D-wave Form Research Articles

Hartree–Fock with Nambu spinors, and d-wave condensation in the 2D Hubbard model

The usual Hartree–Fock approximation to the Hubbard model is based on eigenstates of the electron number operator. But this formulation is not unique. A different (and inequivalent) version, formulated in terms of Nambu two-component spinors, is based on eigenstates of the difference between the numbers of spin up and spin down electrons, with electron density dependent on parameters U,t,t′ and chemical potential μ. The advantage of this formulation is that electron pairing condensates can be directly computed. We show that in the ground states away from half-filling, obtained in this “Nambu” Hartree–Fock approximation, a discrete rotation symmetry is spontaneously broken, and the condensates exhibit the expected d-wave form in momentum space. We also show that the Mott insulator electron configuration is obtained in this formulation at large U/t and half-filling, and roughly locate the boundary, in the hole doping−U/t plane, between a region of local antiferromagnetism and stripe/domain formation, and the region of d-wave condensation.

Single origin of the nodal and antinodal gaps in cuprates

Recent angle-resolved photoemission electron spectroscopy (ARPES) experiments demonstrate that the momentum dependence of the spectral gap in underdoped cuprates does not follow a pure d-wave form (Anzai H. et al., Nat. Commun., 4 (2013) 1815). This deviation is highly controversial. It has often been interpretated as a proof of the non-superconducting origin of the antinodal gap in the underdoped regime. In this paper, we show that the measured angular dependence of the spectral gap can be explained by the basic nature of pairs in high-Tc cuprates. Hole pairs, or pairons, form as a result of the local antiferromagnetic environment on the scale , the magnetic coherence length. The spatial extension of the pairon wave function beyond first nearest neighbours gives rise to the anomalous angular dependence of the gap, in quantitative agreement with experiments. This simple interpretation strongly indicates the common origin of the nodal and antinodal gaps.

Distinct Superconducting Gap on Two Bilayer-Split Fermi Surface Sheets in Bi2Sr2CaCu2O8+δ Superconductor**Supported by the National Natural Science Foundation of China under Grant No 11888101, the National Key Research and Development Program of China under Grant Nos 2016YFA0300300 and 2017YFA0302900, the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (XDB25000000), the Youth Innovation Promotion Association of CAS under

High resolution laser-based angle-resolved photoemission measurements are carried out on an overdoped superconductor Bi2Sr2CaCu2O8+δ with a Tc of 75 K. Two Fermi surface sheets caused by bilayer splitting are clearly identified with rather different doping levels: the bonding sheet corresponds to a doping level of 0.14, which is slightly underdoped while the antibonding sheet has a doping of 0.27 that is heavily overdoped, giving an overall doping level of 0.20 for the sample. Different superconducting gap sizes on the two Fermi surface sheets are revealed. The superconducting gap on the antibonding Fermi surface sheet follows a standard d-wave form while it deviates from the standard d-wave form for the bonding Fermi surface sheet. The maximum gap difference between the two Fermi surface sheets near the antinodal region is ∼2 meV. These observations provide important information for studying the relationship between the Fermi surface topology and superconductivity, and the layer-dependent superconductivity in high temperature cuprate superconductors.

The driving mechanism of the d-wave orbital order in the iron-based superconductors

We study the driving mechanism and the form of the orbital order in the electronic nematic phase of the iron-based superconductors (IBSs) within the random phase approximation of a 5-band model. We find the magnetic correlation energy of the system can be significantly improved when an orbital order of the d-wave form is spontaneously generated. On the other hand, the magnetic correlation energy increases as one introduce either an on-site or an extended s-wave orbital order. More specifically, we find that the on-site orbital order is disfavored by the Hund’s rule coupling and the extended s-wave orbital order is disfavored by the stripy magnetic correlation pattern in the IBSs. Our work indicates that the orbital order and the spin nematic order in the IBSs develop in a cooperative fashion in the electronic nematic phase of the IBSs and should be both understood as a component of a composite order.

Electron spectral functions in a quantum dimer model for topological metals

We study single electron spectral functions in a quantum dimer model introduced by Punk, Allais and Sachdev (Ref. [1]). The Hilbert space of this model is spanned by hard-core coverings of the square lattice with two types of dimers: ordinary bosonic spin-singlets, as well as fermionic dimers carrying charge +e and spin 1/2, which can be viewed as bound-states of spinons and holons in a doped resonating valence bond (RVB) liquid. This model realizes a metallic phase with topological order and captures several properties of the pseudogap phase in hole-doped cuprates, such as a reconstructed Fermi surface with small hole-pockets and a highly anisotropic quasiparticle residue in the absence of any broken symmetries. Using a combination of exact diagonalization and analytical methods we compute electron spectral functions and show that this model indeed exhibits a sizeable antinodal pseudogap, with a momentum dependence deviating from a simple d-wave form, in accordance with experiments on underdoped cuprates.

