More About the Lattice Hamiltonian for Adjoint QCD$_2$

TL;DR

This paper extends a lattice Hamiltonian model for Adjoint QCD$_2$ to arbitrary gauge groups, analyzing gauge invariance, anomalies, and spectra, with explicit calculations for SU(3) and SU(4).

Contribution

It generalizes the lattice Hamiltonian for Adjoint QCD$_2$ to any compact gauge group and develops methods to compute observables and anomalies using Wigner 6j-symbols.

Findings

Gauge invariant state spaces are constructed for general gauge groups.

Exact diagonalization performed for SU(3) up to six sites.

Strong coupling expansions match numerical results for SU(3).

Abstract

In our earlier work arXiv:2311.09334, we introduced a lattice Hamiltonian for Adjoint QCD$_2$ using staggered Majorana fermions. We found the gauge invariant space of states explicitly for the gauge group $\text{SU}(2)$ and used them for numerical calculations of observables, such as the spectrum and the expectation value of the fermion bilinear. In this paper, we carry out a more in-depth study of our lattice model, extending it to any compact and simply-connected gauge group $G$. We show how to find the gauge invariant space of states and use it to study various observables. We also use the lattice model to calculate the mixed 't Hooft anomalies of Adjoint QCD$_2$ for arbitrary $G$. We show that the matrix elements of the lattice Hamiltonian can be expressed in terms of the Wigner 6$j$-symbols of $G$. For $G = \text{SU}(3)$, we perform exact diagonalization for lattices of up to six sites and study the low-lying spectrum, the fermion bilinear condensate, and the string tension. We also show how to write the lattice strong coupling expansion for ground state energies and operator expectation values in terms of the Wigner 6$j$-symbols. For $\text{SU}(3)$ we carry this out explicitly and find good agreement with the exact diagonalizations, and for $\text{SU}(4)$ we give expansions that can be compared with future numerical studies.