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arXiv:cond-mat/0403733 · Physical Review B

Hole-pair symmetry and excitations in the strong-coupling extended t−Jz model: competition between d -wave and p -wave symmetry

Richard M. Fye, George Balster Martins, Elbio R. Dagotto

Published Jun 21, 2004• 5 citations
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Abstract

We analytically calculate the ground state pairing symmetry and excitation spectra of two holes doped into the half-filled $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}{J}_{z}$ model in the strong-coupling limit $({J}_{z}⪢\ensuremath{\mid}t\ensuremath{\mid},\ensuremath{\mid}{t}^{\ensuremath{'}}\ensuremath{\mid},\ensuremath{\mid}{t}^{\ensuremath{''}}\ensuremath{\mid})$. In leading order, this reduces to the ${t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}{J}_{z}$ model, where there are regions of $d$-wave, $s$-wave, and (degenerate) $p$-wave symmetry. We find that the $t\ensuremath{-}{J}_{z}$ model maps in lowest order onto the ${t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}{J}_{z}$ model on the boundary between $d$ and $p$ symmetry, with a flat lower band in the pair excitation spectrum. In higher order, $d$-wave symmetry is selected from the lower pair band. However, we observe that the addition of the appropriate ${t}^{\ensuremath{'}}<0$ and/or ${t}^{\ensuremath{''}}>0$, the signs of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$ found in the hole-doped cuprates, could drive the hole-pair symmetry to $p$-wave symmetry, implying the possibility of competition between $p$-wave and $d$-wave pair ground states. (An added ${t}^{\ensuremath{'}}>0$ and/or ${t}^{\ensuremath{''}}<0$ generally tend to promote $d$-wave symmetry.) We perturbatively construct an extended quasipair for the $t\ensuremath{-}{J}_{z}$ model. In leading order, there are contributions from sites at a distance of $\sqrt{2}$ lattice spacings apart; however, contributions from sites two lattice spacings apart, also of the same order, vanish identically. Finally, we compare our approach with analytic calculations for a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette and with existing numerical work, and discuss possible relevance to the physical parameter regime.

