.spin — Exciton Spin Projection

When using the -s or --spin flag, Xatu outputs the spin characteristics of each computed exciton state. This includes the total spin projection and the individual spin projections of the electron and the hole.

Format

Each line in the .spin file has the format:

n    St    Sh    Se

Where:

  • n: Exciton index (starting from $0$ )

  • St: Total spin projection $S^z$of the exciton

  • Sh: Spin projection $ s^z_h$ of the hole

  • Se: Spin projection $ s^z_e$ of the electron

All values are given in units of \(\hbar\), e.g., \(\pm 1/2,\ \pm 1,\ 0\) , etc.

Spin Projection Output

When running Xatu with the -s or --spin flag, the code outputs a file containing the total spin projection of each excitonic state, along with the spin of the constituent electron and hole states.

Each line in the output contains:

exciton_index   Sz_total   Sz_hole   Sz_electron

Units: \(\hbar\)

Spin Calculation

The total spin projection \(\langle X | \hat{S}_z^T | X \rangle\) for each excitonic state \(|X\rangle\) is computed from the exciton wavefunction coefficients using the expression (see Eq. 31 of the paper):

\[\langle S_z^T \rangle = \sum_{v,c,\mathbf{k}} |A_{vc}^{Q}(\mathbf{k})|^2 (\sigma_c - \sigma_v)\]

where:

  • \(A_{vc}^{Q}(\mathbf{k})\) is the excitonic coefficient in the electron-hole basis

  • \(\sigma_c\), \(\sigma_v \in \{-1/2, +1/2\}\) are the spin projections of the conduction and valence bands, respectively

The spin of the exciton is thus the difference between the electron and hole spin projections, weighted by the probability amplitude of each electron-hole pair in the excitonic state.

Assumptions

  • Spin is assumed to be a good quantum number of the single-particle states.

  • This holds when the Hamiltonian \(H_0\) commutes with \(\hat{S}_z\), i.e., in the absence of spin-orbit coupling or magnetic noncollinearity.

  • Under this condition, spin projections are well-defined and can be treated using scalar labels \(\sigma_n\) for each band.

Reference

For a full derivation, see Section 2.3 and Eq. (30–31) in:

Efficient computation of optical excitations in two-dimensional materials with the Xatu code