Pivoting through the chiral-clock family

TL;DR

The paper develops a pivot-based construction grounded in the Onsager algebra to generate a family of integrable $N$-state chiral clock models, revealing a dihedral symmetry $D_{2N}$ that protects SPT order for even $N$ and representation-SPT (RSPT) physics for odd $N$. It shows that the fixed-point Hamiltonian $A_2$ realizes a cluster-type SPT for even $N$ with a precisely characterized MPS ground state and a symmetry‑fractionalised, doubly degenerate entanglement spectrum, while odd $N$ yields an RSPT with higher-dimensional bond representations. The work maps out rich phase diagrams across $H(oldsymbol heta)= olinebreak olinebreak abla_0 A_0+ olinebreak abla_1 A_1+ olinebreak abla_2 A_2$, identifying special lines (Onsager-Potts, $A_1+ olinebreak \\lambda A_2$, and a $U(1)$ line) and combining analytic pivoting with DMRG numerics to reveal trivial, symmetry-broken, (R)SPT, and gapless phases. Overall, the results elucidate how Onsager-algebra pivots generate topological order in clock models, extendable to broader Onsager families, and they establish concrete diagnostics via string order, symmetry fractionalisation, and entanglement spectra that distinguish the phases.

Abstract

The Onsager algebra, invented to solve the two-dimensional Ising model, can be used to construct conserved charges for a family of integrable $N$-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 $\mathbb{Z}_N$ clock symmetry gives a non-abelian dihedral symmetry group $D_{2N}$. We show that this symmetry gives rise to symmetry-protected topological (SPT) order in this family for all even $N$, and representation-SPT (RSPT) physics for all odd $N$. 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 $N$, the phase transitions are "unnecessarily" critical from the SPT point of view.

Figures (9)

• Figure 1: Web of maps generated by pivoting and Kramers-Wannier duality in the Onsager-integrable chiral-clock family.
• Figure 2: Graphical representations of $U_1(\pi)$ as products $U_{j,j+1}$ from \ref{['U1def']}. The left-hand picture is a depth-two local unitary circuit, the middle a staircase circuit. The latter can be interpreted as an MPU with bond-dimension $N$ (right).
• Figure 3: Schematic phase diagrams for $N=2,3,4$ for the Hamiltonian $H(\alpha,\beta,\gamma)$ parameterised as $\alpha+\beta+\gamma=1$. FM indicates the ferromagnetic phase. The $N$ = 2 phase diagram is known exactly and features direct transitions. An exact MPS ground state occurs on the dotted line. For $N=3,4$ the transitions typically spread out into gapless regions. Within the achieved numerical resolution, we cannot ascertain whether we have a narrow gapless region or a direct transition between the trivial and SPT phases for a certain range along the $\alpha=\gamma$ line for $N$ = 4 (indicated by ${\blacksquare}$). The grey dotted lines are discussed in Section \ref{['sec:speciallines']}.
• Figure 4: The effective central charge $c$ for $H(\alpha,\beta,\gamma)$ with $\alpha=1-\gamma$ for (a) $N$ = 3 and $L$ = 100, (b) $N$ = 4 and $L$ = 40. The value is extracted by fitting the entanglement entropy to the CFT formula \ref{['eq:CardyCalabrese']}. A zero value indicates an area-law ground state. Values of $c$ at the boundary of the gapless region are not meaningful. A unitary transformation relates Hamiltonians with $\gamma$ and $1-\gamma$, and so the data for $N=4$ for $\gamma>0.5$ is that for $\gamma \leq 0.5$.
• Figure 5: DMRG calculations for the $N=3$ (a,b) and $N=4$ (c,d) versions of the Hamiltonian \ref{['Habc']} with $\alpha = 1-\gamma$ and with system sizes $L=200$ and $L=100$ respectively. $\mathcal{O}_Z$ is the local order parameter shown in \ref{['eq:localorder']}, $\mathcal{S}_0$ and $\mathcal{S}_1$ are the trivial and non-trivial string order parameters defined in \ref{['eq:stringtrivial', 'eq:stringspt']} respectively, while $c$ comes from fitting the entanglement entropy to \ref{['eq:CardyCalabrese']}. The Schmidt value $\Lambda_\alpha$ comes from the mid-chain bipartite Schmidt decomposition of the ground state, and the difference between the largest two values $\Delta_\Lambda = \Lambda^2_{1} - \Lambda^2_2$. Dotted lines connecting data points are provided as a guide to the eye.