Temperature Evolution of Energy Gap and Band Structure in the Superconducting and Pseudogap States of Bi2Sr2CaCu2O8+δ Superconductor Revealed by Laser-Based Angle-Resolved Photoemission Spectroscopy**Supported by the National Key Research and Development Program of China under Grant No 2016YFA0300300, the National Natural Science Foundation of China under Grant No 11334010, the National Basic Research Program of China under Grant No 2015CB921300, and

We carry out detailed momentum-dependent and temperature-dependent measurements on Bi2Sr2CaCu2O8+δ (Bi2212) superconductor in the superconducting and pseudogap states by super-high resolution laser-based angle-resolved photoemission spectroscopy. The precise determination of the superconducting gap for the nearly optimally doped Bi2212 ( K) at low temperature indicates that the momentum-dependence of the superconducting gap deviates from the standard d-wave form ()). It can be alternatively fitted by including a high-order term ()) in which the next nearest-neighbor interaction is considered. We find that the band structure near the antinodal region smoothly evolves across the pseudogap temperature without a signature of band reorganization which is distinct from that found in Bi2Sr2CuO6+δ superconductors. This indicates that the band reorganization across the pseudogap temperature is not a universal behavior in cuprate superconductors. These results provide new insights in understanding the nature of the superconducting gap and pseudogap in high-temperature cuprate superconductors.

Interplay between pair density waves and random field disorders in the pseudogap regime of cuprate superconductors

To capture various experimental results in the pseudogap regime of the underdoped cuprate superconductors for temperature $T<T^{*}$, we propose a four-component pair density wave (PDW) state, in which all components compete with each other. Without random field disorders (RFD), only one of the PDW components survives. If the RFD is included, this state could become phase separated and consist of short range PDW stripes, in which two PDW components coexist but differ in magnitudes, resulting in charge density waves (CDW) and a time-reversal symmetry breaking order, in the form of loop current, as secondary composite orders. We call this phase-separated pair nematic (PSPN) state, which could be responsible for the pseudogap. Using a phenomenological Ginzburg-Landau approach and Monte Carlo simulations, we found that in this state, RFD induces short range static CDW with phase-separated patterns in the directional components and the static CDW is destroyed by thermal phase fluctuations at a crossover temperature $T_{CO}<T^{*}$, above which the CDW becomes dynamically fluctuating. The experimentally found CDW with predominantly d-wave form factor constrains the PDW components to have $s^{\prime}\pm id$ pairing symmetries. We also construct a lattice model and compute the spectral functions for the PSPN state and find good agreement with ARPES results.

Normal-state Nernst effect from bidirectional bond density wave state in high- Tc cuprates

The role of charge order in the phase diagram of high temperature cuprate superconductors has been recently re-emphasized by the experimental discovery of an incipient bi-directional charge density wave (CDW) phase in a class of underdoped cuprates. In a subset of the experiments, the CDW has been found to be accompanied by a d-wave intra-unit-cell form factor, indicating modulation of charge density on the oxygen orbitals sandwiched between neighboring Cu atoms on the CuO planes (the so-called bond-density wave (BDW) phase). Here we take a mean field Q_1=(2\pi/3,0) and Q_2=(0,2\pi/3) bi-directional BDW phase with a d-wave form factor, which closely resembles the experimentally observed charge ordered states in underdoped cuprates, and calculate the Fermi surface topology and the resulting quasiparticle Nernst coefficient as a function of temperature and doping. We establish that, in the appropriate doping ranges where the low temperature phase (in the absence of superconductivity) is a BDW, the Fermi surface consists of an electron and a hole pocket, resulting in a low temperature negative Nernst coefficient as observed in experiments.

Nematic fluctuations and their wave vector in two-dimensional metals

We revisit the problem of electrons on a square lattice below half filling close to an Ising-nematic quantum critical point. For Fermi surfaces with sufficiently strong antinodal nesting, the static nematic susceptibility is maximal at the antinodal nesting wave vector within a simple RPA calculation. We present a detailed analysis of the nematic susceptibility within Eliashberg theory and show that the strong interaction between Fermions in the antinodal regions shifts the maximum of the nematic susceptibility to slightly larger wave vectors. The corresponding order is akin to the incommensurate charge-density wave with d-wave form factor found recently in some underdoped cuprate materials. At sufficiently high temperatures around $T/t \sim 0.1$ nematic fluctuations are strongest at zero wave vector.

