Can neural networks learn low-order approximations of the quark propagator?

Why the hopping expansion breaks down for light quarks

The hopping expansion is a power-series approximation of the quark propagator, obtained by solving the discretized Dirac equation. It is the optimal series expansion inside its radius of convergence, but for the physically interesting case of light quarks it diverges. That divergence currently restricts strong-coupling effective theories and finite-density methods to heavy quarks.

The paper frames the problem as a search over low-order truncations rather than over the exact series. Even where the hopping expansion converges, a low-order truncation need not be the best available approximation of the inverse Dirac operator at a fixed number of hop operations. The authors therefore ask whether machine learning can find better truncations, and whether a different expansion exists in the regime where the hopping expansion fails.

The error metric and the equal-cost baseline

Approximation quality is quantified by ε(M,D) = ‖MD − I‖²_F / ‖I‖²_F, where M is the candidate operator and D is the Dirac operator. The denominator removes the dependence on lattice volume. Because the matrices are large, the Frobenius norm is rewritten as an expectation value over random, normally distributed and normalized quark fields, so ε(M,D) = E[‖MDv − v‖²].

The comparison is deliberately cost-matched. Hop operations dominate runtime in lattice QCD because they require communication between parallel processes. An ℓ-layer HC model applies the same set of hop operators per H layer as one step of the hopping expansion, so an ℓ-layer model costs the same as an ℓ-step truncated hopping expansion. GMRES is included as a baseline because it adaptively builds an optimal polynomial approximation of D⁻¹ for each right-hand side, giving a lower bound on what any fixed polynomial in D can achieve.

Two architectures trade expressiveness for trainability

The HC model alternates Clifford layers and hopping layers. A Clifford layer takes n quark fields and builds m linear combinations using trainable complex weights over the 16-element Clifford basis of C⁴ˣ⁴, which makes the weights directly interpretable. A hopping layer applies the identity or forward and backward hop operators along each dimension. Stacking these layers yields arbitrarily deep networks.

The restricted model is closer to the hopping expansion: each restricted hopping layer applies a single trainable combination of identity and γ_μ terms to one quark field, and the full model sums products of these layers with one overall scaling weight. The HC model generalizes the restricted model. The restricted model is easier to train and easier to analyze, while the HC model is more expressive; the paper uses each where it performs best.

Models beat the hopping expansion and GMRES at equal cost

Training used four gauge fields from a quenched Wilson-action ensemble on an 8³ × 16 lattice at β = 6, with the mass held fixed and the Adam optimizer. Models were trained for 36 masses between −5 and 2 in steps of 0.2. Testing used four gauge fields not seen in training, on both 8³ × 16 and 16³ × 32 volumes, with 100 random quark fields per gauge field.

For 1-layer models, the mass-averaged reduction in approximation error was 43% versus the best baseline, split into 55% for m < m_h and 23% for m > m_h. For 4-layer models the improvement rose to 65% for HC models (78% below m_h, 41% above) and 54% for restricted models (64% below, 37% above). The models systematically outperformed both baselines across all mass parameters. The authors attribute this to the models generalizing both the hopping expansion and polynomials in D, so the extra freedom permits better approximations of D⁻¹.

Convergence where the hopping expansion diverges

For m ≳ −0.5 the test score stopped improving reliably beyond 6 layers, which the authors attribute to the many possible good low-order approximations in that regime. For m ≲ −0.5 the test score kept improving up to the 16 layers investigated. At m = 1.4 all approximations showed geometric convergence; at m = −3 the models showed geometric convergence up to 16 layers while the hopping expansion clearly diverged and GMRES stagnated.

Appendix results reported in the paper place the hopping expansion failure below m < −1.2 and GMRES stagnation at m ≲ −3, while the models showed geometric convergence for all masses considered. The paper treats this as a first indication that the models learn novel expansions of the quark propagator beyond the hopping expansion and polynomials of D, not merely better-tuned truncations of the known series.

Reading the learned coefficients

After training, the model is rewritten as M = Σ_p Σ_k c_{p,k} Γ_k T_p, a sum over paths p and Clifford basis elements Γ_k. The hopping expansion can be written in the same form with some c_{p,k} = 0, which makes the two directly comparable. If the model learns a nonzero c_{p,k} that is zero in the hopping expansion, it has found a functional form beyond the hopping expansion. Fitting this form directly by linear regression was rejected because the number of paths grows exponentially with the number of layers.

For the length-0 path, the hopping expansion gives c_{[],1} = 2κ = 1/(m+4) and c_{[],k} = 0 for k ≠ 1. For m ≳ −0.5 the models recover the hopping expansion; for m ≲ −0.5 they deviate systematically. At m = 1 the restricted model yields a 1-layer approximation with coefficients such as 0.193386(29), 0.078651(22) and 0.08896(8), averaged over five training runs with different seeds, differing from the hopping expansion in a statistically significant way. At m = −3 the 16-layer HC model gives c_{[],1} = 0.1882(23) instead of 1, and a staple-path coefficient of −0.0132(9) that is zero in the hopping expansion, showing a structurally new formula.

What transfers and what needs retraining

Approximation quality was independent of lattice volume within statistics, even though all training was done on the smaller 8³ × 16 volume, because the architecture resembles convolutional architectures and was designed to transfer to larger volumes. The models also generalized to unseen gauge fields. The paper states the discovered approximations can serve as plug-in replacements wherever a truncated hopping expansion is currently used.

One boundary matters for deployment: the tuning is partially dependent on the underlying probability distribution of the gauge fields, so a new model should be trained when switching to gauge fields generated with a significantly different action. Tests with dynamical and Haar-distributed gauge fields are reported in an appendix. The practical priority is therefore the light-quark regime m ≲ −0.5, where the HC model performs best and where the hopping expansion diverges and GMRES stagnates; above that threshold both architectures perform similarly and mainly learn improved truncations of the known expansion.

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