Polar Kerr-effect measurements of the high-temperature YBa2Cu3O6+x superconductor: evidence for broken symmetry near the pseudogap temperature

Dynamical interplay between superconductivity and pseudogap in cuprates as revealed by terahertz third-harmonic generation spectroscopy

We report on nonlinear terahertz third-harmonic generation (THG) measurements on YBa2Cu3O6+x thin films. Different from conventional superconductors, the THG signal starts to appear in the normal state, which is consistent with the crossover temperature T* of pseudogap over broad doping levels. Upon lowering the temperature, the THG signal shows an anomaly just below Tc in the optimally doped sample. Notably, we observe a beat pattern directly in the measured real-time waveform of the THG signal. We elaborate that the Higgs mode, which develops below Tc, couples to the mode already developed below T*, resulting in an energy level splitting. However, this coupling effect is not evident in underdoped samples. We explore different potential explanations for the observed phenomena. Our research offers valuable insight into the interplay between superconductivity and pseudogap.

Chiral Charge Density Wave and Backscattering-Immune Orbital Texture in Monolayer 1T-TiTe2

Nontrivial electronic states are attracting intense attention in low-dimensional physics. Though chirality has been identified in charge states with a scalar order parameter, its intertwining with charge density waves (CDW), film thickness, and the impact on the electronic behaviors remain less well understood. Here, using scanning tunneling microscopy, we report a 2 × 2 chiral CDW as well as a strong suppression of the Te-5p hole-band backscattering in monolayer 1T-TiTe2. These exotic characters vanish in bilayer TiTe2 in a non-CDW state. Theoretical calculations prove that chirality comes from a helical stacking of the triple-q CDW components and, therefore, can persist at the two-dimensional limit. Furthermore, the chirality renders the Te-5p bands with an unconventional orbital texture that prohibits electron backscattering. Our study establishes TiTe2 as a promising playground for manipulating the chiral ground states at the monolayer limit and provides a novel path to engineer electronic properties from an orbital degree.

Axion optical induction of antiferromagnetic order

Using circularly polarized light to control quantum matter is a highly intriguing topic in physics, chemistry and biology. Previous studies have demonstrated helicity-dependent optical control of chirality and magnetization, with important implications in asymmetric synthesis in chemistry; homochirality in biomolecules; and ferromagnetic spintronics. We report the surprising observation of helicity-dependent optical control of fully compensated antiferromagnetic order in two-dimensional even-layered MnBi₂Te₄, a topological axion insulator with neither chirality nor magnetization. To understand this control, we study an antiferromagnetic circular dichroism, which appears only in reflection but is absent in transmission. We show that the optical control and circular dichroism both arise from the optical axion electrodynamics. Our axion induction provides the possibility to optically control a family of [Formula: see text]-symmetric antiferromagnets ([Formula: see text], inversion; [Formula: see text], time-reversal) such as Cr₂O₃, even-layered CrI₃ and possibly the pseudo-gap state in cuprates. In MnBi₂Te₄, this further opens the door for optical writing of a dissipationless circuit formed by topological edge states.

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.

Fledgling Quantum Spin Hall Effect in Pseudo Gap Phase of Bi2212

We studied the emergence of the quantum spin Hall (QSH) states for the pseudo-gap (PG) phase of Bi2212 bilayer system, assumed to be D-density wave (DDW) ordered, starting with a strong Rashba spin-orbit coupling (SOC) armed, and the time reversal symmetry (TRS) complaint Bloch Hamiltonian. The presence of strong SOC gives rise to non-trivial, spin-momentum locked spin texture tunable by electric field. The emergence of quantum anomalous Hall effect with TRS broken Chiral DDW Hamiltonian of Das Sarma et al. is found to be possible.

Recent Development of Ultrafast Optical Characterizations for Quantum Materials

The advent of intense ultrashort optical pulses spanning a frequency range from terahertz to the visible has opened a new era in the experimental investigation and manipulation of quantum materials. The generation of strong optical field in an ultrashort time scale enables the steering of quantum materials nonadiabatically, inducing novel phenomenon or creating new phases which may not have an equilibrium counterpart. Ultrafast time-resolved optical techniques have provided rich information and played an important role in characterization of the nonequilibrium and nonlinear properties of solid systems. Here, some of the recent progress of ultrafast optical techniques and their applications to the detection and manipulation of physical properties in selected quantum materials are reviewed. Specifically, the new development in the detection of the Higgs mode and photoinduced nonequilibrium response in the study of superconductors by time-resolved terahertz spectroscopy are discussed.

Magnetic memory and spontaneous vortices in a van der Waals superconductor

Doped Mott insulators exhibit some of the most intriguing quantum phases of matter, including quantum spin liquids, unconventional superconductors and non-Fermi liquid metals^1-3. Such phases often arise when itinerant electrons are close to a Mott insulating state, and thus experience strong spatial correlations. Proximity between different layers of van der Waals heterostructures naturally realizes a platform for experimentally studying the relationship between localized, correlated electrons and itinerant electrons. Here we explore this relationship by studying the magnetic landscape of tantalum disulfide 4Hb-TaS₂, which realizes an alternating stacking of a candidate spin liquid and a superconductor⁴. We report on a spontaneous vortex phase whose vortex density can be trained in the normal state. We show that time-reversal symmetry is broken in the normal state, indicating the presence of a magnetic phase independent of the superconductor. Notably, this phase does not generate ferromagnetic signals that are detectable using conventional techniques. We use scanning superconducting quantum interference device microscopy to show that it is incompatible with ferromagnetic ordering. The discovery of this unusual magnetic phase illustrates how combining superconductivity with a strongly correlated system can lead to unexpected physics.

Sagnac interferometer for time-resolved magneto-optical measurements

We introduce a time-resolved magneto-optical measurement technique based on a zero-area Sagnac interferometer. By replacing a continuous wave light source to a pulsed one, temporal resolution of hundreds of picoseconds is achieved. Because two lights passing through a Sagnac loop always travel the same optical path length, the interference from the phase modulation and Kerr rotation occurs in a pulse mode. For illustration of the apparatus, we present ferromagnetic resonance of a Permalloy film caused by a magnetic field pump. The instrument still possesses the favorable properties of a Sagnac interferometer, such as rejection of all the reciprocal effects, and shows 1μrad/Hz sensitivity at a 3 µW optical power in the pulse mode.

Unconventional chiral charge order in kagome superconductor KV3Sb5

Intertwining quantum order and non-trivial topology is at the frontier of condensed matter physics^1-4. A charge-density-wave-like order with orbital currents has been proposed for achieving the quantum anomalous Hall effect^5,6 in topological materials and for the hidden phase in cuprate high-temperature superconductors^7,8. However, the experimental realization of such an order is challenging. Here we use high-resolution scanning tunnelling microscopy to discover an unconventional chiral charge order in a kagome material, KV₃Sb₅, with both a topological band structure and a superconducting ground state. Through both topography and spectroscopic imaging, we observe a robust 2 × 2 superlattice. Spectroscopically, an energy gap opens at the Fermi level, across which the 2 × 2 charge modulation exhibits an intensity reversal in real space, signalling charge ordering. At the impurity-pinning-free region, the strength of intrinsic charge modulations further exhibits chiral anisotropy with unusual magnetic field response. Theoretical analysis of our experiments suggests a tantalizing unconventional chiral charge density wave in the frustrated kagome lattice, which can not only lead to a large anomalous Hall effect with orbital magnetism, but also be a precursor of unconventional superconductivity.

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.

Field-dependent specific heat of the canonical underdoped cuprate superconductor [Formula: see text]

The cuprate superconductor [Formula: see text], in comparison with most other cuprates, has a stable stoichiometry, is largely free of defects and may be regarded as the canonical underdoped cuprate, displaying marked pseudogap behaviour and an associated distinct weakening of superconducting properties. This cuprate 'pseudogap' manifests as a partial gap in the electronic density of states at the Fermi level and is observed in most spectroscopic properties. After several decades of intensive study it is widely believed that the pseudogap closes, mean-field like, near a characteristic temperature, [Formula: see text], which rises with decreasing hole concentration, p. Here, we report extensive field-dependent electronic specific heat studies on [Formula: see text] up to an unprecedented 400 K and show unequivocally that the pseudogap never closes, remaining open to at least 400 K where [Formula: see text] is typically presumed to be about 150 K. We show from the NMR Knight shift and the electronic entropy that the Wilson ratio is numerically consistent with a weakly-interacting Fermion system for the near-nodal states. And, from the field-dependent specific heat, we characterise the impact of fluctuations and impurity scattering on the thermodynamic properties.

