The Hubbard-Stratonovitch auxiliary-field approach to projection of the ground state of an interacting fermion system from a trial state is examined. It is shown that the method is equivalent to solving a differential equation with diffusion, drift, and branching terms on the manifold of normalized Slater determinants. Explicit general expressions are given for the coefficients of the diffusion equation in terms of the Hubbard-Stratonovitch fields. The form of the equation is somewhat similar to that obtained in the continuum Green's-function Monte Carlo method, although its interpretation and relation to the physical many-body problem is quite different. The character of the representation of the many-body ground state arising in the auxiliary-field projection approach is discussed within this framework. The diffusion process is normally found to concentrate the states representing the ground state near the classical mean-field solutions. The consequences of this picture of the auxiliary-field approach for the ``fermion minus-sign'' problem in this context are discussed, and some conclusions are reached concerning the range of validity of a recently suggested approximation where minus signs are ignored.
We propose a general scheme for a digital construction of lattice gauge theories with dynamical fermions. In this method, the four-body interactions arising in models with
2+1 dimensions and higher are obtained stroboscopically, through a sequence of two-body interactions with ancillary degrees of freedom. This yields stronger interactions than the ones obtained through perturbative methods, as typically done in previous proposals, and removes an important bottleneck in the road towards experimental realizations. The scheme applies to generic gauge theories with Lie or finite symmetry groups, both Abelian and non-Abelian. As a concrete example, we present the construction of a digital quantum simulator for a
Z3 lattice gauge theory with dynamical fermionic matter in
2+1 dimensions, using ultracold atoms in optical lattices, involving three atomic species, representing the matter, gauge, and auxiliary degrees of freedom, that are separated in three different layers. By moving the ancilla atoms with a proper sequence of steps, we show how we can obtain the desired evolution in a clean, controlled way.
This thesis presents the path-integral formulation of the nuclear shell model using the Hubbard-Stratonovich transformation, which linearizes the two-body interaction by auxiliary fields. The path-integral was evaluated via Monte Carlo. The method scales favorably with valence-nucleon number and shell-model basis: full-basis calculations can be done up to the rare-earth region, which cannot be treated by other methods. Observables are calculated for the ground state and in a thermal ensemble. Dynamical correlations are obtained, from which strength functions are extracted through the Maximum Entropy method. Examples in the s-d shell, where exact diagonalization can be carried out, compare well with exact results. The "sign problem", which is generic to fermion Monte Carlo calculations, is proved to be absent in a wide class of interactions including the attractive pairing-plus-multipole interactions. The formulation is general for interacting fermion systems and is well suited for parallel computation. The method has been implemented on the Intel Touchstone Delta System, achieving better than 99% parallelization.