Electronic pair binding and Hund's rule violations in doped ${\text{C}}_{60}$

We calculate the electronic properties of the $t\text{−}J$ model on a ${\text{C}}_{60}$ molecule using the density-matrix renormalization group and show that Hund's first rule is violated and that for an average of three added electrons per molecule, an effective attraction (pair binding) arises for intermediate values of $t/J$. Specifically, it is energetically favorable to put four electrons on one ${\text{C}}_{60}$ and two on a second rather than putting three on each. Our results show that a dominantly electronic mechanism of superconductivity is possible in doped ${\text{C}}_{60}$.

Figure 1

(a) The “spin gap” ${\mathrm{\Delta }}_s$ and (b) orbital gap ${\mathrm{\Delta }}_o$ (defined in Sec. 4) as a function of number of added electrons $n_e$, for different number of DMRG states $m$. Here $t/J=2$. The inset in (a) shows the “spin gap” ${\mathrm{\Delta }}_s$ for $t^{\prime }/t=1.0$ and $t^{\prime }/t=1.2$.

Figure 2

Electronic pair binding energy $E_b\left(n_e\right)$ as a function of $t/J$ with different $n_e$ for both $m=8000$ and $m=10000$ number of DMRG states. The shaded region and lines connecting the data points are guides to the eye only. The inset is the electronic pair binding energy $E_b\left(n_e\right)$ as a function of $n_e$ for both $t^{\prime }/t=1.0$ and $t^{\prime }/t=1.2$ with $t/J=2$ and $m=8000$.

Figure 3

Ground-state energy difference $\delta E=E_0\left(m\right)-E_0$ as a function of DMRG states $m$ for the one-band Hubbard model [see Eq. (A1)] with $N_e$ electrons on the ${\text{C}}_{20}$ molecule at $U/t=2$ in (a) and $U/t=5$ in (b). Here $E_0$ is the ground-state energy obtained by exact diagonalization (see Refs. [32, 33]), while $E_0\left(m\right)$ is the ground-state energy obtained by DMRG simulation with minimal value of $z$-component total spin $S_z=s_{min}$, i.e., $s_{min}=0$ for $N_e=20,22$, and $s_{min}=1/2$ for $N_e=21$. The dashed lines in (a) denote the ground-state energy difference between QMC and exact diagonazation with same $N_e$ and $S_z$ (see Table II in Ref. [33]), for comparison with the DMRG simulation labeled by the same color. The inset in (b) is the electronic pair binding energies $E_b=2E_0(N_e=21)-E_0(N_e=22)-E_0(N_e=20)$ from DMRG (red circle) and ED simulations (solid line). Note that here we use a different definition of the pair binding energy compared with Refs. [32, 33], and a negative $E_b$ means a repulsive interaction between doped electrons.

Figure 4

(a) “Excess spin” $\delta S^2=\vec{S}·\vec{S}-s_{min}(s_{min}+1)$ where $\vec{S}={\sum }_i\overrightarrow{S_i}$ and (b) spin gap ${\mathrm{\Delta }}_s=E_0(s_{min}+1)-E_0\left(s_{min}\right)$ as a function of number of DMRG states $m$, for $t/J=2$ and different numbers of added electrons $n_e$. Here $E_0\left(S_z\right)$ is the ground-state energy for given value of spin $S_z$. Inset in (b): spin gap ${\mathrm{\Delta }}_s$ as a function of $m$ for $n_e=0$ and 1.

Figure 5

Electronic pair binding energy $E_b\left(n_e\right)$ for $t/J=2$ as a function of DMRG states $m$ with different number of added electrons $n_e$. The dashed line indicates the zero. The lines connecting the data points are guides to the eye only.

Figure 6

We map the ${\text{C}}_{60}$ molecule to a one-dimensional chain shown in the figure, where the sequence of sites is represented by the numbers.