Recent progress on superconductors with time-reversal symmetry breaking

Superconductivity and magnetism are adversarial states of matter. The presence of spontaneous magnetic fields inside the superconducting state is, therefore, an intriguing phenomenon prompting extensive experimental and theoretical research. In this review, we discuss recent experimental discoveries of unconventional superconductors which spontaneously break time-reversal symmetry and theoretical efforts in understanding their properties. We discuss the main experimental probes and give an extensive account of theoretical approaches to understand the order parameter symmetries and the corresponding pairing mechanisms, including the importance of multiple bands.

A framework for the description of charge order, pseudo and superconducting gap, critical temperature and pairing interaction in cuprate superconductors

A unified phenomenological description framework is proposed for the evaluation of some of the most important observables of the cuprate superconductors: the pseudogap (PG) ΔPG, the local superconducting amplitudes ΔSC(ri), the critical temperatureTcand charge ordering (CO) parameters. Recent detailed measurements of CO structures and CO wavelengthsλCOare faithfully reproduced by solutions of a Cahn-Hilliard differential equation with a free energy potentialVGLthat produces alternating small charge modulations. The charge oscillations induce atomic fluctuations that mediate the SC pair interaction proportional to theVGLamplitude. The local SC amplitude and phaseθiare connected by Josephson couplingEJ(rij) and the SC long-range order transition occurs whenEJ∼kBTc. The calculated results of the wavelengthλCO, ΔPG,ΔSCandTccalculations are in good agreement with a variety of experiments.

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.

Possible Occurrence of Superconductivity by the π-flux Dirac String Formation Due to Spin-Twisting Itinerant Motion of Electrons

We show that the Rashba spin-orbit interaction causes spin-twisting itinerant motion of electrons in metals and realizes the quantized cyclotron orbits of conduction electrons without an external magnetic field. From the view point of the Berry connection, the cause of this quantization is the appearance of a non-trivial Berry connection A fic = − ℏ 2 e ∇ χ ( χ is an angular variable with period 2 π ) that generates π flux (in the units of ℏ = 1 , e = 1 , c = 1 ) inside the nodal singularities of the wave function (a “Dirac string”) along the centers of spin-twisting. Since it has been shown in our previous work that the collective mode of ∇ χ is stabilized by the electron-pairing and generates supercurrent, the π -flux Dirac string created by the spin-twisting itinerant motion will be stabilized by the electron-pairing and produce supercurrent.

Incoherent strange metal sharply bounded by a critical doping in Bi2212

In normal metals, macroscopic properties are understood using the concept of quasiparticles. In the cuprate high-temperature superconductors, the metallic state above the highest transition temperature is anomalous and is known as the "strange metal." We studied this state using angle-resolved photoemission spectroscopy. With increasing doping across a temperature-independent critical value p _c ~ 0.19, we observed that near the Brillouin zone boundary, the strange metal, characterized by an incoherent spectral function, abruptly reconstructs into a more conventional metal with quasiparticles. Above the temperature of superconducting fluctuations, we found that the pseudogap also discontinuously collapses at the very same value of p _c These observations suggest that the incoherent strange metal is a distinct state and a prerequisite for the pseudogap; such findings are incompatible with existing pseudogap quantum critical point scenarios.

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.

Time Reversal Symmetry Breaking Superconductors: Sr2RuO4 and Beyond

Recent work done on the time reversal symmetry (TRS) breaking superconductors is reviewed in this paper. The special attention is paid to Sr 2 RuO 4 believed to be spin triplet chiral p-wave superconductor which break TRS and is expected to posses non-trivial topological properties. The family of TRS breaking superconductors is growing relatively fast, with many of its newly discovered members being non-centrosymmetric. However not only Sr 2 RuO 4 but also many other superconductors which possess center of inversion also break TRS. The TRS is often identified by means of the muon spin relaxation ( μ SR) and the Kerr effect. Both methods effectively measure the appearance of the spontaneous bulk magnetic field below superconducting transition temperature. This compound provides an example of the material whose many band, multi-condensate modeling has enjoyed a number of successes, but the full understanding has not been achieved yet. We discuss in some details the properties of the material. Among them is the Kerr effect and by understanding has resulted in the discovery of the novel mechanism of the phenomenon. The mechanism is universal and thus applicable to all systems with multi-orbital character of states at the Fermi energy.

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.

