Quantum dimer model on fullerenes: Resonance, scarring, and confinement

The earliest known examples of quantum superposition were found in organic chemistry—in molecules that “resonate” among multiple arrangements of $\pi$-bonds. Small molecules such as benzene resonate among a few bond arrangements. In contrast, a large system that stretches over a macroscopic lattice may resonate among an extensive number of configurations. This is analogous to a quantum spin liquid as described by Anderson's resonating valence bond picture. In this article, we study the intermediate case of somewhat large molecules, the ${\mathrm{C}}_{20}$ and ${\mathrm{C}}_{60}$ fullerenes. We build a minimal description in terms of quantum dimer models (QDMs), allowing for local resonance processes and repulsion between proximate $\pi$-bonds. This allows us to characterize ground states, e.g., with ${\mathrm{C}}_{60}$ forming a superposition of 5828 dimer covers. Despite the large number of contributing dimer covers, the ground state shows strong dimer-dimer correlations. Going beyond the ground state, the full spectrum of ${\mathrm{C}}_{60}$ shows many interesting features. When repulsive terms are neglected, the energy values are placed symmetrically about zero. This reflection symmetry originates from a chiral symmetry of the Hamiltonian, which in turn originates from the local bond geometry of ${\mathrm{C}}_{60}$. This property also leads to a large number of protected zero-energy states. In general, the spectrum contains many scarlike states, corresponding to localized dimer-rearrangement dynamics. Resonance dynamics in QDMs can manifest in the behavior of defects, potentially binding defects via an effective attractive interaction. To test this notion, we introduce pairs of vacancies at all possible separations. Resonance energy reaches its lowest value when the vacancies are closest to one another. This suggests confinement of monomers, albeit within a finite cluster. We discuss qualitative pictures for understanding bonding in fullerenes, and we draw connections with results from quantum chemistry.

Figure 1

Schlegel diagrams for ${\mathrm{C}}_{20}$ (top) and ${\mathrm{C}}_{60}$ (bottom).

Figure 2

Bond correlations in the ground state of ${\mathrm{C}}_{20}$ when $V/t=0$. The reference bond is shown in red. On all other bonds, the bond thickness shown is proportional to the correlation with respect to the reference bond (see text).

Figure 3

Ground-state energy vs distance between vacancies. “Undoped” represents the ${\mathrm{C}}_{20}$ graph with no vacancies. The remaining columns represent vacancies at various distances, e.g., “1nn” represents vacancies on sites that are first neighbors of one another—say on sites 0 and 1 as shown in Fig. 1.

Figure 4

Two dimer covers on the ${\mathrm{C}}_{60}$ graph. Top: a configuration with no flippable loops. Bottom: the dimer cover with the largest number of flippable loops. Each of the 20 hexagons carries three alternating dimers.

Figure 5

Density of states in the ${\mathrm{C}}_{60}$ QDM in the purely kinetic regime, i.e., with $V=0$. The inset shows a limited energy range where three peaks are clearly discernible. The peaks occur at $u_1=2cos(2\pi /5), u_2=2cos(\pi /5)$, and 3.

Figure 6

Graphical representation of a self-contained sector within the Hilbert space. The nodes represent dimer covers, with four shown explicitly. A pair of nodes connected by a bond represents two dimer covers that are connected by a single hexagon-flip process. For example, by flipping one hexagon at a time, we may traverse the following loop: $|{\psi }_0\rangle \to |{\psi }_1\rangle \to |{\psi }_2\rangle \to |{\psi }_3\rangle \to |{\psi }_0\rangle$.

Figure 7

Dimer correlations in the ground state of the ${\mathrm{C}}_{60}$ QDM with $V/t=0$. The top and bottom panels correspond to two choices for the reference bond. In the top figure, the reference bond separates a pentagon from a hexagon. In the bottom figure, it separates two hexagons.

Figure 8

Ground-state energy in ${\mathrm{C}}_{60}$ with two vacancies. The first column (“Undoped”) gives the ground-state energy with no vacancies, for various values of $V/t$. Subsequent columns show the ground-state energy for various relative positions of two vacancies. For example, “1nn” corresponds to two positions that are nearest neighbors on the ${\mathrm{C}}_{60}$ graph. There are two inequivalent choices for nearest neighbors, shown as (0,1) and (0,5). The positions are shown using the site labels given in Fig. 1.