Exact projected entangled pair ground states with topological Euler invariant

Tensor networks constitute arguably the most controllable tool in interacting topological quantum matter. Projected entangled pair states (PEPS) take a special role in this regard. Unfortunately, getting topological invariants in such systems is almost impossible, as phrased by many no-go theories. We profit from recent progress in free-fermion topology to formulate projector Hamiltonians with projected entangled pair ground states that have a finite topological invariant, the Euler class. 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, again the first examples of its kind. The connection to the non-interacting limit allows us to reveal entanglement features characteristic of this phase. Our family of states forms a reference point to check proposed mechanisms in Quantum Hall physics and spin liquids. All our states can be created by shallow quantum circuits from product states and have topological features, which makes them particularly interesting for implementations on noisy intermediate-scale quantum devices and the development of new quantum error correction protocols.