Spin Chains from large-$N$ QCD at strong coupling

Abstract

We study the strong coupling expansion of large $N$ QCD in various dimensions, reformulating the Kogut-Susskind Hamiltonian on a square lattice in terms of (constrained) one dimensional spin chain models. We study the integrability properties of the spin chain obtained this way: there is large class of integrable subsectors, but we show that the full spin chain is not integrable, at least when viewed from a description based on Bethe ansatz. We demonstrate that the spin chains no longer possess integrability due to the constraints arising from the zigzag symmetry of the confining strings. The spin chain description properly estimates the roughening transition point by extrapolating the first-order analytical results based on integrability of some subsectors. The generalization to higher dimensions are also considered, where we also find the small subsectors without the zigzag constraints to be integrable.

Figures (17)

• Figure 1: Possible actions of a single plaquette operator on a string state. The cost to attach a plaquette to a string that is already there is $1/N$ ($1/N$ is proportional to the string coupling constant in the t'Hooft counting), but the plaquette is accompanied by $N/\lambda$ so the factors of $N$ cancel.
• Figure 2: Examples of the type A' deformations where a string shares a corner with the plaquette (figure (a)) and gives a nontrivial overlap with another single string state such as states in (b). These contributions are subleading in $1/N$ due to the change of topology.
• Figure 3: An example of string configurations with excitations on the same links but with different connectivity.
• Figure 4: One example of the type C deformations by a single plaquette action (red in the left figure) generating another degenerate state. Notice that the deformed string has exactly the same string length as the original state. This plaquette action exchanges the $u$ and $r$ letters next to each other.
• Figure 5: An example of evaluating the local orientation of the string. (a) Let us focus on the substring $rrrr\textcolor{red}{urddrrddl}ldd$ (depicted in red). It is bounded by $u$ and $l$. (b) Exchanging the positions of the bulk letters $r$ and $d$, the substring is now $\textcolor{red}{urrrddddl}$. The orientation is determined to be clockwise from this procedure.