etc.). To describe the physics of zero external gauge fields, we work with the choice $\rho^i_{{\rm}{external}}=0$. Working in this subspace means that [*all*]{} physical states must be color singlets, since all gauge transformations are trivial. This means that in this sector the following equations hold as operators identities $$E^i_x-\left( U^{{\rm}{Adj.}}_{x-1}\right)_{ji} E^j_{x-1} = \rho^i_{Fx}. \label{Glaw}$$

Finally we write the lattice Hamiltonian $H$ of the $1+1$-dimensional $SU(N)$ gauge theory with one flavor of fermions. $H$ is given by (in one dimension there is no magnetic field and so the plaquette term is identically zero) $$H=H_E+H_F. \label{eq:H_initial}$$ Here $H_E$ is the electric term, a sum over links $x$ of the $SU(N)$ Casimir operator $$H_G=\frac{g^2}{2}\sum_{x=1}^{L_s}\left(E^i_{x}\right)^2.$$ Next is the fermion Hamiltonian, $$H_F=-\frac{i}{2}\sum_{x=1}^{L_s}\psi^{\dag a\alpha}_x\, \left(\sigma_3\right)_{\alpha\beta} \, U_{x,ab} \, \psi^{b\beta}_{x+1}+h.c. + m\sum_x \psi^{\dag a\alpha}_x \,\left(\sigma_1\right)_{\alpha\beta} \, \psi^{a\beta}_x.  \label{eq:H_F}$$ Here we choose a particular representation of the one-dimensional Dirac matrices using the Pauli matrices $\sigma_{1,3}$ and periodic boundary conditions on the fermions, i.e. $\psi_x=\psi_{x+L_s}$, and $\psi^\dag_x=\psi^\dag_{x+L_s}$.

Superficially, the first term in the Hamiltonian is symmetric under a $U(1)_R\times U(1)_L$ chiral symmetry which is explicitly broken by the mass term to the vector $U(1)$. One can, however, spin diagonalize the fermions by writing $\psi^{\alpha}_x\to \left(e^{-i\pi\, \sigma_2/2 } \sigma_3^x\right)_{\alpha\beta} \psi^{\beta}_x$ 