DOI
10.1103/physrevb.69.224507

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We analytically calculate the ground state pairing symmetry and excitation spectra of two holes doped into the half-filled t\ensuremath−t\ensuremath′\ensuremath−t\ensuremath′′\ensuremath−Jzt\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}{J}_{z}t\ensuremath−t\ensuremath′\ensuremath−t\ensuremath′′\ensuremath−Jz​ model in the strong-coupling limit (Jz⪢\ensuremath∣t\ensuremath∣,\ensuremath∣t\ensuremath′\ensuremath∣,\ensuremath∣t\ensuremath′′\ensuremath∣)({J}_{z}⪢\ensuremath{\mid}t\ensuremath{\mid},\ensuremath{\mid}{t}^{\ensuremath{'}}\ensuremath{\mid},\ensuremath{\mid}{t}^{\ensuremath{''}}\ensuremath{\mid})(Jz​⪢\ensuremath∣t\ensuremath∣,\ensuremath∣t\ensuremath′\ensuremath∣,\ensuremath∣t\ensuremath′′\ensuremath∣). In leading order, this reduces to the t\ensuremath′\ensuremath−t\ensuremath′′\ensuremath−Jz{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}{J}_{z}t\ensuremath′\ensuremath−t\ensuremath′′\ensuremath−Jz​ model, where there are regions of ddd-wave, sss-wave, and (degenerate) ppp-wave symmetry. We find that the t\ensuremath−Jzt\ensuremath{-}{J}_{z}t\ensuremath−Jz​ model maps in lowest order onto the t\ensuremath′\ensuremath−t\ensuremath′′\ensuremath−Jz{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}{J}_{z}t\ensuremath′\ensuremath−t\ensuremath′′\ensuremath−Jz​ model on the boundary between ddd and ppp symmetry, with a flat lower band in the pair excitation spectrum. In higher order, ddd-wave symmetry is selected from the lower pair band. However, we observe that the addition of the appropriate t\ensuremath′<0{t}^{\ensuremath{'}}<0t\ensuremath′<0 and/or t\ensuremath′′>0{t}^{\ensuremath{''}}>0t\ensuremath′′>0, the signs of t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ and t\ensuremath′′{t}^{\ensuremath{''}}t\ensuremath′′ found in the hole-doped cuprates, could drive the hole-pair symmetry to ppp-wave symmetry, implying the possibility of competition between ppp-wave and ddd-wave pair ground states. (An added t\ensuremath′>0{t}^{\ensuremath{'}}>0t\ensuremath′>0 and/or t\ensuremath′′<0{t}^{\ensuremath{''}}<0t\ensuremath′′<0 generally tend to promote ddd-wave symmetry.) We perturbatively construct an extended quasipair for the t\ensuremath−Jzt\ensuremath{-}{J}_{z}t\ensuremath−Jz​ model. In leading order, there are contributions from sites at a distance of 2\sqrt{2}2​ lattice spacings apart; however, contributions from sites two lattice spacings apart, also of the same order, vanish identically. Finally, we compare our approach with analytic calculations for a 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 plaquette and with existing numerical work, and discuss possible relevance to the physical parameter regime.
We analytically calculate the ground state pairing symmetry and excitation spectra of two holes doped into the half-filled t-t'-t''-J_z model in the strong-coupling limit (J_z >> |t|, |t'|, |t''|). For the t'-t''-J_z model, there are regions of d-wave, s-wave, and p-wave symmetry. We find that the t-J_z model maps in lowest order onto the t'-t''-J_z model on the boundary between d and p symmetry, with a flat lower branch of the pair excitation spectrum. In higher order d-wave symmetry is selected; however, we predict that the addition of the appropriate t' and/or t'' should drive the hole-pair symmetry to p-wave. We perturbatively construct an extended quasi-pair for the t-J_z model. We compare with analytic calculations for a 2x2 plaquette and numerical work, and discuss implications for the experimentally relevant parameter regime.
The $t\ensuremath{-}{J}_{z}$ model is the strongly anisotropic limit of the $t\ensuremath{-}J$ model which captures some general properties of doped antiferromagnets (AF's). The absence of spin fluctuations simplifies the analytical treatment of hole motion in an AF background, and allows us to calculate single- and two-hole spectra with a high accuracy using a regular diagram technique combined with a real-space approach. At the same time, numerical studies of this model via exact diagonalization on small clusters show negligible finite-size effects for a number of quantities, thus allowing a direct comparison between analytical and numerical results. Both approaches demonstrate that the holes have a tendency to pair in p- and d-wave channels at realistic values of t/J.t/J.t/J. Interactions leading to pairing and effects selecting p and d waves are thoroughly investigated. The role of transverse spin fluctuations is considered using perturbation theory. Based on the results of the present study, we discuss the pairing problem in the realistic $t\ensuremath{-}J$-like model. Possible implications for preformed pairs formation and phase separation are drawn.
