Exclusion statistics and Rogers-Ramanujan identities

One particularly important property of integrable systems is seen in the spectrum of excitations above the ground state. In all known cases these spectra is of the quasi particle form in which the energies of multiparticle states is additively composed of single particle energies $e_{\alpha}(P)$

$$E_{ex}-E_0=\sum_{\alpha=1}^n\sum_{j=1}^{m_\alpha}e_{\alpha}(P_j^{\alpha})$$ (10)

with the total momentum

$$P=\sum_{\alpha}^n\sum_{j=1}^{m_\alpha}P^{\alpha}_j~~({\rm mod}~2\pi).$$ (11)

Here n is the number of types of quasi-particles and there are $m_{\alpha}$ quasi particles of type $\alpha.$The momenta in the allowed states are quantized in units of $2\pi/M$ and are chosen from the sets  

$$P^{\alpha}_j\in \{P_{\rm min}^{\alpha}({\bf m}), P_{\rm min}... ...({\bf m})+{4\pi\over M},\cdots, P_{\rm max}^{\alpha}({\bf m})\}$$ (12)

with the Fermi exclusion rule  

$$P_{j}^{\alpha}\neq P_{k}^{\alpha}~~{\rm for} j\neq k~~{\rm and~ all}~~ \alpha$$ (13)

and

$$P^{\alpha}_{\rm min}({\bf m})={\pi\over M}[({\bf m}({\bf B}-... ...ha}_{\rm min} +{2\pi\over M}({{\bf u}\over 2}-{\bf A})_{\alpha}$$ (14)

where if some $u_{\alpha}=\infty$ the corresponding $P_{\rm max}^{\alpha}=\infty.$

If some $e_{\alpha}(P)$ vanishes at some momentum (say 0) the system is massless and for $P\sim 0$ a typical behavior is $e_{\alpha}=v\vert P\vert$where v is variously called the speed of light or sound or the spin wave velocity.

The important feature of the momentum selection rules (12) is that in addition to the fermionic exclusion rule (13) is the exclusion of a certain number of momenta at the edge of the momentum zones which is proportional to the number of quasiparticles in the state. For the Ising model at zero field there is only one quasi particle and $P_{\rm min}=0$ so the quasiparticle is exactly the same as a free fermion. However, for all other cases the $P_{\rm min}$ is not zero and exclusion does indeed take place. This is a very explicit characterization of the generalization which general integrable models make over the Ising model.

The exclusion rules (12) lead to what have been called fractional (or exclusion) statistics by Haldane [36]. On the other hand they make a remarkable and beautiful connection with the mathematical theory of Rogers-Ramanujan identities and conformal field theory. We have found that these exclusion rules allow a reinterpretation of all conformal field theories (which are usually discussed in terms of a bosonic Fock space using a Feigen and Fuchs construction [37]) in terms of a set of fermionic quasi-particles [38]. What is most surprising is that there is not just one fermionic representation for each conformal field theory but there are at least as many distinct fermionic representations as thee are integrable perturbations. The search for the complete set of fermionic representations is ongoing and I will only mention here that we have extensive results for the integrable perturbations of the minimal models M(p,p') for the $\phi_{1,3}$ perturbation [39] and the $\phi_{2,1}$ and $\phi_{1,5}$ perturbations [40].