Excitons

Frenkel Excitons

Elaboration on Toyozawa (Eq. 8.1.6)

In lattice systems with localized excitations (e.g., excitons, dipole excitations), the Hamiltonian often has translational invariance. This means the coupling between two sites depends only on their relative positions. This property allows us to diagonalize the Hamiltonian using Bloch-like states, just as in the theory of lattice vibrations or electrons in a periodic potential.

We start with the dipole–dipole interaction Hamiltonian (see Toyozawa, eq 8.1.6):

\[ H_{nm} = D(\mathbf R_{nm}), \quad \mathbf R_{nm} = \mathbf R_n - \mathbf R_m, \]

where

\[ D(\mathbf R_{nm}) = \left[ \frac{\mu^2}{R_{nm}^3} - \frac{3(\mu \cdot \mathbf R_{nm})^2}{R_{nm}^5} \right] \frac{1}{4\pi \epsilon_0}. \]

Equivalently, in the localized basis the Hamiltonian is:

\[ H^{(e)} = \sum_{n,m} H_{nm} |\Phi_n\rangle \langle \Phi_m|. \]

By analogy with lattice vibrations, we consider the trial wavefunction that will help diagonalize this:

\[ \Psi^{(e)}_{\mathbf K} = \frac{1}{\sqrt N} \sum_n e^{i \mathbf K \cdot \mathbf R_n} \Phi_n, \]

where \(\Phi_n\) is a localized excitation at site \(n\). Submitting this Bloch state to the Hamiltonian:

\[ H^{(e)} \Psi^{(e)}_{\mathbf K} = \frac{1}{\sqrt N} \sum_m e^{i \mathbf K \cdot \mathbf R_m} \sum_n H_{nm} \Phi_n. \]

Since \(H_{nm} = D(\mathbf R_{n} - \mathbf R_{m})\) depends only on the difference:

\[ \sum_m H_{nm} e^{i\mathbf K \cdot \mathbf R_m} = \sum_m D(\mathbf R_n - \mathbf R_m)\, e^{i\mathbf K \cdot \mathbf R_m}. \]

Make the substitution \(\mathbf R_m = \mathbf R_n - \mathbf R_{nm}\):

\[ = \sum_m D(\mathbf R_{nm})\, e^{i\mathbf K \cdot (\mathbf R_n - \mathbf R_{nm})}. \]

Factor the \(n\)-dependence:

\[ = e^{i\mathbf K \cdot \mathbf R_n} \sum_m D(\mathbf R_{nm}) e^{-i\mathbf K \cdot \mathbf R_{nm}}. \]

Therefore:

\[ H^{(e)} \Psi^{(e)}_{\mathbf K} = \Bigg(\sum_{m} D(\mathbf R_{nm}) e^{-i \mathbf K \cdot \mathbf R_{nm}} \Bigg) \frac{1}{\sqrt N} \sum_n e^{i \mathbf K \cdot \mathbf R_n} \Phi_n. \]

But the last sum is exactly \(\Psi^{(e)}_{\mathbf K}\).

Thus, the Bloch state is an eigenfunction:

\[ H^{(e)} \Psi^{(e)}_{\mathbf K} = D_{\mathbf K} \Psi^{(e)}_{\mathbf K}, \]

with eigenvalue

\[ D_{\mathbf K} = \sum_{n \neq 0} D(\mathbf R_n) e^{-i \mathbf K \cdot \mathbf R_n}. \]

and thus, the diagonalized energy reads:

\[ E^{(e)}_{\mathbf K} = E^{(g)} + (\varepsilon + D_{\mathbf K}), \]

where \(E^{(g)}\) is the ground-state energy and \(\varepsilon\) is the excitation energy in the absence of interactions.

Note that form of the Bloch state is not arbitrary. It is dictated by the translational invariance of the Hamiltonian, which ensures that wave-like superpositions diagonalize the interaction matrix. Fascinatingly, this is why excitonic, vibrational, and electronic states in lattices share the same mathematical structure.