Bond-order modulated staggered flux phase for the $t{-}J$ model on the square lattice

TL;DR

This paper proposes and analyzes a bond-order modulated staggered flux phase in the $t{-}J$ model, revealing spontaneous $4\times 4$ bond-order patterns near 1/8 doping, and compares its stability to other states using variational Monte Carlo.

Contribution

It introduces a new bond-order modulated staggered flux wave-function for the $t{-}J$ model and studies its stability and properties with an approximation-free variational Monte Carlo approach.

Findings

A $4\times 4$ bond-order modulation appears near 1/8 doping.

Moderate Coulomb repulsion makes the bond-order flux state energetically competitive.

Short-range Coulomb repulsion suppresses the d-wave superconducting state.

Abstract

Motivated by the observation of inhomogeneous patterns in some high-T$_c$ cuprate compounds, several variational Gutzwiller-projected wave-functions with built-in charge and bond order parameters are proposed for the extended $t-J-V$ model on the square lattice at low doping. First, following a recent Gutzwiller-projected mean-field approach by one of us (Phys. Rev. B. {\bf 72}, 060508(R) (2005)), we investigate, as a function of doping and Coulomb repulsion, the stability of the staggered flux phase with respect to small spontaneous modulations of squared unit cells ranging from $2\times 2$ to $\sqrt{32}\times\sqrt{32}$. It is found that a $4\times 4$ bond-order (BO) modulation appears spontaneously on top of the staggered flux pattern for hole doping around 1/8. A related wave-function is then constructed and optimized accurately and its properties studied extensively using an approximation-free variational Monte Carlo scheme. Finally, the competition of the BO-modulated staggered flux wave-function w.r.t. the d-wave RVB wave-function or the commensurate flux state is investigated. It is found that a short range Coulomb repulsion penalizes the d-wave superconductor and that a moderate Coulomb repulsion brings them very close in energy. Our results are discussed in connection to the STM observations in the under-doped regime of some cuprates.