Magnetic Heisenberg-chain/pp-wave correspondence

TL;DR

This paper establishes a correspondence between a decoupling limit of planar N=4 SYM theory with a magnetic field and the ferromagnetic Heisenberg spin chain, analyzing thermodynamics and string duals, including Hagedorn temperatures.

Contribution

It generalizes the decoupling limit of N=4 SYM to include magnetic fields and connects it with the Heisenberg chain and string theory pp-wave backgrounds.

Findings

Decoupling limit relates N=4 SYM with magnetic field to Heisenberg spin chain.

Hagedorn temperature of gauge theory matches that of the spin chain and string spectrum.

Magnetic field effects are geometrically realized in the Penrose limit.

Abstract

We find a decoupling limit of planar N=4 super Yang-Mills (SYM) on R x S^3 in which it becomes equivalent to the ferromagnetic XXX_{1/2} Heisenberg spin chain in an external magnetic field. The decoupling limit generalizes the one found in hep-th/0605234 corresponding to the case with zero magnetic field. The presence of the magnetic field is seen to break the degeneracy of the vacuum sector and it has a non-trivial effect on the low energy spectrum. We find a general connection between the Hagedorn temperature of planar N=4 SYM on R x S^3 in the decoupling limit and the thermodynamics of the Heisenberg chain. This is used to study the Hagedorn temperature for small and large value of the effective coupling. We consider the dual decoupling limit of type IIB strings on AdS_5 x S^5. We find a Penrose limit compatible with the decoupling limit that gives a magnetic pp-wave background. The breaking of the symmetry by the magnetic field on the gauge theory side is seen to have a geometric counterpart in the derivation of the Penrose limit. We take the decoupling limit of the pp-wave spectrum and succesfully match the resulting spectrum to the low energy spectrum on the gauge theory side. This enables us to match the Hagedorn temperature of the pp-wave to the Hagedorn temperature of the gauge theory for large effective coupling. This generalizes the results of hep-th/0608115 to the case of non-zero magnetic field.