Two-Sided Bounds on Ground State Properties in Lattice Gauge Theories
Abstract
Gauge theories describe fundamental interactions in the standard model of particle physics and build effective theories in condensed matter physics. Being notoriously hard to simulate, these theories are commonly regularized as lattice gauge theories which can be evaluated with high precision through Monte Carlo algorithms or variational methods. However, Monte Carlo algorithms are not applicable in all regimes due to the sign problem; while variational methods, quantum and classical alike, only give upper-bounds on the ground-state energy. Their precision crucially depends on the chosen ansatz. Here, we present a framework based on a hierarchy of semidefinite programs to yield increasingly good lower-bounds on the ground-state energy and propagate these bounds to extended observables like Wilson loops and mesonic strings. We demonstrate the numerical capabilities by applying the algorithm to a (1+1)-dimensional ℤ2 theory with dynamic fermionic matter and its pure-gauge version in (2+1)-dimensions. In both cases, we obtain certified intervals for the ground-state energy. For a one-dimensional system with 129 sites, the certified interval has a relative spread of 0.04%. For a two-dimensional system with 97 sites (corresponding to a 7 × 8 lattice), the relative spread is 2.08%. Beyond energy estimates, the proposed method provides rigorous bounds on both short- and long-range observables such as Wilson loops over multiple plaquettes and the magnitude of the mesonic string. The method can be both used as a computational microscope into many-body physics and a way to certify results of quantum simulators.
Related
- Larger physical volume and bounds on the number of matter fields in noncompact gauge theories on a lattice
- Ground-State Properties of a Lattice-Coupled Strongly Correlated Model for Solid Oxygen under Pressure
- Jensen-Mercer two-sided bounds for entropy production rate
- A Note On Lattice Definition of Gauge Theories With Chiral Fermions
- No-go theorems for dual Hamiltonians of Non-Abelian Lattice Gauge Theories
BibTeX
@misc{vallee2026twosided,
title = {{Two-Sided Bounds on Ground State Properties in Lattice Gauge Theories}},
author = {Eloïc Vallée and Ariel Kelman and Julius Mildenberger and Jordi Tura and Patrick Emonts},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06272},
eprint = {2610.06272},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06272},
}
RIS
TY - UNPB
TI - Two-Sided Bounds on Ground State Properties in Lattice Gauge Theories
AU - Eloïc Vallée
AU - Ariel Kelman
AU - Julius Mildenberger
AU - Jordi Tura
AU - Patrick Emonts
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06272
UR - https://arxiv.org/abs/2610.06272
AB - Gauge theories describe fundamental interactions in the standard model of particle physics and build effective theories in condensed matter physics. Being notoriously hard to simulate, these theories are commonly regularized as lattice gauge theories which can be evaluated with high precision through Monte Carlo algorithms or variational methods. However, Monte Carlo algorithms are not applicable in all regimes due to the sign problem; while variational methods, quantum and classical alike, only give upper-bounds on the ground-state energy. Their precision crucially depends on the chosen ansatz. Here, we present a framework based on a hierarchy of semidefinite programs to yield increasingly good lower-bounds on the ground-state energy and propagate these bounds to extended observables like Wilson loops and mesonic strings. We demonstrate the numerical capabilities by applying the algorithm to a (1+1)-dimensional ℤ_2 theory with dynamic fermionic matter and its pure-gauge version in (2+1)-dimensions. In both cases, we obtain certified intervals for the ground-state energy. For a one-dimensional system with 129 sites, the certified interval has a relative spread of 0.04%. For a two-dimensional system with 97 sites (corresponding to a 7 × 8 lattice), the relative spread is 2.08%. Beyond energy estimates, the proposed method provides rigorous bounds on both short- and long-range observables such as Wilson loops over multiple plaquettes and the magnitude of the mesonic string. The method can be both used as a computational microscope into many-body physics and a way to certify results of quantum simulators.
ER -