Symmetry-protected topological (SPT) phases with either internal or spatial symmetries have been studied in different methods within this decade. In this paper, we study the classification and characterization of 3+1d topological phases protected simultaneously by both spatial and internal symmetries from the topological response theory. Physical observables derived from these response theories are discussed and compared with their analogs in 2+1d systems, some of which have been well-studied in quantum Hall (QH) systems, like the Wen-Zee (WZ) term. We then study the bulk topological gauge theories for the corresponding SPT phases. Generalizations to symmetry-enriched topological (SET) phases are also discussed at the end of this paper.Symmetry-protected topological (SPT) phases with either internal or spatial symmetries have been studied in different methods within this decade. In this paper, we study the classification and characterization of 3+1d topological phases protected simultaneously by both spatial and internal symmetries from the topological response theory. Physical observables derived from these response theories are discussed and compared with their analogs in 2+1d systems, some of which have been well-studied in quantum Hall (QH) systems, like the Wen-Zee (WZ) term. We then study the bulk topological gauge theories for the corresponding SPT phases. Generalizations to symmetry-enriched topological (SET) phases are also discussed at the end of this paper.

Jul 27, 20182 citations

Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions

Meng Cheng, Zheng‐Cheng Gu

It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with Zm symmetry and find the classification of SPT phases depends on the parity of m: for even m there are 2m classes, m out of which is intrinsically fermionic SPT phases and cannot be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with Zm symmetry and find the classification of SPT phases depends on the parity of m: for even m there are 2m classes, m out of which is intrinsically fermionic SPT phases and cannot be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.

arXiv:1302.4803Physical Review LettersApr 9, 201467 citations

Classification and Edge States of Symmetry Protected Topological Phases

Xie Chen

Bulletin of the American Physical SocietyMar 3, 20140 citations

Bosonic topological phases of matter: Bulk-boundary correspondence, symmetry protected topological invariants, and gauging

Apoorv Tiwari, Xiao Chen, Ken Shiozaki +1 more

We analyze $2+1d$ and $3+1d$ bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group $G$ by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat $G$ gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat $G$ gauge fields to obtain Dijkgraaf-Witten topological $G$ gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in $2+1d$ and condensing bosonic gauge charges in $3+1d$ . In the dual boundary approach, we study $1+1d$ and $2+1d$ quantum field theories that have $G$ 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct $G$ characters that generate topological data for the bulk topological gauge theory. Finally, we study a $2+1d$ quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where ${\mathbb{Z}}_{2}^{T/R}$ is time-reversal/reflection symmetry, and the $\text{U}(1)$ could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in $3+1d$ that cancels this anomaly. This signals the existence of SPTs in $3+1d$ protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.We analyze $2+1d$ and $3+1d$ bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group $G$ by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat $G$ gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat $G$ gauge fields to obtain Dijkgraaf-Witten topological $G$ gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in $2+1d$ and condensing bosonic gauge charges in $3+1d$. In the dual boundary approach, we study $1+1d$ and $2+1d$ quantum field theories that have $G$ 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct $G$ characters that generate topological data for the bulk topological gauge theory. Finally, we study a $2+1d$ quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where ${\mathbb{Z}}_{2}^{T/R}$ is time-reversal/reflection symmetry, and the $\text{U}(1)$ could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in $3+1d$ that cancels this anomaly. This signals the existence of SPTs in $3+1d$ protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.

arXiv:1710.04730Physical review. B./Physical review. BJun 21, 201869 citations

Minimalist approach to the classification of symmetry protected topological phases

Zhaoxi Xiong

Bulletin of the American Physical SocietyMar 14, 20170 citations

We analyze

2+1d

and

3+1d

bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group

G

by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat

G

gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat

G

gauge fields to obtain Dijkgraaf-Witten topological

G

gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in

2+1d

and condensing bosonic gauge charges in

3+1d

. In the dual boundary approach, we study

1+1d

and

2+1d

quantum field theories that have

G

't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct

G

characters that generate topological data for the bulk topological gauge theory. Finally, we study a

2+1d

quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where

{\mathbb{Z}}_{2}^{T/R}

is time-reversal/reflection symmetry, and the

\text{U}(1)

could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in

3+1d

that cancels this anomaly. This signals the existence of SPTs in

3+1d

protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.