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arXiv:cond-mat/9902287 · Physical review. B, Condensed matter

Pairing via index theorem

D. H. Lee

Published Nov 1, 1999• 7 citations
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Abstract

This work is motivated by a specific point of view: at short distances and high energies the undoped and underdoped cuprates resemble the \ensuremath{\pi}-flux phase of the $t\ensuremath{-}J$ model. The purpose of this paper is to present a mechanism by which pairing grows out of the doped \ensuremath{\pi}-flux phase. According to this mechanism pairing symmetry is determined by a parameter controlling the quantum tunneling of gauge flux quanta. For zero tunneling the symmetry is ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}{+id}_{\mathrm{xy}},$ while for large tunneling it is ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}.$ A zero-temperature critical point separates these two limits.

DOI
10.1103/physrevb.60.12429

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This work is motivated by a specific point of view: at short distances and high energies the undoped and underdoped cuprates resemble the \ensuremath{\pi}-flux phase of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model. The purpose of this paper is to present a mechanism by which pairing grows out of the doped \ensuremath{\pi}-flux phase. According to this mechanism pairing symmetry is determined by a parameter controlling the quantum tunneling of gauge flux quanta. For zero tunneling the symmetry is dx2\ensuremath−y2+idxy,{d}_{{x}^{2}\ensuremath{-}{y}^{2}}{+id}_{\mathrm{xy}},dx2\ensuremath−y2​+idxy​, while for large tunneling it is dx2\ensuremath−y2.{d}_{{x}^{2}\ensuremath{-}{y}^{2}}.dx2\ensuremath−y2​. A zero-temperature critical point separates these two limits.
The flux phase predicted by the $t\text{\ensuremath{-}}J$ model in the large-NNN limit exhibits features that make it a candidate for describing the pseudogap regime of cuprates. However, certain properties, such as, for instance, the prediction of well-defined quasiparticle peaks, speak against this scenario. We have addressed these issues by computing self-energy renormalizations in the vicinity of flux phase. The calculated spectral functions show features similar to those observed in experiments. At low doping, near the flux phase, the spectral functions are anisotropic on the Fermi surface and very incoherent near the hot spots. The temperature and doping evolution of self-energy and spectral functions are discussed and compared with the experiment.
We outline the basic ideas involved in a recently proposed [17] derivation of a gauge theory for underdoped cuprates in the “spin gap phase”, performed essentially step by step starting from the t — J model, considered as a model Hamiltonian for the CuO2 layers. The basic tool is the U(1) x SU(2) Chern-Simons bosonization, to which it is dedicated a somewhat detailed discussion. The basic output is a “spingap” not vanishing in any direction and an antiferromagnetic correlation length proportional to the inverse square root of doping concentration, in agreement with data deduced from the neutron experiments. The model also exhibits a small half-pocket Fermi surface around (±π/2, ±π/2) and a linear in temperature dependence of in-plane resistivity in certain temperature range.
We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping which respects an SU(2)\mathrm{SU}(2)SU(2) symmetry---a symmetry previously known to be important at half filling. The mean field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain ddd-wave superconductor, spin gap, strange metal, and Fermi liquid phases. The spin gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The electron spectral function shows small Fermi pockets at low doping which continuously evolve into the large Fermi surface at high doping concentrations.
Spin-charge split pairing in underdoped cuprate superconductors: support from low-TTT specific heat
We calculate the specific heat of a weakly interacting dilute system of bosons on a lattice and show that it is consistent with the measured electronic specific heat in the superconducting state of underdoped cuprates with boson concentration ρ∼x/2ρ\sim x/2ρ∼x/2, where xxx is the hole (dopant) concentration. As usual, the T3T^3T3 term is due to Goldstone phonons. The zero-point energy, through its dependence on the condensate density ρ0(T)ρ_0(T)ρ0​(T), accounts for the anomalous TTT-linear term. These results support the split-pairing mechanism, in which spinons (pure spin) are paired at T∗T^*T∗ and holons (pure charge) form real-space pairs at Tp<T∗T_p < T^*Tp​<T∗, creating a gauge-coupled physical pair of charge +2e+2e+2e and concentration x/2x/2x/2 which Bose condenses below TcT_cTc​, accounting for the observed phases.
There are two seemingly unrelated puzzles about the cuprate superconductors. The first puzzle concerns the strong non-BCS behavior around xc{x}_{c}xc​, the end point of the superconducting dome on the overdoped side, where the cuprate is believed to be well described by the Fermi-liquid theory. This is the most evident in the observed ${\ensuremath{\rho}}_{s}(0)\ensuremath{-}{T}_{c}$ scaling and the large amount of uncondensed optical spectral weight at low energy. The second puzzle concerns the remarkable robustness of the ddd-wave pairing against the inevitable disorder effect in such a doped system, which is also totally unexpected from the conventional BCS picture. Here we show that these two puzzles are deeply connected to the origin of a third puzzle about the cuprate superconductors, namely, the mysterious quantum critical behavior observed around x∗{x}^{*}x∗, the so called pseudogap end point. Through a systematic variational Monte Carlo (VMC) study of the disordered 2D $t\ensuremath{-}J$ model from the resonating valence bond (RVB) perspective, we find that the ddd-wave pairing in this model is remarkably more robust against the disorder effect than that in a conventional ddd-wave BCS superconductor. We find that such remarkable robustness can be attributed to the spin-charge separation mechanism in the RVB picture, through which the ddd-wave RVB pairing of the charge-neutral spinons becomes essentially immune to the disorder potential except for the secondary effect related to the modulation of the local doping level by the disorder. We propose that there exists a Mott transition at x∗{x}^{*}x∗, where the RVB pairing in the underdoped regime is transmuted into the increasingly more BCS-like pairing for x>x∗x>{x}^{*}x>x∗, whose increasing fragility against the disorder effect leads to the non-BCS behavior and the ultimate suppression of superconductivity around xc{x}_{c}xc​.
A tale of two pairs: on the origin of the pseudogap end point in the high-Tc_{c}c​ cuprate superconductors
There are two seemingly unrelated puzzles about the cuprate superconductors. The first puzzle concerns the strong non-BCS behavior around xcx_{c}xc​, the end point of the superconducting dome on the overdoped side, where the cuparte is believed to be well described by the fermi liquid theory. This is the most evident in the observed ρs(0)−Tcρ_{s}(0)-T_{c}ρs​(0)−Tc​ scaling and the large amount of uncondensed optical spectral weight at low energy. The second puzzle concerns the remarkable robustness of the d-wave pairing against the inevitable disorder effect in such a doped system, which is also totally unexpected from the conventional BCS picture. Here we show that these two puzzles are deeply connected to the origin of a third puzzle about the cuprate superconductors, namely, the mysterious quantum critical behavior observed around x∗x^{*}x∗, the so-called pseudogap end point. Through a systematic variational optimization of the disordered 2D t−Jt-Jt−J model from the resonating valence bond(RVB) perspective, we find that the d-wave pairing in this model is remarkably more robust against the disorder effect than that in a conventional d-wave BCS superconductor. We find that such remarkable robustness can be attributed to the spin-charge separation mechanism in the RVB picture, through which the d-wave RVB pairing of the charge neutral spinons becomes essentially immune to the impurity potential except for the secondary effect related to the modulation of the local doping level by the disorder. We propose that there exists a Mott transition at x∗x^{*}x∗, where the RVB pairing in the underdoped regime is transmuted into the increasingly more BCS-like pairing for x>x∗x>x^{*}x>x∗, whose increasing fragility against the disorder effect leads to the non-BCS behavior and the ultimate suppression of superconductivity around xcx_{c}xc​.