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arXiv:2601.10065

Minimally Truncated SU(3) Lattice Gauge Theory and String Tension

Vincent Chen, Berndt Müller, Xiaojun Yao

hep-lathep-phhep-thquant-phSubmitted Jan 15, 2026• Announced Jan 16, 2026
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Abstract

We study SU(3) gauge theory on small lattices in the minimal (qutrit) electric field truncation retaining only the ${\bf 1}, {\bf 3}, {\bf \overline{3}}$ representations for the link variables. Explicit expressions are given for the Kogut-Susskind Hamiltonian for the square plaquette chain and the two-dimensional honeycomb lattice. Our formalism can be easily extended to the minimally truncated general SU($N_c$) gauge theory. The addition of (static) quarks is discussed. We present results for the energy spectrum of the gauge field on these lattices by exact diagonalization of the Hamiltonian and analyze its statistical properties. We also compute the SU(3) string tension and discuss how it is modified by vacuum fluctuations. Finally, we calculate the potential energies of a static quark-antiquark pair and three static quarks and study their screening at finite temperature.

Comments
15 pages, 26 figures

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We study SU(3) gauge theory on small lattices in the minimal (qutrit) electric field truncation retaining only the 1,3,3‾{\bf 1}, {\bf 3}, {\bf \overline{3}}1,3,3 representations for the link variables. Explicit expressions are given for the Kogut-Susskind Hamiltonian for the square plaquette chain and the two-dimensional honeycomb lattice. Our formalism can be easily extended to the minimally truncated general SU(NcN_cNc​) gauge theory. The addition of (static) quarks is discussed. We present results for the energy spectrum of the gauge field on these lattices by exact diagonalization of the Hamiltonian and analyze its statistical properties. We also compute the SU(3) string tension and discuss how it is modified by vacuum fluctuations. Finally, we calculate the potential energies of a static quark-antiquark pair and three static quarks and study their screening at finite temperature.
We show that a simple qubit-regularized SU(3)\mathrm{SU}(3)SU(3) lattice gauge theory on a plaquette chain admits a continuum limit with massive gauge-invariant excitations, providing a minimal toy model of strong interactions without quarks. By mapping the plaquette-chain Hamiltonian to the three-state quantum clock model in a magnetic field, we demonstrate that the theory can be tuned to a continuum limit governed at short distances by the Z3\mathbb{Z}_3Z3​ parafermion conformal field theory, which serves as the ultraviolet fixed point. A small relevant magnetic perturbation then drives the system to a massive continuum quantum field theory in the infrared, featuring relativistic particles. In this continuum theory we compute two universal ratios: the mass ratio of the lightest massive particles with opposite charge conjugation, m−/m+=1.459(2)m^{-}/m^{+} = 1.459(2)m−/m+=1.459(2), and σ/m+=0.2648(2)\sqrtσ/m^{+} = 0.2648(2)σ​/m+=0.2648(2), where σσσ denotes the string tension between a static quark and antiquark.
Quantum simulations of lattice gauge theories are anticipated to directly probe the real time dynamics of QCD, but scale unfavorably with the required truncation of the gauge fields. Improved Hamiltonians are derived to correct for the effects of gauge field truncations on the SU(3) Kogut-Susskind Hamiltonian. It is shown in 1+1D1+1D1+1D that this enables low chromoelectric field truncations to quantitatively reproduce features of the untruncated theory over a range of couplings and quark masses. In 3+1D3+1D3+1D, an improved Hamiltonian is derived for lattice QCD with staggered massless fermions. It is shown in the strong coupling limit that the spectrum qualitatively reproduces aspects of two flavor QCD and simulations of a small system are performed on IBM's Perth quantum processor.
A Hamiltonian lattice formulation of lattice gauge theories opens the possibility for quantum simulations of the non-perturbative dynamics of QCD. By parametrizing the gauge invariant Hilbert space in terms of plaquette degrees of freedom, we show how the Hilbert space and interactions can be expanded in inverse powers of N. At leading order in this expansion, the Hamiltonian simplifies dramatically, both in the required size of the Hilbert space as well as the type of interactions involved. Adding a truncation of the resulting Hilbert space in terms of local electric energy states we give explicit constructions that allow simple representations of SU(3) gauge fields on qubits and qutrits to leading order in large N.
We give computer-assisted, two-sided spectral-gap bounds for the untruncated SU(3) lattice gauge Hamiltonian on open chains of one, two, and three elementary cubes. All vertices satisfy the neutral Gauss law, and all elementary plaquettes, including shared interior faces, enter the Hamiltonian. The normalization is K = T + 3F − M/2, with C2(3) = 4/3. Complete electric-energy windows of dimensions 141, 531, and 1205, exact Haar Gram matrices, and certified complementary-space estimates yield gap enclosures [2.915605103, 4.978456040], [0.777213074, 5.394807087], and [0.328406186, 5.692597307]. For three cubes, retaining the complete Wilson output resolved into 20 electric-energy layers closes the Schur inertia test. We quantify the energy-window ceiling, prove an unconditional exponential dimension obstruction for complete absolute electric windows, and obtain positive gap bounds throughout 0.45 ≤ β ≤ 0.502 for Kβ = T + β(6F − M). Larger-window rational trial vectors sharpen both endpoints; an additional certified minimal-truncation benchmark and independent Haar audits accompany the proof. The results concern finite lattices; no thermodynamic or continuum mass gap is inferred.
We argue that a simple qubit-regularized SU(2)\mathrm{SU}(2)SU(2) lattice gauge theory (LGT) on a plaquette chain serves as a pseudo-one-dimensional toy model for Yang-Mills (YM) theory in three spatial dimensions. We map the chain Hamiltonian to the Transverse Field Ising Model (TFIM) in a uniform magnetic field and demonstrate that it can be tuned to a continuum limit in which the short-distance physics is governed by the asymptotically free Ising conformal field theory (CFT) describing free Majorana fermions, while the long-distance regime contains massive excitations of the E8E_8E8​ quantum field theory (QFT) that can be interpreted as one-dimensional analogues of glueballs. Furthermore, we find σ/m1=0.249(1)\sqrtσ/m_1 = 0.249(1)σ​/m1​=0.249(1) where σσσ is the string tension between two static quarks and m1m_1m1​ is the mass of the lightest glueball.
Quantum simulations of lattice gauge theories offer the potential to directly study the nonperturbative dynamics of quantum chromodynamics, but naive analyses suggest that they require large computational resources. Large N c expansions are performed to order 1 / N c to simplify the Hamiltonian of pure SU ( N c ) lattice gauge theories. A reformulation of the electric basis is introduced with a truncation strategy based on the construction of local Krylov subspaces with plaquette operators. Numerical simulations show that these truncated Hamiltonians are consistent with traditional lattice calculations at relatively small couplings. It is shown that the computational resources required for quantum simulation of time evolution generated by these Hamiltonians is 17–19 orders of magnitude smaller than previous approaches, provided that the truncations in this work can reach lattice spacings in three-dimensional simulations comparable to the two-dimensional simulations performed.