Matrix-product state skeletons in Onsager-integrable quantum chains

TL;DR

This work generalizes the concept of MPS skeletons from free-fermion classifications to interacting Onsager-integrable chains, specifically the $N$-state chiral clock models. By factoring the Laurent polynomial $f(z)$ as $f(z)=\pm z^p g(z)^2$ and exploiting the Onsager algebra, the authors construct exact MPS ground states in gapped regions surrounding fixed-point generators $A_k$, and identify corresponding exact MPS eigenstates throughout the skeleton, with a dense set of Hamiltonians $H_A$ enabling finite-bond-dimension representations. They derive a closed-form for the disorder parameter along the skeleton, analyze the spectrum via Onsager-sector decompositions and Baxter-type energy densities, and provide explicit d=1 constructions and excited-state results, all while highlighting the broad applicability beyond the clock representation. The results offer an analytic framework to approximate ground states in a wide region of the phase diagram and connect algebraic structures to concrete MPS representations, with potential implications for parafermionic generalizations and numerics. Overall, the paper advances understanding of exact MPS structures in interacting integrable chains and furnishes practical tools for exploring their phase diagrams and observable properties.

Abstract

Matrix-product state (MPS) skeletons are connected networks of Hamiltonians with exact MPS ground states that underlie a phase diagram. Such skeletons have previously been found in classes of free-fermion models. For the translation-invariant BDI and AIII free-fermion classes, it has been shown that the underlying skeleton is dense, giving an analytic approach to MPS approximation of ground states anywhere in the class. In this paper, we partially expose the skeleton in certain interacting spin chains: the $N$-state Onsager-integrable chiral clock families. We construct MPS that form a dense MPS skeleton in the gapped regions surrounding a sequence of fixed-point Hamiltonians (the generators of the Onsager algebra). Outside these gapped regions, these MPS remain eigenstates, but no longer give the many-body ground state. Rather, they are ground states in particular sectors of the spectrum. Our methods also allow us to find further MPS eigenstates; these correspond to low-lying excited states within the aforementioned gapped regions. This set of MPS excited states goes beyond the previous analysis of ground states on the $N=2$ free-fermion MPS skeleton. As an application of our results, we find a closed form for the disorder parameter in a family of interacting models. Finally, we remark that many of our results use only the Onsager algebra and are not specific to the chiral clock model representation.

Figures (2)

• Figure 1: Schematic phase diagrams for the Hamiltonian $H = \alpha A_0+\beta A_1 + \gamma A_2$ with normalisation $\alpha+\beta+\gamma=1$. The winding number of $f(z)$ is denoted by $\omega$. The coloured regions are gapped and connected to fixed point $A_k$, therefore corresponding to the intersection of the set $\mathcal{S}$ with the phase diagram. The white regions are gapless and outside $\mathcal{S}$. For $N=2$, the phase diagram is well-understood Wolf06Smith22. Diagrams for $N=3,4$ were found numerically in Ref. Jones24. For $N=4$, it is unclear whether the trivial and SPT phases have a direct transition at the black square. The dashed line represents a leg of the skeleton parameterised by $H = (1-\lambda)^2A_0 + 2\lambda (1-\lambda)A_1 + \lambda^2A_2$. For $N=2$, the ground state is an MPS along the entire path. We show that the analogous state is an eigenstate for general $N$, and is the ground state in the gapped regions. The crosses represent points where the MPS construction given in this paper has a singularity; this coincides with a gap closing in the sector $\mathcal{S}$.
• Figure 2: (a) The circuit diagram for the $d=1$ ground state. This can be interpreted as an MPO acting on a product state, giving an MPS. The blue boxes represent each unitary gate $U_{j,j+1}$, and the $\ket{v_j}$ are the single-site states. (b) The MPS tensor from \ref{['eqn:MPS-tensor-d=1']}.