Hamiltonian Truncation Framework for Gauge Theories on the Interval

TL;DR

This work develops and tests a Hamiltonian truncation framework for 1+1D gauge theories on a spatial interval, fixing axial gauge to remove gauge-field degrees of freedom and truncating the interacting Hamiltonian in the free-Dirac basis via an energy cutoff. It validates the method by benchmarking the Schwinger model against exact bosonization results and extends the formalism to SU($N$) gauge theory with a massless fermion, uncovering confinement features and a light color-singlet meson. The approach provides a real-time, lattice-free nonperturbative tool that complements lattice field theory and serves as a controlled platform for exploring richer theories and observables in gauge dynamics. The demonstrated convergence and qualitative infrared behavior motivate future studies of mass effects, theta parameters, additional gauge groups, flavor content, and extensions to higher dimensions or quantum-computing implementations.

Abstract

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.

Figures (8)

• Figure 1: The left plot shows the excited energy gaps $\Delta E_n L$ of the Schwinger model at moderate coupling $gL = 8$. The result from truncation is denoted by dots and connected by red or orange lines. The exact result from bosonization is given in black dashed lines. The right plot shows vacuum energy $E_0 L$ plotted against $1/(E_{\rm max} L)^2$ in black dots, along with its linear fit given by a dashed red line.
• Figure 2: The convergence of the first few excited states in the spectrum for the Schwinger model at moderate coupling $gL = 8$. The energy eigenvalues $E_n L$ are shown versus $1/(E_{\rm max} L)^2$. Values obtained from truncation are denoted by black dots, and a linear fits of the energy levels versus $1/(E_{\rm max} L)^2$ are shown as red dashed lines. The expected scaling with $E_{\rm max} L$ holds for the low-lying eigenstates.
• Figure 3: The left plot shows the excited energy gaps $\Delta E_n L$ of the Schwinger model at strong coupling $gL = 24$. The result from truncation is denoted by dots and connected by colored lines. The exact result from bosonization is given in black dashed lines. The numerical data converges towards the exact result. The right plot shows the $1/(E_{\rm max} L)^2$ convergence of the vacuum energy $E_0 L$ as $E_{\rm max} L$ is increased.
• Figure 4: The convergence of the first few excited states in the spectrum for the Schwinger model at moderate coupling $gL = 24$. The energy eigenvalues $E_n L$ are shown versus $1/(E_{\rm max} L)^2$. Values obtained from truncation are denoted by black dots, and the $1/(E_{\rm max} L)^2$ fits are shown as red dashed lines. The domain of the fit lines indicate the regions over which the fits were performed. Here we see that for higher excited states, the expected scaling emerges only in the high $E_{\rm max} L$ tails.
• Figure 5: The spectrum as $gL$ is varied. On the left hand side, we plot the deviation in energy gaps of $H_{\rm eff}$ from the predicted bosonized result. To guide the eye, we show the bosonized result from $gL =8$ with black dashed lines. We see that as $gL$ grows, the agreement with the exact bosonized result deteriorates. On the right hand side, we plot the spectrum against $(gL)^4$. We demonstrate its scaling using a linear fit, shown in red. Only points with the smallest $gL$ values are used in the fit, indicated by the range of the solid red line, while the dashed red line shows the fit's extrapolation to larger $gL$ values.