Scale-invariant magnetoresistance in a cuprate superconductor

The anomalous metallic state in the high-temperature superconducting cuprates is masked by superconductivity near a quantum critical point. Applying high magnetic fields to suppress superconductivity has enabled detailed studies of the normal state, yet the direct effect of strong magnetic fields on the metallic state is poorly understood. We report the high-field magnetoresistance of thin-film La_2-x Sr _x CuO₄ cuprate in the vicinity of the critical doping, 0.161 ≤ p ≤ 0.190. We find that the metallic state exposed by suppressing superconductivity is characterized by magnetoresistance that is linear in magnetic fields up to 80 tesla. The magnitude of the linear-in-field resistivity mirrors the magnitude and doping evolution of the well-known linear-in-temperature resistivity that has been associated with quantum criticality in high-temperature superconductors.

Three-Dimensional Fermi Surface of Overdoped La-Based Cuprates

We present a soft x-ray angle-resolved photoemission spectroscopy study of overdoped high-temperature superconductors. In-plane and out-of-plane components of the Fermi surface are mapped by varying the photoemission angle and the incident photon energy. No k_{z} dispersion is observed along the nodal direction, whereas a significant antinodal k_{z} dispersion is identified for La-based cuprates. Based on a tight-binding parametrization, we discuss the implications for the density of states near the van Hove singularity. Our results suggest that the large electronic specific heat found in overdoped La_{2-x}Sr_{x}CuO_{4} cannot be assigned to the van Hove singularity alone. We therefore propose quantum criticality induced by a collapsing pseudogap phase as a plausible explanation for observed enhancement of electronic specific heat.

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.

Intertwining Topological Order and Broken Symmetry in a Theory of Fluctuating Spin-Density Waves

The pseudogap metal phase of the hole-doped cuprate superconductors has two seemingly unrelated characteristics: a gap in the electronic spectrum in the "antinodal" region of the square lattice Brillouin zone and discrete broken symmetries. We present a SU(2) gauge theory of quantum fluctuations of magnetically ordered states which appear in a classical theory of square lattice antiferromagnets, in a spin-density wave mean field theory of the square lattice Hubbard model, and in a CP^{1} theory of spinons. This theory leads to metals with an antinodal gap and topological order which intertwines with the observed broken symmetries.

Mottness at finite doping and charge-instabilities in cuprates

The influence of the Mott physics on the doping-temperature phase diagram of copper oxides represents a major issue that is subject of intense theoretical and experimental effort. Here, we investigate the ultrafast electron dynamics in prototypical single-layer Bi-based cuprates at the energy scale of the O-2p→Cu-3d charge-transfer (CT) process. We demonstrate a clear evolution of the CT excitations from incoherent and localized, as in a Mott insulator, to coherent and delocalized, as in a conventional metal. This reorganization of the high-energy degrees of freedom occurs at the critical doping p_cr ≈0.16 irrespective of the temperature, and it can be well described by dynamical mean field theory calculations. We argue that the onset of the low-temperature charge instabilities is the low-energy manifestation of the underlying Mottness that characterizes the p < p_cr region of the phase diagram. This discovery sets a new framework for theories of charge order and low-temperature phases in underdoped copper oxides.

Neutron scattering by Dirac multipoles

Scattering by magnetic charge formed by Dirac multipoles that are magnetic and polar is examined in the context of materials with properties that challenge conventional concepts. An order parameter composed of Dirac quadrupoles has been revealed in the pseudo-gap phase of ceramic, high-T _c superconductors on the basis of Kerr effect and magnetic neutron Bragg diffraction measurements. Construction of Dirac quadrupoles that emerge from centrosymmetric sites used by Cu ions in the ceramic superconductor Hg1201 is illustrated, together with selection rules for excitations that will feature in neutron inelastic scattering, and RIXS experiments. We report magnetic scattering amplitudes for diffraction by polar multipoles that have universal value, because they are not specific to ceramic superconductors. To illustrate this attribute, we consider neutron Bragg diffraction from a magnetically ordered iridate (Sr₂IrO₄) and discuss shortcomings in published interpretations of diffraction data.

