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Recently, it was realized that quantum states of matter can be classified aslong-range entangled (LRE) states (i.e. with non-trivial topological order) andshort-range entangled (SRE) states (\ie with trivial topological order). We canuse group cohomology class ${\cal H}^d(SG,R/Z)$ to systematically describe theSRE states with a symmetry $SG$ [referred as symmetry-protected trivial (SPT)or symmetry-protected topological (SPT) states] in $d$-dimensional space-time.In this paper, we study the physical properties of those SPT states, such asthe fractionalization of the quantum numbers of the global symmetry on somedesigned point defects, and the appearance of fractionalized SPT states on somedesigned defect lines/membranes. Those physical properties are SPT invariantsof the SPT states which allow us to experimentally or numerically detect thoseSPT states, i.e. to measure the elements in ${\cal H}^d(G, R/Z)$ that labeldifferent SPT states. For example, 2+1D bosonic SPT states with $Z_n$ symmetryare classified by a $Z_n$ integer $m \in {\cal H}^3(Z_n, R/Z)=Z_n$. We findthat $n$ identical monodromy defects, in a $Z_n$ SPT state labeled by $m$,carry a total $Z_n$-charge $2m$ (which is not a multiple of $n$ in general).

Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions