We study interaction effects on the topological crystalline insulators protected by time-reversal $(T)$ and reflection symmetry $(R)$ in two and three spatial dimensions. From the stability analysis of the edge states with bosonization, we find that the classification of the two-dimensional symmetry-protected topological (SPT) phases protected by ${Z}_{2}\ifmmode\times\else\texttimes\fi{}[\text{U(1)}\ensuremath{\rtimes}T]$ symmetry is reduced from $\mathbb{Z}$ to ${\mathbb{Z}}_{4}$ by interactions where the ${Z}_{2}$ symmetry denotes the reflection whose mirror plane is the two-dimensional plane itself. By extending the approach recently proposed by Isobe and Fu, we show that the classification of the three-dimensional SPT phases (i.e., topological crystalline insulators) protected by $R\ifmmode\times\else\texttimes\fi{}[\text{U(1)}\ensuremath{\rtimes}T]$ symmetry is reduced from $\mathbb{Z}$ to ${\mathbb{Z}}_{8}$ by interactions.We study interaction effects on the topological crystalline insulators protected by time-reversal $(T)$ and reflection symmetry $(R)$ in two and three spatial dimensions. From the stability analysis of the edge states with bosonization, we find that the classification of the two-dimensional symmetry-protected topological (SPT) phases protected by ${Z}_{2}\ifmmode\times\else\texttimes\fi{}[\text{U(1)}\ensuremath{\rtimes}T]$ symmetry is reduced from $\mathbb{Z}$ to ${\mathbb{Z}}_{4}$ by interactions where the ${Z}_{2}$ symmetry denotes the reflection whose mirror plane is the two-dimensional plane itself. By extending the approach recently proposed by Isobe and Fu, we show that the classification of the three-dimensional SPT phases (i.e., topological crystalline insulators) protected by $R\ifmmode\times\else\texttimes\fi{}[\text{U(1)}\ensuremath{\rtimes}T]$ symmetry is reduced from $\mathbb{Z}$ to ${\mathbb{Z}}_{8}$ by interactions.

arXiv:1505.06598Physical Review BAug 7, 201528 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

Topological phases protected by reflection symmetry and cross-cap states

Gil Young Cho, Chang-Tse Hsieh, Takahiro Morimoto +1 more

Twisting symmetries provides an efficient method to diagnose symmetry-protected topological (SPT) phases. In this paper, edge theories of (2+1)-dimensional topological phases protected by reflection as well as other symmetries are studied by twisting reflection symmetry, which effectively puts the edge theories on an unoriented space-time, such as the Klein bottle. A key technical step taken in this paper is the use of the so-called cross-cap states, which encode entirely the unoriented nature of space-time, and can be obtained by rearranging the space-time geometry and exchanging the role of space and time coordinates. When the system is in a nontrivial SPT phase, we find that the corresponding cross-cap state is noninvariant under the action of the symmetries of the SPT phase, but acquires an anomalous phase. This anomalous phase, with a proper definition of a reference state, on which symmetry acts trivially, reproduces the known classification of (2+1)-dimensional bosonic and fermionic SPT phases protected by reflection symmetry, including in particular the ${\mathbb{Z}}_{8}$ classification of topological crystalline superconductors protected by reflection and time-reversal symmetries.Twisting symmetries provides an efficient method to diagnose symmetry-protected topological (SPT) phases. In this paper, edge theories of (2+1)-dimensional topological phases protected by reflection as well as other symmetries are studied by twisting reflection symmetry, which effectively puts the edge theories on an unoriented space-time, such as the Klein bottle. A key technical step taken in this paper is the use of the so-called cross-cap states, which encode entirely the unoriented nature of space-time, and can be obtained by rearranging the space-time geometry and exchanging the role of space and time coordinates. When the system is in a nontrivial SPT phase, we find that the corresponding cross-cap state is noninvariant under the action of the symmetries of the SPT phase, but acquires an anomalous phase. This anomalous phase, with a proper definition of a reference state, on which symmetry acts trivially, reproduces the known classification of (2+1)-dimensional bosonic and fermionic SPT phases protected by reflection symmetry, including in particular the ${\mathbb{Z}}_{8}$ classification of topological crystalline superconductors protected by reflection and time-reversal symmetries.

arXiv:1501.07285Physical Review BMay 26, 201546 citations

Symmetry-Protected Topological Phases

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

Quantum science and technologyJan 1, 20197 citations

Hidden-symmetry–protected topological phases on a one-dimensional lattice

Linhu Li, Shu Chen

We demonstrate the existence of a topologically nontrivial phase in a one-dimensional fermionic lattice system subjected to synthetic gauge fields, which is beyond the standard Altland-Zirnbauer classification of topological insulators. The topological phase can be characterized by the presence of degenerate zero-mode edge states or a quantized Berry phase of the occupied Bloch band. By analyzing symmetries of the system, we identify that the topological phase and zero-mode edge states are protected by two hidden symmetries. An extended model with hidden symmetry breaking is also studied in order to reveal the effect of hidden symmetries on the symmetry-protected topological phase.We demonstrate the existence of a topologically nontrivial phase in a one-dimensional fermionic lattice system subjected to synthetic gauge fields, which is beyond the standard Altland-Zirnbauer classification of topological insulators. The topological phase can be characterized by the presence of degenerate zero-mode edge states or a quantized Berry phase of the occupied Bloch band. By analyzing symmetries of the system, we identify that the topological phase and zero-mode edge states are protected by two hidden symmetries. An extended model with hidden symmetry breaking is also studied in order to reveal the effect of hidden symmetries on the symmetry-protected topological phase.

arXiv:1411.0260Europhysics Letters (EPL)Feb 1, 201525 citations

We study interaction effects on the topological crystalline insulators protected by time-reversal

(T)

and reflection symmetry

(R)

in two and three spatial dimensions. From the stability analysis of the edge states with bosonization, we find that the classification of the two-dimensional symmetry-protected topological (SPT) phases protected by ${Z}_{2}\ifmmode\times\else\texttimes\fi{}[\text{U(1)}\ensuremath{\rtimes}T]$ symmetry is reduced from

\mathbb{Z}

{\mathbb{Z}}_{4}

by interactions where the

{Z}_{2}

symmetry denotes the reflection whose mirror plane is the two-dimensional plane itself. By extending the approach recently proposed by Isobe and Fu, we show that the classification of the three-dimensional SPT phases (i.e., topological crystalline insulators) protected by $R\ifmmode\times\else\texttimes\fi{}[\text{U(1)}\ensuremath{\rtimes}T]$ symmetry is reduced from

\mathbb{Z}

{\mathbb{Z}}_{8}

by interactions.

{\mathbb{Z}}_{8}

classification of topological crystalline superconductors protected by reflection and time-reversal symmetries.