The Segment Picture Solver

The TRIQS-based hybridization-expansion segment picture solver (CTSEG-J) can tackle the generic problem of a quantum impurity coupled to an external environement (bath). The “impurity” can be any set of orbitals, on one or several atoms. The CTSEG-J 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-J is of the form

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

CTSEG-J 3.2.0

This is the homepage of CTSEG 3.2.0. For changes see the changelog page.

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. \(\Delta_{ab}^{\sigma \sigma'}(\tau)\) 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 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.