Hagedorn Temperature in Large $N$ Majorana Quantum Mechanics

We discuss two types of quantum mechanical models that couple large numbers of Majorana fermions and have orthogonal symmetry groups. In models of vector type, only one of the symmetry groups has a large rank. The large $N$ limit is taken keeping $gN=\lambda$ fixed, where $g$ multiplies the quartic Hamiltonian. We introduce a simple model with $O(N)\times SO(4)$ symmetry, whose energies are expressed in terms of the quadratic Casimirs of the symmetry groups. This model may be deformed so that the symmetry is $O(N)\times O(2)^2$, and the Hamiltonian reduces to that studied in arXiv:1802.10263. We find analytic expressions for the large $N$ density of states and free energy. In both vector models, the large $N$ density of states varies approximately as $e^{-|E|/\lambda}$ for a wide range of energies. This gives rise to critical behavior as the temperature approaches the Hagedorn temperature $T_{\rm H} = \lambda$. In the formal large $N$ limit, the specific heat blows up as $(T_H- T)^{-2}$, which implies that $T_H$ is the limiting temperature. However, at any finite $N$, it is possible to reach arbitrarily large temperatures. Thus, the finite $N$ effects smooth out the Hagedorn transition. We also study models of matrix type, which have two $O(N)$ symmetry groups with large rank. An example is provided by the Majorana matrix model with $O(N)^2\times O(2)$ symmetry, which was studied in arXiv:1802.10263. In contrast with the vector models, the density of states is smooth and nearly Gaussian near the middle of the spectrum.