\psi^{a}_{x+1}+h.c. + m\sum_x \psi^{\dag a}_x \sigma_1 \, \psi^{a}_x.\label{H'0}$$ Here we denote by $e^{i\varphi_a}$ the $a^{\rm th}$ eigenvalues of the constant spatial gauge field operator. Clearly, to get from one needs to remove all but the eigenvalues of the spatial Polyakov loops operator from the system. Using a sequence of a change of variables, this is easy to do in the path integral approach. But in the Hamiltonian there seems to be a conceptual difficulty with such a process : the degrees of freedom we wish to gauge away are represented by quantum operators and it is not clear how to ‘gauge them away’.

To proceed we choose to generalize, to the lattice, the formalism constructed in the seminal Ref. [@LNT]. The general idea is to find a Hilbert space realization of a unitary operator ${\cal F}$ that will rotate quantum states into a basis where the Hamiltonian looks like . To do so we follow the path integral picture as a guide : first we define the spatial Polyakov loop operator $P$ to be $$P_{ab}= \sum_{c,d,\cdots, z=1}^N U_{1,ac} U_{2,cd} \cdots U_{L_s,zb}\,,$$ and second we define the ‘eigenvalue operators’ $\varphi_a$ through $$P_{ab}= \sum_{c} S^\dag_{ac}(P) \, e^{i\,L_s\,\varphi_c(P)} \, S_{cb}(P).\label{P}$$ Here the matrix 