Uniform D-wave Research Articles

Enhanced Pair-Density-Wave Vertices in a Bilayer Hubbard Model at Half Filling.

Motivated by the pair-density-wave (PDW) state found in the one-dimensional Kondo-Heisenberg chain, we report on a determinant quantum MonteCarlo study of pair fields for a two-dimensional half-filled Hubbard layer coupled to an itinerant, noninteracting layer with one electron per site. In a specific range of interlayer hopping, the pairing vertex associated with PDW order becomes more attractive than that for uniform d-wave pairing, although both remain subdominant to the leading antiferromagnetic correlations at half filling. Our result sheds light on where one potentially may find a PDW state in such a model.

Mechanism for fluctuating pair density wave

In weakly coupled BCS superconductors, only electrons within a tiny energy window around the Fermi energy, EF, form Cooper pairs. This may not be the case in strong coupling superconductors such as cuprates, FeSe, SrTiO3 or cold atom condensates where the pairing scale, EB, becomes comparable or even larger than EF. In cuprates, for example, a plausible candidate for the pseudogap state at low doping is a fluctuating pair density wave, but no microscopic model has yet been found which supports such a state. In this work, we write an analytically solvable model to examine pairing phases in the strongly coupled regime and in the presence of anisotropic interactions. Already for moderate coupling we find an unusual finite temperature phase, below an instability temperature Ti, where local pair correlations have non-zero center-of-mass momentum but lack long-range order. At low temperature, this fluctuating pair density wave can condense either to a uniform d-wave superconductor or the widely postulated pair-density wave phase depending on the interaction strength. Our minimal model offers a unified framework to understand the emergence of both fluctuating and long range pair density waves in realistic systems.

Traces of electron-phonon coupling in one-dimensional cuprates

The appearance of certain spectral features in one-dimensional (1D) cuprate materials has been attributed to a strong, extended attractive coupling between electrons. Here, using time-dependent density matrix renormalization group methods on a Hubbard-extended Holstein model, we show that extended electron-phonon (e–ph) coupling presents an obvious choice to produce such an attractive interaction that reproduces the observed spectral features and doping dependence seen in angle-resolved photoemission experiments: diminished 3kF spectral weight, prominent spectral intensity of a holon-folding branch, and the correct holon band width. While extended e–ph coupling does not qualitatively alter the ground state of the 1D system compared to the Hubbard model, it quantitatively enhances the long-range superconducting correlations and suppresses spin correlations. Such an extended e–ph interaction may be an important missing ingredient in describing the physics of the structurally similar two-dimensional high-temperature superconducting layered cuprates, which may tip the balance between intertwined orders in favor of uniform d-wave superconductivity.

Topological Doping and Superconductivity in Cuprates: An Experimental Perspective

Hole doping into a correlated antiferromagnet leads to topological stripe correlations, involving charge stripes that separate antiferromagnetic spin stripes of opposite phases. The topological spin stripe order causes the spin degrees of freedom within the charge stripes to feel a geometric frustration with their environment. In the case of cuprates, where the charge stripes have the character of a hole-doped two-leg spin ladder, with corresponding pairing correlations, anti-phase Josephson coupling across the spin stripes can lead to a pair-density-wave order in which the broken translation symmetry of the superconducting wave function is accommodated by pairs with finite momentum. This scenario is now experimentally verified by recently reported measurements on La2−xBaxCuO4 with x=1/8. While pair-density-wave order is not common as a cuprate ground state, it provides a basis for understanding the uniform d-wave order that is more typical in superconducting cuprates.

Atomic-Scale Visualization of the Cuprate Pair Density Wave State

A recent discovery of the pair density wave (PDW) order creates a stir in understanding an underlying physics in the pseudogap states of the cuprate high temperature superconductors. We have performed a Spectroscopic Imaging Scanning Tunneling Microscopy (SI-STM) measurements on Bi$_2$Sr$_2$CaCu$_2$O$_8$+$\delta$ and identified that spatially uniform d-wave superconductivity (DSC), d-symmetry charge density wave (CDW), and the PDW play important roles in an intertwined fashion. In this review, we show how these different electronic degrees of freedom emerges in electronic structures of the cuprate and discuss how they are related to each other.

