Some results of our investigation of the scaling behaviour of the alternating spin chain will be presented. We argue that in the critical regime it is governed by a conformal field theory possessing a continuous spectrum of scaling dimensions. One of the first questions that needs to be addressed concerns the density of states of the continuous theory. Using the powerful method of the ODE/IQFT correspondence we determined the phase shifts, which appear in the quantization condition for the spectrum of the chain for large, but finite lattice size L. As usual, the phase shifts are simply related to the density of states of the continuous theory.
Firstly, the concept of the Renormalization Group flow will be discussed in order to understand how we assign the scale dependence to the states. Then mostly half of the talk will be devoted to the simpler XXZ spin chain. We will see how one can assign the scaling to the observable quantities such as the energy and the quasi-shift operator. I will also try to touch some aspects of the ODE/IQFT correspondence in order to explain how the spin chain behaves when the size of the system goes to infinity.

Then I will turn to the alternating spin chain. We will focus on the large L behaviour, using the techniques introduced in the simpler context of the XXZ model. And as the end result, the formulae for the phase shifts and density of states of the continuous theory will be presented and discussed.

The talk is based on joint work with V. Bazhanov, S. Lukyanov and G. Kotousov, arXiv:1903.05033.