We study the behavior of a simple string bit model at finite temperature. We use thermal perturbation theory to analyze the high temperature regime. But at low temperatures we rely on the large $N$ limit of the dynamics, for which the exact energy sp
ectrum is known. Since the lowest energy states at infinite $N$ are free closed strings, the $N=infty$ partition function diverges above a finite temperature $beta_H^{-1}$, the Hagedorn temperature. We argue that in these models at finite $N$, which then have a finite number of degrees of freedom, there can be neither an ultimate temperature nor any kind of phase transition. We discuss how the discontinuous behavior seen at infinite $N$ can be removed at finite $N$. In this resolution the fundamental string bit degrees of freedom become more active at temperatures near and above the Hagedorn temperature.
We develop superstring bit models, in which the lightcone transverse coordinates in D spacetime dimensions are replaced with d=D-2 double-valued flavor indices $x^k-> f_k=1,2$; $k=2,...,d+1$. In such models the string bits have no space to move. Lett
ing each string bit be an adjoint of a color group U(N), we then analyze the physics of t Hoofts limit $N->infty$, in which closed chains of many string bits behave like free lightcone IIB superstrings with d compact coordinate bosonic worldsheet fields $x^k$, and s pairs of Grassmann fermionic fields $theta_{L,R}^a$, a=1,..., s. The coordinates $x^k$ emerge because, on the long chains, flavor fluctuations enjoy the dynamics of d anisotropic Heisenberg spin chains. It is well-known that the low energy excitations of a many-spin Heisenberg chain are identical to those of a string worldsheet coordinate compactified on a circle of radius $R_k$, which is related to the anisotropy parameter $-1<Delta_k<1$ of the corresponding Heisenberg system. Furthermore there is a limit of this parameter, $Delta_k->pm 1$, in which $R_k->infty$. As noted in earlier work [Phys.Rev.D{bf 89}(2014)105002], these multi-string-bit chains are strictly stable at $N=infty$ when d<s and only marginally stable when d=s. (Poincare supersymmetry requires d=s=8, which is on the boundary between stability and instability.)
In string bit models, the superstring emerges as a very long chain of bits, in which s fermionic degrees of freedom contribute positively to the ground state energy in a way to exactly cancel the destabilizing negative contributions of d=s bosonic de
grees of freedom. We propose that the physics of string formation be studied nonperturbatively in the class of string bit models in which s>d, so that a long chain is stable, in contrast to the marginally stable (s=d=8) superstring chain. We focus on the simplest of these models with s=1 and d=0, in which the string bits live in zero space dimensions. The string bit creation operators are N X N matrices. We choose a Hamiltonian such that the large N limit produces string moving in one space dimension, with excitations corresponding to one Grassmann lightcone worldsheet field (s=1) and no bosonic worldsheet field (d=0). We study this model at finite N to assess the role of the large N limit in the emergence of the spatial dimension. Our results suggest that string-like states with large bit number M may not exist for N<(M-1)/2. If this is correct, one can have finite chains of string bits, but not continuous string, at finite N. Only for extremely large N can such chains behave approximately like continuous string, in which case there will also be the (approximate) emergence of a new spatial dimension. In string bit models designed to produce critical superstring at N=infinity, we can then expect only approximate Lorentz invariance at finite N, with violations of order 1/N^2.
We propose boundary conditions on a two dimensional 6-vertex model, which is defined on the lightcone lattice for an open string worldsheet. We show that, in the continuum limit, the degrees of freedom of this 6-vertex model describe a target space c
oordinate compactified on a circle of radius R, which is related to the vertex weights. This conclusion had already been established for the case of a 6-vertex model on the worldsheet lattice for the propagator of a closed string. This exercise illustrates how the Bethe ansatz works in the presence of boundaries, at least of this particular type.