We report results of large-scale ground-state density matrix renormalization group (DMRG) calculations on t-[Formula: see text]-J cylinders with circumferences 6 and 8. We determine a rough phase diagram that appears to approximate the two-dimensional (2D) system. While for many properties, positive and negative [Formula: see text] values ([Formula: see text]) appear to correspond to electron- and hole-doped cuprate systems, respectively, the behavior of superconductivity itself shows an inconsistency between the model and the materials. The [Formula: see text] (hole-doped) region shows antiferromagnetism limited to very low doping, stripes more generally, and the familiar Fermi surface of the hole-doped cuprates. However, we find [Formula: see text] strongly suppresses superconductivity. The [Formula: see text] (electron-doped) region shows the expected circular Fermi pocket of holes around the [Formula: see text] point and a broad low-doped region of coexisting antiferromagnetism and d-wave pairing with a triplet p component at wavevector [Formula: see text] induced by the antiferromagnetism and d-wave pairing. The pairing for the electron low-doped system with [Formula: see text] is strong and unambiguous in the DMRG simulations. At larger doping another broad region with stripes in addition to weaker d-wave pairing and striped p-wave pairing appears. In a small doping region near [Formula: see text] for [Formula: see text], we find an unconventional type of stripe involving unpaired holes located predominantly on chains spaced three lattice spacings apart. The undoped two-leg ladder regions in between mimic the short-ranged spin correlations seen in two-leg Heisenberg ladders.
We study dynamical properties of two holes in the t-J, t-Jz{\mathit{J}}_{\mathit{z}}Jz​, and Hubbard models using exact diagonalization techniques on small clusters. For the three models we found that the ground state of two holes has d-wave symmetry. Studying the dynamical d-wave pairing susceptibility at zero momentum, we observed a quasiparticle-like peak at the bottom of the spectrum in a broad region of parameter space. The lowest-energy p-wave state is close in energy to the d-wave ground state, and they both behave qualitatively similarly, while the lowest-energy states of the s-wave and extended-s-wave subspaces have higher energies and present no quasiparticle peak. The ground-state energy of two holes in the t-J model scales with J as a power law. Binding energies for different symmetries as well as dynamical properties for nonzero momentum are also discussed.
Superconductivity in a partially filled flat band presents a vexing conceptual hurdle because the absence of a Fermi surface precludes a weak-coupling regime where one can extend insights from the Bardeen-Cooper-Schrieffer picture of a Fermi surface instability. We approach the strongly correlated problem of flat band superconductivity from the strong coupling limit of local attractive interactions on line-graph lattices, whose non-interacting bandstructures host exactly flat bands due to frustrated hopping. In this limit, the pair kinetic energy which sets the superfluid stiffness is expected to scale inversely with the pair binding interaction. Here we demonstrate a striking counterexample. We show that when doped charges propagate on the line-graph of a lattice with strong pairing interaction and broken time-reversal symmetry, they bind into obstructed Cooper pairs whose motion is frustrated by destructive interference. As a result, the leading-order pair kinetic energy vanishes identically in the strong-coupling expansion, producing a flat bosonic band of compact localised pair states, zero superfluid stiffness at leading order, and an extensively degenerate many-body ground state manifold. At quarter filling, the frustrated pair dynamics maps onto a quantum dimer model which has a ddd-wave resonating-valence-bond ground state when time-reversal is broken. The pairing Hamiltonian in this limit thus has a topologically ordered spin liquid ground state which becomes exact at the analytically solvable Rokhsar-Kivelson point with long-range entanglement and deconfined holon excitations. Interestingly, we find exact compact localised eigenstates and extensive degeneracies in the many-body eigenstates of this emergent dimer model. Our results establish a disorder-free mechanism for interaction-driven localisation, in which strong pairing collapses the kinetic energy of Cooper pairs.
We have numerically investigated the doped t-J ladder using exact diagonalization. We have studied both the limit of strong interchain coupling and isotropic coupling. The ladder scales to the Luther-Emery liquid regime in the strong interchain coupling limit. In this strong coupling limit there is a simple picture of the excitation spectrum that can be continued to explain the behavior at isotropic coupling. At J=0 we have indications of a ferromagnetic ground state. At a large J/t the ladder is phase separated into holes and a Heisenberg ladder. At intermediate coupling the ground state shows hole pairing with a modified d-wave symmetry. The excitation spectrum separates into a limited number of quasiparticles which carry charge +\ensuremath{\Vert}e\ensuremath{\Vert} and spin 1/2 and a triplet magnon mode. At half filling the former vanish but the latter evolves continuously into the magnon band of the spin liquid. At low doping the quasiparticles form a dilute Fermi gas with a strong attraction but simultaneously the Fermi wave vector, as would be measured in photoemission, is large. The dynamical structure factors are calculated and are found to be very similar to calculations on two-dimensional clusters. \textcopyright{} 1996 The American Physical Society.