Pivoting through the chiral-clock family

The Onsager algebra, invented to solve the two-dimensional Ising model, can be used to construct conserved charges for a family of integrable N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> -state chiral clock models. We show how it naturally gives rise to a “pivot” procedure for this family of chiral Hamiltonians. These Hamiltonians have an anti-unitary CPT symmetry that when combined with the usual \mathbb{Z}_N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mi>N</mml:mi> </mml:msub> </mml:math> clock symmetry gives a non-Abelian dihedral symmetry group D_{2N} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:math> . We show that this symmetry gives rise to symmetry-protected topological (SPT) order in this family for all even N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> , and representation-SPT (RSPT) physics for all odd N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> . The simplest such example is a next-nearest-neighbour chain generalising the spin-1/2 cluster model, an SPT phase of matter. We derive a matrix-product state representation of its fixed-point ground state along with the ensuing entanglement spectrum and symmetry fractionalisation. We analyse a rich phase diagram combining this model with the Onsager-integrable chiral Potts chain, and find trivial, symmetry-breaking and (R)SPT orders, as well as extended gapless regions. For odd N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> , the phase transitions are “unnecessarily” critical from the SPT point of view.