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 $\hat{\mathcal{U}}$ describes the non-interacting lattice (phonon) degrees of freedom and $\hat{\mathcal{K}}$ and $\hat{\mathcal{V}}$ are the total electron kinetic and potential energies, respectively. Note that both $\hat{\mathcal{K}}$ and $\hat{\mathcal{V}}$ can depend on the dynamical lattice coordinates, leading to an electron-phonon ($e$-ph) coupling that is either diagonal or off-diagonal in the orbital basis. $\hat{\mathcal{V}}$ also includes any contributions from local intra- and inter-orbital Hubbard repulsion or extended Hubbard interactions.

The non-interacting lattice terms are further subdivided into the sum of three terms

\[\begin{align*} \hat{\mathcal{U}} = \hat{\mathcal{U}}_{\rm qho} + \hat{\mathcal{U}}_{\rm anh} + \hat{\mathcal{U}}_{\rm disp}. \end{align*}\]

The first term

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

describes the placement of local quantum harmonic oscillator (QHO) modes in a cluster, i.e. an Einstein solid, while the second term

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

introduces anharmonic contributions to the oscillator potential. The third term

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

introduces coupling (or dispersion) between the QHO modes. The sums over $\mathbf{i} \ (\mathbf{j})$ and $\nu \ (\gamma)$ run over unit cells in the lattice and phonon modes within each unit cell.

The position and momentum operators for each QHO mode are $\hat{X}_{\mathbf{i},\nu}$ and $\hat{P}_{\mathbf{i},\nu}$ respectively, with corresponding phonon mass $M_{\mathbf{i},\nu}$. The spring constant is $K_{\mathbf{i},\nu} = M_{\mathbf{i},\nu} \Omega_{0,\mathbf{i},\nu}^2$, with $\Omega_{0,\mathbf{i},\nu}$ specifying the phonon frequency. $\hat{\mathcal{U}}_{\rm anh}$ then introduces an anharmonic $\hat{X}_{\mathbf{i},\nu}^4$ contribution to the QHO potential energy that is controlled by the parameter $\Omega_{a,\mathbf{i},\nu}$. Similarly, $\tilde{\Omega}_{0,(\mathbf{i},\alpha),(\mathbf{j},\gamma)} \ (\tilde{\Omega}_{a,(\mathbf{i},\alpha),(\mathbf{j},\gamma)})$ is the coefficient controlling harmonic (anharmonic) dispersion between QHO modes. The parameter $\zeta_{(\mathbf{i},\alpha),(\mathbf{j},\gamma)} = \pm 1$ determines whether the difference or sum appears of pairs of phonon displacements appears the dispersive coupling terms. Note that unlike harmonic parameters $\Omega_{0,\mathbf{i},\nu}$ and $\tilde{\Omega}_{0,(\mathbf{i},\alpha),(\mathbf{j},\gamma)}$, the anharmonic parameters $\Omega_{a,\mathbf{i},\nu}$ and $\tilde{\Omega}_{a,(\mathbf{i},\alpha),(\mathbf{j},\gamma)}$ do not have units of frequency, but instead include an additional factor of inverse length squared.

The electron kinetic energy is conveniently expressed as

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

The first term describes the non-interacting spin-$\sigma$ electron kinetic energy

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

where $t_{\sigma,(\mathbf{i},\nu),(\mathbf{j},\gamma)}$ is the spin-$\sigma$ hopping integral from orbital $\gamma$ in unit cell $\mathbf{j}$ to orbital $\nu$ in unit cell $\mathbf{i}$, and may be real or complex. The second term 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

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

Here, the modulations of the spin-$\sigma$ hopping integrals to $m$\textsuperscript{th} $(=1-4)$ order in displacement are controlled by the parameters $\alpha_{\sigma,m,(\mathbf{i},\nu,\eta),(\mathbf{j},\gamma,\rho)}$. Note that if the corresponding bare hopping amplitude $t_{\sigma,(\mathbf{i},\eta),(\mathbf{j},\rho)}$ is complex, then the $\alpha_{\sigma,m,(\mathbf{i},\nu,\eta),(\mathbf{j},\gamma,\rho)}$ parameter is defined to share the same complex phase. This convention ensures that $e$-ph interaction only modulates the magnitude of the hopping amplitude and not the phase.

