Possible charge instabilities in two-dimensional doped Mott insulators

Motivated by the growing evidence of the importance of charge fluctuations in the pseudogap phase in high-temperature cuprate superconductors, we apply a large-$N$ expansion formulated in a path integral representation of the two-dimensional $t$-$J$ model on a square lattice. We study all possible charge instabilities of the paramagnetic state in leading order of the $1/N$ expansion. While the $d$-wave charge density wave (flux phase) becomes the leading instability for various choices of model parameters, we find that a $d$-wave Pomeranchuk (electronic nematic phase) instability occurs as a next leading one. In particular, the nematic state has a strong tendency to become inhomogeneous. In the presence of a large second nearest-neighbor hopping integral, the flux phase is suppressed and the electronic nematic instability becomes leading in a high doping region. Besides these two major instabilities, bond-order phases occur as weaker instabilities close to half-filling. Phase separation is also detected in a finite temperature region near half-filling.

Figure 1

Sketch of various phases appearing in our work. Commensurate orders for (a) $d$CDW, (b) $d$PI, (c) BOP${}_x$, and (d) BOP${}_{xy}$ in real space. The commensurate $d$PI has a momentum $\mathbf{q}=(0,0)$ whereas the commensurate $d$CDW, BOP${}_x$, and BOP${}_{xy}$ have $\mathbf{q}=(\pi ,\pi )$. Solid and dashed lines in (b)–(d) represent the strong and weak bonds, respectively. (e) Fermi surface deformations (black line) associated with the commensurate $d$PI; the original Fermi surface is sketched by gray lines.

Figure 2

Critical temperature versus doping rate for $d$CDW, $d$PI, BOP${}_x$, BOP${}_{xy}$, and PS for $t^{\prime }=0$ and $J=0.3$. Thick (thin) lines describe commensurate (incommensurate) transitions. The critical line for PS is shown in a larger scale of $T$ in the inset.

Figure 3

Modulation vectors for $d$CDW (left panel), $d$PI (middle panel), and BOP${}_x$ (right panel) along the corresponding outer critical line in Fig. 2. For each critical temperature, the critical doping rate can be read off from Fig. 2. Because of symmetry, the results along the direction of $(0,\pi )$-$(\pi ,\pi )$ are the same as those along the $(\pi ,0)$-$(\pi ,\pi )$ direction for the $d$CDW and $d$PI.

Figure 4

Critical temperature and doping rate for $t^{\prime }=-0.20$ (upper panel) and $t^{\prime }=-0.30$ (lower panel). The notation is the same as Fig. 2.

Figure 5

Modulation vectors for $d$CDW (left panels), $d$PI (middle panels), and BOP${}_x$ (right panels) for $t^{\prime }=-0.20$ (upper row), $t^{\prime }=-0.30$ (middle row), and $t^{\prime }=-0.40$ (lower row) along the outer critical line of the corresponding order for each $t^{\prime }$ in Figs. 4 and 6 (middle).

Figure 6

The same plot as Fig. 4, but for different choices of $t^{\prime }$: $t^{\prime }=-0.35$ (top panel), $t^{\prime }=-0.40$ (middle panel), and $t^{\prime }=-0.45$ (bottom panel). In the latter two cases the phase diagram in a high doping region is magnified in the inset.