Integrable models on Rydberg atom chains

TL;DR

This work develops and applies an integrability framework for spin chains with Rydberg constraints, extending medium-range methods to constrained Hilbert spaces. By inserting projection operators into the Lax/RLL formalism and employing a constrained boost-like construction, the authors classify all time- and space-reflection symmetric integrable Hamiltonians of ranges 3 and 4, identifying RSOS (range 3) and constrained XXZ plus a novel range-4 family (the double golden chain). They further connect these 1D constrained models to 2D IRF/RSOS statistical systems and provide detailed Lax, R-matrix, and G-operator realizations, including a perturbative Lax description for the double golden chain and a Temperley-Lieb structure at a golden-ratio critical point. The results sharpen the landscape around the PXP model, showing no exact nearby integrable point at range 4 and offering a structured path toward higher-range classifications and potential Bethe-ansatz formulations. Overall, the paper demonstrates that constrained integrable models on Rydberg chains host rich algebraic structures, golden-chain criticality, and deep connections to 2D statistical physics and conformal field theories.

Abstract

We initiate a systematic study of integrable models for spin chains with constrained Hilbert spaces; we focus on spin-1/2 chains with the Rydberg constraint. We extend earlier results for medium-range spin chains to the constrained Hilbert space, and formulate an integrability condition. This enables us to construct new integrable models with fixed interaction ranges. We classify all time- and space-reflection symmetric integrable Rydberg-constrained Hamiltonians of range 3 and 4. At range 3, we find a single family of integrable Hamiltonians: the so-called RSOS quantum chains, which are related to the well-known RSOS models of Andrews, Baxter, and Forrester. At range 4 we find two families of models, the first of which is the constrained XXZ model. We also find a new family of models depending on a single coupling $z$. We provide evidence of two critical points related to the golden ratio $φ$, at $z=φ^{-1/2}$ and $z=φ^{3/2}$. We also perform a partial classification of integrable Hamiltonians for range 5.

Figures (11)

• Figure 1: Depiction of a plaquette. The variables $a, b, c, d$ can take values $0$ and $1$. We have the Rydberg constraints for the pairs $ab$, $ac$, $bd$ and $cd$. The weight of the plaquette is denoted as $w_{abcd}$.
• Figure 2: Yang-Baxter relation for the plaquette weights. It is understood that there is a summation for the middle vertex, while keeping the boundary vertices fixed. The weights $G$ depend on $u$ and $v$, but this is not denoted explicitly.
• Figure 3: Monodromy matrix for total length $L+1$, acting from the lower set of variables to the upper set. The value of a certain component is the product of the plaquette weights $w(u)$. The periodic transfer matrix is obtained by allowing configurations only with $a_0=a_L$ and $b_0=b_L$.
• Figure 4: The definition of the inverse of $G$. The dashed line signals that those values are identical.
• Figure 5: Yang-Baxter relation with quantum spaces identified.