Quantum phase transitions between bosonic symmetry-protected topological phases in two dimensions: Emergent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">QED</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>and anyon superfluid

Abstract

Inspired by the Chern-Simons effective theory description of symmetry protected topological (SPT) phases in two dimensions, we present a projective construction for many-body wave functions of SPT phases. Using this projective construction, we can systematically write down trial wave functions of SPT phases on a lattice. An explicit example of SPT phase with U(1) symmetry is constructed for two types of bosons with filling ${\ensuremath{\nu}}_{{b}_{1}}={\ensuremath{\nu}}_{{b}_{2}}=\frac{1}{2}$ per site on a square lattice. We study continuous phase transitions between different U(1)-SPT phases based on projective construction. The effective theory around the critical point is an emergent ${\mathrm{QED}}_{3}$ with fermion number ${N}_{f}=2$. Such a continuous phase transition, however, needs fine tuning, and in general there are intermediate phases between different U(1)-SPT phases. We show that such an intermediate phase has the same response as an anyon superconductor, and hence dub it ``anyon superfluid.'' A schematic phase diagram of interacting bosons with U(1) symmetry is depicted.