Fig. 1: Logic chart and flux realizations of projective algebraic relations.

a Gauge flux in a lattice requires the space group symmetries to be projectively represented. b Work flow diagram of this work. Given a space group G, the classification of projective representations are given by \({H}^{2}(G,{{\mathbb{Z}}}_{2})\). These representations are captured by projective symmetry algebras (PSAs) with a complete set of cohomology invariants, from which we construct a canonical model that can realize all possible PSAs, and derive nontrivial physical consequences. c–h Illustrate the construction method for realizing the five basic classes of PSAs (cohomology invariants). Specifically, (c, d, e, h) are for nontrivial σ, α, β, τ, respectively, and (f and g) are for nontrivial η. In (g), the bond connecting the two black sites can be either present or absent, and the total flux through the vertical and horizontal rectangular plaquettes is required to be π. Here, the π flux in a plaquette is realized by a negative hopping amplitude (marked by red color) on an edge of that plaquette.