Fig. 2: Cartoon depicting AGL preserving and violating LSH states.

(a) A graphical representation of the strong-coupling vacuum (odd sites completely filled and even sites empty) for a 4-site lattice using the LSH quantum numbers that satisfy AGLs on all the links. The kets corresponds to the SU(3) LSH basis defined as \({\vert {n}_{P},{n}_{Q},{\nu }_{\underline{1}},{\nu }_{0},{\nu }_{1}\rangle }_{r}\). (b) A graphical representation of another state, charaterized by on-site LSH quantum numbers. This state do not satisfy on-link AGLs as depicted by discontinuity of directed flux lines across the links. The erroneous hopping terms \({H}_{I}^{{\prime} }\) in (13) would cause a non-zero matrix element between the states given in (a) and (b). (LSH representation of \({H}_{I}^{{\prime} }\) is given in equation (39)). As depicted in the cartoon, the total number of fermions on the lattice is still conserved (the filled black circles correspond to fully-filled occupation numbers, as shown in Fig. 1). The discontinuity of directed flux lines are connected to violation of individual global charges q_f defined in (11).