Proposed Chiral Texture of the Magnetic Moments of Unit-Cell Loop Currents in the Pseudogap Phase of Cuprate Superconductors

We propose a novel chiral order parameter to explain the unusual polar Kerr effect in underdoped cuprates. It is based on the loop-current model by Varma, which is characterized by the in-plane anapole moment $\mathit{N}$ and exhibits the magnetoelectric effect. We propose a helical structure where the vector ${\mathit{N}}^{(n)}$ in the layer $n$ is twisted by the angle $\pi /2$ relative to ${\mathit{N}}^{(n-1)}$, thus breaking inversion symmetry. We show that coupling between magnetoelectric terms in the neighboring layers for this structure produces optical gyrotropy, which results in circular dichroism and the polar Kerr effect.

Figure 1

(a) Loop-current order in a ${\mathrm{CuO}}_2$ plane [16]. Black arrows show directions of microscopic persistent currents between copper and oxygen atoms. Green arrow shows the anapole moment $\mathit{N}$. (b) Chiral order constructed on a series of parallel ${\mathrm{CuO}}_2$ planes. The vector $\mathit{N}$ rotates by the angle $\pi /2$ from one layer to another, and the period of the structure is fourfold. The blue and red curves are the two magnetic field lines that intertwine in a double helix.

Figure 2

(a) Left: Feynman diagram for the effective action of electromagnetic fields (wavy lines), obtained by integrating out the electron field (solid lines with arrows). Right: The magnetoelectric term in the effective action, Eq. (5), where the double line represents the anapole moment $\mathit{N}$. (b) Coupling between the magnetoelectric terms at the neighboring layers produces the effective action for the electric field in Eq. (9). The dashed wavy line represents the magnetic field propagator, and the double solid line represents the interlayer correlator of the anapole moments in Eq. (2).

Figure 3

Schematic illustration of the rotation of electric polarization for the chiral state shown in Fig. 1b. The electric field ${\mathit{E}}^{(n)}$ at the bottom layer induces the magnetization ${\mathit{M}}^{(n)}$, which couples to the top layer and induces the electric polarization ${\mathit{P}}^{(n+1)}\perp {\mathit{E}}^{(n)}$ because of the twist in the anapole moments ${\mathit{N}}^{(n)}\perp {\mathit{N}}^{(n+1)}$.