# Bosonic SPT phases with time reversal and a $Z_2$ symmetry

Consider a bosonic system with time reversal symmetry $$\mathcal{T}$$ and a unitary on-site $$\mathbb{Z}_2$$ symmetry. Suppose the symmetry is realized in a special way such that $$\mathcal{T}^2= (-1)^B$$ acting on any physical local operator, where $$B=0$$ if the operator is $$\mathbb{Z}_2$$ even, and $$B=1$$ if the operator is $$\mathbb{Z}_2$$ odd. How many SPT phases are there, in $$1+1$$ and $$3+1$$ dimensions?

$$\newcommand{\Z}{\mathbb{Z}}$$
I'm not entirely sure that this is what you're looking for, but it seems like the total symmetry is a $$G=\Z_2\rtimes \Z_2^T$$, with antiunitary $$\Z_2^T$$, so the number of SPT phases is inside  $$\mathrm{H}^{d+1}(G,U(1)) = \begin{cases} \Z_2^{(d+2)/2} & d = 0 \mod 2 \\ \Z^{(d+1)/2}_{2} & d = 1 \mod 2 \end{cases}$$ So it would be two SPT phases in (1+1)D and four SPT phases in (3+1)D.
• thanks! I think you mean $G= Z_2\rtimes Z_2^T$, and the cohomology is in Eq.(J122) of the paper cited above? If so, there are certain typo above, because d/2 is fractional for d=1 mod 4 etc. Jul 9, 2020 at 14:23