Flat bands in Creutz ladder with supertranslation invariance

Using nilpotent matrices, we have developed a mechanism for constructing Compact Localized States (CLS), and hence flat-bands for non-interacting fermions on a lattice. The ensuing Hamiltonian is invariant under arbitrary space-dependent temporal translations, a.k.a. supertranslation transformations otherwise associated with asymptotic symmetries for flat space-time. This results in an infinite number of global charges. We prove that the following properties seen in flat band models are direct consequences of the infinite number of conserved supertranslation charges: (i) ultra-locality (zero correlation length) in correlation, (ii) subsystem size independence of entanglement entropy in the ground state, (iii) absence of Aharonov Bohm effect, (iv) decay of CLS to Bloch states when quenched by perturbative deformation that breaks supertranslation. For practical purposes, we add interactions to the Hamiltonian on the well-known flat band model in 1 dimension, the Creutz ladder. Interaction terms were constructed using the CLS as building blocks, keeping the supertranslation symmetry manifest. The existence of the infinite number of commuting charges directly dictates that the interacting theory is integrable without the need to resort to Bethe ansatz. The model has a rich phase diagram in addition to the two topological phases of the free Creutz ladder. In the third phase, which emerges in the half-filling case, has a degenerate ground state and choosing states of definite crystal momentum causes supertranslation symmetries to be spontaneously broken. However, if the chemical potential is turned to zero, the third phase will not show symmetry breaking. Ultra-locality in this phase is robust under perturbative deformations, which break the supertranslation. However, near the critical points in the other phases, such perturbative deformations give rise to divergent correlation length, as expected traditionally.