Supported Hamiltonians

This section describes the class of Hamiltonians SmoQyDQMC.jl currently supports, and how the various terms appearing in the Hamiltonian are parameterized within the code. We start by partitioning the full Hamiltonian as

\[\begin{align*} \hat{\mathcal{H}} = \hat{\mathcal{U}} + \hat{\mathcal{K}} + \hat{\mathcal{V}}, \end{align*}\]

where $\hat{\mathcal{U}}$ is the bare lattice energy, $\hat{\mathcal{K}}$ the total electron kinetic energy, and $\hat{\mathcal{V}}$ the total electron potential energy. In the discussion that follows we apply the normalization $\hbar = 1$ throughout.

The bare lattice term is further decomposed into

\[\begin{align*} \hat{\mathcal{U}} = \hat{\mathcal{U}}_{\rm ph} + \hat{\mathcal{U}}_{\rm disp}, \end{align*}\]

where

\[\begin{align*} \hat{\mathcal{U}}_{\rm ph} =& \sum_{\mathbf{i},\nu}\sum_{n_{\mathbf{i},\nu}} \left[ \frac{1}{2M_{n_{\mathbf{i},\nu}}}\hat{P}_{n_{\mathbf{i},\nu}} + \frac{1}{2}M_{n_{\mathbf{i},\nu}}\Omega_{0,n_{\mathbf{i},\nu}}^2\hat{X}_{n_{\mathbf{i},\nu}}^2 + \frac{1}{24}M_{n_{\mathbf{i},\nu}}\Omega_{a,n_{\mathbf{i},\nu}}^2\hat{X}_{n_{\mathbf{i},\nu}}^4 \right] \end{align*}\]

describes the placement of local dispersionless phonon (LDP) modes in the lattice, i.e. an Einstein solid, and

\[\begin{align*} \hat{\mathcal{U}}_{\rm disp} =& \sum_{\substack{\mathbf{i},\nu \\ \mathbf{j},\gamma}}\sum_{\substack{n_{\mathbf{i},\nu} \\ n_{\mathbf{j},\gamma}}} \frac{M_{n_{\mathbf{i},\alpha}}M_{n_{\mathbf{j},\gamma}}}{M_{n_{\mathbf{i},\alpha}}+M_{n_{\mathbf{j},\gamma}}}\left[ \tilde{\Omega}^2_{0,n_{\mathbf{i},\alpha},n_{\mathbf{j},\gamma}}(\hat{X}_{n_{\mathbf{i},\nu}}-\hat{X}_{n_{\mathbf{j},\gamma}})^2 + \frac{1}{12}\tilde{\Omega}^2_{a,n_{\mathbf{i},\alpha},n_{\mathbf{j},\gamma}}(\hat{X}_{n_{\mathbf{i},\nu}}-\hat{X}_{n_{\mathbf{j},\gamma}})^4 \right] \end{align*}\]

introduces dispersion between the LDP modes. The sums over $\mathbf{i} \ (\mathbf{j})$ and $\nu \ (\gamma)$ run over unit cells in the lattice and orbitals within each unit cell respectively. A sum over $n_{\mathbf{i},\nu} \ (n_{\mathbf{j},\gamma})$ then runs over the LDP modes placed on a given orbital in the lattice.

The position and momentum operators for each LPD mode are given by $\hat{X}_{n_{\mathbf{i},\nu}}$ and $\hat{P}_{n_{\mathbf{i},\nu}}$ respectively, with corresponding phonon mass $M_{n_{\mathbf{i},\nu}}$. The spring constant is $K_{n_{\mathbf{i},\nu}} = M_{n_{\mathbf{i},\nu}} \Omega_{0,n_{n_{\mathbf{i},\nu}}}^2$, with $\Omega_{0,n_{n_{\mathbf{i},\nu}}}$ specifying the phonon frequency. The $U_{\rm ph}$ also supports an anharmonic $\hat{X}_{n_{\mathbf{i},\nu}}^4$ contribution to the LDP potential energy that is controlled by the parameter $\Omega_{a,n_{n_{\mathbf{i},\nu}}}$. Similary, $\tilde{\Omega}_{0,n_{\mathbf{i},\alpha},n_{\mathbf{j},\gamma}} \ (\tilde{\Omega}_{a,n_{\mathbf{i},\alpha},n_{\mathbf{j},\gamma}})$ is the coefficient controlling harmonic (anhmaronic) dispersion between LDP modes.

Next we trace out the phonon degrees of freedom

The electron kinetic energy is decomposed as

\[\begin{align*} \hat{\mathcal{K}} = \sum_{\sigma=\uparrow,\downarrow} \left[ \hat{\mathcal{K}}_{\sigma,0} + \hat{\mathcal{K}}_{\sigma,{\rm ssh}} \right], \end{align*}\]

where

\[\begin{align*} \hat{\mathcal{K}}_{\sigma,0} =& -\sum_{\substack{\mathbf{i},\nu \\ \mathbf{j},\gamma}} \left[ t_{\sigma,(\mathbf{i},\nu),(\mathbf{j},\gamma)} \hat{c}^\dagger_{\sigma,\mathbf{i},\nu}\hat{c}_{\sigma,\mathbf{j},\gamma} + {\rm h.c.} \right] \end{align*}\]

