Surface field theories of point group symmetry protected topological phases

Abstract

We identify field theories that describe the surfaces of three-dimensional\nbosonic point group symmetry protected topological (pgSPT) phases. The\nanomalous nature of the surface field theories is revealed via a dimensional\nreduction argument. Specifically, we study three different surface field\ntheories. The first field theory is quantum electrodynamics in three space-time\ndimensions (QED3) with four flavors of fermions. We show this theory can\ndescribe the surfaces of a majority of bosonic pgSPT phases protected by a\nsingle mirror reflection, or by $C_{nv}$ point group symmetry for $n=2,3,4,6$.\nThe second field theory is a variant of QED3 with charge-1 and charge-3 Dirac\nfermions. This field theory can describe the surface of a reflection symmetric\npgSPT phase built by placing an $E_{8}$ state on the mirror plane. The third\nfield theory is an ${\\rm O}(4)$ non-linear sigma model with a topological\ntheta-term at $\\theta=\\pi$, or, equivalently, a non-compact ${\\rm CP}^1$ model.\nUsing a coupled wire construction, we show this is a surface theory for bosonic\npgSPT phases with ${\\rm U}(1) \\times \\mathbb{Z}_{2}^{P}$ symmetry. For the\nlatter two field theories, we discuss the connection to gapped surfaces with\ntopological order. Moreover, we conjecture that the latter two field theories\ncan describe surfaces of more general bosonic pgSPT phases with $C_{nv}$ point\ngroup symmetry.\n