Classification of topological insulators and superconductors with multiple order-two point group symmetries

We present a method for computing the classification groups of topological insulators and superconductors in the presence of \mathbb{Z}_2^{× n} <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:mrow> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> point group symmetries, for arbitrary natural numbers n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> . Each symmetry class is characterized by four possible additional symmetry types for each generator of \mathbb{Z}_2^{× n} <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:mrow> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> , together with bit values encoding whether pairs of generators commute or anticommute. We show that the classification is fully determined by the number of momentum- and real-space variables flipped by each generator, as well as the number of variables simultaneously flipped by any pair of generators. As a concrete illustration, we provide the complete classification table for the case of \mathbb{Z}_2^{× 2} <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:mrow> <mml:mo>×</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> point group symmetry.