D-density Wave Research Articles

A model of d-wave superconductivity, antiferromagnetism, and charge order on the square lattice

We describe the confining instabilities of a proposed quantum spin liquid underlying the pseudogap metal state of the hole-doped cuprates. The spin liquid can be described by a SU(2) gauge theory of Nf = 2 massless Dirac fermions carrying fundamental gauge charges-this is the low-energy theory of a mean-field state of fermionic spinons moving on the square lattice with π-flux per plaquette in the ℤ2 center of SU(2). This theory has an emergent SO(5)f global symmetry and is presumed to confine at low energies to the Néel state. At nonzero doping (or smaller Hubbard repulsion U at half-filling), we argue that confinement occurs via the Higgs condensation of bosonic chargons carrying fundamental SU(2) gauge charges also moving in π ℤ2-flux. At half-filling, the low-energy theory of the Higgs sector has Nb = 2 relativistic bosons with a possible emergent SO(5)b global symmetry describing rotations between a d-wave superconductor, period-2 charge stripes, and the time-reversal breaking "d-density wave" state. We propose a conformal SU(2) gauge theory with Nf = 2 fundamental fermions, Nb = 2 fundamental bosons, and a SO(5)f×SO(5)b global symmetry, which describes a deconfined quantum critical point between a confining state which breaks SO(5)f and a confining state which breaks SO(5)b. The pattern of symmetry breaking within both SO(5)s is determined by terms likely irrelevant at the critical point, which can be chosen to obtain a transition between Néel order and d-wave superconductivity. A similar theory applies at nonzero doping and large U, with longer-range couplings of the chargons leading to charge order with longer periods.

Higher-order topological state induced by d -wave competing orders in high- Tc superconductor based heterostructures

We introduce a two-dimensional Chern insulator in proximity to a $d$-wave pseudogap state of the high-T$_c$ superconducting material as an effective platform to realize the higher order topological system. The proximity-induced $d$-density-wave (DDW) order in the Chern insulator layer serves as an effective mass. The edge states will be fully gapped by this DDW order. In the real space, the sign of the DDW order parameter changes at the system corners due to the $d$-wave factor, leading to the gapless corner states, indicating that this system may be in a higher order topological state. The higher order topology in this coupled system is confirmed based on the calculation of the edge polarization and the quadrupole moment. In the superconducting state where the superconducting order and the DDW order coexist, the Majorana corner states emerge.

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.

Enhanced superconductivity due to d-density waves in cuprates

Abstracts We present a strong-coupling study of the competition and interplay between the d -density wave (DDW) and d -wave superconducting (DSC) states within the fluctuation-exchange approximation for the two-dimensional Hubbard model. We find the occurrence of a new DDW state with phase fluctuations induced by spin fluctuations above T C . It is shown that in this state the density of states has a pseudogap, which is strong in the underdoped regime, but is weak in the overdoped regime. We obtain new results for the full reduction of T C in the strong DDW state, a strong enhancement of T C in the weak DDW state, and the coexistence of the ordered DDW and DSC states below T C . This sheds new light on a comprehensive explanation of both the pseudogap and high- T C superconductivity in cuprates.

Unconventional superconductivity in the strong-coupling limit for the heavy fermion system CeCoIn5

We present scanning tunneling spectroscopy measurements of the local quasiparticles' excitation spectra of the heavy fermion CeCoIn5 between 440mK and 3K in samples with a bulk Tc=2.25K. The spectral shape of our low-temperature tunneling data, quite textbook nodal-Δ conductance, allow us to confidently fit the spectra with a d-wave density of states considering also a shortening of quasiparticles' lifetime term Γ. The Δ(0) value obtained from the fits yields a BCS ratio 2Δ/kTc=7.73 suggesting that CeCoIn5 is an unconventional superconductor in the strong coupling limit. The fits also reveal that the height of coherence peaks in CeCoIn5 is reduced with respect to a pure BCS spectra and therefore the coupling of quasiparticles with spin excitations should play a relevant role. The tunneling conductance shows a depletion at energies smaller than Δ for temperatures larger than the bulk Tc, giving further support to the existence of a pseudogap phase that in our samples span up to T*∼1.2Tc. The phenomenological scaling of the pseudogap temperature observed in various families of cuprates, 2Δ/kT*∼4.3, is not fulfilled in our measurements. This suggests that in CeCoIn5 the strong magnetic fluctuations might conspire to close the local superconducting gap at a smaller pesudogap temperature-scale than in cuprates.

