Strong coupling from the Hubbard model

TL;DR

This paper explores the spectrum of the Hubbard model in relation to N=4 super Yang-Mills theory, deriving explicit formulas for operator dimensions at strong coupling and comparing with string theory results.

Contribution

It provides a detailed analysis of the operator spectrum at large 't Hooft coupling using the Hubbard model, including explicit solutions and comparisons with string theory.

Findings

Derived polynomial equations for operator dimensions at strong coupling

Expressed 't Hooft parameter as a function of operator dimension

Found agreement with string theory results for certain R-charges

Abstract

It was recently observed that the one dimensional half-filled Hubbard model reproduces the known part of the perturbative spectrum of planar N=4 super Yang-Mills in the SU(2) sector. Assuming that this identification is valid beyond perturbation theory, we investigate the behavior of this spectrum as the 't Hooft parameter \lambda becomes large. We show that the full dimension \Delta of the Konishi superpartner is the solution of a sixth order polynomial while \Delta for a bare dimension 5 operator is the solution of a cubic. In both cases the equations can be solved easily as a series expansion for both small and large \lambda and the equations can be inverted to express \lambda as an explicit function of \Delta. We then consider more general operators and show how \Delta depends on \lambda in the strong coupling limit. We are also able to distinguish those states in the Hubbard model which correspond to the gauge invariant operators for all values of \lambda. Finally, we compare our results with known results for strings on AdS_5\times S^5, where we find agreement for a range of R-charges.