Time-Reversal Symmetry Breaking by a ($d+id$) Density-Wave State in Underdoped Cuprate Superconductors

It was proposed that the $i{d}_{x^2-y^2}$ density-wave state (DDW) may be responsible for the pseudogap behavior in the underdoped cuprates. Here we show that the admixture of a small $d_{xy}$ component to the DDW state breaks the symmetry between the counterpropagating orbital currents of the DDW state and, thus, violates the macroscopic time-reversal symmetry. This symmetry breaking results in a nonzero polar Kerr effect, which has recently been observed in the pseudogap phase.

Figure 1

Left: The real-space picture of the $i{d}_{x^2-y^2}$ density-wave state. Spontaneous currents are shown by the arrows. Clockwise and counterclockwise circulations are indicated by the $-$ and the $+$ signs, respectively. Right: The same for the $d_{xy}+i{d}_{x^2-y^2}$ density-wave state. Now the neighboring plaquettes have different amplitudes of the next-nearest-neighbor tunneling, indicated by the solid and dashed diagonal lines.

Figure 2

The Fermi surface of the $d_{xy}+i{d}_{x^2-y^2}$ density-wave state, consisting of the electron and hole pockets. The hole pockets are located near ($\pi /2$, $\pm \pi /2$), and the electron pocket near ($\pi$, 0) in the reduced Brillouin zone. The band-structure parameters are given at the end of the Letter, and $W_0=20\mathrm{meV}$ is taken for this plot.

Figure 3

The calculated Kerr angle as a function of the hole concentration $x$. We use the linear dependence $W_0(x)=W_0(1-x/x_c)$ to model roughly the pseudogap variation with $x$, where $x_c=0.175$ and $W_0=30\mathrm{meV}$. Although the admixture ${\Delta }_0=0.001W_0/2$ of the $d_{xy}$ component is small, it leads to a nonzero Kerr angle of the same order of magnitude as in the experiments [9].