Session Q23: Superconductivty Theory III: Mainly Cuprates

Grand-canonical variational study for the Gutzwiller-projected BCS wave function

We study the Gutzwiller-projected BCS wave function in the 2D extended t-J model using a variational Monte Carlo method. To take into account the effect of Gutzwiller projection, a fugacity factor proposed by Laughlin and Anderson has been included into the coherence factor of the BCS state. We show that the ground-state energy and excitation spectra calculated in the grand-canonical picture are essentially the same as previous results in the canonical scheme if the free energy is used for minimization. Except for La-214 materials, we find that the doping dependence of chemical potential is consistent with experimental findings on several cuprates. We have investigated the asymmetry of tunneling conductance observed by scanning tunneling spectroscopy becomes much stronger as decreasing doping. A very huge enhancement of phase fluctuation in the underdoped regime has been found.   

Fluctuating Bond Model of cuprate superconductivity.

The fluctuating bond model(FBM),an empirical model based on a strong,local coupling of electrons to the square of the planar oxygen vibrator amplitudes, provides explainations of the d-wave pairing mechanism\footnote{Nature phys. 3, 184-191 (2007)} and the pseudogap\footnote{Phys. Rev. B 83, 144503 (2011)} of cuprate superconductivity. The d-wave pairing is mediated by the anharmonic phonon (planar oxygen vibrator) which is also responsible for the pseudogap when the C4 symmetry of the oxygen vibrator is broken. Here we present calculations of gap and T{\_}c in a unitary framework involving pseudogap as a competing order parameter, with the help of ab initio simulations.   

d-wave superconductivity from correlated hopping interactions determined by angle-resolved photoemission spectroscopy

Starting from a generalized Hubbard model, in which correlated-hopping interactions are considered in addition to the on-site repulsive Coulomb one, we solve numerically two coupled integral equations [1] within the Bardeen-Cooper-Schrieffer formalism, in order to quantify the doping effects on the critical temperature ($T_{c})$, $d$-wave superconducting gap, and the electronic specific heat. Within the mean-field approximation, we have determined the single and correlated electron hopping parameters for La$_{2-x}$Sr$_{x}$CuO$_{4}$ compound by using angle-resolved photoemission spectroscopy (ARPES) data [2]. The resulting parametrized Hubbard model is able to explain the experimental $T_{c}$ variation as a function of the doping level ($x)$. In addition, the observed power-law behavior of the superconducting specific heat is reproduced by this correlated-hopping Hubbard model without adjustable parameters. [1] L.A. P\'{e}rez, J.S. Mill\'{a}n, C. Wang, \textit{Int. J. Mod. Phys. B} \textbf{24}, 5229-5239 (2010). [2] T. Yoshida, \textit{et al}., \textit{Phys. Rev. B} \textbf{74,} 224510 (2006).   

Role of the charge reservoir layer in cuprate superconductors

The high pressure measurements of Jorgensen et al., which show that T$_c$ increases as the apical oxygen distance from the CuO$_2$ plane decreases, is at odds with current theoretical predictions of T$_c$. Furthermore, the unusually close T$_c$ (and similar NMR signatures) of Hg and Tl based families of cuprates, in addition to the facts that their T$_c$ is significantly higher than superconductors without charge reservoir layers (CRLs), suggest that CRLs play a significant role in determining T$_c$. Based on recent studies of S. Raghu et al. on the effects of longer ranged interactions on unconventional superconductivity, we discuss the possibility of the generic role that the CRLs may have in enhancing T$_c$ by considering their role in screening short-range repulsive Coulomb interactions within the CuO$_2$ plane.   

Phase separation instabilities and pairing modulations in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$

There is a growing evidence that the unconventional spatial inhomogeneities accompanied with the pairing of electrons, subsequent quantum phase transitions (QPTs), and condensation control coherent states in doped high-Tc cuprate superconductors, iron pnictide and telluride materials. We show that these superconducting states can be obtained from the phase separation instabilities near the quantum critical points. We examine electron coherent and incoherent pairing instabilities using our results on exact diagonalization in pyramidal and octahedron Hubbard-like clusters under variation of chemical potential (or doping), interaction strength, temperature and magnetic field. We also evaluate the dependence of the energy gap function in the vicinity of the sign change (nodes) as a function of position of the apical oxygen atom, due to vibration of apical atom and variation of inter-site coupling. The developed approach provides a simple microscopic explanation of (correlation) supermodulation of the coherent pairing gap observed recently in the scanning tunneling microscopy experiments at atomic scale in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$.   

