Bond-order-modulated staggered-flux phase of the $tJ$ model on a square lattice

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 $t\text{−}J\text{−}V$ model on the square lattice at low doping. First, following a recent Gutzwiller-projected mean-field approach by one of us [D. Poilblanc, Phys. Rev. B 72, 060508(R) (2005)], we investigate, as a function of doping and Coulomb repulsion the relative stability of a wide variety of modulated structures with square unit cells of size $2\times 2$, $\sqrt{8}\times \sqrt{8}$, $4\times 4$, and $\sqrt{32}\times \sqrt{32}$. It is found that the $4\times 4$ bond-order wave function with staggered-flux pattern (and small charge and spin current density wave) is a remarkable competitive candidate for hole doping around $1∕8$ in agreement with scanning tunneling microscopy observations in the underdoped regime of some cuprates. This wave function is then optimized accurately and its properties studied extensively using a variational Monte Carlo scheme. Moreover, we find that under increasing the Coulomb repulsion, the $d$-wave superconducting RVB wave function is rapidly destabilized with respect to the $4\times 4$ bond-order wave function. The stability of the bond-modulated wave function is connected to a gain of Coulomb and exchange energies. We suggest that such ordering patterns could be dynamical or could spontaneously appear in the vicinity of an impurity or a vortex in the mixed phase of the cuprates. Finally, we consider also a commensurate flux phase, but this wave function turns out not to be competitive because of its rather poor kinetic energy. However, we find it has very competitive exchange and Coulomb energies.

Figure 1

(Color online) $4\times 4\text{unit}$ cell used in both the MF approach and the variational wave function. Note the existence of six independent bonds (bold bonds), that for convenience are labeled from 1–6, and of three a priori nonequivalent sites. The center of the dashed plaquette is the center of the (assumed) $C_{4V}$ symmetry. Other sizes of the same type of structure are considered in the MF case, respectively: $2\times 2$, $\sqrt{8}\times \sqrt{8}$, and $\sqrt{32}\times \sqrt{32}$ unit cells.

Figure 2

Mean-field phase diagrams obtained by solving self-consistently the mean-field equations on a $128\times 128$ lattice (for ${\ell }_0=4$) vs hole doping $x$ and repulsion $V_0$ (in units of $J$). Top: results obtained assuming a $4\times 4\text{unit}$ cell; bottom: the same with a $\sqrt{32}\times \sqrt{32}$ tilted unit cell. In both cases, a $C_{4v}$ symmetry is assumed (see text).

Figure 3

(Color online) (a) Energy per site (in units of $J$ and for $t=3J$) obtained by solving the mean-field equations using the initial $4\times 4\text{unit}$ cell (see text) for a moderate value of $V_0$. The SFP energy is also shown for comparison. The FL energy has been substracted from all data for clarity. (b) Comparison of the energies (for $V_0=0$) using different initial conditions (see text), a $4\times 4$ or a $\sqrt{32}\times \sqrt{32}$ unit cell; due to very small energy differences, the SFP energy is used as a reference for an easier comparison. The different phases specified by arrows and characterized by the number of sites $N_{SC}$ of their actual supercells refer to the ones in Fig. 2. For doping $x=0.14$, the minimization leads to a solution with small imaginary parts (of order ${10}^{-4}$) very similar to a FL phase, which we call ${\mathrm{FL}}^{*}$.

Figure 4

Energy per lattice site of the $\mathrm{RVB}∕\mathcal{J}$, $\mathrm{SFP}∕\mathcal{J}$, and $\mathrm{BO}∕\mathcal{J}$ wave functions minus the energy of the projected Fermi-sea wave function.

Figure 5

Kinetic and exchange energy per site of the $\mathrm{RVB}∕\mathcal{J}$ wave function minus the respective exchange and kinetic energy of the simple projected Fermi sea. Inset: value of the variational $d$-wave gap.

Figure 6

Total energy per site of the $\mathrm{BO}∕\mathcal{J}$ minus the energy of the $\mathrm{SFP}∕\mathcal{J}$ wave functions. For ${\ell }_0=2$ We show results for both lattices with 64 and 256 sites at the same $1∕8$ doping.

Figure 7

Total energy per site of the $\mathrm{BO}∕\mathcal{J}$ variational wave function with variational parameters $\mathrm{Im}\left({\tilde{t}}_{i,j}\right)=\pm \theta$ on the bonds 1, 2, 3, and $\mathrm{Im}\left({\tilde{t}}_{i,j}\right)=\pm 0.149$ on the bonds 4, 5, 6. The sign of $\mathrm{Im}\left({\tilde{t}}_{i,j}\right)$ is oriented according to the staggered flux pattern. We have chosen for all the links $\mathrm{Re}\left({\tilde{t}}_{i,j}\right)=0.988$. Results for $V_0=0$ and $V_0=5$ with ${\ell }_0=4$ are shown.

Figure 8

Kinetic, exchange and Coulomb energy per site of the $\mathrm{BO}∕\mathcal{J}$ wave function minus the respective associated energy of the $\mathrm{SFP}∕\mathcal{J}$ wave function.

Figure 9

(Color online) Local expectation values (a),(b),(c) of the kinetic and exchange energies of the projected $\mathrm{BO}∕\mathcal{J}$ wave functions on each of the bonds within the unit cell. Width of filled square symbols is proportional to the (a) real and (b) imaginary part of $⟨c_i^{+}{c}_j⟩$, and (c) to the local exchange energy $⟨{\mathbf{S}}_i\bullet {\mathbf{S}}_j⟩$. The sign of the imaginary part of the hopping bonds is according to the staggered-flux pattern (arrows). The wave function has small charge-density variations (d), therefore we subtract the mean value $n$ to the local density: size of circles are proportional to $⟨n_i-n⟩$, and circles are open (filled) for negative (positive) sign. The biggest circle corresponds to an on-site charge deviation of 2%. All the above results are for ${\ell }_0=4$ and $V_0=5$.