Density wave instabilities in a correlated two-dimensional metal

Motivated by recent experimental evidence of charge order in the pseudogap phase of cuprates, we perform a variational analysis of spin-singlet density wave ordering in metals with antiferromagnetic interactions on the square lattice, using a wave function with double occupancy projected out. We examine ordering with and without time-reversal symmetry, with an arbitrary wave vector and a tunable form factor. Depending on parameters, we find $d$-form factor density wave ordering, with a wave vector either parallel to the lattice generators or diagonally oriented, or a ground state which carries a time-reversal breaking pattern of spontaneous currents.

Figure 1

Fermi surface with $t_1=1$, $t_2=-0.32$, $t_3=0.128$, and $\mu =-1.11856$. For this dispersion we have $Q_0=4\pi /11$.

Figure 2

Schematic illustration of the ordering patterns in real space. (a) Diagonal $d$-form factor CDW [$P_3\ne 0$, $\mathit{Q}=(0.25,0.25)\pi$]. (b) Horizontal, predominantly $d$-form factor CDW [$P_3\ne 0$, $\mathit{Q}=(0.25,0)\pi$]. (c) Staggered flux state [$P_5\ne 0$, $\mathit{Q}=(\pi ,\pi )$]. For the time-reversal invariant orders (a) and (b) we show the fluctuation of $\langle c_{\mathbf{x}+\mathbf{a}}^{†}{c}_{\mathbf{x}}\rangle$ about the average with positive (red) and negative (blue) values. For the time-reversal breaking order (c) we show the pattern of permanent currents.

Figure 3

In the top row, the gain in variational energy (per site, times 100) by allowing ordering at wave vector $\mathit{Q}$ is displayed. Each diagram is for a different Fermi surface, shown in the inset. The local minimum at $\mathit{Q}=(\pi ,\pi )$ is the staggered flux state $P_5\ne 0$. The local minima at $\mathit{Q}=(Q_0,Q_0)$ and $(Q_0,0)$ are predominantly $d$-form factor CDWs ($P_3\ne 0$). The red dots denote wave vectors $\mathit{Q}$ that connect hot spots. In the bottom row, we show approximate diagrams of the ground-state order, as a function of $J$ and $V$. The setup corresponding to the plot above is indicated with a black dot. Model parameters (values in the brackets correspond to plots from left to right): $x=10\%$, $t_1=1$, $t_2\in \{0.5,0.16,0.18\}$, $t_3\in \{0.6,0.9,1.6\}$, $J\in \{0.09,0.235,0.4\}$, $V\in \{1.,1.5,0.5\}$. Variational parameters: $T_1=1$, $T_2\in \{-0.1,-0.32,-0.5\}$, $T_3=0.128$. As described in the text the Fermi surface is determined by the $T_a$ parameters.