Pairing Correlations in the two-layer attractive Hubbard Model

Quantum Monte Carlo (QMC) is used to study pairing correlations in a two-layer Hubbard Model in which one layer is attractive ($U<0$), and the other uncorrelated ($U=0$). We provide a detailed characterization of how superconductivity in the $U<0$ layer induces pairing in the $U=0$ layer, as a function of the interlayer hopping, density, temperature, and on-site attraction strength in the $U<0$ layer. QMC data are complemented by calculations within the Bogoliubov - de Gennes approximation.   

Pair-Density-Wave Superconducting Order in Two-Leg Ladders

We show, using bosonization and Renormalization Group methods, that Pair Density Wave (PDW) state happens in the system of two extended Hubbard-Heisenberg models on two leg ladders. Two different models are discussed in detail. In even legged ladder system we find that PDW state is the dominant instability for certain filling factors and some range of parameters. We show that phase diagram of the spin gap regime is composed of two dual phases: PDW and uniform Superconducting (SC) states. The phase transition between uniform SC and PDW in this model is shown to be in the Ising universality class. The idea has been generalize to the case of other commensurate fillings where we find higher order commensurate PDW states. Then we consider a two leg ladder system with nonzero flux $\Phi$ per plaquette. This system is commensurate for wide range of the electron fillings when $\Phi=\pi$. We show that commensurate PDW and incommensurate CDW are some of the phases present in the phase diagram of this model. We show how formation of PDW order in the ladder embodies the notion of intertwined orders.   

Numerical Investigations of Spontaneous Orbital Currents in the Three-Orbital Hubbard Model

Recent experiments show that the pseudogap regime of the cuprate superconductors could be characterized by a phase where the time-reversal symmetry is spontaneously broken but the translational symmetry remains intact. One possible but still highly disputed theory involves a circulating current looping around the Cu and O atoms. To address this issue, we perform large-scale exact diagonalization using a three-orbital Hubbard model on a Cu8O16 cluster. We find that the current-current correlations fall off quickly and show no signs of particular orbital current patterns. We also extend our calculations by adding explicitly a perturbative orbital current term into the Hamiltonian to study its experimental consequences. Using the magnetic moment measured in neutron scattering to constrain the strength of this perturbation, we compute the dichroic signals of various photon-spectroscopies to provide experimental benchmarks to test the existence of such a circulating current phase.   

Momentum-Resolved d-wave Eliashberg Calculation Using The Spin Excitation Spectrum for LSCO Superconductors

We solve the momentum resolved d-wave Elishberg equation employing the magnetic excitation spectrum from the inelastic neutron scattering on the LSCO superconductors reported by Vignolle et al. [1]. The magnetic excitation spectrum exhibits 2 peaks: a sharp incommensurate peak at 18 meV at momentum ($\pi,\pi\pm\delta$) and ($\pi\pm\delta,\pi$), and another broad peak near 40$\sim$70 meV at momentum ($\pi,\pi$). Above 70 meV, the magnetic excitation spectrum has a long tail that is shaped into a circle centered at ($\pi,\pi$) with $\delta '$. The sign of the real part of the total self-energy $\Sigma(\vec{k},\omega)+X(\vec{k},\omega)$ is determined by the momentum position of the peaks of the magnetic excitation spectrum and bare dispersion $\xi(\vec{k})$. We will discuss the effects of the each component of the magnetic excitation spectrum on the self-energy $\Sigma(\vec{k},\omega)$ the renormalization of the band dispersion $X(\vec{k},\omega)$, the pairing function $\phi(\vec{k},\omega)$, and the spectral function $A(\vec{k},\omega)$.\\[4pt] [1] B.Vignolle et.al., Nature Physics 3,163-167 (2007)   

Superconducting Instabilities in Electronic Liquid Crystal Phases

We discuss the superconducting instabilities in a two dimensional fermionic system in uniform electronic liquid crystal phases which cause the Fermi surfaces (FS) to become distorted. We study the case of fermions in a time-reversal broken nematic-like state with $l=3$ in the charge channel, and the $l=2$ state in the spin channel. We discuss the competition between singlet and triplet superconducting states with pair density wave states.   