Lastly, the electron potential energy is expressed as

\[\begin{align*} \hat{\mathcal{V}} = \hat{\mathcal{V}}_0 + \hat{\mathcal{V}}_{\rm hol} + \hat{\mathcal{V}}_{\rm hub} + \hat{\mathcal{V}}_\text{exh} = \sum_\sigma \hat{\mathcal{V}}_{0,\sigma} + \sum_\sigma \hat{\mathcal{V}}_{{\rm hol},\sigma} + \hat{\mathcal{V}}_{\rm hub}+ \hat{\mathcal{V}}_\text{exh}, \end{align*}\]

where

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

is the non-interacting spin-$\sigma$ electron potential energy. Here, $\mu_\sigma$ is the spin-$\sigma$ chemical potential and $\epsilon_{\sigma, \mathbf{i},\nu}$ is the spin-$\sigma$ on-site energy for orbital $\nu$ in unit cell ${\bf i}$.

The second term

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

is the contribution to the spin-$\sigma$ electron potential energy that results from a Holstein- or Fr{\"o}hlich-like coupling to the lattice degrees of freedom. The parameter $\kappa_{\sigma,m,(\mathbf{i},\nu),(\mathbf{j},\gamma)}$ controls the strength of this coupling in the $\hat{\mathcal{V}}_{\text{hol},\sigma}$ term. It is important to note that the two parametrizations that are available in SmoQyDQMC.jl are inequivalent, with the first being particle-hole symmetric in the atomic limit.

The third term

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

defines the intra-orbtial/local Hubbard interaction, where $U_{\mathbf{i},\nu}$ is the interaction strength. Note that SmoQyDQMC.jl allows the user to parameterize the Hubbard interaction using either functional form for $\hat {\mathcal V}_{\rm hub}$. The top-most is particle-hole symmetric and is often useful at half-filling.

Lastly, the fourth term

\[\begin{equation*} \begin{aligned} \hat{\mathcal{V}}_\text{exh} & = \begin{cases} \sum_{\substack{\mathbf{i},\nu,\sigma \\ \mathbf{j},\gamma,\sigma'}} V_{(\mathbf{i},\nu),(\mathbf{j},\gamma)}\big(\hat{n}_{\sigma,\mathbf{i},\nu}-\frac{1}{2}\big)\big(\hat{n}_{\sigma',\mathbf{j},\gamma}-\frac{1}{2}\big)\\ \sum_{\substack{\mathbf{i},\nu,\sigma \\ \mathbf{j},\gamma,\sigma'}} V_{(\mathbf{i},\nu),(\mathbf{j},\gamma)}\hat{n}_{\sigma,\mathbf{i},\nu}\hat{n}_{\sigma',\mathbf{j},\gamma} \end{cases}\\ \end{aligned} \end{equation*}\]

introduces extended Hubbard interactions with $V_{(\mathbf{i},\nu),(\mathbf{j},\gamma)}$ subject to the constraint $V_{(\mathbf{i},\nu),(\mathbf{i},\nu)} = 0$. Note, however, that local inter-orbital Hubbard interactions can still be treated by defining an interaction within a single unit cell $\mathbf{i} = \mathbf{j}$ between a pair of orbitals $\nu \ne \gamma$ that share the same basis vector $\mathbf{r}_\nu = \mathbf{r}_\gamma$. In this case, the parameter $V_{(\mathbf{i},\nu),(\mathbf{j},\gamma)}$ is typically denoted $U_{\mathbf{i}, \nu, \gamma}$. As with the local Hubbard interaction, users can parameterize the extended Hubbard interaction using either a particle-hole symmetric (top) or asymmetric (bottom) form.