Kramers Degeneracy in a Magnetic Field and Zeeman Spin-Orbit Coupling in Antiferromagnetic Conductors

In this Letter, I study the magnetic response of electron wave functions in a commensurate collinear antiferromagnet. I show that, at a special set of momenta, hidden antiunitary symmetry protects Kramers degeneracy of Bloch eigenstates against a magnetic field, pointing transversely to staggered magnetization. Hence, a substantial momentum dependence of the transverse $g$-factor in the Zeeman term, turning the latter into a spin-orbit coupling that may be present in materials from chromium to borocarbides, cuprates, pnictides, as well as organic and heavy fermion conductors.

Figure 1

Doubly commensurate collinear antiferromagnet on a simple rectangular lattice. In the absence of magnetism, time reversal $\theta$ and primitive translations ${\mathbf{T}}_x$ and ${\mathbf{T}}_y$, shown by dashed arrows, are symmetry operations. In the antiferromagnetic state, neither of the three remains a symmetry, but the products $\theta {\mathbf{T}}_x$ and $\theta {\mathbf{T}}_y$, shown by solid arrows, do, as illustrated by filled spin arrows. Small dashed rectangle at the center is the Wigner-Seitz cell boundary in the paramagnetic state, while the shaded hexagon is its antiferromagnetic counterpart. Notice that neither of the point group operations interchanges the two sublattices; hence, any point symmetry of the lattice, including inversion $\mathcal{I}$, remains a symmetry of the antiferromagnetic state.

Figure 2

Relative orientation of ${\mathbf{\Delta }}_{\mathbf{r}}$, ${\mathbf{\Delta }}_{\mathbf{r}+\mathbf{a}}$, ${\mathbf{H}}_{\parallel }$, and ${\mathbf{H}}_{\perp }$. Notice that $\theta$ flips both ${\mathbf{\Delta }}_{\mathbf{r}}$ and $\mathbf{H}$, while ${\mathbf{T}}_{\mathbf{a}}$ leaves $\mathbf{H}$ intact, but inverts ${\mathbf{\Delta }}_{\mathbf{r}}$.

Figure 3

The paramagnetic ($\mathbf{p}=\pm \frac{\pi }{a}$), and the antiferromagnetic ($\mathbf{p}=\pm \frac{\pi }{2a}$) Brillouin zone boundaries of a one-dimensional Néel antiferromagnet. In the antiferromagnetic state, the two points $\mathbf{p}=\pm \frac{\pi }{2a}$ are identical modulo the antiferromagnetic reciprocal lattice vector $\mathbf{Q}=\frac{\pi }{a}$. At these two points, antiunitary symmetry ${\mathbf{U}}_{\mathbf{n}}(\pi ){\mathbf{T}}_{\mathbf{a}}\theta$ protects Kramers degeneracy against transverse magnetic field.

Figure 4

Geometry of the problem. (a) The Brillouin zone for a simple rectangular lattice (the rectangle) and its antiferromagnetic counterpart (MBZ, shaded hexagon). Thick curve (red online), passing through point $\Sigma$, shows a typical degeneracy line $g_{\perp }(\mathbf{p})=0$. At the MBZ boundary, only momentum $\mathbf{p}$ at point $\Sigma$ is equivalent to $-\mathbf{p}$ modulo the reciprocal lattice vector of the antiferromagnetic state. For a generic ${\mathbf{p}}^{\prime }$, shown by the dashed arrow, this is no longer true. (b) The Brillouin zone of a simple square lattice and its antiferromagnetic counterpart (shaded diagonal square). The degeneracy line must contain the entire MBZ boundary, shown in red online.