Positive-definite functional integrals for interacting fermions

Heinz J. Schulz

G. Puddu

We obtain exact functional integrals expressions for many-body matrix elements of the type ⟨ψ, J ,π, N , Z |e -βĤ |ψ, J ,π, N , Z ⟩, between states having good angular-momentum, parity and particle number, in the case of particle-number-preserving functional integrals expressions for e -βĤ are used. Compact expressions are obtained without using angular-momentum projectors. In the case of an even number of particles, a new proof of positivity for the pairing plus quadrupole model is given, which is angular-momentum and parity projected to J π = 0 + . It is shown that the propagator in the functional integral is reduced to a Hermitian positive-definite propagator by the use of angular-momentum eigenstates.

Joel Feldman, Horst Knörrer, Eugene Trubowitz B

G. Puddu

We discuss exact functional integral expressions for many-body matrix elements of the type ⟨ψ, A |e −βĤ |ψ, A ⟩, between particle number projected Hartree–Fock–Bogoliubov (HFB) wavefunctions with time-reversal symmetry. A proof of positivity is given for a class of Hamiltonians and when the HFB wavefunctions with time-reversal symmetry are particle number projected to an even number of particles. We show explicitly how to reduce the propagator in the functional integral to a Hermitian positive definite propagator for particle pairs. This result generalizes that previously obtained using Bardeen–Cooper–Schrieffer wavefunctions.

Barak Hirshberg, Michele Invernizzi, Michele Parrinello

We present a method for performing path integral molecular dynamics (PIMD) simulations for fermions and address its sign problem. PIMD simulations are widely used for studying many-body quantum systems at thermal equilibrium. However, they assume that the particles are distinguishable and neglect bosonic and fermionic exchange effects. Interacting fermions play a key role in many chemical and physical systems, such as electrons in quantum dots and ultracold trapped atoms. A direct sampling of the fermionic partition function is impossible using PIMD since its integrand is not positive definite. We show that PIMD simulations for fermions are feasible by employing our recently developed method for bosonic PIMD and reweighting the results to obtain fermionic expectation values. The approach is tested against path integral Monte Carlo (PIMC) simulations for up to seven electrons in a two-dimensional quantum dot for a range of interaction strengths. However, like PIMC, the method suffers from the sign problem at low temperatures. We propose a simple approach for alleviating it by simulating an auxiliary system with a larger average sign and obtaining an upper bound to the energy of the original system using the Bogoliubov inequality. This allows fermions to be studied at temperatures lower than would otherwise have been feasible using PIMD, as demonstrated in the case of a three-electron quantum dot. Our results extend the boundaries of PIMD simulations of fermions and will hopefully stimulate the development of new approaches for tackling the sign problem.