Toy models for wrapping effects

TL;DR

This paper explores simplified spin chain models, including the Hubbard model and algebraic Bethe ansatz based Hamiltonians, to understand wrapping effects and finite size corrections in integrable systems related to N=4 SYM.

Contribution

It introduces toy models that shed light on wrapping effects, demonstrating limitations of effective spin chain descriptions and relating wrapping corrections to Luscher-like approaches.

Findings

Hubbard model shows effective spin chains are insufficient for finite size effects

Wrapping corrections can be computed and related to Luscher-like formulas

Long range Hamiltonians reveal connections between wrapping interactions and transcendentality

Abstract

The anomalous dimensions of local single trace gauge invariant operators in N=4 supersymmetric Yang-Mills theory can be computed by diagonalizing a long range integrable Hamiltonian by means of perturbative asymptotic Bethe ansatz. This formalism breaks down when the number of fields of the composite operator is smaller than the range of the Hamiltonian which coincides with the order in perturbation theory at study. We analyze two spin chain toy models which might shed some light on the physics behind these wrapping effects. One of them, the Hubbard model, is known to be closely related to N=4 SYM. In this example, we find that the knowledge of the effective spin chain description is insufficient to reconstruct the finite size effects of the underlying electron theory. We compute the wrapping corrections for generic states and relate them to a Luscher like approach. The second toy models are long range integrable Hamiltonians built from the standard algebraic Bethe anstatz formalism. This construction is valid for any symmetry group. In particular, for non-compact groups it exhibits an interesting relation between wrapping interactions and transcendentality.