Odd-frequency triplet pairing in mixed-parity superconductors

We show that mixed-parity superconductors may exhibit equal-spin pair correlations that are odd-in-time and can be tuned by means of an applied field. The direction and the amplitude of the pair correlator in the spin space turn out to be strongly dependent on the symmetry of the order parameter, and thus provide a tool to identify different types of singlet-triplet mixed configurations. We suggest that odd-in-time spin-polarized pair correlations can be generated without magnetic inhomogeneities in superconducting/ferromagnetic hybrids with non-centrosymmetric superconductor or when parity mixing is induced at the interface. Paola Gentile, Canio Noce, Alfonso Romano, Gaetano Annunziata, Jacob Linder, Mario Cuoco, arXiv:1109.4885   

Common Fermi-liquid origin of T2 resistivity and superconductivity in n-type SrTiO3

SrTiO3 is a semiconductor which, when doped with a low density of electrons, becomes a good conductor with relatively high mobility and strong temperature dependence of the electrical resistivity and the infrared optical conductivity. At low temperatures the material becomes superconducting with a maximum reported Tc below 1 K with a dome-shaped doping dependence of Tc, both in the 3D bulk material and at the 2D LaAlO3/SrTiO3 interface. The DC resistivity below 100 K has a T$^2$ temperature dependence. The quasiparticles are in the anti-adiabatic limit with respect to electron-phonon interaction, which renders the interaction mediated through phonons effectively non-retarded. We apply Fermi-liquid theory for the T$^2$ term in the resistivity, and combine this with expressions for Tc and with the Brinkman-Platzman-Rice (BPR) sum-rule to obtain Landau parameters of n-type SrTiO3. These parameters are comparable to those of liquid 3He, indicating interesting parallels between these Fermi-liquids despite the differences between the composite fermions from which they are formed.   

Time Reversal Invariant Topological Superconductor in Rashba System

Recently, topological superconductors (TSCs) have been intensively studied as one of the topologically nontrivial phases. TSCs are characterized by topological numbers classified by their symmetries and dimensions. In the previous proposals on TSCs in Rashba systems, Zeeman splitting is necessary and therefore time reversal symmetry (TRS) is broken. We consider instead a bilayer Rashba system, e.g., a two-interface system, where hybridization causes a band gap. As for the single layer case, it has been shown experimentally that at the interface of SrTiO$_{3}$/LaAlO$_{3}$, two dimensional electron gas with Rashba spin-orbit interaction and superconductivity is formed. We find that the hybridization in bilayer system leads to topological phase without breaking TRS. This system belongs to the class characterized by Z$_{2}$ index. We obtain the conditions for odd-parity pair potentials by analyzing relations between strength of interactions and types of pair potentials. TSCs are attained in the case when the system has an odd-parity pair potential and the Fermi energy lies in the hybridization gap. We analytically calculate a topological number in a bulk system, and explicitly confirm the bulk-edge correspondence by performing numerical calculation in a finite system with edges.   

Pairing Theories in Very High Magnetic Fields: Effects of Condensed and Non-condensed Pairs

In this talk we focus on the interplay of pseudogap effects with high magnetic fields. Our work is built on reformulating the Gor'kov equations in the non-linear regime into a Landau level basis. Here we explore two distinct fermionic pairing schemes associated with the Abrikosov lattice in real and reciprocal space. We show how both have antecedents in the literature and both can be extended to accommodate non-condensed pairs. With this formulation, we can calculate quantities including the local density of states and $H_{c2}$ in this regime. Importantly, both pairing theories yield gapless excitations which should be relevant to cuprate magneto-oscillatory experiments. The additional implications for the high field diamagnetic susceptibility are discussed.   

Quantum oscillations in vortex-liquids

Motivated by observations of quantum oscillations in underdoped cuprates [1], we examine the electronic density of states (DOS) in a {\em vortex-liquid} state, where long-range phase coherence is destroyed by an external magnetic field $H$ but the local pairing amplitude survives. We note that this regime is distinct from that studied in most of the recent theories, which have focused on either a Fermi liquid with a competing order parameter or on a d-wave vortex lattice. The cuprate experiments are very likely in a resistive vortex-liquid state. We generalize the $s$-wave analysis of Maki and Stephen [2] to $d$-wave pairing and examine various regimes of the chemical potential, gap and field. We find that the $(1/H)$ oscillations of the DOS at the chemical potential in a $d$-wave vortex-liquid are much more robust, i.e., have a reduced damping, compared to the $s$-wave case. We critically investigate the conventional wisdom relating the observed frequency to the area of an underlying Fermi surface. We also show that the oscillations in the DOS cross over to a $\sqrt{H}$ behavior in the low field limit, in agreement with the recent specific heat measurements. [1] L.~Taillefer, J.~Phys.~Cond.~Mat.~{\bf 21}, 164212 (2009). [2] M.~J.~Stephen, Phys.~Rev.~B {\bf 45}, 5481 (1992).