Information geometry of continuum limits in lattice field theory: The Gross-Neveu model with Wilson fermions
Abstract
When the Boltzmann weight is nonnegative, the actions of a lattice field theory define a family of probability distributions that forms a statistical manifold, and its Fisher metric gives the number of measurements needed to distinguish two nearby actions. In the large-N limit of the two-dimensional lattice Gross-Neveu model with Wilson fermions, we study the continuum limit, in which the lattice spacing a is taken to zero with the mass gap Λ fixed, through the Fisher metric of the distribution of the auxiliary field σ. The renormalization condition fixes an expectation coordinate, and dimensional transmutation becomes an exponential relation between it and a natural coordinate. The main result is that the a→0 limits differ between the Fisher metric of the long-distance modes, whose momenta are fixed in physical units, and the Fisher metric of the full P(σ). In the former, the lengths of the infinitely many irrelevant directions that change the kinetic term shrink, relative to the mass-term direction, to order aΛlog(1/(aΛ)) or smaller, and when several modes are combined, only the two directions (Λ,m) of the continuum theory remain. In the latter, the kinetic-term directions shrink at the same rate as the mass-term direction, and the difference in the regularization remains in correlations at distances of the order of the lattice spacing. An exception is the deformation that only adds a contact term to the propagator: as a→0 its length goes to zero in both metrics, and it becomes indistinguishable from a change of the coupling 1/g2 (a lattice version of Kutasov's result). The curvature of the coupling plane at fixed kinetic term diverges as (1/2)log2(1/(aΛ)), not as the correlation volume often assumed in thermodynamic geometry, and the critical point is infinitely far in Fisher distance from any point at finite a.
Related
- Progress on Sp(4) lattice theories with Grid: Möbius domain wall fermions and continuum extrapolations
- The Standard Model on a finite bosonic lattice with continuum Weyl fermions
- Symplectic lattice gauge theories in the Grid framework: domain wall fermions and continuum extrapolations
- Adaptive sensing beyond non-adaptive information limits: End-to-end co-design of geometry, policy, and inference
- A gauge-invariant measure for lattice chiral fermions
BibTeX
@misc{tomiya2026information,
title = {{Information geometry of continuum limits in lattice field theory: The Gross-Neveu model with Wilson fermions}},
author = {Akio Tomiya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34277},
eprint = {2610.34277},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34277},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Information geometry of continuum limits in lattice field theory: The Gross-Neveu model with Wilson fermions
AU - Akio Tomiya
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34277
UR - https://arxiv.org/abs/2610.34277
AB - When the Boltzmann weight is nonnegative, the actions of a lattice field theory define a family of probability distributions that forms a statistical manifold, and its Fisher metric gives the number of measurements needed to distinguish two nearby actions. In the large-N limit of the two-dimensional lattice Gross-Neveu model with Wilson fermions, we study the continuum limit, in which the lattice spacing a is taken to zero with the mass gap Λ fixed, through the Fisher metric of the distribution of the auxiliary field σ. The renormalization condition fixes an expectation coordinate, and dimensional transmutation becomes an exponential relation between it and a natural coordinate. The main result is that the a→0 limits differ between the Fisher metric of the long-distance modes, whose momenta are fixed in physical units, and the Fisher metric of the full P(σ). In the former, the lengths of the infinitely many irrelevant directions that change the kinetic term shrink, relative to the mass-term direction, to order aΛlog(1/(aΛ)) or smaller, and when several modes are combined, only the two directions (Λ,m) of the continuum theory remain. In the latter, the kinetic-term directions shrink at the same rate as the mass-term direction, and the difference in the regularization remains in correlations at distances of the order of the lattice spacing. An exception is the deformation that only adds a contact term to the propagator: as a→0 its length goes to zero in both metrics, and it becomes indistinguishable from a change of the coupling 1/g^2 (a lattice version of Kutasov's result). The curvature of the coupling plane at fixed kinetic term diverges as (1/2)log^2(1/(aΛ)), not as the correlation volume often assumed in thermodynamic geometry, and the critical point is infinitely far in Fisher distance from any point at finite a.
ER -