Incommensurate charge ordered states in the t–t′–J model

We study the incommensurate charge ordered states in the model using the Gutzwiller mean field theory on large systems. In particular, we explore the properties of incommensurate charge modulated states referred to as nodal pair density waves (nPDW) in the literature. nPDW states intertwine site and bond charge order with modulated d-wave pair order, and are characterized by a nonzero amplitude of uniform pairing; they also manifest a dominant intra-unit cell d-density wave form factor. To compare with a recent scanning tunneling microscopy (STM) study (Hamidian et al 2015 Nat. Phys. 12 150) of the cuprate superconductor BSCCO-2212, we compute the continuum local density of states (LDOS) at a typical STM tip height using the Wannier function based approach. By Fourier transforming Cu and O sub-lattice LDOS we also obtain bias-dependent intra-unit cell form factors and spatial phase difference. We find that in the nPDW state the behavior of form factors and spatial phase difference as a function of energy agrees remarkably well with the experiment.This is in contrast to commensurate charge modulated states, which we show do not agree with experiment. We propose that the nPDW states are good candidates for the charge density wave phase observed in the superconducting state of underdoped cuprates.

Staggered Flux State in Two-Dimensional Hubbard Models

The stability and other properties of a staggered flux (SF) state or a correlated d-density wave state are studied for the Hubbard (t-t'-U) model on extended square lattices, as a low-lying state that competes with the d(x2-y2)-wave superconductivity (d-SC) and possibly causes the pseudogap phenomena in underdoped high-Tc cuprates and organic kappa-BEDT-TTF salts. In calculations, a variational Monte Carlo method is used. In the trial wave function, a configuration-dependent phase factor, which is vital to treat a current-carrying state for a large U/t, is introduced in addition to ordinary correlation factors. Varying U/t, t'/t, and the doping rate (delta) systematically, we show that the SF state becomes more stable than the normal state (projected Fermi sea) for a strongly correlated (U/t\gtrsim 5) and underdoped (delta\lesssim 0.16) area. The decrease in energy is sizable, particularly in the area where Mott physics prevails and the circular current (order parameter) is strongly suppressed. These features are consistent with those for the t-J model. The effect of the frustration t'/t plays a crucial role in preserving charge homogeneity and appropriately describing the behavior of hole- and electron-doped cuprates and kappa-BEDT-TTF salts. We argue that the SF state does not coexist with d-SC and is not a `normal state' from which d-SC arises. We also show that a spin current (flux or nematic) state is never stabilized in the same regime.

D-Density Wave Scenario Description of the New Hidden Charge Order in Cuprates

In this paper, we show that the theory of high $T_c$ superconductivity based on a microscopic model with $d$-density wave (DDW) scenario in the pseudogap phase is able to reproduce some of the most important features of the recent experimentally discovered hidden charge order in several families of Cuprates. In particular, by computing and comparing energies of charge orders of different modulation directions derived from a full microscopic theory with $d$-density wave scenario, the axial charge order $\phi_{X(Y)}$ with wavevector $\mathbf{Q}=(Q_0,0)((0,Q_0))$ is shown to be unambiguously energetically more favorable over the diagonal charge order $\phi_{X\pm Y}$ with wavevector $\mathbf{Q}=(Q_0,\pm Q_0)$ at least in commensurate limit, to be expected also to hold even to more general incommensurate case, in agreement with experiment. The two types of axial charge order $\phi_X$ and $\phi_Y$ are degenerate by symmetry. We find that within the superconducting background, biaxial (checkerboard) charge order is energetically more favorable than uniaxial (stripe) charge order, and therefore checkerboard axial charge order should be the one observed in experiments, assuming a single domain of charge ordered state on each CuO$_2$ plane.

Superconducting and DDW states of high-Tccuprates with Rashba and Dresselhaus spin-orbit couplings

Abstractauthoren The effects of the spin–orbit couplings (SOC) including the Rashba and the Dresselhaus interactions are considered for the high cuprates with superconducting gap (DSC) and the pseudogap (d-density wave, DDW) phase. The sole Rashba or the Dresselhaus interaction is shown to have a stronger effect on the superconducting phase when compared to the mixed SOC. The temperature dependencies of the DSC and DDW parameters along with the specific heat of both the underdoped and the overdoped cuprates were considered.

Calculation for polar Kerr effect in high-temperature cuprate superconductors

A mechanism is proposed for the tantalizing evidence of polar Kerr effect in a class of high temperature superconductors--the signs of the Kerr angle from two opposite faces of the same sample are identical and magnetic field training is non-existent. The mechanism does not break global time reversal symmetry, as in an antiferromagnet, and results in zero Faraday effect. It is best understood in a phenomenological model of bilayer cuprates, such as YBCO, in which intra-bilayer tunneling nucleates a chiral d-density wave such that the individual layers have opposite chirality. Although specific to the chiral d-density wave, the mechanism may be more general to any quasi-two-dimensional orbital antiferromagnet in which time reversal symmetry is broken in each plane, but not when averaged macroscopically.