The form and origin of orbital ordering in the electronic nematic phase of iron-based superconductors

We investigated the form of orbital ordering in the electronic nematic phase of iron-based superconductors by applying a group theoretical analysis on a realistic five-band model. We find the orbital order can be either of the inter-orbital s-wave form or intra-orbital d-wave form. From the comparison with existing ARPES measurements of band splitting, we find the orbital ordering in the 122 system is dominated by an intra-orbital d-wave component, while that of the 111 system is dominated by an inter-orbital s-wave component. We find both forms of orbital order are strongly entangled with the nematicity in the spin correlation of the system. The condensation energy of the magnetic ordered phase is found to be significantly improved (by more than 20%) when the degeneracy between the (π, 0) and (0, π) ordering pattern is lifted by the orbital order. We argue there should be a large difference in both the scattering rate and the size of the possible pseudogap on the electron pocket around the X = (π, 0) and Y = (0, π) point in the electronic nematic phase. We propose this as a possible origin for the observed nematicity in resistivity measurements.

Signatures of a momentum independent pseudogap in the electronic density of states and Raman spectroscopy of the underdoped cuprates

We propose a hybridization phenomenology to describe the pseudogap state of underdoped cuprates. We show how a momentum independent pseudogap opens asymmetrically from the Fermi-surface but symmetric to the zeroes of the hybridized bonding dispersion, which results in false d-wave characteristics of the pseudogap at the Fermi level. By comparing against a d-wave form factor we illustrate the difficulty in identifying a momentum independent order in momentum averaged quantities such as the electronic Raman response. We identify a suppression in the single-particle density of states which produces a hump feature which should be observable experimentally in tunnelling spectra and distinguishes the s-wave and d-wave ordering scenarios.

Spontaneous Fermi Surface Deformation in the Three-Band Hubbard Model: A Variational Monte Carlo Study

We perform a variational Monte Carlo study on spontaneous d-wave form Fermi surface deformation ($d$FSD) within the three-band Hubbard model. It is found that the variational energy of a projected Fermi sea is lowered by introducing an anisotropy between the hopping integrals along the x and y directions. Our results show that the $d$FSD state has the strongest tendency at half-filling in the absence of magnetism, and disappears as the hole concentration increases to $n_h\approx 1.15$. This is qualitatively in agreement with the mean field analysis and the exact diagonalization calculation for the one-band models, and provides a qualitative explanation to the "intra-unit-cell" electronic nematicity revealed by the scanning tunneling microscopy. An analysis of the dependence of $d$FSD on the parameters of the three-band model indicates that the copper on-site Coulomb interaction, the nearest-neighbor copper-oxygen repulsion, and the charge-transfer energy have a remarkable positive effect on $d$FSD.

Pseudogap, Superconducting Gap, and Fermi Arc in High-TcCuprates Revealed by Angle-Resolved Photoemission Spectroscopy

We present an overview of angle-resolved photoemission spectroscopy (ARPES) studies of high-temperature cuprate superconductors aiming at elucidating the relationship between the superconductivity, the pseudogap, and the Fermi arc. ARPES studies of underdoped samples show a momentum dependence of the energy gap below Tc which deviates from a simple d-wave form, suggesting the coexistence of multiple energy scales in the superconducting state. Hence, two distinct energy scales have been introduced, namely, the gap near the node (characterized by Delta_0) and in the anti-nodal region (characterized by Delta^*). Dichotomy between them has been demonstrated from the material, doping, and temperature dependence of the energy gap. While Delta^* at the same doping level is approximately material independent, Delta_0 shows a strong material dependence tracking the magnitude of Tc,max. The anti-nodal gap does not close at Tc in contrast to the gap near the node which follows something closer to a BCS-like temperature dependence. An effective superconducting gap Delta_sc defined at the end point of the Fermi arc is found to be proportional to Tc's in various materials.

Effect of pairing fluctuations on low-energy electronic spectra in cuprate superconductors

We describe here a minimal theory of tight binding electrons moving on the square planar Cu lattice of the hole-doped cuprates and mixed quantum mechanically with pairs of them (Cooper pairs). Superconductivity occurring at the transition temperature T_c is the long-range, d-wave symmetry phase coherence of these Cooper pairs. Fluctuations necessarily associated with incipient long-range superconducting order have a generic large distance behaviour near T_c. We calculate the spectral density of electrons coupled to such Cooper pair fluctuations and show that features observed in Angle Resolved Photo Emission Spectroscopy (ARPES) experiments on different cuprates above T_c as a function of doping and temperature emerge naturally in this description. These include `Fermi arcs' with temperature-dependent length and an antinodal pseudogap which fills up linearly as the temperature increases towards the pseudogap temperature. Our results agree quantitatively with experiment. Below T_c, the effects of nonzero superfluid density and thermal fluctuations are calculated and compared successfully with some recent ARPES experiments, especially the observed `bending' or deviation of the superconducting gap from the canonical d-wave form.