is the non-interacting spin-$\sigma$ electron kinetic energy, and

\[\begin{align*} \hat{\mathcal{K}}_{\sigma,{\rm ssh}} =& \sum_{\substack{\mathbf{i},\nu \\ \mathbf{j},\gamma}}\sum_{\substack{n_{\mathbf{i},\nu} \\ n_{\mathbf{j},\gamma}}}\sum_{m=1}^4 (\hat{X}_{n_{\mathbf{i},\nu}}-\hat{X}_{n_{\mathbf{j},\gamma}})^m\left[ \alpha_{\sigma,m,n_{\mathbf{i},\nu},n_{\mathbf{j},\gamma}} \hat{c}^\dagger_{\sigma,\mathbf{i},\nu}\hat{c}_{\sigma,\mathbf{j},\gamma} + {\rm h.c.} \right] \end{align*}\]

is describes the interaction between the lattice degrees of freedom and the spin-$\sigma$ electron kinetic energy via a Su-Schrieffer-Heeger (SSH)-like coupling mechanism. The hopping integral between from orbital $\gamma$ in unit cell $\mathbf{j}$ to orbital $\nu$ in unit cell $\mathbf{i}$ is given by $t_{(\mathbf{i},\nu),(\mathbf{j},\gamma)}$, and may in general be complex. The modulations to this hopping integral are controlled by the parameters $\alpha_{m,(\mathbf{i},\nu),(\mathbf{j},\gamma)}$, where $m\in [1,4]$ specifies the order of the difference in the phonon positions that modulates the hopping integral.

Lastly, the electron potential energy is broken down into the three terms

\[\begin{align*} \hat{\mathcal{V}} = \sum_{\sigma=\uparrow,\downarrow} \left[ \hat{\mathcal{V}}_{\sigma,0} + \hat{\mathcal{V}}_{\sigma,{\rm hol}} \right] + \hat{\mathcal{V}}_{\rm hub}, \end{align*}\]

where

\[\begin{align*} \hat{\mathcal{V}}_{\sigma,0} =& \sum_{\mathbf{i},\nu} \left[ (\epsilon_{\sigma,\mathbf{i},\nu} - \mu) \hat{n}_{\sigma,\mathbf{i},\nu} \right] \end{align*}\]

is the non-interacting spin-$\sigma$ electron potential energy,

\[\begin{align*} \hat{\mathcal{V}}_{\sigma,{\rm hol}} =& \begin{cases} \sum_{\mathbf{i},\nu} \sum{\mathbf{j},\gamma} \sum_{n_{\mathbf{i},\nu}} \left[\sum_{m=1,3}\tilde{\alpha}_{\sigma,m,n_{\mathbf{i},\nu},(\mathbf{j},\gamma)} \ \hat{X}^m_{n_{\mathbf{i},\nu}}(\hat{n}_{\sigma,\mathbf{j},\gamma} - \tfrac{1}{2}) + \sum_{m=2,4}\tilde{\alpha}_{\sigma,m,n_{\mathbf{i},\nu},(\mathbf{j},\gamma)} \ \hat{X}^m_{n_{\mathbf{i},\nu}}\hat{n}_{\sigma,\mathbf{j},\gamma}\right] \\ \sum_{\mathbf{i},\nu} \sum{\mathbf{j},\gamma} \sum_{n_{\mathbf{i},\nu}} \sum_{m=1}^4 \tilde{\alpha}_{\sigma,m,n_{\mathbf{i},\nu},(\mathbf{j},\gamma)} \ \hat{X}^m_{n_{\mathbf{i},\nu}} \hat{n}_{\sigma,\mathbf{j},\gamma} \end{cases} \end{align*}\]

is the contribution to the spin-$\sigma$ electron potential energy that results from a Holstein-like coupling to the lattice degrees of freedom, and

\[\begin{align*} \hat{\mathcal{V}}_{{\rm hub}}=& \begin{cases} \sum_{\mathbf{i},\nu}U_{\mathbf{i},\nu}\big(\hat{n}_{\uparrow,\mathbf{i},\nu}-\tfrac{1}{2}\big)\big(\hat{n}_{\downarrow,\mathbf{i},\nu}-\tfrac{1}{2}\big)\\ \sum_{\mathbf{i},\nu}U_{\mathbf{i},\nu}\hat{n}_{\uparrow,\mathbf{i},\nu}\hat{n}_{\downarrow,\mathbf{i},\nu} \end{cases} \end{align*}\]

is the on-site Hubbard interaction contribution to the electron potential energy. In $\hat{\mathcal{V}}_0$ the chemical potential is given by $\mu$, and $\epsilon_{\mathbf{i},\nu}$ is the on-site energy, the parameter $\tilde{\alpha}_{m,n_{\mathbf{i},\nu},(\mathbf{j},\gamma)}$ controls the strength of the Holstein-like coupling in $\hat{\mathcal{V}}_{\rm ph}$, and $U_{\mathbf{i},\nu}$ is the on-site Hubbard interaction strength in $\hat{\mathcal{V}}_{\rm hub}$. Note that either functional form for $\hat{\mathcal{V}}_{\rm hub}$ and $\hat{\mathcal{V}}_{\sigma, {\rm hub}}$ can be used in the code. Note that the two possible parameterizations for $\hat{\mathcal{V}}_{\sigma, {\rm hub}}$ are inequivalent!