From Bands and Orbitals to Effective Lattice Hamiltonians

A low-energy lattice Hamiltonian is obtained by choosing a target band subspace, constructing localized orbitals, projecting the one-body dynamics, and matching screened interactions after higher-energy degrees of freedom are removed. The result is basis- and window-dependent but scientifically controlled when target observables and parameter drift are reported.

Required background. Use the general microscopic-to-continuum matching logic. Helpful background. Lattice regulators and continuum targets supplies the regulator distinction.

From Bloch bands to localized orbitals

Choose an isolated or disentangled set of Bloch states and a smooth unitary frame $U_{mn}(\mathbf k)$. Wannier orbitals are

$$|w_{\mathbf R a}\rangle=\frac1{N_k}\sum_{\mathbf k n} e^{-i\mathbf k\cdot\mathbf R}U_{na}(\mathbf k)|\psi_{n\mathbf k}\rangle.$$

The hopping tensor is

$$t_{\mathbf R a,\mathbf R'b} =-\langle w_{\mathbf R a}|h_0|w_{\mathbf R'b}\rangle$$

in the sign convention $H_t=-\sum t_{ij}^{ab}c_{ia}^\dagger c_{jb}$. A smooth change of frame rotates both $t$ and every interaction tensor. Individual onsite energies and hoppings are not gauge invariant; the reconstructed band subspace and physical predictions are. Maximally localized constructions and their topology limitations are reviewed in Marzari et al. 2012, §§ II–VI.

Screened interactions and downfolding

Project a screened interaction $W_{\mathrm r}$ into the same orbitals:

$$U^{abcd}_{ijkl}=\int\mathrm d^3r\,\mathrm d^3r' w^*_{ia}(\mathbf r)w_{jb}(\mathbf r) W_{\mathrm r}(\mathbf r,\mathbf r') w^*_{kc}(\mathbf r')w_{ld}(\mathbf r').$$

The screening $W_{\mathrm r}$ must exclude polarization processes that the low-energy model will reproduce, or those processes are double counted. Constrained RPA is one controlled prescription once the target polarization subspace is specified Aryasetiawan et al. 2004, pp. 195104-1–195104-12.

Formally, a projector $P$ onto the target space gives an energy-dependent effective Hamiltonian

$$H_{\mathrm{eff}}(E)=PHP+PHQ\frac{1}{E-QHQ}QHP, \qquad Q=1-P.$$

Expanding the resolvent requires separation from discarded states. Truncating its frequency dependence to static hoppings and interactions is another approximation, not a change of notation.

Checks on the model reduction

Report the band window, orbital gauge, retained hopping and interaction range, screening construction, double-counting correction, omitted phonons or spin–orbit terms, and covariance of fitted parameters. Validate by reconstructing target bands and symmetries and by repeating the calculation across admissible windows. A Hubbard model is an effective coordinate choice, not a unique microscopic observable.

Exercises

Show that a unitary rotation within the target orbital subspace leaves the reconstructed one-particle spectrum invariant.

Solution

At each momentum, $h(\mathbf k)\mapsto U^\dagger(\mathbf k)h(\mathbf k)U(\mathbf k)$. Unitary similarity preserves eigenvalues. The real-space hopping entries change, which is why their numerical values are basis-dependent even though the target bands are not.

References

• Ferdi Aryasetiawan, Masatoshi Imada, Antoine Georges, Gabriel Kotliar, Silke Biermann, and Alexander I. Lichtenstein, “Frequency-Dependent Local Interactions and Low-Energy Effective Models from Electronic Structure Calculations,” Physical Review B 70 (2004) 195104, doi:10.1103/PhysRevB.70.195104.
• Nicola Marzari, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza, and David Vanderbilt, “Maximally Localized Wannier Functions: Theory and Applications,” Reviews of Modern Physics 84 (2012) 1419–1475, doi:10.1103/RevModPhys.84.1419.