Auxiliary-Field Monte Carlo

Auxiliary-field Monte Carlo (AFQMC) replaces a two-body interaction by a sum or integral over one-body propagators coupled to auxiliary fields. Fermions can then be traced out analytically, leaving determinant weights to sample. This page develops finite-temperature determinant AFQMC for a Hubbard interaction; projector and constrained-phase variants require additional choices.

Required background. Fermionic basis conventions, MCMC, and imaginary-time discretization supply the sign, sampling, and Trotter controls.

A discrete decoupling with a four-state check

For $U>0$, use the shifted on-site interaction $U(n_\uparrow-\frac12)(n_\downarrow-\frac12)$. The identity

$$e^{-\epsilon U(n_\uparrow-\frac12)(n_\downarrow-\frac12)} =\frac12e^{-\epsilon U/4} \sum_{s=\pm1}e^{\lambda s(n_\uparrow-n_\downarrow)}, \qquad \cosh\lambda=e^{\epsilon U/2}$$

is exact on a single site. In the empty and doubly occupied states both sides equal $e^{-\epsilon U/4}$. In either singly occupied state the right side becomes $e^{-\epsilon U/4}\cosh\lambda=e^{\epsilon U/4}$. This four-state verification should precede any lattice code.

The full kinetic-interaction splitting still has a time-step error. Exact decoupling of one interaction factor does not make the product formula exact.

Determinants and equal-time Green functions

For a sampled field configuration $s_{i\ell}$, build a one-particle propagator $B_{\ell\sigma}$ for every time slice and spin. With a declared time ordering, the fermion trace gives

$$w(s)\propto\prod_{\sigma} \det\left[I+B_{M\sigma}\cdots B_{1\sigma}\right].$$

A corresponding equal-time convention is $G_{ij,\sigma}=\langle c_{i\sigma}c_{j\sigma}^\dagger\rangle_s$, with

$$G_\sigma=(I+B_{M\sigma}\cdots B_{1\sigma})^{-1}$$

at the chosen slice boundary. In this convention $\langle n_i\rangle_s=1-G_{ii}$. The cyclic product changes when the measurement slice changes; update the formula and indexing together.

The determinant weight may be signed or complex. Sampling its modulus requires reweighting and the associated sign-problem diagnostics.

Stabilization is part of the algorithm

At large $\beta$, products of $B$ matrices contain exponentially growing and decaying directions. Direct multiplication can lose the small singular directions long before overflow occurs. Periodically factor products using a scale-aware QR or SVD scheme, retain separate magnitude factors, and evaluate determinants through phase/sign and logarithmic magnitude.

Fast rank-one field updates reduce cost but accumulate roundoff. Recompute the Green function from stabilized factors periodically and compare the refreshed result with the incrementally updated one. Vary the stabilization interval; unchanged Monte Carlo statistics with drifting refresh residuals do not establish reliability.

Benchmarks and sign-free conditions

At $U=0$, auxiliary fields decouple and the result must equal the free-fermion thermal matrix. With one orbital of energy $e$ per spin, measured relative to the chemical potential, $G=1/(1+e^{-\beta e})$ and the occupation is $1/(1+e^{\beta e})$.

For a repulsive Hubbard model at half filling on a bipartite lattice with the appropriate real hopping and particle-hole symmetry, the spin determinant product is nonnegative in this formulation. Changing chemical potential, frustration, or hopping structure can remove that property. State the actual symmetry conditions instead of labeling all Hubbard calculations sign free.

Compare a two-site interacting calculation against full Fock-space diagonalization at fixed $\beta$, then extrapolate the time step and confirm statistical consistency.

Exercises

Why does the single-orbital Green function above approach one for large positive $e$ at low temperature?

Solution

The orbital becomes empty, so $\langle c c^\dagger\rangle=1-\langle n\rangle$ tends to one. Confusing this Green-function convention with $\langle c^\dagger c\rangle$ would reverse the occupation and can produce an apparently smooth but unphysical density curve.

References

• Blankenbecler, Richard, Douglas J. Scalapino, and Robert L. Sugar, “Monte Carlo calculations of coupled boson-fermion systems. I,” Physical Review D 24, 2278–2286 (1981), doi:10.1103/PhysRevD.24.2278 — determinant sampling.
• Hirsch, J. E., “Discrete Hubbard-Stratonovich transformation for fermion lattice models,” Physical Review B 28, 4059–4061 (1983), doi:10.1103/PhysRevB.28.4059 — discrete interaction decoupling.
• Troyer, Matthias, and Uwe-Jens Wiese, “Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations,” Physical Review Letters 94, 170201 (2005), arXiv:cond-mat/0408370 — generic sign-problem complexity and its limits.