.. _cthyb:

A word on the algorithm
=======================

The continuous-time quantum Monte Carlo (CTQMC) algorithm
is based on a hybridization expansion of the partition function
[#ctqmc1]_, [#ctqmc2]_. The principle of the algorithm is to sample
stochastically the diagrams of this expansion with the correct Monte Carlo
weights and to compute the Green's function. A Monte Carlo configuration
:math:`\mathcal{C}` is a set of fermionic operators (in interaction
representation) at different imaginary times:

.. math::

  \mathcal{C} = d^\dagger_{\alpha_1}(\tau_1) d_{\alpha'_1}(\tau'_1) d^\dagger_{\alpha_2}(\tau_2)
                d^\dagger_{\alpha_3}(\tau_3) \ldots d_{\alpha}(\tau_N)

The algorithm samples new configurations by inserting/removing pairs of
operators, or by moving operators in the configuration. Note that it is a
finite-temperature algorithm, and so :math:`\tau \in [0,\beta]`, where
:math:`\beta` is the inverse temperature. The Monte Carlo weight of a
configuration is essentially the product of the trace :math:`\mathrm{Tr} \,
\mathcal{C}` and the determinant of a matrix, whose elements are the
hybridization functions :math:`\Delta_{\alpha_i \alpha_j'} (\tau_i - \tau_j')`.

The main inputs of the solver are the hybridization functions
:math:`\Delta(i\omega_n)` and the local Hamiltonian
:math:`\mathcal{H}_\mathrm{loc}` of the impurity. The solver then computes the
Green's function on the imaginary-time interval :math:`[0,\beta]`.  This can be
done in the imaginary time representation, as well as on a basis of Legendre
polynomials, as described in Ref. [#legendre]_.
Note that our implementation of the algorithm uses a *matrix* representation
[#ctqmc3]_ of the operators :math:`d^\dagger_{\alpha}`. This allows the use
of any local Hamiltonian :math:`\mathcal{H}_\mathrm{loc}` in the algorithm.

.. [#ctqmc1] P. Werner, A. Comanac, L. de' Medici, M. Troyer, and
             A. J. Millis, Phys. Rev. Lett. 97, 076405 (2006).
.. [#ctqmc2] E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov,
             M. Troyer, and P.Werner, Rev. Mod. Phys. 83, 349 (2011).
.. [#legendre] L. Boehnke, H. Hafermann, M. Ferrero, F. Lechermann, and O. Parcollet,
               Phys. Rev. B 84, 075145 (2011).
.. [#ctqmc3] P. Werner and A. J. Millis,
             Phys. Rev. B 74, 155107 (2006).