.. _ctseg:

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 :math:`\Delta(tau)` and perprendicular spin-spin interaction 
:math:`J_{\perp} (\tau)`. Details can be found in [#ctqmc1]_ [#ctqmc2]_. The partition function is :math:`Z = \int [Dc][D\overline{c}]e^{-\mathcal{S}[c,\overline{c}]}`,
with the action :math:`\mathcal{S}` given by 

.. math::

  \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}

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

This action can be recast as 

.. math::

  \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}

where 

.. math::

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

:math:`s^{\pm} = s_x \pm i s_y` and :math:`\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 :math:`S` in powers of :math:`\Delta(\tau)`
and :math:`\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 :math:`\mathcal{J}^{\perp}(\tau)` expansion only for a single orbital. For multiple 
    orbitals, an expansion only in :math:`\Delta(\tau)` is possible. For a single orbital, it is also possible to carry out an 
    expansion in :math:`\mathcal{J}^{\perp}(\tau)` only (i.e., with :math:`\Delta(\tau) = 0`). 

Configuration
**************

Sign
*****

.. [#ctqmc1] Otsuki, J. Spin-boson coupling in continuous-time quantum Monte Carlo. 
             Phys. Rev. B 87, 125102 (2013).

.. [#ctqmc2] Ayral, T. PhD thesis: Nonlocal Coulomb Interactions and Electronic
            Correlations: Novel Many-Body Approaches. Ecole Polytechnique (2015). `tel-01247625 <https://hal.archives-ouvertes.fr/tel-01247625>`_