The state space of a generic string bit model is spanned by N×N matrix creation operators acting on a vacuum state. Such creation operators transform in the adjoint representation of the color group U(N) [or SU(N) if the matrices are traceless]. We consider a system of b species of bosonic bits and f species of fermionic bits. The string, emerging in the N→∞ limit, identifies P+=mM2 where M is the bit number operator and P-=H2 where H is the system Hamiltonian. We study the thermal properties of this string bit system in the case H=0, which can be considered the tensionless string limit: the only dynamics is restricting physical states to color singlets. Then the thermal partition function Tre-βmM can be identified, putting x=e-βm, with a generating function χ0bf(x), for which the coefficient of xn in its expansion about x=0 is the number of color singlets with bit number M=n. This function is a purely group theoretic object, which is well studied in the literature. We show that at N=∞ this system displays a Hagedorn divergence at x=1/(b+f) with ultimate temperature TH=m/ln(b+f). The corresponding function for finite N is perfectly finite for 0<x<1, so the N=∞ system exhibits a phase transition at temperature TH which is absent for any finite N. We demonstrate that the low temperature phase is unstable above TH. The lowest-order 1/N asymptotic correction, for x→1 in the high temperature phase, is computed for large N. Remarkably, this is related to the number of labeled Eulerian digraphs with N nodes. Systematic methods to extend our results to higher orders in 1/N are described.
ASJC Scopus subject areas
- Physics and Astronomy (miscellaneous)