Lattice Hamiltonian for Adjoint QCD$_2$

We introduce a Hamiltonian lattice model for the $(1+1)$-dimensional $\text{SU}(N_c)$ gauge theory coupled to one adjoint Majorana fermion of mass $m$. The discretization of the continuum theory uses staggered Majorana fermions. We analyze the symmetries of the lattice model and find lattice analogs of the anomalies of the corresponding continuum theory. An important role is played by the lattice translation by one lattice site, which in the continuum limit involves a discrete axial transformation. On a lattice with periodic boundary conditions, the Hilbert space breaks up into sectors labeled by the $N_c$-ality $p=0, \ldots N_c-1$. Our symmetry analysis implies various exact degeneracies in the spectrum of the lattice model. In particular, it shows that, for $m=0$ and even $N_c$, the sectors $p$ and $p'$ are degenerate if $|p-p'| = N_c/2$. In the $N_c = 2$ case, we explicitly construct the action of the Hamiltonian on a basis of gauge-invariant states, and we perform both a strong coupling expansion and exact diagonalization for lattices of up to $12$ lattice sites. Upon extrapolation of these results, we find good agreement with the spectrum computed previously using discretized light-cone quantization. One of our new results is the first numerical calculation of the fermion bilinear condensate.