Kerr Effect from Diffractive Skew Scattering in Chiral p_{x}±ip_{y} Superconductors

We calculate the temperature dependent anomalous ac Hall conductance σ_{H}(Ω,T) for a two-dimensional chiral p-wave superconductor. This quantity determines the polar Kerr effect, as it was observed in Sr_{2}RuO_{4} [J. Xia et al., Phys. Rev. Lett. 97, 167002 (2006)]. We concentrate on a single band model with an arbitrary isotropic dispersion relation subjected to rare, weak impurities treated in the Born approximation. As we explicitly show by detailed computation, previously omitted contributions to the extrinsic part of an anomalous Hall response, physically originating from diffractive skew scattering on quantum impurity complexes, appear to the leading order in the impurity concentration. By direct comparison with published results from the literature we demonstrate the relevance of our findings for the interpretation of the Kerr effect measurements in superconductors.

Nanoscale electrodynamics of strongly correlated quantum materials

Electronic, magnetic, and structural phase inhomogeneities are ubiquitous in strongly correlated quantum materials. The characteristic length scales of the phase inhomogeneities can range from atomic to mesoscopic, depending on their microscopic origins as well as various sample dependent factors. Therefore, progress with the understanding of correlated phenomena critically depends on the experimental techniques suitable to provide appropriate spatial resolution. This requirement is difficult to meet for some of the most informative methods in condensed matter physics, including infrared and optical spectroscopy. Yet, recent developments in near-field optics and imaging enabled a detailed characterization of the electromagnetic response with a spatial resolution down to 10 nm. Thus it is now feasible to exploit at the nanoscale well-established capabilities of optical methods for characterization of electronic processes and lattice dynamics in diverse classes of correlated quantum systems. This review offers a concise description of the state-of-the-art near-field techniques applied to prototypical correlated quantum materials. We also discuss complementary microscopic and spectroscopic methods which reveal important mesoscopic dynamics of quantum materials at different energy scales.

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.

Electrical and thermal transport properties of the electron-doped cuprate Sm2-x Ce x CuO4-y system

Electrical and thermal transport measurements were performed on thin films of the electron-doped superconductor Sm_2-x Ce _x CuO_4-y (x  =  0.13  -  0.19) in order to study the evolving nature of the charge carriers from the under-doped to over-doped regime. A temperature versus cerium content (T  -  x) phase diagram has been constructed from the electrical transport measurements, yielding a superconducting region similar to that found for other electron-doped superconductors. Thermopower measurements show a dramatic change from the underdoped region (x  <  0.15) to the overdoped region (x  >  0.15). Application of the Fisher-Fisher-Huse (FFH) vortex glass scaling model to the magnetoresistance data was found to be insufficient to describe the data in the region of the vortex-solid to vortex-liquid transition. It was found instead that the modified vortex glass scaling model of Rydh, Rapp, and Anderson provided a good description of the data, indicating the importance of the applied field on the pinning landscape. A magnetic field versus temperature (H  -  T) phase diagram has also been constructed for the films with [Formula: see text], displaying the evolution of the vortex glass melting lines H _g (T) across the superconducting regime.

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

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

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.

Quantum-critical fluctuations in 2D metals: strange metals and superconductivity in antiferromagnets and in cuprates

The anomalous transport and thermodynamic properties in the quantum-critical region, in the cuprates, and in the quasi-two dimensional Fe-based superconductors and heavy-fermion compounds, have the same temperature dependences. This can occur only if, despite their vast microscopic differences, a common statistical mechanical model describes their phase transitions. The antiferromagnetic (AFM)-ic models for the latter two, just as the loop-current model for the cuprates, map to the dissipative XY model. The solution of this model in (2+1)D reveals that the critical fluctuations are determined by topological excitations, vortices and a variety of instantons, and not by renormalized spin-wave theories of the Landau-Ginzburg-Wilson type, adapted by Moriya, Hertz and others for quantum-criticality. The absorptive part of the fluctuations is a separable function of momentum [Formula: see text], measured from the ordering vector, and of the frequency ω and the temperature T which scale as [Formula: see text] at criticality. Direct measurements of the fluctuations by neutron scattering in the quasi-two-dimensional heavy fermion and Fe-based compounds, near their antiferromagnetic quantum critical point, are consistent with this form. Such fluctuations, together with the vertex coupling them to fermions, lead to a marginal fermi-liquid, with the imaginary part of the self-energy [Formula: see text] for all momenta, a resistivity [Formula: see text], a [Formula: see text] contribution to the specific heat, and other singular fermi-liquid properties common to these diverse compounds, as well as to d-wave superconductivity. This is explicitly verified, in the cuprates, by analysis of the pairing and the normal self-energy directly extracted from the recent high resolution angle resolved photoemission measurements. This reveals, in agreement with the theory, that the frequency dependence of the attractive irreducible particle-particle vertex in the d-wave channel is the same as the irreducible particle-hole vertex in the full symmetry of the lattice.

