The endpoint of partial deconfinement

TL;DR

This paper investigates the microcanonical ensemble of a simple matrix quantum mechanics model, revealing a transition from partial confinement with Hagedorn density of states to a non-Hagedorn phase at a specific energy scale.

Contribution

It introduces a detailed analysis of the confinement/deconfinement transition and partial confinement in a matrix model, highlighting the role of Young diagram shapes and maximal depth in the transition.

Findings

Identification of a Hagedorn density of states in the model.

The transition occurs at energy proportional to N^2/4.

The system stops exhibiting Hagedorn behavior when Young diagrams reach maximal depth.

Abstract

We study the matrix quantum mechanics of two free hermitian $N\times N$ matrices subject to a singlet constraint in the microcanonical ensemble. This is the simplest example of a theory that at large $N$ has a confinement/deconfinement transition. In the microcanonical ensemble, it also exhibits partial confinement with a Hagedorn density of states. We argue that the entropy of these configurations, calculated by a counting of states based on the fact that Young diagrams are dominated by Young diagrams that have the VKLS shape. When the shape gets to the maximal depth allowed for a Young diagram of $SU(N)$, namely $N$, we argue that the system stops exhibiting the Hagedorn behavior. The number of boxes (energy) at the transition is $N^2/4$, independent of the charge of the state.