Relative anomalies in (2+1)D symmetry enriched topological states
Maissam Barkeshli, Meng Cheng
Abstract
Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetry-enriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionalization classes of a given (2+1)D topological order. This method applies to all types of symmetry actions, including anyon-permuting symmetries and general space-time reflection symmetries. We demonstrate compatibility of the relative anomaly formula with previous results for diagnosing anomalies for \mathbb{Z}_2^{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:math> space-time reflection symmetry (e.g. where time-reversal squares to the identity) and mixed anomalies for U(1) \times \mathbb{Z}_2^{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> and U(1) \rtimes \mathbb{Z}_2^{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>⋊</mml:mo> <mml:msubsup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> symmetries. We also study a number of additional examples, including cases where space-time reflection symmetries are intertwined in non-trivial ways with unitary symmetries, such as \mathbb{Z}_4^{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>4</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:math> and mixed anomalies for \mathbb{Z}_2 \times \mathbb{Z}_2^{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> symmetry, and unitary \mathbb{Z}_2 \times \mathbb{Z}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> symmetry with non-trivial anyon permutations.