Quantum geometry beyond single flat bands and Euler exact projected entangled pair ground states

University of Manchester

The past few years have seen a revived interest in quantum geometrical characterizations. Although the metric tensor has been connected to many geometrical concepts for single bands, the exploration of these concepts to a multi-band paradigm still promises a new field of interest. I will discuss a new route involving Plücker embeddings to represent arbitrary classifying spaces, being the essential objects that encode all the relevant topology for any multi-band system. While I will argue that this tool can be applied in contexts that range from response theories to finding quantum volumes and bounds on superfluid densities as well as possible quantum computations, I will in particular also show that they can be used to formulate projector Hamiltonians with projected entangled pair ground (PEPS) states that have a finite topological invariant, the Euler class, circumventing many no-go theorems. We further demonstrate the versatility of our model states by applying a shallow quantum circuit, producing interacting PEPS and simple parent Hamiltonians in the Euler phase. These model states moreover pinpoint to new interacting physics.