Intertwined orders from symmetry projected wavefunctions of repulsively interacting Fermi gases in optical lattices

Unconventional strongly correlated phases of the repulsive Fermi–Hubbard model, which could be emulated by ultracold vapors loaded in optical lattices, are investigated by means of energy minimizations with quantum number projection before variation and without any assumed order parameter. Using a tube-like geometry of optical plaquettes to realize the four-leg ladder Hubbard Hamiltonian, we highlight the intertwining of spin-, charge-, and pair-density waves embedded in a uniform d-wave superfluid background. As the lattice filling increases, this phase emerges from homogenous states exhibiting spiral magnetism and evolves towards a doped antiferromagnet. A concomitant enhancement of long-ranged d-wave pairing correlations is also found. Numerical tests of the approach for two-dimensional clusters are carried out, too.

Topological pair-density-wave superconducting states.

We show that the pair-density-wave (PDW) superconducting state emergent in extended Heisenberg-Hubbard models in two-leg ladders is topological in the presence of an Ising spin symmetry and supports a Majorana zero mode (MZM) at an open boundary and at a junction with a uniform d-wave one-dimensional superconductor. Similarly to a conventional finite-momentum paired state, the order parameter of the PDW state is a charge-2e field with finite momentum. However, the order parameter here is a quartic electron operator and conventional mean-field theory cannot be applied to study this state. We use bosonization to show that the 1D PDW state has a MZM at a boundary. This superconducting state is an exotic topological phase supporting Majorana fermions with finite-momentum pairing fields and charge-4e superconductivity.

Exploring intertwined orders in cuprate superconductors

The concept of intertwined orders has been introduced to describe the cooperative relationship between antiferromagnetic spin correlations and electron (or hole) pair correlations that develop in copper-oxide superconductors. This contrasts with systems in which, for example, charge-density-wave (CDW) order competes for Fermi surface area with superconductivity. La2−xBaxCuO4 with x=0.125 provides an example in which the ordering of spin stripes coincides with the onset of two-dimensional superconducting correlations. The apparent frustration of the interlayer Josephson coupling has motivated the concept of the pair-density-wave superconductor, a state that theoretical calculations show to be energetically competitive with the uniform d-wave superconductor. Even at x=0.095, where there is robust superconductivity below 32K in zero field, the coexistence of strong, low-energy, incommensurate spin excitations implies a spatially modulated and intertwined pair wave function. Recent observations of CDW order in YBa2Cu3O6+x and other cuprate families have raised interesting questions regarding the general role of charge modulations and the relation to superconductivity. While there are differences in the doping dependence of the modulation wave vectors in YBa2Cu3O6+x and La2−xBaxCuO4, the maximum ordering strength is peaked at the hole concentration of 1/8 in both cases. There are also possible connections with the quantum oscillations that have been detected about the same hole concentration but at high magnetic fields. Resolving these relationships remains a research challenge.

Competing States in thet-JModel: Uniformd-Wave State versus Stripe State

Variational studies of the t-J model on the square lattice based on infinite projected-entangled pair states confirm an extremely close competition between a uniform d-wave superconducting state and different stripe states. The site-centered stripe with an in-phase d-wave order has an equal or only slightly lower energy than the stripe with antiphase d-wave order. The optimal stripe filling is not constant but increases with J/t. A nematic anisotropy reduces the pairing amplitude and the energies of stripe phases are lowered relative to the uniform state with increasing nematicity.

