Toric-code insulator enriched by translation symmetry

As was suggested twenty years ago, double-vortex condensation can drive a superconductor into a fractionalized insulator with \ensuremath{\mathbb{Z}}${}₂$ topological order. With the advent of symmetry-protected topological phases, the authors explore here the effect of strong correlation in a weak topological superconductor, showing that double-vortex condensation gives rise to a translation symmetry-enriched \ensuremath{\mathbb{Z}}${}₂$ topological order. This $t\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c$-$c\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}e$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}r$ features two types of visons with mutual semionic statistics, which transform into each other under discrete translation. This work also sheds light on realizing topological order and fractionalization by gapping semimetals.