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Physical Review Letters

Superconductivity from commensurate flux phases

Pascal Lederer, Didier Poilblanc, Thomas Maurice Rice

Published Oct 2, 1989• 191 citations
Publisher

Abstract

We use a renormalized mean-field theory of the t-J model to show that flux states closely related to those recently proposed by Anderson, Shastry, and Hristopoulos are stabilized, at values of J/t\ensuremath{\gtrsim}1, when the flux per plaquette is commensurate with the electron density in a way similar to that found recently for noninteracting electrons. These states (in the interacting t-J model) are characterized by a collective gauge variable which leads to superconductivity. At not too small values of J/t the state with half a flux quantum per plaquette is stable at low doping.

DOI
10.1103/physrevlett.63.1519

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We use a renormalized mean-field theory of the t-J model to show that flux states closely related to those recently proposed by Anderson, Shastry, and Hristopoulos are stabilized, at values of J/t\ensuremath{\gtrsim}1, when the flux per plaquette is commensurate with the electron density in a way similar to that found recently for noninteracting electrons. These states (in the interacting t-J model) are characterized by a collective gauge variable which leads to superconductivity. At not too small values of J/t the state with half a flux quantum per plaquette is stable at low doping.
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We study the stability of the staggered-flux phase in the t-J model away from half-filling using a systematic large-N slave-boson approach. Found below a critical doping concentration ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$=a(J/t)2{)}^{2})2, is a flux-density-wave instability with an incommensurability wave number \ensuremath{\sim}${\mathrm{\ensuremath{\delta}}}^{3/4}$. The instability towards a modulated-flux state is due to low-lying phase fluctuations of the valence bonds. When the doping parameter exceeds ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$, we find a fully gapped d-wave superconducting state due to residual quasiparticle interactions.
A renormalized Hamiltonian approach is applied to construct commensurate flux phases for the t-J model. The assumption of nonuniform phases and modulus for the bond hopping amplitudes is necessary and (sufficient) condition of self-consistency. The supercell of the modulation is closely related to the filling. The self-consistent coupled equations are solved numerically on finite square clusters of sizes as large as 20\ifmmode\times\else\texttimes\fi{}20. The differences in kinetic energy and in spin-spin correlation with respect to the t=0 uniform commensurate flux state show \ensuremath{\sim}-(t/J)1/2{)}^{1/2})1/2 and \ensuremath{\sim}t/J behaviors, respectively. When t/J\ensuremath{\ge}1, time-reversal symmetry is only ``weakly broken'' and the Fermi level lies in a pseudogap with a depressed density of state.
We study the phase diagram of the $t\ensuremath{-}J$ model using a mean field type approximation within the Baym-Kadanoff perturbation expansion for Hubbard X operators. The line separating the normal state from a d-wave flux or bond-order state starts near optimal doping at T=0T=0T=0 and rises quickly with decreasing doping. The transition temperature Tc{T}_{c}Tc​ for d-wave superconductivity increases monotonically in the overdoped region towards optimal doping. Near optimal doping a strong competition between the two d-wave order parameters sets in leading to a strong suppression of Tc{T}_{c}Tc​ in the underdoped region. Treating for simplicity the flux phase as commensurate the superconducting and flux phases coexist in the underdoped region below Tc,{T}_{c},Tc​, whereas a pure flux phase exists above Tc{T}_{c}Tc​ with a pseudogap of d-wave symmetry in the excitation spectrum. We also find that incommensurate charge-density-wave ground states due to Coulomb interactions do not modify strongly the above phase diagram near the superconducting phase, at least, as long as the latter exists at all.