compounds. We introduce new theoretical developments following the study of the antiferromagnetic quantum critical point in two dimensions, in which preemptive orders in both charge and superconducting (SC) sectors emerge, that are in turn related by an SU(2) symmetry. We consider the implications of this proliferation of orders in the underdoped region, and provide a study of the type of fluctuations which characterize the SU(2) symmetry. We identify an intermediate energy scale where the SC fluctuations are dominant and argue that they are unstable towards the formation of a resonant excitonic state at the pseudogap temperature T (*). We discuss the implications of this scenario for a few key experiments.

Commensurate antiferromagnetic excitations as a signature of the pseudogap in the tetragonal high-Tc cuprate HgBa2CuO(4+δ)

Antiferromagnetic correlations have been argued to be the cause of the d-wave superconductivity and the pseudogap phenomena exhibited by the cuprates. Although the antiferromagnetic response in the pseudogap state has been reported for a number of compounds, there exists no information for structurally simple HgBa₂CuO_4+δ. Here we report neutron-scattering results for HgBa₂CuO_4+δ (superconducting transition temperature T_c≈71 K, pseudogap temperature T*≈305 K) that demonstrate the absence of the two most prominent features of the magnetic excitation spectrum of the cuprates: the X-shaped ‘hourglass' response and the resonance mode in the superconducting state. Instead, the response is Y-shaped, gapped and significantly enhanced below T*, and hence a prominent signature of the pseudogap state.

In the cuprates, antiferromagnetic correlations might be the cause of the pseudogap phenomenon. Here the authors use neutron scattering on the tetragonal cuprate HgBa₂CuO_4+δ revealing commensurate antiferromagnetic excitations as a signature of the pseudogap 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.

Intra-unit-cell magnetic correlations near optimal doping in YBa2Cu3O6.85

The pseudo-gap phenomenon in copper oxide superconductors is central to any description of these materials as it prefigures the superconducting state itself. A magnetic intra-unit-cell order was found to occur just at the pseudo-gap temperature in four cuprate high-T_c superconducting families. Here we present polarized neutron-scattering measurements of nearly optimally doped YBa₂Cu₃O_6.85, carried out on two different spectrometers, that reveal several features. The intra-unit-cell order consists of finite-sized planar domains that are very weakly correlated along the c axis. At high temperature, only the out-of-plane magnetic components correlate, indicating a strong Ising anisotropy. An aditional in-plane response develops at low temperature, giving rise to an apparent tilt of the magnetic moment. The discovery of these two regimes puts stringent constraints, which are tightly bound to the pseudo-gap physics, on the intrinsic nature of intra-unit-cell order.

The pseudo-gap phenomenon is central to the description of high-T_c superconductivity in copper oxides. Here, the authors investigate nearly optimally doped YBCO using polarized neutron scattering to characterize intra-unit-cell magnetic correlations in relation with the pseudo-gap temperature.

Phase diagram of cuprate high-temperature superconductors described by a field theory based on anharmonic oxygen degrees of freedom

In high temperature superconductors, although some phenomena such as the Mott transition (MT) at low doping are clearly driven by electron correlations, recent experimental data imply that anharmonic oxygen degrees of freedom-characteristic of perovskite materials-are playing a significant role. A key test of the role of anharmonic oxygen is to reproduce the complex cuprate phase diagram from a simple model. Here, we show that a field theory based on nonlinear coupling to anharmonic oxygens, parametrized from ab initio calculations, quantitatively reproduces the cuprate phase diagram for dopings above the MT. Pairing is mediated by renormalized oxygen vibrations transmuted into excitations of the pseudogap. The observed strong dependence of gap to transition temperature ratio on Tc also emerges from this field theory. This work suggests that including vibrational degrees of freedom is key to developing a complete understanding of the cuprates.

Direct spectroscopic evidence for phase competition between the pseudogap and superconductivity in Bi2Sr2CaCu2O(8+δ)

In the high-temperature (T(c)) cuprate superconductors, a growing body of evidence suggests that the pseudogap phase, existing below the pseudogap temperature T*, is characterized by some broken electronic symmetries distinct from those associated with superconductivity. In particular, recent scattering experiments have suggested that charge ordering competes with superconductivity. However, no direct link of an interplay between the two phases has been identified from the important low-energy excitations. Here, we report an antagonistic singularity at T(c) in the spectral weight of Bi2Sr2CaCu2O(8+δ) as compelling evidence for phase competition, which persists up to a high hole concentration p ~ 0.22. Comparison with theoretical calculations confirms that the singularity is a signature of competition between the order parameters for the pseudogap and superconductivity. The observation of the spectroscopic singularity at finite temperatures over a wide doping range provides new insights into the nature of the competitive interplay between the two orders and the complex phase diagram near the pseudogap critical point.

