Local spin structure of large spin fermions

We show that large spin fermions have very rich spin structures. The local spin order of a spin-$f$ Fermi gas is a linear combination of $2f$ (particle-hole) angular momentum states, $L=1,...,2f$. $L=1,2$ represent ferromagnetic and nematic spin order, while $L\ge 3$ are higher spin orders that have no analog in spin-1/2 systems. Each $L$ spin sector is characterized as $L$ pairs of antipodal points on a sphere. Model calculations show that some of these spin orders have the symmetry of Platonic solid, and many of them have non-Abelian line defects.

Figure 1

Inset (i): Ferromagnetic order ${\Phi }^{\left(1\right)}$ represented by the Majorana pair $\left[\widehat{\mathbf{n}}\right]\equiv (\widehat{\mathbf{n}},-\widehat{\mathbf{n}})$. $\left[\widehat{\mathbf{n}}\right]$ and $[-\widehat{\mathbf{n}}]$ are distinct. Inset (ii): Uniaxial nematic state ${\Phi }^{\left(2\right)}$ represented by two identical Majorana pairs. It has the symmetry of a rod. $\left[\widehat{\mathbf{n}}\right]\left[\widehat{\mathbf{n}}\right]$ and $[-\widehat{\mathbf{n}}][-\widehat{\mathbf{n}}]$ are identical. (iii) Biaxial nematic state ${\Phi }^{\left(2\right)}$ with two distinct Majorana pairs: The quantum state is invariant under $\pi$ rotation about $\widehat{\mathbf{n}}\times \widehat{\mathbf{m}},\widehat{\mathbf{n}}+\widehat{\mathbf{m}},\widehat{\mathbf{n}}-\widehat{\mathbf{m}}$. It has the symmetry of a brick with different edge lengths. Inset (iv): The Majorana points cannot form a tetrahedron because its vertices do not form antipodal pairs. The numerals $(1,2,...,7)$ denote the spin order $({\Phi }^{\left(1\right)},{\Phi }^{\left(2\right)},...,{\Phi }^{\left(7\right)})$ of a spin $f=7/2$ Fermi gas along the loop $\mathcal{C}$ in real space. ${\Phi }_M^{\left(L\right)}\left(\mathbf{R}\right)$ is represented by a sphere $S^{\left(L\right)}$ of radius $|{\lambda }^{\left(L\right)}\left(\mathbf{R}\right)|$ marked with $L$ pairs of antipodal points. Here, ${\lambda }^{\left(7\right)}=0$, and ${\Phi }^{\left(3\right)},{\Phi }^{\left(4\right)},{\Phi }^{\left(6\right)}$ form an octahedron, a cubic, and an icosahedron. ${\Phi }^{\left(5\right)}$ is a polyhedron with five vertices forming a pentagon. The texture of ${\Phi }_M^{\left(2\right)}\left(x\right)$ depicted implies a line defect inside loop $C$, whereas the texture of ${\Phi }_M^{\left(1\right)}\left(x\right)$ is defect free. Our model calculations for spin $f=21/2$ Fermi gas reveal Platonic solid configurations like (c), (d), (f), and dodecahedron (g) in certain parameter regimes.

Figure 2

The transition temperature for spin order $|{\Phi }^{\left(L\right)}\rangle$ for $L=1,2,6,3,5,4,7$ (from bottom to top), for a spin $f=21/2$ Fermi gas.