We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group ${Z}_{8}$ . In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries.We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group ${Z}_{8}$. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries.

arXiv:1502.06962Physical Review BAug 10, 2015111 citations

Real-space construction of crystalline topological superconductors and insulators in 2D interacting fermionic systems

Jian-Hao Zhang, Shuo Yang, Yang Qi +1 more

The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this paper, we systematically classify the crystalline topological superconductors (TSC) and topological insulators (TI) in 2D interacting fermionic systems by using an explicit real-space construction. In particular, we discover an intriguing fermionic crystalline topological superconductor that can only be realized in interacting fermionic systems (i.e., not in free-fermion or interacting bosonic systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for generic 2D interacting fermionic systems.The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this paper, we systematically classify the crystalline topological superconductors (TSC) and topological insulators (TI) in 2D interacting fermionic systems by using an explicit real-space construction. In particular, we discover an intriguing fermionic crystalline topological superconductor that can only be realized in interacting fermionic systems (i.e., not in free-fermion or interacting bosonic systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for generic 2D interacting fermionic systems.

arXiv:2012.15657Physical Review ResearchJul 29, 202237 citations

Symmetry-protected quantum phase transition in topological insulators

L. -F. Liu, X. L. Zhang, Su-Peng Kou

The European Physical Journal BAug 1, 20121 citation

Correlation effects on topological crystalline insulators

Tsuneya Yoshida, Akira Furusaki

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

General Theory of interacting Topological insulators

Shoucheng Zhang

Bulletin of the American Physical SocietyMar 25, 20114 citations

{Z}_{8}

. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries.

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.