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 [I. R. Klebanov et al., Phys. Rev. D 97, 106023 (2018)]. 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 $T_{\mathrm{H}}=\lambda$. 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 Klebanov et al. In contrast with the vector models, the density of states is smooth and nearly Gaussian near the middle of the spectrum.

Figure 1

The logarithm of the density of states of the $O(N)\times O(2{)}^2$ vector model, shown for $N=100$. For comparison, the large $N$ result (2.13) is shown with a dashed line.

Figure 2

The cactus diagrams, which are of order $N$, vanish due to the Majorana nature of the variables. The “necklace” diagrams, are not equal to zero and give the leading contributions in the large $N$ limit, which are of order $N^0$.

Figure 3

The plot of specific heat $C$ for the $O(N)\times O(2{)}^2$ model, as a function of temperature $T/\lambda$, for $N=50$, 100, 150. The specific heat has a pronounced peak which gets closer to $T/\lambda =1$ as $N$ grows.

Figure 4

The logarithm of the density of states for the $O(200)\times SO(4)$ (on the left) and $O(300)\times SO(4)$ (on the right) models with Hamiltonian (1.2). For comparison, the large $N$ result (3.3) is shown with a dashed line.

Figure 5

The plot of specific heat $C$ for the $O(N)\times SO(4)$ model, as a function of temperature $T/\lambda$, for $N=50$, 100, 150. The peak in specific heat gets closer to $T/\lambda =1$ as $N$ increases.

Figure 6

The spectrum for $N_1=N_2=8$ and $N_1=N_2=10$ on the top and the bottom row. One can see that the spectrum is Gaussian, but split into two branches. The fit is quite close to the theoretical predictions.

Figure 7

The spectrum for $N_1=N_2=9$. As one can see it has the same features as for $N_1=N_2=8$ and $N_1=N_2=10$, but there is no separation between the even and the odd energy sectors. It could indicate that this difference has a purely group theoretic explanation.

Figure 8

The specific heat as a function of temperature for the $O(N{)}^2\times O(2)$ matrix model with $N=10$. The low-temperature peak is due to the discreteness of the spectrum. At higher $T$, the specific heat falls off polynomially with the power $\alpha =\frac{d\log C}{d\log T}=-1.98$, close to that predicted by the analytic result (4.15).