Geometric entropy of nonrelativistic fermions and two-dimensional strings.

We consider the geometric entropy of free nonrelativistic fermions in two dimensions and show that it is ultraviolet finite for finite fermi energies, but divergent in the infrared. In terms of the corresponding collective field theory this is a nonperturbative effect and is related to the soft behaviour of the usual thermodynamic entropy at high temperatures. We then show that thermodynamic entropy of the singlet sector of the one dimensional matrix model at high temperatures is governed by nonperturbative effects of the underlying string theory. In the high temperature limit the “exact” expression for the entropy is regular but leads to a negative specific heat, thus implying an instability. We speculate that in a properly defined two dimensional string theory, the thermodynamic entropy could approach a constant at high temperatures and lead to a geometric entropy which is finite in the ultraviolet. Recently the entropy of entanglement between different regions of space in quantum field theories have been intensively studied [1]-[6]. The motivation for this is its direct connection to the question of information loss due to black holes and black hole entropy [7]-[12]. A significant feature of this entanglement entropy, or “geometric entropy” is that it is ultraviolet divergent in typical field theories. This has been interpreted to imply that at least at the semiclassical level information loss due to the formation of a horizon is inevitable in quantum field theories. The divergence of the entropy is a reflection of short distance singularities in quantum field theories. Alternatively [13]-[16] the divergence is related to the behaviour of the usual thermodynamic entropy at high temperatures since, as we shall see below, the geometric entropy effectively involves an integral of the thermodynamic entropy density over all temperatures. One may hope that in string theories this divergence disppears because of a soft ultraviolet behaviour [11]. However, to leading order in the string perturbation expansion, the thermodynamic free energy of a string is equal to the sum of the free energies of the physical modes of the string and one would obtain the same divergence in each term of the sum. Furthermore, unlike in a field theory of a finite number of fields, the thermodynamic free energy of free strings is itself divergent at the Hagedorn temperature. This is an infrared diveregence and might signal an instability of the theory and one might obtain a finite answer once one takes into account interactions and shift to a stable vacuum [17]. As argued in [13, 14] the geometric entropy in string theory is afflicted by this Hagedorn transition and what appears as an ultraviolet divergence in each term of the sum over all string modes may be interpreted as an infrared problem in the full answer. The need to include string interactions calls for a formulation of the problem in some well defined and tractable string field theory. While this appears to be an almost impossible task at present, there is one string theory where a tractable nonperturbative formulation exists, at least for some bulk quantities. This is the two dimensional string defined via the one dimensional matrix model [18]. The singlet sector of the matrix model 1 may be written as a two dimensional collective field theory of the density variable.The fluctuations of the collective field represent a massless particle which is the only 1 As we will see soon the nonsinglet contributions are irrelevant for a calculation of the geometric entropy in the ground state