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 $\mathrm{O}(4)$ nonlinear sigma model with a topological theta term at $\ensuremath{\theta}=\ensuremath{\pi}$, or, equivalently, a noncompact ${\mathrm{CP}}^{1}$ model. Using a coupled wire construction, we show this is a surface theory for bosonic pgSPT phases with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{P}$ 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.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 $\mathrm{O}(4)$ nonlinear sigma model with a topological theta term at $\ensuremath{\theta}=\ensuremath{\pi}$, or, equivalently, a noncompact ${\mathrm{CP}}^{1}$ model. Using a coupled wire construction, we show this is a surface theory for bosonic pgSPT phases with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{P}$ 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.

arXiv:1711.03649Physical review. B./Physical review. BFeb 26, 201815 citations

Symmetry-Protected Topological Phases

Bei Zeng, Xie Chen, D. L. Zhou +1 more

Quantum science and technologyJan 1, 20197 citations

Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries

Ken Shiozaki, Hassan Shapourian, Shinsei Ryu

Finding suitable nonlocal order parameters that distinguish various symmetry-protected topological (SPT) phases is an important subject in view of experimental and numerical detection of SPT phases. By ``simulating'' the generating manifold of cobordism group in the operator formalism, the authors here propose nonlocal operations as diagnoses for SPT phases protected by point group symmetries. The nonlocal operations involve ``partial point group transformations'', which are obtained by point group transformations restricted to a spatial subregion on a given ground-state wave function. Through analytical and numerical calculations, the authors show that the complex $U(1)$ phase of the ground state expectation value of such partial point group transformations may serve as an order parameter for those SPT phases. The examples in the paper include the ${Z}_{8}$ and ${Z}_{16}$ invariants of topological superconductors protected by inversion symmetry in $(1+1)$ and $(3+1)$ dimensions, respectively, as well as the lens space topological invariants in $(2+1)$ -dimensional fermionic topological phases.Finding suitable nonlocal order parameters that distinguish various symmetry-protected topological (SPT) phases is an important subject in view of experimental and numerical detection of SPT phases. By ``simulating'' the generating manifold of cobordism group in the operator formalism, the authors here propose nonlocal operations as diagnoses for SPT phases protected by point group symmetries. The nonlocal operations involve ``partial point group transformations'', which are obtained by point group transformations restricted to a spatial subregion on a given ground-state wave function. Through analytical and numerical calculations, the authors show that the complex $U(1)$ phase of the ground state expectation value of such partial point group transformations may serve as an order parameter for those SPT phases. The examples in the paper include the ${Z}_{8}$ and ${Z}_{16}$ invariants of topological superconductors protected by inversion symmetry in $(1+1)$ and $(3+1)$ dimensions, respectively, as well as the lens space topological invariants in $(2+1)$-dimensional fermionic topological phases.

arXiv:1609.05970Physical review. B./Physical review. BMay 25, 201773 citations

Minimalist approach to the classification of symmetry protected topological phases

Zhaoxi Xiong

Bulletin of the American Physical SocietyMar 14, 20170 citations

Classification of fermionic symmetry-protected topological phases

Qing-Rui Wang, Zheng‐Cheng Gu

Bulletin of the American Physical SocietyMar 4, 20190 citations

{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

\mathrm{O}(4)

nonlinear sigma model with a topological theta term at $\ensuremath{\theta}=\ensuremath{\pi}$, or, equivalently, a noncompact

{\mathrm{CP}}^{1}

model. Using a coupled wire construction, we show this is a surface theory for bosonic pgSPT phases with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{P}$ 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.

U(1)

phase of the ground state expectation value of such partial point group transformations may serve as an order parameter for those SPT phases. The examples in the paper include the

{Z}_{8}

and

{Z}_{16}

invariants of topological superconductors protected by inversion symmetry in

(1+1)

and

(3+1)

dimensions, respectively, as well as the lens space topological invariants in

(2+1)

-dimensional fermionic topological phases.