Preface

w2dynamics is a hybridization-expansion continuous-time quantum Monte Carlo package, developed jointly in Vienna and Würzburg.

The algorithm is conceptually the same as in the implementation of cthyb based on the TRIQS library. For an overview look here: A word on the algorithm.

However there are some mayor differences in implementation, especially in the fermionic trace \(\mathrm{Tr} \, \mathcal{C}\), which will be described below.

A Monte Carlo configuration \(\mathcal{C}\) is a set of fermionic operators (in interaction representation) at different imaginary times:

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

triqs_cthyb calculates \(\mathrm{Tr} \, \mathcal{C} = \sum_n <n|\mathcal{C}|n>\) in a matrix-matrix algorithm, i.e. it multiplies the matrix representations of the operators to save them in a binary tree, and in the very end contracting one matrix with the eigenstates \(|n>\). w2dyn_cthyb performs matrix-vector operations by multiplying the matrix representations of the operators repeatedly on the states.

w2dyn_cthyb employs an importance sampling of the sum over the outer states \(|n>\) (superstate-sampling or state-sampling algorithm [3]), whereas triqs_cthyb performs this sum explicitely.

triqs_cthyb uses the lazy trace evaluation of [1] not to waste time calculating the exact trace for very unlikely configurations, whereas w2dyn_cthyb uses the sliding window technique by Hiroshi Shinaoka to propose updates local in imaginary time [2].