auxiliaryfield

An auxiliary field is a field introduced into a theoretical framework to simplify calculations or to expose collective degrees of freedom, without being a primary physical observable of interest. In quantum many-body and quantum field theories, auxiliary fields are often used to convert interacting problems into more tractable ones by decoupling nonlinear interactions.

A common method is the Hubbard–Stratonovich transformation, which introduces an auxiliary (bosonic) field to replace a quadratic or quartic fermion interaction with a bilinear fermion term coupled to the auxiliary field. After integrating over the auxiliary field, the original interaction is recovered. This approach enables the use of one-body techniques and Monte Carlo sampling that would be difficult for the original interacting system.

In condensed matter and lattice gauge theories, auxiliary fields underpin various decoupling strategies. For example, in the Hubbard model, spin or charge channels may be chosen for decoupling, leading to different representations of the same physics. In auxiliary-field quantum Monte Carlo methods, configurations of the auxiliary field are sampled to evaluate observables, effectively transforming a many-body problem into a statistical average over field configurations.

Dynamic versus static variants exist: auxiliary fields can depend on imaginary time (dynamic) or be time-independent (static), and they can be discrete or continuous in their values. While powerful, these methods can introduce technical challenges such as sign or phase problems, depending on the model and decoupling scheme.

Overall, auxiliary fields are a versatile mathematical tool that enable tractable formulations and numerical approaches for interacting systems, while the fields themselves are auxiliary to the primary physical quantities of interest.