pythtb.Wannier.maxloc

Wannier.maxloc(alpha=0.5, max_iter=1000, tol=1e-05, grad_min=0.001, verbose=False)

Unitary transformation to minimize the gauge-dependent spread.

This procedure implements the Marzari-Vanderbilt maximal localization algorithm [1]. Given a (disentangled) subspace (.tilde_states), it finds the optimal unitary transformation that minimizes the gauge-dependent part of the Wannier spread \(\widetilde{\Omega}\). The algorithm proceeds iteratively, applying gradient-descent updates to the unitary matrices at each k-point until convergence.

Parameters:

alphafloat, optional

Step size for gradient descent. Typical values are between 0 and 1.

max_iterint, optional

Maximum number of iterations for the optimization. Default is 1000.

tolfloat, optional

Convergence tolerance for the change in spread. Default is 1e-5.

grad_minfloat, optional

Minimum gradient magnitude for convergence. Default is 1e-3.

verbosebool, optional

If True, print progress messages. Default is False.

Notes

• The gauge-dependent contribution to the total Wannier spread is

  \[\widetilde{\Omega} = \Omega - \Omega_I,\]

  where \(\Omega\) is the total quadratic spread functional and \(\Omega_I\) is the gauge-invariant part obtained during the disentanglement step.

• The minimization is achieved by unitary rotations of the form

  \[| u_{m\mathbf{k}}^{\text{new}} \rangle = \sum_{n} U_{nm}(\mathbf{k}) \, | u_{n\mathbf{k}}^{\text{old}} \rangle,\]

  with \(U(\mathbf{k}) \in U(N)\), where \(N\) is the dimension of the disentangled subspace at each k-point.

• The gradient of \(\widetilde{\Omega}\) with respect to an infinitesimal anti-Hermitian generator \(A(\mathbf{k})\) is computed, and the unitary matrices are updated via

  \[U(\mathbf{k}) \;\to\; \exp[-\epsilon A(\mathbf{k})] \, U(\mathbf{k}),\]

  where \(\epsilon = \alpha/4 \sum_{\mathbf{b}}w_b\) is the step size (given alpha).

• Iteration proceeds until the gradient norm falls below grad_min and the change in spread is smaller than tol, or the maximum number of iterations is reached.

References

[1]

Marzari, N., & Vanderbilt, D. Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev. B 56, 12847 (1997).