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The interplay of symmetry, topology, and many-body effects in theclassification of possible phases of matter poses a formidable challenge thatis attracting great attention in condensed-matter physics. Such many-bodyeffects are typically induced by inter-particle interactions involving anaction at a distance, such as the Coulomb interaction between electrons in asymmetry-protected topological (SPT) phase. In this work, we show that similarphenomena also appear in high-energy physics, where inter-particle interactionsare mediated by gauge bosons, and constrained by a local gauge symmetry. Weintroduce a variant of the so-called Schwinger model, which describes quantumelectrodynamics in (1+1) dimensions (QED$_2$), and show that it can host SPTphases with a topological contribution to the vacuum {\theta} angle, whichleads to a new type of topological QED$_2.$ We use bosonization anddensity-matrix renormalization group techniques to study its rich phase diagramin great detail, and present a scheme for its realization in experiments ofultra-cold atoms in optical lattices.

Symmetry-protected topological phases in lattice gauge theories: topological QED$_2$