Matrix-product state skeletons in Onsager-integrable quantum chains

TL;DR

This paper explores matrix-product state skeletons in certain interacting quantum chains, revealing dense structures in gapped regions and identifying new excited states, thus extending understanding beyond free-fermion models.

Contribution

It constructs MPS skeletons in Onsager-integrable spin chains, including excited states, and provides a closed form for the disorder parameter, broadening the scope beyond free-fermion models.

Findings

Dense MPS skeletons in gapped regions surrounding fixed-point Hamiltonians.

Identification of MPS excited states beyond ground states.

Closed form for the disorder parameter in interacting models.

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.