Toric-code insulator enriched by translation symmetry

As was suggested twenty years ago, double-vortex condensation can drive a superconductor into fractionalized insulator with \ensuremath{\mathbb{Z}}${}_{2}$ topological order. With the advent of symmetry-protected phases, authors explore here effect strong correlation in weak superconductor, showing that gives rise to translation symmetry-enriched 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 visons mutual semionic statistics, which transform each other under discrete translation. work also sheds light on realizing order and fractionalization by gapping semimetals.