Vestigial chiral and charge orders from bidirectional spin-density waves: Application to the iron-based superconductors

Recent experiments in optimally hole-doped iron arsenides have revealed a novel magnetically ordered ground state that preserves tetragonal symmetry, consistent with either a charge-spin density wave (CSDW), which displays a non-uniform magnetization, or a spin-vortex crystal (SVC), which displays a non-collinear magnetization. Here we show that, similarly to the partial melting of the usual stripe antiferromagnet into a nematic phase, either of these phases can also melt in two stages. As a result, intermediate paramagnetic phases with vestigial order appears: a checkerboard charge density-wave for the CSDW ground state, characterized by an Ising-like order parameter, and a remarkable spin-vorticity density-wave for the SVC ground state -- a triplet d-density wave characterized by a vector chiral order parameter. We propose experimentally detectable signatures of these phases, show that their fluctuations can enhance the superconducting transition temperature, and discuss their relevance to other correlated materials.

Instability towards staggered loop currents in the three-orbital model for cuprate superconductors

We present evidence for the existence of a spontaneous instability towards an orbital loop-current phase in a multiorbital Hubbard model for the CuO$_2$ planes in cuprates. Contrary to the previously proposed $\theta_{II}$ phase with intra-unit cell currents, the identified instability is towards a staggered pattern of intertwined current loops. The orbitally resolved current pattern thereby shares its staggered character with the proposal of d-density wave order. The current pattern will cause a Fermi surface reconstruction and the opening of a pseudogap. We argue that the pseudogap phase with time-reversal symmetry breaking currents is susceptible to further phase transitions and therefore offers a route to account for axial incommensurate charge order and a polar Kerr effect in underdoped cuprates.

Stability of Staggered Flux State in d-p Model Studied Using Variational Monte Carlo Method

The stability of a staggered flux (SF) or d-density wave state is studied in a d-p model using a variational Monte Carlo (VMC) method. This state possesses a local circular current and possibly causes the pseudogap phase in high-Tc cuprate superconductors. We introduce into the trial function a configuration dependent phase factor, which was recently shown to be indispensable to stabilize current-carrying states. We pay attention to the energy gain as a function of adjacent O-O hopping ▪ and Cu-site coulomb repulsion U. It is found the SF state becomes stable for hole doping rates of ▪ and ▪, owing to the reduction in kinetic energy, but seems fragile because the energy gain rapidly decays on doping holes for any ▪. The loop current tends to increase as ▪ increases.

Hartree-Fock computation of the high-Tccuprate phase diagram

A computation of the cuprate phase diagram is presented. Adiabatic deformability back to the density function band structure is assumed. Symmetry constraints lead to a fermi liquid theory with 5 interaction parameters. Two of these are forced to zero by experiment. The remaining 3 are fit to (1) the moment of the antiferromagnetic state at half filling, (2) the superconducting gap at optimal doping, and (3) the maximum pseudogap, which I identify as d-density wave. Solution of the Hartree-Fock equations gives, in quantitative agreement with experiment, (1) quantum phase transitions at 5% and 16% p-type doping, (2) insulation below 5%, (3) a d-wave pseudogap quasiparticle spectrum, (4) pseudogap and superconducting gap values as a function of doping, (5) superconducting Tc versus doping, (6) London penetration depth versus doping, (7) spin wave velocity. The fit points to superexchange mediated by the bonding O atom in the Cu-O plane as the causative agent of all three ordering phenomena.

Amplitude mode of thed-density-wave state and its relevance to high-Tccuprates

We calculate the spectrum of the amplitude mode, the analog of the Higgs mode in high energy physics, for the $d$-density wave (DDW) state proposed to describe the pseudogap phase of the high $T_c$ cuprates. Even though the state breaks translational symmetry by a lattice spacing and is described by a particle-hole singlet order parameter at the wave vector $q=Q=(\pi,\pi)$, remarkably, we find that the amplitude mode spectrum can have peaks at both $q=(0,0)$ and $q=Q=(\pi,\pi)$; we shall lattice spacing to unity. In general, the spectrum is non-universal, and, depending on the microscopic parameters, can have one or two peaks in the Brillouin zone, signifying existence of two kinds of magnetic excitations. Our theory sheds important light on how multiple inelastic neutron peaks at different wave vectors can, in principle, arise even with an order parameter that condenses at $Q=(\pi,\pi)$.

Gordian knot in the pseudo‐gap phase of hole‐underdoped cuprates

AbstractWe consider the coexistent d‐density wave (DDW) order, at the anti‐ferromagnetic (AF) wave vector Q = (π,π), representing the pseudo‐gap (PG) state, and d‐wave superconductivity (DSC), driven by an assumed attractive interaction, for the two‐dimensional (2D) fermion system on a square lattice starting with a mean‐field Hamiltonian involving the singlet DDW and the DSC pairings. The two energy gaps, corresponding to PG and DSC, in the excitation spectrum do not merge into one ‘quadrature’ gap if the normal state dispersion is imperfectly nested. With the pinning of the van Hove singularities (vHSs) of the dispersion close to the chemical potential (µ), we show that the PG and DSC are representing two competing orders as the former brings about a depletion of the spectral weight (a contour plot of the spectral weight density for 60 K at the doping level ∼10% is shown in the abstract figure) available for pairing in the anti‐nodal region of momentum space in the coexistent state. The calculation of quasi‐particle thermal conductivity αxx in the DDW and normal phases via the Boltzmann equation in the relaxation‐time approximation shows that there is a mild discontinuity in αxx at the PG transition temperature T*. magnified imageContour plot of the spectral weight density on the first Brillouin zone for the coexisting pseudo‐gap (PG) and superconducting (SC) phases at 60 K. The gap amplitudes are $\Delta _{{\rm PG}}^{{\rm (0)}} /t_{1} $ ∼ 0.024 and $\Delta _{{\rm SC}}^{{\rm (0)}} /t_{1} $ ∼ 0.01, where t1 corresponds to first‐neighbour hopping.

Impurity effects on the d-density wave and superconductivity phase of cuprates

The effect of non-magnetic impurity-scattering is considered on the asymmetric superconducting gap (SC) and the d-density wave (DDW) phase of the high Tc cuprates. The phase diagrams for the cuprates demonstrate that Tc decreases with scattering potential and is most affected in the coexisting phase of DDW and SC orders. Moreover, the SC gap to Tc ratio increases with doping and impurity scattering. The transition temperature of the DDW state, T∗ also decreases with scattering. The density of states in the mixed SC+DDW state for the underdoped (UD) phase, SC state in the overdoped phase and the DDW state in the UD phase, also suffers a modification. Its effect on specific heat results is also discussed. The increase in the gap to Tc ratio and the specific heat are also observed in the presence of the non-magnetic impurities. A suppression in the superfluid density is found with the increase in impurity.

Spin dynamics in the pseudogap state: phase fluctuation versus d-density wave

The dynamical spin susceptibility in the pseudogap phase of high-Tc cuprates is examined for two different scenarios: the phase fluctuation (PF) of pairing and the d-density wave (DDW) order. In the PF scenario, while the resonant feature of the spin response at Q = (π,π) is preserved at weak PF, the feature is lost as the fluctuation is increased. The dispersion of the spin response in the PF scenario is simply a broadened version of the spectrum in the d-wave superconducting (DSC) state at all the strengths of PFs being discussed. In the DDW scenario, there is no spin resonant mode found at the AFM wave vector in nature, and the spin response exhibits the almost dispersionless feature markedly different from that in the PF scenario. As far as the spin dynamics in the pseudogap state is concerned, our results support the scenario that the pseudogap phenomenon is derived from a distinct normal-state order.

Diamagnetic Susceptibility Obtained from the Six-Vertex Model and Its Implications for the High-Temperature Diamagnetic State of Cuprate Superconductors

We study the diamagnetism of the six-vertex model with the arrows as directed bond currents. To our knowledge, this is the first study of the diamagnetism of this model. A special version of this model, called the F model, describes the thermal disordering transition of an orbital antiferromagnet, known as d-density wave, a proposed state for the pseudogap phase of the high-T(c) cuprates. We find that the F model is strongly diamagnetic and the susceptibility may diverge in the high-temperature critical phase with power-law arrow correlations. These results may explain the surprising recent observation of a diverging low-field diamagnetic susceptibility seen in some optimally doped cuprates within the d-density wave model of the pseudogap phase.

Effective field theory and integrability in two-dimensional Mott transition

We study the Mott transition in a two-dimensional lattice spinless fermion model with nearest neighbors density–density interactions. By means of a two-dimensional Jordan–Wigner transformation, the model is mapped onto the lattice XXZ spin model, which is shown to possess a quantum group symmetry as a consequence of a recently found solution of the Zamolodchikov tetrahedron equation. A projection (from three to two space–time dimensions) property of the solution is used to identify the symmetry of the model at the Mott critical point as U q ( sl ( 2 ) ^ ) ⊗ U q ( sl ( 2 ) ^ ) , with deformation parameter q = −1. Based on this result, the low-energy effective field theory for the model is obtained and shown to be a lattice double Chern–Simons theory with coupling constant k = 1 (with the standard normalization). By further employing the effective filed theory methods, we show that the Mott transition that arises is of topological nature, with vortices in an antiferromagnetic array and matter currents characterized by a d-density wave order parameter. We also analyze the behavior of the system upon weak coupling, and conclude that it undergoes a quantum gas–liquid transition which belongs to the Ising universality class.