Charge order and its connection with Fermi-liquid charge transport in a pristine high-T(c) cuprate

Electronic inhomogeneity appears to be an inherent characteristic of the enigmatic cuprate superconductors. Here we report the observation of charge-density-wave correlations in the model cuprate superconductor HgBa2CuO(4+δ) (T(c)=72 K) via bulk Cu L3-edge-resonant X-ray scattering. At the measured hole-doping level, both the short-range charge modulations and Fermi-liquid transport appear below the same temperature of about 200 K. Our result points to a unifying picture in which these two phenomena are preceded at the higher pseudogap temperature by q=0 magnetic order and the build-up of significant dynamic antiferromagnetic correlations. The magnitude of the charge modulation wave vector is consistent with the size of the electron pocket implied by quantum oscillation and Hall effect measurements for HgBa2CuO(4+δ) and with corresponding results for YBa2Cu3O(6+δ), which indicates that charge-density-wave correlations are universally responsible for the low-temperature quantum oscillation phenomenon.

Pseudogap in cuprates in the loop-current ordered state

Scanning tunneling microscopy (STM) has revealed that the magnitude of the pseudo-gap in under-doped cuprates varies spatially and is correlated with disorder. The loop-current order, characterized by the anapole vector Ω, discovered in under-doped cuprates occurs in the same region of the temperature and doping as the pseudo gap observed in STM and ARPES experiments. Since translational symmetry remains unchanged in the pure limit, no gap occurs at the chemical potential. On the other hand for disorder coupling linearly to the different possible orientations of Ω, there can only be a finite temperature dependent static correlation length for the loop-current state at any temperature. This leads to formation of domains of the ordered state with different orientation and magnitude of Ω in each. For the characteristic size of the domains much larger than the Fermi-vectors [Formula: see text], the boundary of the domains leads to forward scattering of the Fermions. Such forward scattering is shown to push states near the chemical potential to energies both above and below it leading to a pseudo-gap with an angular dependence which is maximum in the [Formula: see text] directions because the single-particle energies are degenerate in these directions for all domains. The magnitude of the average gap systematically increases with the square of the average loop order parameter measured by polarized neutron scattering. This result is tested. A unique result of the gap due to forward scattering is the lack of a bump in the density of states at the 'edge' of the pseudo-gap so that the depletion of states near the chemical potential is recovered only in integration up to the edge of the band. This is also in agreement with a variety of experiments. Some predictions for further experiments are provided. Due to the finite correlation length, low frequency excitations are expected at long wavelength at all temperatures in the 'ordered' phase. Such fluctuations motionally average over the shifts in frequencies of local probes such as NMR and muon resonance expected for a truly static order.

In-plane magnetoresistance obeys Kohler's rule in the pseudogap phase of cuprate superconductors

We report in-plane resistivity (ρ) and transverse magnetoresistance (MR) measurements for underdoped HgBa(2)CuO(4+δ) (Hg1201). Contrary to the long-standing view that Kohler's rule is strongly violated in underdoped cuprates, we find that it is in fact satisfied in the pseudogap phase of Hg1201. The transverse MR shows a quadratic field dependence, δρ/ρ(0)=aH(2), with a(T)∝T(-4). In combination with the observed ρ∝T(2) dependence, this is consistent with a single Fermi-liquid quasiparticle scattering rate. We show that this behavior is typically masked in cuprates with lower structural symmetry or strong disorder effects.

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.

Quantum spin Hall effect in a three-orbital tight-binding Hamiltonian

We consider the quantum spin hall (QSH) state in a three-orbital model, with a certain spin loop current order which is induced by spin-dependent interactions. This type of order is motivated by the loop current model, which was proposed long ago to describe the pseudogap phase of cuprates. It is shown that this model has nontrivial Chern parity by directly counting the zeros of the Pfaffian of the time reversal operator. By connecting to the second Chern number, we explicitly show the property of singularities of the wave-functions, and also their dependance on the gauge choices. Also it is shown that the Berry phase of the QSH state can be mapped to a non-Abelian instanton.

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.

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.

