Spin rotationally symmetric domain flux phases in underdoped cuprate superconductors

We propose a different form of inhomogeneous phase consisting of out-of-phase staggered flux domains separated by diagonal charged domain walls centered either on bonds or on sites. Remarkably, such domain flux phases are spin rotationally symmetric and exhibit conelike quasiparticle dispersion near the Fermi energy, as well as incommensurate order of orbital currents. Such features are consistent with the pseudogap behavior and the diagonal stripes observed experimentally in lightly doped cuprates. A renormalized mean-field theory shows that these solutions with coexisting charge modulation and charge currents are competitive ground-state candidates within the $t\text{−}J$ model.

Figure 1

(Color online) Spatial modulation of the hole density $n_{hi}$ (circles), bond-order parameter ${\chi }_{ij}$ (lines with arrows indicating the direction of charge currents), and flux ${\Phi }_{◻}$ defined by Eq. (6) (positive/negative flux indicated by symbols + and ${(-)}_3$ distribution found in two FDF phases at hole doping $x=1∕8$ and $t∕J=3$. Circle diameters are proportional to the doped hole densities; widths of bond lines connecting them are proportional to the magnitudes of the bond-order parameters ${\chi }_{ij}$, while the magnitude of flux flowing through each plaquette is represented by the size of the + and − symbols. Two distinct phases are (a) bond-centered FDF phase with a vanishing current (dashed lines) at the DW bonds and (b) site-centered FDF phase with a vanishing flux (indicated by 0) at the DW plaquettes. (c) shows the self-consistent CF phase $(t=0)$ characterized by the uniform fictitious flux ${\Phi }_{◻}=\frac{1}{2}(1-x)$, by the phases of ${\chi }_{ij}$ given by Eq. (7), as well as by the homogeneous charge distribution.

Figure 2

(Color online) [(a) and (e)] Hole density $n_{hi}$ (4), [(b) and (f)] spin correlation $S_i$ (7), [(c) and (g)] bond charge $T_i$ (9), and [(d) and (h)] modulated flux ${\Phi }_{\pi i}$ (11) in the bond-centered (left) and site-centered (right) FDF phases at $x=1∕16$ for $t∕J=1$ (triangles) and $t∕J=3$ (squares). For comparison, circles depict the related $t∕J\to 0$ solutions: the CF phase with uniform fictitious flux $\Phi =15∕32$ (left) and a two-domain $\mid \Phi \mid =\frac{1}{2}$ SF phase (right).

Figure 3

(Color online) Electronic structure of the site-centered FDF phase (solid lines) and SF phase (dashed lines) along the main directions of the Brillouin zone for $x=1∕16$ and $t∕J=3$. Inset shows a pseudogap between the FDF bands along the $X\text{−}Y$ direction near the Fermi energy $\mu$ (thin dashed line).