SO(5) symmetry in the quantum Hall effect in graphene

Electrons in graphene have four flavors associated with low-energy spin and valley degrees of freedom. The fractional quantum Hall effect in graphene is dominated by long-range Coulomb interactions, which are invariant under rotations in spin-valley space. This SU(4) symmetry is spontaneously broken at most filling factors, and also weakly broken by atomic scale valley-dependent and valley-exchange interactions with coupling constants $g_z$ and $g_{\perp }$. In this paper, we demonstrate that when $g_z=-g_{\perp }$, an exact SO(5) symmetry survives which unifies the Néel spin order parameter of the antiferromagnetic state and the $XY$ valley order parameter of the Kekulé distortion state into a single five-component order parameter. The proximity of the highly insulating quantum Hall state observed in graphene at $\nu =0$ to an ideal SO(5) symmetric quantum Hall state remains an open experimental question. We illustrate the physics associated with this SO(5) symmetry by studying the multiplet structure and collective dynamics of filling factor $\nu =0$ quantum Hall states based on exact-diagonalization and low-energy effective theory approaches. This allows to illustrate how manifestations of the SO(5) symmetry would survive even when it is weakly broken.

Figure 1

Schematic illustration of the five-component $(T_{x,y},N_{x,y,z})$ order parameter space, and of rotations in this vector space produced by the SO(5) generators.

Figure 2

Low-energy spectrum on the torus geometry for zero total momentum, filling factor $\nu =0$, and orbital Landau level degeneracy $N_{\varphi }=8$ as a function of ${\theta }_g$ in the range $[\pi /2,5\pi /4]$. $E_{\text{v}}$ is defined as the difference between the eigenvalues of $H_{\text{C}}+H_{\text{v}}$ and the Coulomb-only ground state energy. All plotted eigenvalues are degenerate in the absence of $H_{\text{v}}$. (a) Ground-state energies in a series of $(T_z,S)$ sectors. The solid lines show the lowest $T_z=0$ energies for different total spin $S$ values. Similarly, the dashed lines show the lowest spin singlet ($S=0$) energies in different $T_z$ sectors. The ground state has $S=0$ and $T_z=0$ throughout the plotted ${\theta }_g$ range. The inset shows the mean-field phase diagram over the full ${\theta }_g$ range from Ref. [20]. (b) Low-energy states in the $T_z=0$ sector for a series of total spin $S$ quantum numbers. Note that at ${\theta }_g=3\pi /4$, states with different $S$ values are degenerate because of the hidden SO(5) symmetry.

Figure 3

Geometric representation of SU(4) multiplet structures. (a) The octahedron in ($S_z,N_z,T_z$) space represents the SU(4) multiplet structure of Coulomb ground states at $\nu =0$. (b) A $T_z$-constant plane in the octahedron displayed for $T_z=N_{\varphi }-4$ reached by applying lowering operators to the CDW state with $T_z=N_{\varphi }$. The size of the symbols indicates the degeneracy at each point in the ($S_z,N_z$) plane. (c) Multiplet structures of the first three levels of SO(5).

Figure 4

Finite size scaling analysis. (a) $E_{\text{v}}/g$ at ${\theta }_g=3\pi /4$ as a function of ${\Gamma }^2$ for $N_{\varphi }$ ranging from 4 to 8. (b) In a given ($T_z=0,S$) sector, the lowest energy at ${\theta }_g=4\pi /7\in [\pi /2,3\pi /4]$ as a function of $S^2$. (c) In a given ($S=0,T_z$) sector, the lowest energy at ${\theta }_g=\pi$ as a function of $T_z^2$. The inset in each figure shows the inverse of slope vs $N_{\varphi }$. See text for a more detailed description.