Pair binding and Hund's rule breaking in high-symmetry fullerenes

TL;DR

The study addresses whether Hund's rule holds and whether pair binding occurs in high-symmetry fullerenes under the Hubbard model. Using large-scale DMRG with $SU(2)$ spin, $U(1)$ charge, and optional $\,\bZ_N$ symmetry, it maps out spin states, Mott transitions, and pair-binding tendencies across $C_{20}$, $C_{28}$, $C_{40}$, and $C_{60}$. Key findings include a Mott transition at $U_c \sim 2.2 t$ for $C_{20}$ with repulsive pair binding across $U$, a spin-2 Hund magnet phase in $C_{28}$ transitioning to lower spins before Mott localization, Hund's rule breaking at half filling in $C_{40}$ (but restoration upon one-electron doping), and Hund's rule breaking in $C_{60}$ with minimum-spin states upon two or three-electron doping, all consistent with an electronic mechanism for superconductivity in $C_{60}$ lattices and suggesting geometric frustration suppresses pairing in smaller fullerenes.

Abstract

Highly-symmetric molecules often exhibit degenerate tight-binding states at the Fermi edge. This typically results in a magnetic ground state if small interactions are introduced in accordance with Hund's rule. In some cases, Hund's rule may be broken, which signals pair binding and goes hand-in-hand with an attractive pair-binding energy. We investigate pair binding and Hund's rule breaking for the Hubbard model on high-symmetry fullerenes C$_{20}$, C$_{28}$, C$_{40}$, and C$_{60}$ by using large-scale density-matrix renormalization group calculations. We exploit the SU(2) spin symmetry, the U(1) charge symmetry, and optionally the Z(N) spatial rotation symmetry of the problem. For C$_{20}$, our results agree well with available exact-diagonalization data, but our approach is numerically much cheaper. We find a Mott transition at $U_c\sim2.2t$, which is much smaller than the previously reported value of $U_c\sim4.1t$ that was extrapolated from a few datapoints. We compute the pair-binding energy for arbitrary values of $U$ and observe that it remains overall repulsive. For larger fullerenes, we are not able to evaluate the pair binding energy with sufficient precision, but we can still investigate Hund's rule breaking. For C$_{28}$, we find that Hund's rule is fulfilled with a magnetic spin-2 ground state that transitions to a spin-1 state at $U_{c,1}\sim5.4t$ before the eventual Mott transition to a spin singlet takes place at $U_{c,2}\sim 11.6t$. For C$_{40}$, Hund's rule is broken in the singlet ground state, but is restored if the system is doped with one electron. Hund's rule is also broken for C$_{60}$, and the doping with two or three electrons results in a minimum-spin state. Our results are consistent with an electronic mechanism of superconductivity for C$_{60}$ lattices. We speculate that the high geometric frustration of small fullerenes is detrimental to pair binding.

Figures (10)

• Figure 1: Schlegel diagrams (nearest-neighbor graphs) of the high-symmetry fullerenes studied in this work. The hexagon faces are shaded grey for clarity, illustrating the full separation of the frustrated pentagon faces for $\mathrm{C}_{12}$ and $\mathrm{C}_{60}$.
• Figure 2: Energy levels of the tight-binding (Hückel) model for the high-symmetry fullerenes studied in this work (see Fig. \ref{['fig:fullerenes']}) along with their irreducible representations. The red dotted line marks the Fermi energy at half filling.
• Figure 3: DMRG data for the ground-state energy per site for $\mathrm{C}_{20}$ at $U=2$ in various sectors of the particle number $N_{\text{tot}}$ and the total spin $S_{\text{tot}}$. Small circles: SU(2)$\times$U(1)-symmetric calculation, bond dimensions $\chi_{\text{SU(2)}} \leq 10000.0$. Triangles: SU(2)$\times$U(1)$\times$$\mathbb{Z}_5$-symmetric calculation, $\chi_{\text{SU(2)}} = 15000.0$ (one datapoint in selected sectors). The DMRG values are linearly extrapolated in the variance per site (which decreases with increasing bond dimension); the error bars associated with the fit are also shown but are barely visible here. The crosses show the reference values from exact diagonalization. The pictures below the plots schematically visualize the filling of the HOMO (arrows: electrons, black circles: empty levels). Hund's rule holds true at any filling.
• Figure 4: Triplet gap Eq. \ref{['eq:gap:triplet']} and pair-binding energy Eq. \ref{['eq:Eb']} for $\mathrm{C}_{20}$ as a function of $U$ at $N_\text{tot}=20$. $U_c\approx2.2$ marks the metal-insulator (Mott) transition. Bullets indicate the raw data, lines are a result of Akima spline interpolation (for the triplet gap) or linear interpolation (for $E_b$).
• Figure 5: Ground-state energy per site and HOMO filling for $\mathrm{C}_{28}$ at $U=2$. The bond dimensions are $\chi_{\text{SU(2)}} \leq 10000.0$ for the SU(2)$\times$U(1)-symmetric calculation (small circles) and $\chi_{\text{SU(2)}} = 10000.0, 15000.0$ for the SU(2)$\times$U(1)$\times$$\mathbb{Z}_3$-symmetric calculation (triangles).