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arXiv:2509.17890 · Journal of High Energy Physics

Hamiltonian truncation framework for gauge theories on the interval

R. Houtz, James Ingoldby

Published Oct 28, 2026• 0 citations
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Abstract

A bstract In this work, we investigate gauge theories in two dimensions nonperturbatively using the Hamiltonian truncation approach. Working on a spatial interval and adopting the axial gauge, we remove all gauge field degrees of freedom and express the interacting Hamiltonian in the eigenbasis of the free Dirac theory, truncated at a finite energy. As a benchmark we analyse the Schwinger model, where our numerical spectra agree closely with the exact results from bosonization across a wide range of couplings, validating the construction of the Hamiltonian. We then generalize the formulation to nonabelian gauge groups and apply it to SU(3) gauge theory with a single massless Dirac fermion. These results demonstrate that gauge theories can be explored nonperturbatively using a truncated Hamiltonian that generates evolutions in ordinary time, offering a complementary alternative to lattice field theory.

DOI
10.1007/jhep10(2026)276

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A bstract In this work, we investigate gauge theories in two dimensions nonperturbatively using the Hamiltonian truncation approach. Working on a spatial interval and adopting the axial gauge, we remove all gauge field degrees of freedom and express the interacting Hamiltonian in the eigenbasis of the free Dirac theory, truncated at a finite energy. As a benchmark we analyse the Schwinger model, where our numerical spectra agree closely with the exact results from bosonization across a wide range of couplings, validating the construction of the Hamiltonian. We then generalize the formulation to nonabelian gauge groups and apply it to SU(3) gauge theory with a single massless Dirac fermion. These results demonstrate that gauge theories can be explored nonperturbatively using a truncated Hamiltonian that generates evolutions in ordinary time, offering a complementary alternative to lattice field theory.
Maxwell-Chern-Simons (MCS) theory in 2+12+12+1 dimensions provides a paradigmatic example of a topological gauge theory with both dynamical and topological degrees of freedom. Its Euclidean formulation suffers from a sign problem, making Hamiltonian numerical approaches particularly attractive. As a first step toward the non-perturbative Hamiltonian study of MCS theory, we investigate the constant mode sector on a spatial torus. Being analytically solvable in the continuum, it provides an ideal benchmark for understanding how the topological properties of the theory are encoded in a finite-dimensional lattice Hilbert space. We construct a finite-dimensional discretization of the torus of flat connections and show that the resulting lattice problem maps onto a generalized Harper-Hofstadter model with twisted boundary conditions. We identify the commensurability conditions under which the finite lattice exactly reproduces the magnetic translation algebra and the topological degeneracy of the continuum theory. A systematic analysis of gauge field truncation and its convergence toward the continuum limit is then presented.
Gauge theories establish the standard model of particle physics, and lattice gauge theory (LGT) calculations employing Markov Chain Monte Carlo (MCMC) methods have been pivotal in our understanding of fundamental interactions. The present limitations of MCMC techniques may be overcome by Hamiltonian-based simulations on classical or quantum devices, which further provide the potential to address questions that lay beyond the capabilities of the current approaches. However, for continuous gauge groups, Hamiltonian-based formulations involve infinite-dimensional gauge degrees of freedom that can solely be handled by truncation. Current truncation schemes require dramatically increasing computational resources at small values of the bare couplings, where magnetic field effects become important. Such limitation precludes one from `taking the continuous limit' while working with finite resources. To overcome this limitation, we provide a resource-efficient protocol to simulate LGTs with continuous gauge groups in the Hamiltonian formulation. Our new method allows for calculations at arbitrary values of the bare coupling and lattice spacing. The approach consists of the combination of a Hilbert space truncation with a regularization of the gauge group, which permits an efficient description of the magnetically-dominated regime. We focus here on Abelian gauge theories and use 2 + 1 dimensional quantum electrodynamics as a benchmark example to demonstrate this efficient framework to achieve the continuum limit in LGTs. This possibility is a key requirement to make quantitative predictions at the field theory level and offers the long-term perspective to utilise quantum simulations to compute physically meaningful quantities in regimes that are precluded to quantum Monte Carlo.
