Möbius molecules and fragile Mott insulators

Motivated by the concept of Möbius aromatics in organic chemistry, we extend the recently introduced concept of fragile Mott insulators (FMI) to ring-shaped molecules with repulsive Hubbard interactions threaded by a half-quantum of magnetic flux $(hc/2e)$. In this context, an FMI is the insulating ground state of a finite-size molecule that cannot be adiabatically connected to a single Slater determinant, i.e., to a band insulator, provided that time-reversal and lattice translation symmetries are preserved. Based on exact numerical diagonalization for finite Hubbard interaction strength $U$ and existing Bethe-ansatz studies of the one-dimensional Hubbard model in the large-$U$ limit, we establish a duality between Hubbard molecules with $4n$ and $4n+2$ sites, with $n$ integer. A molecule with $4n$ sites is an FMI in the absence of flux but becomes a band insulator in the presence of a half-quantum of flux, while a molecule with $4n+2$ sites is a band insulator in the absence of flux but becomes an FMI in the presence of a half-quantum of flux. Including next-nearest-neighbor hoppings gives rise to new FMI states that belong to multidimensional irreducible representations of the molecular point group, giving rise to a rich phase diagram.

Figure 1

Orbital topology of (a) an aromatic molecule and (b) a Möbius aromatic.

Figure 2

Ground-state energy of the four-site (red diamonds) and six-site (blue circles) half-filled Hubbard model with zero flux (open markers) and $\pi$ flux (filled markers), obtained by ED. The BIs are lower in energy than the FMIs.

Figure 3

Single-particle energy levels for the four- and six-site rings at $U=0$. The insets give the $U>0$ ground-state eigenvalues $\lambda$ and $\tilde{\lambda }$ of the translation operators ${\mathcal{R}}_0$ and ${\mathcal{R}}_{\varphi }$, respectively.

Figure 4

Phase diagram for $N=6$ at 0 flux with nearest-neighbor hopping $t_1$ and NNN hopping $t_2$, obtained by ED. Ground states are labeled by a symbol ${}^{2S+1}\Gamma$, where $\Gamma$ denotes the irreducible representation of the molecular point group $C_{6v}$ according to which the ground state transforms and $S$ is the total spin in the ground state.

Figure 5

Phase diagram for $N=6$ at $\pi$-flux with nearest-neighbor hopping $t_1$ and NNN hopping $t_2$, obtained by ED. Ground states are labeled by a symbol ${}^{2S+1}\tilde{\Gamma }$ where $\tilde{\Gamma }$ denotes the irreducible representation of the molecular point group $C_{6v}$ according to which the ground state transforms and $S$ is the total spin in the ground state. $\tilde{\Gamma }$ differs from $\Gamma$ in the definition of the translation operator (${\mathcal{R}}_{\pi /N}$ for $\pi$ flux and ${\mathcal{R}}_0$ for zero flux, see Sec. 3).