Anomalous Hall Effect in a 2D Rashba Ferromagnet

Skew scattering on rare impurity configurations is shown to dominate the anomalous Hall effect in a 2D Rashba ferromagnet. The mechanism originates in scattering on rare impurity pairs separated by distances of the order of the Fermi wavelength. The corresponding theoretical description goes beyond the conventional noncrossing approximation. The mechanism provides the only contribution to the anomalous Hall conductivity in the most relevant metallic regime and strongly modifies previously obtained results for lower energies in the leading order with respect to impurity strength.

Necessity of Time-Reversal Symmetry Breaking for the Polar Kerr Effect in Linear Response

We show that, measured in a backscattering geometry, the polar Kerr effect is absent if the nonlocal electromagnetic response function respects Onsager symmetry, characteristic of thermodynamic states that preserve time-reversal symmetry. A key element is an expression for the reflectivity tensor in terms of the retarded Green's function.

Magnetostriction and Magnetostructural Domains in Antiferromagnetic YBa2Cu3O6

We use high-resolution neutron Larmor diffraction and capacitative dilatometry to investigate spontaneous and forced magnetostriction in undoped, antiferromagnetic YBa_{2}Cu_{3}O_{6.0}, the parent compound of a prominent family of high-temperature superconductors. Upon cooling below the Néel temperature T_{N}=420  K, Larmor diffraction reveals the formation of magnetostructural domains of characteristic size ∼240  nm. In the antiferromagnetic state, dilatometry reveals a minute (4×10^{-6}) orthorhombic distortion of the crystal lattice in external magnetic fields. We attribute these observations to exchange striction and spin-orbit coupling induced magnetostriction, respectively, and show that they have an important influence on the thermal and charge transport properties of undoped and lightly doped cuprates.

Ferro-type order of magneto-electric quadrupoles as an order-parameter for the pseudo-gap phase of a cuprate superconductor

There is general agreement within the community of researchers that investigate high-Tc materials that it is most important to understand the pseudo-gap phase. To this end, many experiments on various cuprates have been reported. Two prominent investigations-Kerr effect and neutron Bragg diffraction-imply that underdoped YBCO samples possess long-range magnetic order of an unusual kind. However, other measurements do not support the existence of magnetic order. Here we show that the Kerr effect and magnetic Bragg diffraction data are individual manifestations of ordered magneto-electric quadrupoles at Cu sites. While the use of magneto-electric multipoles is new in studies of the electronic properties of cuprates, they are not unknown in other materials, including an investigation with x-rays of the parent compound CuO. We exploit the recent prediction that neutrons are deflected by magneto-electric multipoles. The outcome of our study is a theory for the order-parameter of the pseudo-gap phase without the aforementioned conflict with other measurements, and the first experimental evidence that neutrons interact with multipoles belonging to a state of magnetic charge.

Specific heat and sound velocity at the relevant competing phase of high-temperature superconductors

Recent highly accurate sound velocity measurements reveal a phase transition to a competing phase in YBa2Cu3O6+δ that is not identified in available specific heat measurements. We show that this signature is consistent with the universality class of the loop current-ordered state when the free-energy reduction is similar to the superconducting condensation energy, due to the anomalous fluctuation region of such a transition. We also compare the measured specific heat with some usual types of transitions, which are observed at lower temperatures in some cuprates, and find that the upper limit of the energy reduction due to them is about 1/40th the superconducting condensation energy.

Coexistence of charge-density-wave and pair-density-wave orders in underdoped cuprates

We analyze incommensurate charge-density-wave (CDW) and pair-density-wave (PDW) orders with transferred momenta (±Q,0)/(0,±Q) in underdoped cuprates within the spin-fermion model. Both orders appear due to an exchange of spin fluctuations before magnetic order develops. We argue that the ordered state with the lowest energy has nonzero CDW and PDW components with the same momentum. Such a state breaks C_{4} lattice rotational symmetry, time-reversal symmetry, and mirror symmetries. We argue that the feedback from CDW/PDW order on fermionic dispersion is consistent with ARPES data. We discuss the interplay between the CDW/PDW order and d_{x^{2}-y^{2}} superconductivity and make specific predictions for experiments.

