Many-Body Invariants for Super Spin Chains with Antiunitary Symmetries

Proposals for many-body invariants for super-spin chains with anti-unitary symmetries are evaluated. Symmetry protected phases are modeled as homotopy classes of gapped ground states. The formalism of matrix product states is systematically extended to fermionic systems with anti-unitary symmetries. A basis-independent diagrammatic approach capable of handling anti-unitary symmetries is developed. Suggestions from the literature for observables of a twisted entanglement entropy type are calculated and proven to be topological invariants of fermionic matrix product states. The viability of classifications via these invariants is discussed as well as the connection to the cohomology classification of one-dimensional fermionic symmetry protected phases. Taking the limit of diverging bond dimension while controlling the correlation length, the homotopy invariance is proved to persist

Supersimplicity and quadratic extensions

International audienceAn elliptic curve over a supersimple field with exactly one extension of degree 2 has an s-generic point

SUPERSIMPLICITY AND QUADRATIC EXTENSIONS

Abstract. Elliptic curves over a supersimple field with exactly one extension of degree 2 have s-generic rational points