Auxiliary-field Monte Carlo and the sign problem

For a repulsive Hubbard interaction written in particle-hole-symmetric form, introduce an Ising-valued field at each spatial site and imaginary-time slice. A discrete Hubbard–Stratonovich identity expresses the local interaction exponential as a sum of exponentials linear in the spin occupations. Check it separately for empty, doubly occupied, and singly occupied states. The field’s coupling is dimensionless and grows with the product of interaction energy and time slice. This identity itself is exact for that local factor.

The full lattice still contains a noncommuting kinetic operator. A Trotter breakup used to separate kinetic and interaction parts introduces a systematic slice error distinct from the exact decoupling identity. For each complete auxiliary configuration, the fermion Hamiltonian is quadratic, and its thermal trace becomes a determinant of a one-particle propagation product. Two spin species often contribute separate determinants. Their signs are not independent physical probabilities; the product with the auxiliary-field measure defines the configuration weight.

A typical finite-temperature weight contains determinants of identity plus a product of one-particle matrices along imaginary time. At low temperature, multiplying these matrices directly can produce extremely different singular scales and lose meaningful digits. Stabilized factorizations and independently verified Green functions are needed. Numerical overflow is a conditioning problem; curing it does not remove cancellation between physically positive and negative weights. Conversely, a symmetry-protected nonnegative weight still needs stable arithmetic and an equilibrated sampler.

Certain models and parameter choices guarantee a nonnegative product through particle-hole or antiunitary symmetries. For example, the repulsive Hubbard model at particle-hole-symmetric half filling on a bipartite lattice with real spin-independent hopping between opposite sublattices has a familiar sign-free determinant formulation. Doping, frustrating hopping, fields, or a changed decoupling can invalidate the needed relation. Thus sign freedom is a theorem about a specified representation and symmetry conditions, not a universal property of all fermions or of all Hubbard-model calculations.

For real signed weights, sample their absolute values and restore the sign in a ratio of expectations. When the average sign is small, both numerator and denominator can be small differences of much larger contributions. The denominator’s relative uncertainty grows roughly as the inverse average sign at fixed independent sample count. The observable ratio has correlated numerator and denominator errors, so its uncertainty requires their covariance or resampling of complete paired records. A denominator error alone is not the full observable error bar.

In many problematic regimes, average sign falls exponentially with inverse temperature and system size, causing an exponential sampling cost for fixed relative precision. A generic efficient cure is not known, and the NP-hardness result concerns a general class of sign problems rather than proving every instance impossible. Constrained-path or fixed-phase approximations may be useful but introduce model-dependent bias. Real-time integrals can have complex phase cancellations even where an imaginary-time representation is sign free. None of these cancellations is resolved by simply increasing arithmetic precision.