The Analytic Bethe Ansatz for a Chain with Centrally Extended su(2|2) Symmetry

TL;DR

Beisert develops an analytic Bethe Ansatz framework for spin chains with centrally extended $rak{su}(2|2)$ and its extension to $rak{psu}(2,2|4)$, motivated by planar AdS/CFT and Hubbard-model connections. He derives a simple Yang–Baxter proof from representation theory, identifies a hidden supersymmetry linking to Shastry’s Hubbard R-matrix, and constructs transfer-matrix eigenvalues to obtain Bethe equations. The work also sketches how transfer matrices for $rak{psu}(2,2|4)$ symmetry might be assembled from two $rak{su}(2|2)$ chains, paving a path toward an exact spectrum at finite coupling. The results illuminate the common integrable structure underlying AdS/CFT, the Hubbard model, and related spin chains, with a robust analytic Bethe Ansatz toolkit and fusion structure guiding future rigorous treatment.

Abstract

We investigate the integrable structure of spin chain models with centrally extended su(2|2) and psu(2,2|4) symmetry. These chains have their origin in the planar AdS/CFT correspondence, but they also contain the one-dimensional Hubbard model as a special case. We begin with an overview of the representation theory of centrally extended su(2|2). These results are applied in the construction and investigation of an interesting S-matrix with su(2|2) symmetry. In particular, they enable a remarkably simple proof of the Yang-Baxter relation. We also show the equivalence of the S-matrix to Shastry's R-matrix and thus uncover a hidden supersymmetry in the integrable structure of the Hubbard model. We then construct eigenvalues of the corresponding transfer matrix in order to formulate an analytic Bethe ansatz. Finally, the form of transfer matrix eigenvalues for models with psu(2,2|4) symmetry is sketched.

Figures (14)

• Figure 1: Young tableaux for long and short representations $\{m,n\}$ and $\langle m,n\rangle$. The label $\ast$ represents a stack of arbitrarily many boxes. A single box represents the fundamental representation $\langle 0,0\rangle$.
• Figure 2: Diagrammatic representation of multiplet splitting (\ref{['eq:Splitting ']}) using Young tableaux. The filled boxes on the right hand side are to be removed leaving short representations.
• Figure 3: Scattering process and transformation of central charges.
• Figure 4: Factorised scattering.
• Figure 5: Unitarity condition and Yang-Baxter equation.