Conformal Invariance in the Ising Quantum Chain

Abstract

At last, we shall give the first explicit example of the application of conformal invariance methods in the context of a specific lattice model. We choose the 2D Ising model as an example. Of course, the model was solved by Onsager back in 1944 and we shall not learn anything about this model which has not yet been deduced long ago by different means [G22, G19]. However, since all quantities of interest can be worked out explicitly, we get in this example absolute control of the critical behaviour of the model. This is useful to see how the techniques of conformal invariance work and one is not disturbed by convergence problems to reach the N → ∞ limit. In particular, recalling the treatment of the same model in the language of continuum field theory in Chap. 6, this example further illustrates the general machinery used for applying conformal invariance and useful insight will be obtained by comparing these two different approaches to the same problem. For the convenience of the reader, the main steps to be followed applying conformal invariance to any given lattice model will be summarized at the end of this chapter.