Spin-orbit coupling and strong electronic correlations in cyclic molecules

In atoms spin-orbit coupling (SOC) cannot raise the angular momentum above a maximum value or lower it below a minimum. Here we show that this need not be the case in materials built from nanoscale structures including multinuclear coordination complexes, materials with decorated lattices, or atoms on surfaces. In such cyclic molecules the electronic spin couples to currents running around the molecule. For odd-fold symmetric molecules (e.g., odd-membered rings) the SOC is highly analogous to the atomic case; but for even-fold symmetric molecules every angular momentum state can be both raised and lowered. These differences arise because for odd-fold symmetric molecules the maximum and minimum molecular orbital angular momentum states are time-reversal conjugates, whereas for even-fold symmetric molecules they are aliases of the same single state. We show, from first-principles calculations, that in suitable molecules this molecular SOC is large, compared to the energy differences between frontier molecular orbitals. Finally, we show that, when electronic correlations are strong, molecular SOC can cause highly anisotropic exchange interactions and discuss how this can lead to effective spin models with compass Hamiltonians.

Figure 1

Allowed matrix elements of $H_{SO}$ for systems with cyclic symmetry, ${\tilde{\mathcal{C}}}_N$. (a) For odd $N$ there is a maximum (minimum) molecular angular momentum state $\left|L\right.〉$ ($\left|-L\right.〉$) that cannot be raised (lowered) by SMOC. For spherically symmetric systems (e.g., atoms) all shells contain an odd number of states, $2l+1=1,3,5,\cdots$ and have maximum (minimum) $m_l$ values, thus the odd $N$ cyclic and spherically symmetric cases are highly analogous. (b) In contrast, for even $N$ all states couple to a state with equal total angular momentum about $z, j=k+\sigma$, e.g., $\left|L;\uparrow \right.〉$ couples to $\left|1-L;\downarrow \right.〉$.

Figure 2

Illustration of the aliasing of the maximum and minimum angular momentum states in even-fold symmetric molecules. Lines show the molecular orbital angular momentum states defined in Eq. (3) in the $N\to \infty$ limit. We plot snapshots of the real part of the wave function as it evolves under the trivial Schrödinger time evolution. Here we show the wave functions for ${\omega }_{k}t=0.1$, where $\hslash {\omega }_k$ is the energy of the state $\left|k\right.〉$. (a) For three sites the maximum ($k=+1$, red) and minimum ($k=-1$, blue) angular momentum states are distinguishable when sampled on the three sites (marked on the abscissa; values sampled are marked by triangles). (b) On four sites the $k=+1$ (red) and $k=-1$ (blue) angular momentum states remain distinguishable when sampled on the four sites (values sampled are marked by diamonds/squares). (c) However, the maximum ($k=+2$, red) and minimum ($k=-2$, blue) angular momentum states are indistinguishable when sampled on the four sites (values sampled are marked by diamonds/squares); i.e., $\left|+2\right.〉\equiv \left|-2\right.〉$. For simplicity the phase factors, ${\eta }_k$, are not included in the figure, but, clearly, an overall phase factor cannot remove the aliasing. Animations of the full time evolution, shown in the Supplemental Material [56], underscore that this argument holds at all times.

Figure 3

The low-energy physics of a single ${\mathrm{Mo}}_3{\mathrm{S}}_7$(dmit)${}_3$ molecule can be understood in terms of six Wannier spinors (three Kramers pairs). The large components of one are shown above; the others are related by the ${\tilde{\mathcal{C}}}_3$ and/or time-reversal symmetry. The four panels display the (a) real and (b) imaginary parts of the spin-up large component and the (c) real and (d) imaginary parts of the spin-down large component. Note that the isosurface in (a) corresponds to a contour value 50 times smaller than those in (b)–(d).

Figure 4

The exchange anisotropies vary significantly in molecular materials with different packing motifs. (a) Sketch of a pair of nearest neighbors in the $t-J$ model for ${\mathcal{C}}_2$ molecules. Spheres indicate the Wannier orbitals and the curves connecting them show the molecular symmetry. The nearest-neighbor intermolecular hopping, $t$, is marked. The local $x$ and $y$ axes are uniquely determined by the SMOC via the phase convention chosen in Eq. (3). Thus we parametrize the packing motif by the angles between the local axes on neighboring molecules: $\theta$ ($\varphi$) is the relative rotation about the $y$ ($z$) axes; the effective Hamiltonian is independent of rotations about the $x$ axes. The local coordinate system is shown in black and the angles are marked relative to the gray axes, which point in the same directions on both molecules. As we only consider pairwise interactions we write the effective Hamiltonian in the local coordinates of the $i\mathrm{th}$ molecule, cf. Eq. (10). (b) Parallel stacking ($\theta =\varphi =0$) leads to Ising anisotropy. (c) Perpendicular packing ($\theta =\pi /2, \varphi =0$) gives XY anisotropy. (d), (e) More complicated packing leads to lower symmetry exchange Hamiltonians [here we plot (d) $\theta =\varphi =1$ and (e) and $\varphi =-\theta =2\pi /3]$. In all plots $J=0$ and ${\mathcal{J}}_0=t^2{t}_c^2{J}_c/\left\{2[t_c^2+{\lambda }^2][2(t_c^2+{\lambda }^2)-J_c\sqrt{t_c^2+{\lambda }^2}]\right\}$. Analytical expressions for ${\mathcal{J}}_{\alpha \beta }$ and ${\mathit{D}}^{\pm }$ are given in the Supplemental Material [56].

Figure 5

Effective spin-one model for half-filled ${\mathcal{C}}_2$ molecules. (a)–(c) Spectrum of a single ${\mathcal{C}}_2$ molecule with two electrons in two orbitals with (a) $J_F=0, \lambda \ne 0$, (b) $J_F\ne 0, \lambda =0$, and (c) $J_F\ne 0$ and $\lambda \ne 0$. Numbers on the right-hand side of these panels label the degeneracies of the states. In cases (a) and (b) the low-energy states are a singlet and a triplet. Only when both $J_F$ and $\lambda$ are nonzero is the degeneracy of the triplet lifted, (c). (d)–(f) Parameters of the effective spin-one model [Eq. (12)]. As in the spin-1/2 case (Fig. 4) SMOC causes large anisotropic interactions. These are most pronounced when the lowest-energy spin-singlet excitation on a single molecule is at low energies.