Symbolic determinant construction of perturbative expansions

Abstract

We present a symbolic algorithm for treating perturbative expansions of Hamiltonians with general two-body interactions. The method, formally equivalent to determinant Monte Carlo methods, merges well-known analytics with the recently developed symbolic integration tool, algorithmic Matsubara integration (AMI) that allows for the evaluation of the imaginary frequency/time integrals. By explicitly performing Wick contractions at each order of the perturbative expansion we order-by-order construct the fully analytic solution of the Green's function and self energy expansions. A key component of this process is the assignment of momentum/frequency conserving labels for each contraction that motivates us to present a fully symbolic Fourier transform procedure which accomplishes this feat. These solutions can be applied to a broad class of quantum chemistry problems and are valid at arbitrary temperatures and on both the real- and Matsubara-frequency axis. To demonstrate the utility of this approach, we present results for simple molecular systems as well as model lattice Hamiltonians. We highlight the case of molecular problems where our results at each order are numerically exact with no stochastic uncertainty.