Long-Range GL(n) Integrable Spin Chains and Plane-Wave Matrix Theory

TL;DR

Beisert and Klose analyze the most general perturbatively integrable long-range spin chain with spins in the fundamental representation of $\mathfrak{gl}(n)$. They derive the Hamiltonian and an asymptotic Bethe ansatz up to fourth order, and propose all-order Bethe equations whose moduli are tied to rapidity maps, dressing phases, and normalization and similarity transformations. Applying the framework to plane-wave matrix theory, they demonstrate that a closed $\mathfrak{su}(3)$ sector is not of the general form, implying PWMT is not integrable beyond the first perturbative order and that the full model is non-integrable. They further discuss gauge-theory and matrix-model constraints, provide explicit model coefficients, and highlight the universality of the perturbative integrable structure along with directions for generalization to other representations and sectors.

Abstract

Quantum spin chains arise naturally from perturbative large-N field theories and matrix models. The Hamiltonian of such a model is a long-range deformation of nearest-neighbor type interactions. Here, we study the most general long-range integrable spin chain with spins transforming in the fundamental representation of gl(n). We derive the Hamiltonian and the corresponding asymptotic Bethe ansatz at the leading four perturbative orders with several free parameters. Furthermore, we propose Bethe equations for all orders and identify the moduli of the integrable system. We finally apply our results to plane-wave matrix theory and show that the Hamiltonian in a closed sector is not of this form and therefore not integrable beyond the first perturbative order. This also implies that the complete model is not integrable.

Figures (3)

• Figure 1: Three planar Feynman diagrams of range $R=4$. The left interaction contains at least 3 four-vertices and appear at three loops, i.e. $\mathcal{O}(\lambda^2)$. The middle one contains at least 4 four-vertices and can appear only at four loops, i.e. $\mathcal{O}(\lambda^3)$. The right one may (in principle) represent a fermionic loop and could also arise at four loops.
• Figure 2: Structure of closed sectors of PWMT. When moving along a vertical/horizontal line, the index range of $\alpha$/$a$ is reduced by one, i.e. one index value is deactivated. When moving along a diagonal line, one index $a$ is fixed to a particular value.
• Figure 3: Graphical representation of the permutation symbol $\{2,1,0,2\}$.