At half-filling the repulsive Hubbard model is in a Mott insulating phase. The charge degrees of freedom are gapped, whereas the spin degrees of freedom remain gapless. At low energies the spin sector is actually scale invariant (apart from logarithmic corrections) and Conformal Field Theory (CFT) methods may be applied to determine the low-energy behaviour of correlation functions involving only the spin sector. On the other hand, the charge sector is not scale invariant and CFT does not provide any information for correlators involving the charge degrees of freedom. In this chapter we will show that there exists a particular continuum limit of the half filled Hubbard model, in which it is possible to calculate dynamical correlation functions by means of methods of integrable quantum field theory. We first construct a Lorentz invariant scaling limit starting from the results for the excitation spectrum and the S-matrix discussed in Chapter 7. This scaling limit is identified as the SU(2) Thirring model, which is an integrable relativistic quantum field theory. Next we discuss a continuum limit, which is obtained directly from the Hubbard Hamiltonian and describes the vicinity of the scaling limit.

Construction of the scaling limit

The simplest way of constructing the scaling limit is to start with the results for the dispersions of the elementary excitations and the S-matrix derived in Chapter 7 and then look for a particular limit in which Lorentz invariance is recovered.