Gauge theories establish the standard model of particle physics, and lattice gauge theory (LGT) calculations employing Markov Chain Monte Carlo (MCMC) methods have been pivotal in our understanding of fundamental interactions. The present limitations of MCMC techniques may be overcome by Hamiltonian-based simulations on classical or quantum devices, which further provide the potential to address questions that lay beyond the capabilities of the current approaches. However, for continuous gauge groups, Hamiltonian-based formulations involve infinite-dimensional gauge degrees of freedom that can solely be handled by truncation. Current truncation schemes require dramatically increasing computational resources at small values of the bare couplings, where magnetic field effects become important. Such limitation precludes one from `taking the continuous limit' while working with finite resources. To overcome this limitation, we provide a resource-efficient protocol to simulate LGTs with continuous gauge groups in the Hamiltonian formulation. Our new method allows for calculations at arbitrary values of the bare coupling and lattice spacing. The approach consists of the combination of a Hilbert space truncation with a regularization of the gauge group, which permits an efficient description of the magnetically-dominated regime. We focus here on Abelian gauge theories and use 2+12+12+1 dimensional quantum electrodynamics as a benchmark example to demonstrate this efficient framework to achieve the continuum limit in LGTs. This possibility is a key requirement to make quantitative predictions at the field theory level and offers the long-term perspective to utilise quantum simulations to compute physically meaningful quantities in regimes that are precluded to quantum Monte Carlo.
We examine the Kogut-Susskind formulation of lattice gauge theories under the light of fermionic and bosonic degrees of freedom that provide a description useful to the development of quantum simulators of gauge-invariant models. We consider both discrete and continuous gauge groups and adopt a realistic multicomponent Fock space for the definition of matter degrees of freedom. In particular, we express the Hamiltonian of the gauge theory and the Gauss law in terms of Fock operators. The gauge fields are described in two different bases based on either group elements or group representations. This formulation allows for a natural scheme to achieve a consistent truncation of the Hilbert space for continuous groups, and provides helpful tools to study the connections of gauge theories with topological quantum double and string-net models for discrete groups. Several examples, including the case of the discrete D3{D}_{3}D3​ gauge group, are presented.
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.
We study quantum simulation of SU(3) non-Abelian gauge theory dynamically coupled with fundamental fermions in 3+13+13+1 dimensions by employing the lattice Hamiltonian in axial gauge that avoids Gauss's law constraints. The temporal component of the gauge field is analytically solved in terms of independent field degrees of freedom and a lattice regulated Green's function. The axial gauge condition is trivially maintained in time evolution, even under Trotterization. The gauge field degrees of freedom are expressed in the local field basis and can be efficiently transformed into the canonical conjugate momentum basis by local quantum Fourier transforms. We prove the number of qubits needed for describing all states up to an energy EEE with an accuracy εεε on a lattice of volume VVV at bare coupling ggg is bounded as 16nAV+12nfV16n_A V + 12n_f V16nA​V+12nf​V, where $n_A \approx \log_2 (\frac{64 E' V^{4/3}}{π^2ε} + \frac{32\sqrt{2}g n_f E'^{1/2} V^{7/6}}{\sqrt{3}π^3ε^{1/2}} ) $ is the number of qubits needed for each independent gauge field per site with a shifted energy E′E'E′, and nfn_fnf​ denotes the number of fermion flavors. We then analyze a quantum algorithm for time evolution that is based on Trotterization, quantum Fourier transform, and Jordan-Wigner transformation, for which quantum circuits can be explicitly constructed under arbitrary gauge field truncation and digitization. We find the numbers of CNOT and single-qubit rotation gates both scale as O(nA4V4/3)+O(V5/3)O(n_A^4 V^{4/3}) + O(V^{5/3})O(nA4​V4/3)+O(V5/3) per Trotter step for fixed nf≤6n_f\leq 6nf​≤6. We conclude that quantum resources needed for simulating real-time dynamics of lattice QCD scale polynomially with volume, energy, time, accuracy, and bare Hamiltonian parameters.