The CTSEG-J algorithm

Double expansion

The double-expansion segment-picture continuous-time quantum Monte Carlo algorithm (CTSEG-J) is based on an expansion of the partition function in powers of the hybridization function \(\Delta(tau)\) and perprendicular spin-spin interaction \(J_{\perp} (\tau)\). Details can be found in [1] [2]. The partition function is \(Z = \int [Dc][D\overline{c}]e^{-\mathcal{S}[c,\overline{c}]}\), with the action \(\mathcal{S}\) given by

\[\begin{split}\begin{split} \mathcal{S} &= \iint_0^{\beta} \mathrm{d} \tau \mathrm{d} \tau' \sum_{a,b} \left\{ \overline{c}_{a\sigma} (\tau) \left( (\partial_{\tau} + \epsilon_{a\sigma})\delta_{ab}^{\sigma \sigma'} \delta_{\tau - \tau'} + \Delta_{ab}^{\sigma \sigma'}(\tau - \tau')\right) c_{b\sigma'}(\tau') \right\} \\ &+ \frac{1}{2} \iint_0^{\beta} \mathrm{d} \tau \mathrm{d} \tau' \sum_{a,b} \mathcal{U}_{ab}(\tau - \tau') n_a(\tau) n_b(\tau') + \frac{1}{2} \iint_0^{\beta} \mathrm{d} \tau \mathrm{d} \tau' \sum_{a, \xi = x, y, z} s_a^{\xi}(\tau) \mathcal{J}_a^{\xi}(\tau - \tau') s_a^{\xi} (\tau') \end{split}\end{split}\]

Here \(\beta\) is the inverse temperature, \(a\) denote orbital indices, \(\sigma\) spin indices (\(\sigma = \uparrow, \downarrow\)), \(n_a \equiv \sum_{\sigma} n_{a\sigma}\), \(s_a^{\xi} \equiv \frac{1}{2} \sum_{\sigma \sigma'} \overline{c}_{a\sigma} \sigma_{\sigma \sigma'}^{\xi} c_{a \sigma'}\) and \(\sigma^{\xi}\) are the Pauli matrices. \(\overline{c}_{a\sigma}(\tau)\) and \(c_{a\sigma}(\tau)\) are the \(\beta\)-antiperiodic Grassman fields corresponding to the fermion creation and annihilation operators on the impurity, respectively.

This action can be recast as

\[\begin{split}\begin{split} \mathcal{S} &= \iint_0^{\beta} \mathrm{d} \tau \mathrm{d} \tau' \sum_{a,b} \left\{ \overline{c}_{a\sigma} (\tau) \left( (\partial_{\tau} + \epsilon_{a\sigma})\delta_{ab}^{\sigma \sigma'} \delta_{\tau - \tau'} + \Delta_{ab}^{\sigma \sigma'}(\tau - \tau')\right) c_{b\sigma'}(\tau') \right\} \\ &+ \frac{1}{2} \iint_0^{\beta} \mathrm{d} \tau \mathrm{d} \tau' \sum_{a,b} \mathcal{U}_{uv}(\tau - \tau') n_a(\tau) n_b(\tau') + \frac{1}{2} \iint_0^{\beta} \mathrm{d} \tau \mathrm{d} \tau' \sum_{a} \mathcal{J}_a^{\perp}(\tau - \tau') s_a^{+}(\tau) s_a^{-} (\tau') \end{split}\end{split}\]

where

\[\mathcal{U}_{uv}(\tau - \tau') = \mathcal{U}_{ab}(\tau - \tau') + (-1)^{\sigma \sigma'} \frac{1}{4} \mathcal{J}_a^z(\tau) \delta_{ab},\]

\(s^{\pm} = s_x \pm i s_y\) and \(\mathcal{J}^{\perp} \equiv \mathcal{J}^x = \mathcal{J}^y\). The CTSEG-J solver stochastically explores the terms (or configurations) generated by the expansion of \(S\) in powers of \(\Delta(\tau)\) and \(\mathcal{J}^{\perp}(\tau)\) and samples the observables of interest (e.g. the Green’s function) every few configurations.

Note

Our CTSEG implementation supports the \(\mathcal{J}^{\perp}(\tau)\) expansion only for a single orbital. For multiple orbitals, an expansion only in \(\Delta(\tau)\) is possible. For a single orbital, it is also possible to carry out an expansion in \(\mathcal{J}^{\perp}(\tau)\) only (i.e., with \(\Delta(\tau) = 0\)).

Configuration

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