Interplay between superconductivity and flux phase in the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>J</mml:mi></mml:math>model

Abstract

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 $T=0$ and rises quickly with decreasing doping. The transition temperature ${T}_{c}$ 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 ${T}_{c}$ in the underdoped region. Treating for simplicity the flux phase as commensurate the superconducting and flux phases coexist in the underdoped region below ${T}_{c},$ whereas a pure flux phase exists above ${T}_{c}$ 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.