Critical bifurcation and deconfined quantum criticality in an interacting cluster Ising chain

We investigate the cluster Ising chain with an additional nearest-neighbor interaction using tensor-network methods and a weak-coupling field theory. Without the interaction, the Jordan-Wigner transformation decomposes the model into a triplet of identical Majorana chains related by an exact O(3) flavor symmetry. Their common mass vanishes at the SU(2)₂ Wess-Zumino-Novikov-Witten critical point with central charge c = 3/2 separating the symmetry-protected topological cluster phase from a ferromagnet. The interaction reduces O(3) to its cyclic subgroup C₃, splitting the triplet into a singlet and a doublet whose gaps close separately, producing a critical bifurcation into Ising $c = 1/2$ and Gaussian $c = 1$ critical lines. A second ferromagnetic phase opens between these critical lines for repulsive interactions and a disordered phase for attractive ones. The Gaussian line then separates two Landau-incompatible ferromagnets, realizing a deconfined quantum critical line with emergent O(2) symmetry, along which our independently extracted exponents vary continuously yet satisfy the parameter-free relation β= (2ν- 1)/4 of the eight-vertex weak universality class. At stronger repulsion this line opens into an extended gapless floating phase with incommensurate algebraic correlations, entered through Berezinskii-Kosterlitz-Thouless transitions. All of these phases and the transitions between them are captured by the weak-coupling theory. Beyond its regime of validity, our simulations reveal a translation-symmetry-breaking antiferromagnet reached through first-order transitions.