Surface field theories of point group symmetry protected topological phases

We identify field theories that describe the surfaces of three-dimensional bosonic point group symmetry protected topological (pgSPT) phases. The anomalous nature of the surface field theories is revealed via a dimensional reduction argument. Specifically, we study three different surface field theories. The first field theory is quantum electrodynamics in three space-time dimensions (QED3) with four flavors of fermions. We show this theory can describe the surfaces of a majority of bosonic pgSPT phases protected by a single mirror reflection, or by $C_{nv}$ point group symmetry for $n=2,3,4,6$. The second field theory is a variant of QED3 with charge-1 and charge-3 Dirac fermions. This field theory can describe the surface of a reflection symmetric pgSPT phase built by placing an $E_{8}$ state on the mirror plane. The third field theory is an ${\rm O}(4)$ non-linear sigma model with a topological theta-term at $\theta=\pi$, or, equivalently, a non-compact ${\rm CP}^1$ model. Using a coupled wire construction, we show this is a surface theory for bosonic pgSPT phases with ${\rm U}(1) \times \mathbb{Z}_{2}^{P}$ symmetry. For the latter two field theories, we discuss the connection to gapped surfaces with topological order. Moreover, we conjecture that the latter two field theories can describe surfaces of more general bosonic pgSPT phases with $C_{nv}$ point group symmetry.Comment: 16 pages, 2 figure

New classes of systematic effects in gas spin co-magnetometers

Atomic co-magnetometers are widely used in precision measurements searching for spin interactions beyond the Standard Model. We describe a new $^3$He-$^{129}$Xe co-magnetometer probed by Rb atoms and use it to identify two general classes of systematic effects in gas co-magnetometers, one associated with diffusion in second-order magnetic field gradients and another due to temperature gradients. We also develop a general and practical approach for calculating spin relaxation and frequency shifts due to arbitrary magnetic field gradients and confirm it experimentally.Comment: 5 pages, 4 figure

The dollar and policy options

Dollar, American ; Foreign exchange - Law and legislation

Topological phases protected by point group symmetry

We consider symmetry protected topological (SPT) phases with crystalline point group symmetry, dubbed point group SPT (pgSPT) phases. We show that such phases can be understood in terms of lower-dimensional topological phases with on-site symmetry, and can be constructed as stacks and arrays of these lower-dimensional states. This provides the basis for a general framework to classify and characterize bosonic and fermionic pgSPT phases, that can be applied for arbitrary crystalline point group symmetry and in arbitrary spatial dimension. We develop and illustrate this framework by means of a few examples, focusing on three-dimensional states. We classify bosonic pgSPT phases and fermionic topological crystalline superconductors with $Z_2^P$ (reflection) symmetry, electronic topological crystalline insulators (TCIs) with ${\rm U}(1) \times {Z}_2^P$ symmetry, and bosonic pgSPT phases with $C_{2v}$ symmetry, which is generated by two perpendicular mirror reflections. We also study surface properties, with a focus on gapped, topologically ordered surface states. For electronic TCIs we find a $Z_8 \times Z_2$ classification, where the $Z_8$ corresponds to known states obtained from non-interacting electrons, and the $Z_2$ corresponds to a "strongly correlated" TCI that requires strong interactions in the bulk. Our approach may also point the way toward a general theory of symmetry enriched topological (SET) phases with crystalline point group symmetry.Comment: v2: Minor changes/additions to introduction and discussion sections, references added, published version. 21 pages, 11 figure