The Segment Picture Solver

The TRIQS-based hybridization-expansion segment picture solver (CTSEG) can tackle the generic problem of a quantum impurity coupled to an external environment (bath). The “impurity” can be any set of orbitals, on one or several atoms. The CTSEG solver supports (possibly retarded) density-density and spin-spin interactions on the impurity. Under these restrictions, it provides better performance than the generic CTHYB solver, that supports generic local interaction vertices. The imaginary time action solved by CTSEG is of the form

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

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 Grassmann fields corresponding to the fermion creation and annihilation operators on the impurity, respectively. \(\Delta_{ab}^{\sigma \sigma'}(\tau)\) is the hybridization function, that accounts for particle exchange between the impurity and the bath, and \(\mathcal{U}_{ab} (\tau)\) and \(\mathcal{J}_{a}^{\xi} (\tau)\) are the (dynamical) density-density and spin-spin interactions, respectively. The \(\epsilon_{a\sigma}\) are orbital energies and \(\mu\) is the chemical potential.

The CTSEG solver carries out a double expansion in the hybridization term and in the perpendicular spin-spin interaction term to obtain the fully interacting impurity Green’s function \(G(\tau)\) and a range of other observables. Learn how to use it in the Documentation.