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arXiv:2405.15848 · SciPost Physics

Integrable models on Rydberg atom chains

Luke Corcoran, Marius de Leeuw, Balázs Pozsgay

Published Apr 25, 2025• 5 citations
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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 z . We provide evidence of two critical points related to the golden ratio \phi ϕ , at z=\phi^{-1/2} z = ϕ − 1 / 2 and z=\phi^{3/2} z = ϕ 3 / 2 . We also perform a partial classification of integrable Hamiltonians for range 5.

DOI
10.21468/scipostphys.18.4.139

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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 z . We provide evidence of two critical points related to the golden ratio \phi ϕ , at z=\phi^{-1/2} z = ϕ − 1 / 2 and z=\phi^{3/2} z = ϕ 3 / 2 . We also perform a partial classification of integrable Hamiltonians for range 5.
We study integrable spin chains and quantum and classical cellular automata with interaction range ℓ≥3. This is a family of integrable models for which there was no general theory so far. We develop an algebraic framework for such models, generalizing known methods from nearest-neighbor interacting chains. This leads to a new integrability condition for medium-range Hamiltonians, which can be used to classify such models. A partial classification is performed in specific cases, including U(1)-symmetric three-site interacting models, and Hamiltonians that are relevant for interaction-round-a-face models. We find a number of models which appear to be new. As an application we consider quantum brickwork circuits of various types, including those that can accommodate the classical elementary cellular automata on light cone lattices. In this family we find that the so-called Rule150 and Rule105 models are Yang-Baxter integrable with three-site interactions. We present integrable quantum deformations of these models, and derive a set of local conserved charges for them. For the famous Rule54 model we find that it does not belong to the family of integrable three-site models, but we cannot exclude Yang-Baxter integrability with longer interaction ranges.
We review the notion of Yang-Baxter integrability for spin chains that have Hilbert spaces with constraints, such as a Rydberg blockade. We focus on anyonic chains, whose constraints arise from the fusion rules of the fusion categories on which they are based. We discuss the emergence of Temperley-Lieb algebras and present a new result on which types of anyonic chains exhibit them. We then give an overview of known results for integrable anyonic chains and extend them to several fusion categories up to rank 777. Using a modification of the boost operator formalism, we find several new integrable anyonic chains and discuss some of their properties. These include spin-32\frac3223​ models for su(2)k\mathfrak{su}(2)_ksu(2)k​ fusion categories, anyonic chains based on the Tambara-Yamagami fusion categories TY(Zn)(\mathbb{Z}_n)(Zn​), and product fusion categories Fib×\times×Fib and Fib×\times×Ising. We review recent results for spin chains based on the Haagerup-Izumi fusion category HI(Z3)(\mathbb{Z}_3)(Z3​), and present preliminary numerics for a HI(Z5)(\mathbb{Z}_5)(Z5​) model.
In this paper, we review the notion of Yang–Baxter integrability for spin chains that have Hilbert spaces with constraints, such as a Rydberg blockade. We focus on anyonic chains, whose constraints arise from the fusion rules of the fusion categories on which they are based. We discuss the emergence of Temperley–Lieb algebras and present a new result on which types of anyonic chains exhibit them. We then give an overview of known results for integrable anyonic chains and extend them to several fusion categories up to rank [Formula: see text]. By using a modification of the boost operator formalism, we find several new integrable anyonic chains and discuss some of their properties. These include spin-[Formula: see text] models for [Formula: see text] fusion categories, anyonic chains based on the Tambara–Yamagami fusion categories TY[Formula: see text] and product fusion categories Fib[Formula: see text]Fib and Fib[Formula: see text]Ising. We review recent results for spin chains based on the Haagerup–Izumi fusion category HI[Formula: see text] and present preliminary numerics for a HI[Formula: see text] model.
Abstract We classify all fundamental integrable spin chains with two-dimensional (2D) local Hilbert space which have regular R -matrices of difference form. This means that the R -matrix underlying the integrable structures is of the form and reduces to the permutation operator at some particular point. We find a total of 14 independent solutions, eight of which correspond to well-known eight or lower vertex models. The remaining six models appear to be new and some have peculiar properties such as not being diagonalizable or being nilpotent. Furthermore, for even R -matrices, we find a bijection between solutions of the Yang–Baxter equation (YBE) and the graded YBE which extends our results to the graded 2D case.
We construct the first integrable models based on the Haagerup fusion category H 3 . We introduce a Haagerup version of the anyonic spin chain and use the boost operator formalism to identify two integrable Hamiltonians of PXP type on this chain. The first of these is an analog of the golden chain; it has a topological symmetry based on H 3 and satisfies the Temperley-Lieb algebra with parameter δ = ( 3 + 13 ) / 2 . We prove its integrability using a Lax formalism, and construct the corresponding solution to the Yang-Baxter equation. We present numerical evidence that this model is gapless with a dynamical critical exponent z ≠ 1 . The second integrable model we find breaks the topological symmetry. We present numerical evidence that this model reduces to a CFT in the large volume limit with central charge c ∼ 3 / 2 .
We construct new families of spin chain Hamiltonians that are local, integrable, and translationally invariant. To do so, we make use of the Clifford group that arises in quantum information theory. We consider translation invariant Clifford group transformations that can be described by matrix product operators (MPOs). We classify translation invariant Clifford group transformations that consist of a shift operator and an MPO of bond dimension two—this includes transformations that preserve locality of all Hamiltonians and those that lead to non-local images of particular operators but, nevertheless, preserve locality of certain Hamiltonians. We characterize translation invariant Clifford group transformations that take single-site Pauli operators to local operators on at most five sites—examples of Quantum Cellular Automata—leading to a discrete family of Hamiltonians that are equivalent to the canonical XXZ model under such transformations. For spin chains solvable by the algebraic Bethe ansatz, we explain how conjugating by an MPO affects the underlying integrable structure. This allows us to relate our results to the usual classifications of integrable Hamiltonians. We also treat the case of spin chains solvable by free fermions.