Mixed state of aπ-striped superconductor

A model of an anti-phase modulated d-wave superconductor has been proposed to describe the decoupling between Cu-O planes in 1/8 doped La$_{2-x}$Ba$_{x}$CuO$_{4}$. Unlike a uniform d-wave superconductor, this model exhibits an extended Fermi surface. Within Bogoliubov-de Gennes theory, we study the mixed state of this model and compare it to the case of a uniform d-wave superconductor. We find a periodic structure of the low-energy density of states, with a period that is proportional to $B$, corresponding to Landau levels that are a coherent mixture of particles and holes. These results are also discussed in the context of experiments which observe quantum oscillations in the cuprates, and are compared to those for models in which the Fermi surface is reconstructed due to translational symmetry breaking in the non-superconducting state and to a model of a Fermi-arc metal.

Unidirectional Charge Instability of the d-Wave Resonating Valence Bond Superconductor

Starting from a uniform d-wave superconducting phase we study the energy cost due to imposed unidirectional defects with a vanishing pairing amplitude. Both renormalized mean-field theory and variational Monte Carlo calculations within the t–J model yield that the energies of inhomogeneous and uniform phases are very close to each other. This suggests that small perturbations in the microscopic Hamiltonian might lead to inhomogeneous superconducting phases in real materials as observed in recent scanning tunneling microscopy on Ca2−xNaxCuO2Cl2.

Signature of the staggered flux state around a superconducting vortex in underdoped cuprates

Based on the SU(2) lattice-gauge-theory formulation of the $t\ensuremath{-}J$ model, we discuss a possible signature of the unit-cell doubling associated with the staggered-flux (SF) state in the lightly doped spin liquid. Although the SF state appears only dynamically in a uniform d-wave superconducting state, a topological defect [SU(2) vortex] freezes the SF state inside the vortex core. Consequently, the unit-cell doubling shows up in the hopping $({\ensuremath{\chi}}_{\mathrm{ij}})$ and pairing $({\ensuremath{\Delta}}_{\mathrm{ij}})$ order parameters of physical electrons. We find that whereas the center in the vortex core is a SF state, as one moves away from the core center, a correlated staggered modulation of ${\ensuremath{\chi}}_{\mathrm{ij}}$ and ${\ensuremath{\Delta}}_{\mathrm{ij}}$ becomes predominant. We predict that over the region outside the core and inside the internal gauge-field-penetration depth around a vortex center, the local density of states exhibits a staggered peak-dip (SPD) structure inside the V-shaped profile when measured on the bonds. The SPD structure has its direct origin in the unit-cell doubling associated with the SF core and the robust topological texture, which has little to do with the symmetry of the d-wave order parameter. Therefore the structure may survive the tunneling-matrix-element effects and easily be detected by the scanning-tunnel-microscope experiment.

ANTIFERROMAGNETISM, STRIPES, AND SUPERCONDUCTIVITY IN THE t–J MODEL WITH COULOMB INTERACTION

We study mean-field phases of the t–J model with long-range Coulomb interaction. In the order of increasing doping density we find a classical antiferromagnet, charge and spin stripes, and a uniform d-wave superconductor, at the realistic doping parameters. Both in-phase and anti-phase stripes exist as metastable configurations, but the in-phase stripes have a slightly lower energy. The dependence of the stripe width and the inter-stripe spacing on the doping is examined. Effects of fluctuations around the mean-field states are discussed.

Ginzburg-Landau theory of the Abrikosov lattice in a d-wave superconductor

We consider the Ginzburg-Landau free energy for the case of coexisting s- and d-wave order parameters, coupled to a gauge field, for a system in which the minimum energy state in the absence of magnetic fields is uniform d-wave. The eigenvalue equation arising from the coupled Ginzburg-Landau equations, in the presence of a uniform magnetic field, is implicitly non-linear and has no obvious closed-form solution. We employ a simple variational two-component wave function to minimize the quadratic free energy, and use linear combinations of these wave functions to construct variational Abrikosov lattice solutions which minimize the total free energy. The resulting lattice has an oblique structure, similar to that observed by small angle neutron scattering.