Quantum Deformations of the One-Dimensional Hubbard Model

TL;DR

This paper develops a quantum deformation U_q(psu(2|2)⋉ℝ^3) of the central-extended symmetry underlying the one-dimensional Hubbard model and derives its fundamental R-matrix. Using this R-matrix, the authors construct an integrable spin chain with three independent parameters and obtain the corresponding nested Bethe equations, thereby providing a quantum-deformed analogue of the Hubbard spectrum. They establish a precise connection to the Alcaraz–Bariev models and show how the standard Hubbard Hamiltonian arises in appropriate limits, illuminating how quantum deformations modify scattering, spectrum, and integrability. The work lays a foundation for q-deformed AdS/CFT integrability structures and potential condensed-matter applications, while suggesting further exploration of quantum-affine extensions and alternative coalgebras. Overall, the paper demonstrates that U_q(h) can host a rich, quasi-triangular Hopf-algebraic framework yielding a tractable, three-parameter quantum Hubbard family with deep links to known integrable models.

Abstract

The centrally extended superalgebra psu(2|2)xR^3 was shown to play an important role for the integrable structures of the one-dimensional Hubbard model and of the planar AdS/CFT correspondence. Here we consider its quantum deformation U_q(psu(2|2)xR^3) and derive the fundamental R-matrix. From the latter we deduce an integrable spin chain Hamiltonian with three independent parameters and the corresponding Bethe equations to describe the spectrum on periodic chains. We relate our Hamiltonian to a two-parametric Hamiltonian proposed by Alcaraz and Bariev which can be considered a quantum deformation of the one-dimensional Hubbard model.

Figures (4)

• Figure 1: Distinguished Dynkin diagram of $\mathfrak{su}(2|2)$.
• Figure 2: Multiplet Splitting. The long multiplet $\{m,n\}$ consists of two short multiplets $\langle m+1,n\rangle$ and $\langle m,n+1\rangle$ and a representation of $\mathrm{U}_q(\mathfrak{h})$ maps between the short multiplets (arrows). Generically, the short multiplets are connected in all possible ways (left). When the shortening condition holds one (middle) or both (right) arrows between the short multiplets are broken. In the second figure from the left, $\langle m+1,n\rangle$ is a subrepresentation while $\langle m,n+1\rangle$ is a factor representation, and the long multiplet is indecomposable. The long multiplet in the right figure is fully decomposable.
• Figure 3: Tensor product structure of two fundamental representations along a chain. $\langle 1,0\rangle$ contains the state $\mathopen{|}\phi^1\phi^1\mathclose{\rangle}$ while $\langle 0,1\rangle$ contains the state $\mathopen{|}\psi^1\psi^1\mathclose{\rangle}$. The arrows indicate the action of the long representation. The long arrows break when $x^{+}_{2}=x^{-}_{1}$ or $x^{-}_{2}=x^{+}_{1}$, respectively.
• Figure 4: Scattering process and transformation of central charges.