Hagedorn Temperature in Large N Majorana Quantum Mechanics

Abstract

We discuss two types of quantum mechanical models that couple large numbers of Ma- jorana 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 = λ fixed, where g multiplies the quartic Hamiltonian. We introduce a simple model with O(N )×SO(4) symmetry, whose energies are expressed in terms of the quadratic Casimirs of the symme- try groups. This model may be deformed so that the symmetry is O(N ) × O(2)2, and the Hamiltonian reduces to that studied in [1]. 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|/λ for a wide range of energies. This gives rise to critical behavior as the temperature approaches the Hagedorn temperature TH = λ. In the formal large N limit, the specific heat blows up as (TH − T )−2, which implies that TH 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 × O(2) symmetry, which was studied in [1]. In contrast with the vector models, the density of states is smooth and nearly Gaussian near the middle of the spectrum.