Noninvertible Symmetry-Protected Topological Order in a Group-Based Cluster State

Despite growing interest in beyond-group symmetries in quantum condensed matter systems, there are relatively few microscopic lattice models explicitly realizing these symmetries, and many phenomena have yet to be studied at the microscopic level. We introduce a one-dimensional stabilizer Hamiltonian composed of group-based Pauli operators whose ground state is a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>G</a:mi><a:mo>×</a:mo><a:mi>Rep</a:mi><a:mo stretchy="false">(</a:mo><a:mi>G</a:mi><a:mo stretchy="false">)</a:mo></a:math>-symmetric state: the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>G</e:mi></e:math>-cluster state introduced by Brell []. We show that this state lies in a symmetry-protected topological (SPT) phase protected by <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>G</g:mi><g:mo>×</g:mo><g:mi>Rep</g:mi><g:mo stretchy="false">(</g:mo><g:mi>G</g:mi><g:mo stretchy="false">)</g:mo></g:math> symmetry, distinct from the symmetric product state by a duality argument. We identify several signatures of SPT order, namely, protected edge modes, string order parameters, and topological response. We discuss how <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>G</k:mi></k:math>-cluster states may be used as a universal resource for measurement-based quantum computation, explicitly working out the case where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>G</m:mi></m:math> is a semidirect product of Abelian groups.