Determination of the superconducting gap in near optimally doped Bi2Sr2−xLaxCuO6+δ(x~0.4) from low-temperature specific heat

Low-temperature specific heat of the monolayer high-Tc superconductor Bi_2Sr_{2-x}La_xCuO_{6+\delta} has been measured close to the optimal doping point (x ~ 0.4) in different magnetic fields. The identification of both a T^2 term in zero field and a \sqrt{H} dependence of the specific heat in fields is shown to follow the theoretical prediction for d-wave pairing, which enables us to extract the slope of the superconducting gap in the vicinity of the nodes (v_{\Delta}, which is proportional to the superconducting gap \Delta_0 at the antinodes according to the standard d_{x^2-y^2} gap function). The v_{\Delta} or \Delta_0 (~ 12 meV) determined from this bulk measurement shows close agreement with that obtained from spectroscopy or tunneling measurements, which confirms the simple d-wave form of the superconducting gap.

Exotic superconducting state embedded in the hidden order state of URu2Si2

The heavy-fermion compound URu2Si2 has mystified researchers since the superconducting state (Tc=1.45K) is embedded within the enigmatic ‘‘hidden order” phase (Th=17.5K). Here, we report charge and thermal transport measurements on ultraclean single crystals of URu2Si2 with very large residual-resistivity-ratio down to 30mK (∼Tc/50), which reveal a number of unprecedented superconducting properties. The results provide strong evidence for a new type of unconventional superconductivity with two distinct gaps having different nodal topology. We propose a gap function with chiral d-wave form Δ(k)=Δ0kz(kx+iky). We also demonstrate that a distinct flux line lattice melting transition with outstanding characters occurs well below the upper critical fields even at sub-Kelvin temperature. The intriguing superconducting state of URu2Si2 adds a unique and exciting example to the list of unconventional superconductors.

Promoting a core electron to fill a d-shell in negative ions: Shape versus Feshbach resonances and a novel threshold law

Two new results emerging from inner-shell photodetachment of transition metal negative ions are obtained, following studies in Pt−. First, the d-wave form of the Wigner threshold law is observed for the first time in single-photon measurements. Second, single-vacancy valence shells are filled with a core electron. While this should result in stabilization of the core excited state, producing Feshbach resonances, we find that stabilization does not occur for some core excitations, dramatically demonstrating the importance of core-valence interactions.

Momentum dependence of pseudogap and superconducting gap in variation theory

To consider the origin of a pseudogap and a superconducting (SC) gap found in the high- T c cuprates, we evaluated the momentum dependence of the singlet gap corresponding to the pseudogap and the SC gap in the t– J model, using an optimization variational Monte Carlo (VMC) method. In the underdoped regime, the singlet gap is significantly modified from the simple d x 2 - y 2 ( d)-wave gap (∝ cos k x − cos k y ) by the contribution of long-range pairings. Its angular dependence at the quasi Fermi surface is qualitatively consistent with those experimentally observed in both hole and electron-doped cuprates. On the other hand, a SC gap is almost unchanged, preserving the original simple d-wave form. Thus, it seems that the incoherent part of the singlet gap mainly influences the forms of observed gaps.

Gap structure and exotic superconducting state of URu2Si2

The heavy-fermion compound URu2Si2 has mystified researchers since the superconducting state coexists with enigmatic "hidden order" phase. In order to elucidate the superconducting gap function of URu2Si2, we performed the thermal transport measurements at low temperatures on ultraclean single crystals of URu2Si2. The results provide strong evidence for a new type of unconventional superconducting state, having two distinct gaps with different nodal topologies. We propose a gap function with chiral d-wave form Δk ∝ kz(kx ± iky).

Energy gaps in the failed high-Tc superconductor La1.875Ba0.125CuO4

A central issue on high-Tc superconductivity is the nature of the normal-state gap (pseudogap) in the underdoped regime and its relationship with superconductivity. Despite persistent efforts, theoretical ideas for the pseudogap evolve around fluctuating superconductivity, competing order and spectral weight suppression due to many-body effects. Recently, while some experiments in the superconducting state indicate a distinction between the superconducting gap and pseudogap, others in the normal state, either by extrapolation from high-temperature data or directly from La1.875Ba0.125CuO4 (LBCO-1/8) at low temperature, suggest the ground-state pseudogap is a single gap of d-wave form. Here we report angle-resolved photoemission (ARPES) data from LBCO-1/8, collected with improved experimental conditions, that reveal the ground-state pseudogap has a pronounced deviation from the simple d-wave form. It contains two distinct components: a d-wave component within an extended region around the node and the other abruptly enhanced close to the antinode, pointing to a dual nature of the pseudogap in this failed high-Tc superconductor which involves a possible precursor pairing energy scale around the node and another of different but unknown origin near the antinode.