TY - JOUR

T1 - Bosonic topological phases of matter

T2 - Bulk-boundary correspondence, symmetry protected topological invariants, and gauging

AU - Tiwari, Apoorv

AU - Chen, Xiao

AU - Shiozaki, Ken

AU - Ryu, Shinsei

N1 - Funding Information:
X.C. was supported by a postdoctoral fellowship from the Gordon and Betty Moore Foundation, under the EPiQS initiative, Grant GBMF4304, at the Kavli Institute for Theoretical Physics. K.S. is supported by RIKEN Special Postdoctoral Researcher Program. This work is supported in part by the NSF under Grant No. DMR-1455296.
Funding Information:
A.T. would like to thank Michael Stone, Srinidhi Ramamurthy, Michael V. Pak, Xueda Wen, Lakshya Bhardwaj, and Itziar Ochoa de Alaiza for several helpful discussions. X.C. was supported by a postdoctoral fellowship from the Gordon and Betty Moore Foundation, under the EPiQS initiative, Grant GBMF4304, at the Kavli Institute for Theoretical Physics. K.S. is supported by RIKEN Special Postdoctoral Researcher Program. This work is supported in part by the NSF under Grant No. DMR-1455296.
Publisher Copyright:
© 2018 American Physical Society.

PY - 2018/6/21

Y1 - 2018/6/21

N2 - 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 Z2T/R×U(1) anomaly where Z2T/R is time-reversal/reflection symmetry, and the 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 U(1)×Z2T,R.

AB - 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 Z2T/R×U(1) anomaly where Z2T/R is time-reversal/reflection symmetry, and the 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 U(1)×Z2T,R.

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U2 - 10.1103/PhysRevB.97.245133

DO - 10.1103/PhysRevB.97.245133

M3 - Article

AN - SCOPUS:85049164716

SN - 2469-9950

VL - 97

JO - Physical Review B

JF - Physical Review B

IS - 24

M1 - 245133

ER -