Phase diagram of a three-orbital model for high-Tc cuprate superconductors

We study the phase diagram of an effective three-orbital model of the cuprates using variational Monte Carlo calculations on asymptotically large lattices and exact diagonalization on a 24-site cluster. States with ordered orbital current loops (LC), itinerant antiferromagnetism, d-wave superconductivity, and the Fermi liquid are investigated using appropriate Slater determinants refined by Jastrow functions for on-site and intersite correlations. We find an LC state stable in the thermodynamic limit for a range of parameters compatible with the Fermi surface of a typical hole doped superconductor provided the transfer integrals between the oxygen atoms have signs determined by the effects of indirect transfer through the Cu-4s orbitals as suggested by Andersen. The results of the calculations are that the LC phase gives way at lower dopings to an antiferromagnetism phase, and at larger dopings to superconductivity and Fermi liquid phases.

Evidence of chiral order in the charge-ordered phase of superconducting La1.875Ba0.125Cuo4 single crystals using polar Kerr-effect measurements

High resolution polar Kerr effect measurements were performed on La1.875Ba0.125CuO4 single crystals revealing that a finite Kerr signal is measured below an onset temperature TK that coincides with the charge ordering transition temperature TCO. We further show that the sign of the Kerr signal cannot be trained with the magnetic field, is found to be the same on opposite sides of the same crystal, and is odd with respect to strain in the diagonal direction of the unit cell. These observations are consistent with a chiral "gyrotropic" order above Tc for La1.875Ba0.125CuO4; similarities to other cuprates suggest that it is a universal property in the pseudogap regime.

Charge order driven by Fermi-arc instability in Bi2Sr(2-x)La(x)CuO(6+δ)

The understanding of the origin of superconductivity in cuprates has been hindered by the apparent diversity of intertwining electronic orders in these materials. We combined resonant x-ray scattering (REXS), scanning-tunneling microscopy (STM), and angle-resolved photoemission spectroscopy (ARPES) to observe a charge order that appears consistently in surface and bulk, and in momentum and real space within one cuprate family, Bi2Sr(2-x)La(x)CuO(6+δ). The observed wave vectors rule out simple antinodal nesting in the single-particle limit but match well with a phenomenological model of a many-body instability of the Fermi arcs. Combined with earlier observations of electronic order in other cuprate families, these findings suggest the existence of a generic charge-ordered state in underdoped cuprates and uncover its intimate connection to the pseudogap regime.

Superconducting pairing and the pseudogap in the nematic dynamical stripe phase of La2-xSrxCuO4

Fully absorption coefficient corrected Raman spectra were obtained in La2-xSrxCuO4. The B1g spectra have a Fleury-Loudon type two-magnon peak (resonant term) whose energy decreases from 3180 cm(-1) (394 meV) to 440 cm(-1) (55 meV) on increasing the carrier density from x = 0 to 0.25, while the B2g spectra have a 1000-3500 cm(-1) (124-434 meV) hump (hill) whose lower-edge energy increases from x = 0 to 0.115 and then stays constant to x = 0.25. The B2g hump is assigned to the electronic scattering (non-resonant term) of the spectral function with magnetic self-energy. The completely different carrier density dependence arises from anisotropic magnetic excitations of spin-charge stripes. The B1g spectra were assigned to the sum of k ∥  and k⊥ stripe excitations and the B2g spectra to k⊥ stripe excitations according to the calculation by Seibold and Lorenzana (2006 Phys. Rev. B 73 144515). The k ∥  and k⊥ stripe excitations in fluctuating spin-charge stripes were separately detected for the first time. The appearance of only k⊥ stripe excitations in the electronic scattering arises from the charge hopping perpendicular to the stripe. This is the same direction as the Burgers vector of the edge dislocation in metal. The successive charge hopping in the Burgers vector direction across the charge stripes may cause Cooper pairs as predicted by Zaanen et al (2004 Ann. Phys. 310 181). Indeed, this is supported by the experimental fact that the superconducting coherent length coincides with the inter-charge stripe distance in the wide carrier density range. The one-directional charge hopping perpendicular to the stripe causes the flat Fermi surface and the pseudogap near (π,0) and (0,π), but the states around (π/2,π/2) cannot be produced. The low-energy Raman scattering disclosed that the electronic states at the Fermi arc around (π/2,π/2) are coupled to the A1g soft phonon of the tetragonal-orthorhombic phase transition. This suggests that the Fermi arc is produced by the electron-phonon interaction. All the present Raman data suggest that Cooper pairs are formed at moving edge dislocations of dynamical charge stripes.