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.

Enhancement of superconductivity near a nematic quantum critical point

We consider a low T_{c} metallic superconductor weakly coupled to the soft fluctuations associated with proximity to a nematic quantum critical point (NQCP). We show that (1) a BCS-Eliashberg treatment remains valid outside of a parametrically narrow interval about the NQCP, (2) the symmetry of the superconducting state (d wave, s wave, p wave) is typically determined by the noncritical interactions, but T_{c} is enhanced by the nematic fluctuations in all channels, and (3) in 2D, this enhancement grows upon approach to criticality up to the point at which the weak coupling approach breaks down, but in 3D, the enhancement is much weaker.

From quantum matter to high-temperature superconductivity in copper oxides

The discovery of high-temperature superconductivity in the copper oxides in 1986 triggered a huge amount of innovative scientific inquiry. In the almost three decades since, much has been learned about the novel forms of quantum matter that are exhibited in these strongly correlated electron systems. A qualitative understanding of the nature of the superconducting state itself has been achieved. However, unresolved issues include the astonishing complexity of the phase diagram, the unprecedented prominence of various forms of collective fluctuations, and the simplicity and insensitivity to material details of the 'normal' state at elevated temperatures.

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.

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.

Unconventional symmetries of Fermi liquid and Cooper pairing properties with electric and magnetic dipolar fermions

The rapid experimental progress of ultra-cold dipolar fermions opens up a whole new opportunity to investigate novel many-body physics of fermions. In this article, we review theoretical studies of the Fermi liquid theory and Cooper pairing instabilities of both electric and magnetic dipolar fermionic systems from the perspective of unconventional symmetries. When the electric dipole moments are aligned by the external electric field, their interactions exhibit the explicit d(r(2)-3z(2)) anisotropy. The Fermi liquid properties, including the single-particle spectra, thermodynamic susceptibilities and collective excitations, are all affected by this anisotropy. The electric dipolar interaction provides a mechanism for the unconventional spin triplet Cooper pairing, which is different from the usual spin-fluctuation mechanism in solids and the superfluid (3)He. Furthermore, the competition between pairing instabilities in the singlet and triplet channels gives rise to a novel time-reversal symmetry breaking superfluid state. Unlike electric dipole moments which are induced by electric fields and unquantized, magnetic dipole moments are intrinsic proportional to the hyperfine-spin operators with a Lande factor. Its effects even manifest in unpolarized systems exhibiting an isotropic but spin-orbit coupled nature. The resultant spin-orbit coupled Fermi liquid theory supports a collective sound mode exhibiting a topologically non-trivial spin distribution over the Fermi surface. It also leads to a novel p-wave spin triplet Cooper pairing state whose spin and orbital angular momentum are entangled to the total angular momentum J = 1 dubbed the J-triplet pairing. This J-triplet pairing phase is different from both the spin-orbit coupled (3)He-B phase with J = 0 and the spin-orbit decoupled (3)He-A phase.

Pairing and the phase diagram of the normal coherence length ξN(T, x) above Tc of La(2-x)Sr(x)CuO4 thin films probed by the Josephson effect

The long range proximity effect in high-T_c c-axis Josephson junctions with a high-T_c barrier of lower T_c is still a puzzling phenomenon. It leads to supercurrents in junctions with much thicker barriers than would be allowed by the conventional proximity effect. Here we measured the T − x (Temperature-doping level) phase diagram of the barrier coherence length ξ_N(T, x), and found an enhancement of ξ_N at moderate under-doping and high temperatures. This indicates that a possible origin of the long range proximity effect in the cuprate barrier is the conjectured pre-formed pairs in the pseudogap regime, which increase the length scale over which superconducting correlations survive in the seemingly normal barrier. In more details, we measured the supercurrents I_c of Superconducting - Normal - Superconducting SNS c-axis junctions, where S was optimally doped Y Ba₂Cu₃O_7−δ below T_c (90 K) and N was La_2−xSr_xCuO₄ above its T_c (<25 K) but in the pseudogap regime. From the exponential decay of I_c(T) ∝ exp[−d/ξ_N(T)], where d is the barrier thickness, the ξ_N(T) values were extracted. By repeating these measurements for different barrier doping levels x, the whole phase diagram of ξ_N(T, x) was obtained.

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.