Absence of static loop-current magnetism at the apical oxygen site in HgBa2CuO4+δ from NMR

The simple structure of HgBa(2)CuO(4+δ) (Hg1201) is ideal among cuprates for study of the pseudogap phase as a broken symmetry state. We have performed (17)O nuclear magnetic resonance on an underdoped Hg1201 crystal with a transition temperature of 74 K to look for circulating loop currents proposed theoretically and inferred from neutron scattering. The narrow spectra preclude static local fields in the pseudogap phase at the apical site, suggesting that the moments observed with neutrons are fluctuating. The nuclear magnetic resonance frequency shifts are consistent with a dipolar field from the Cu(2+) site.

Proposed chiral texture of the magnetic moments of unit-cell loop currents in the pseudogap phase of cuprate superconductors

We propose a novel chiral order parameter to explain the unusual polar Kerr effect in underdoped cuprates. It is based on the loop-current model by Varma, which is characterized by the in-plane anapole moment N and exhibits the magnetoelectric effect. We propose a helical structure where the vector N(n) in the layer n is twisted by the angle π/2 relative to N(n-1), thus breaking inversion symmetry. We show that coupling between magnetoelectric terms in the neighboring layers for this structure produces optical gyrotropy, which results in circular dichroism and the polar Kerr effect.

Universal sheet resistance and revised phase diagram of the cuprate high-temperature superconductors

Upon introducing charge carriers into the copper-oxygen sheets of the enigmatic lamellar cuprates, the ground state evolves from an insulator to a superconductor and eventually to a seemingly conventional metal (a Fermi liquid). Much has remained elusive about the nature of this evolution and about the peculiar metallic state at intermediate hole-carrier concentrations (p). The planar resistivity of this unconventional metal exhibits a linear temperature dependence (ρ ∝ T) that is disrupted upon cooling toward the superconducting state by the opening of a partial gap (the pseudogap) on the Fermi surface. Here, we first demonstrate for the quintessential compound HgBa2CuO4+δ a dramatic switch from linear to purely quadratic (Fermi liquid-like, ρ ∝ T(2)) resistive behavior in the pseudogap regime. Despite the considerable variation in crystal structures and disorder among different compounds, our result together with prior work gives insight into the p-T phase diagram and reveals the fundamental resistance per copper-oxygen sheet in both linear (ρ = A1T) and quadratic (ρ = A2T(2)) regimes, with A1 ∝ A2 ∝ 1/p. Theoretical models can now be benchmarked against this remarkably simple universal behavior. Deviations from this underlying behavior can be expected to lead to new insight into the nonuniversal features exhibited by certain compounds.

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.

Bounding the pseudogap with a line of phase transitions in YBa2Cu3O6+δ

Close to optimal doping, the copper oxide superconductors show 'strange metal' behaviour, suggestive of strong fluctuations associated with a quantum critical point. Such a critical point requires a line of classical phase transitions terminating at zero temperature near optimal doping inside the superconducting 'dome'. The underdoped region of the temperature-doping phase diagram from which superconductivity emerges is referred to as the 'pseudogap' because evidence exists for partial gapping of the conduction electrons, but so far there is no compelling thermodynamic evidence as to whether the pseudogap is a distinct phase or a continuous evolution of physical properties on cooling. Here we report that the pseudogap in YBa2Cu3O6+δ is a distinct phase, bounded by a line of phase transitions. The doping dependence of this line is such that it terminates at zero temperature inside the superconducting dome. From this we conclude that quantum criticality drives the strange metallic behaviour and therefore superconductivity in the copper oxide superconductors.

Time-resolved optical reflectivity of the electron-doped Nd(2-x)Ce(x)CuO(4+δ) cuprate superconductor: evidence for an interplay between competing orders

We use pump-probe spectroscopy to measure the photoinduced reflectivity ΔR of the electron-doped cuprate superconductor Nd(2-x)Ce(x)CuO(4+δ) at a value of x near optimal doping, as a function of time, temperature, and laser fluence. We observe the onset of a negative ΔR signal at T(*)≈75 K, above the superconducting transition temperature, T(c), of 23 K. The relatively slow decay of ΔR, compared to the analogous signal in hole doped compounds, allows us to resolve time-temperature scaling consistent with critical fluctuations. A positive ΔR signal onsets at T(c) that we associate with superconducting order. We find that the two signals are strongly coupled below T(c), in a manner that suggests a repulsive interaction between superconductivity and another fluctuating order.