The Ising Model Correlation Functions

Abstract

We have now all the necessary tools to completely discuss a specific example: the celebrated critical Ising model. We recall that the content in primary operators, read from the Kac table (Table 5), reduces to the identity operator 1 = ø_(1,1) (conformal weights $\Delta _{1,2} = \bar \Delta _{1,2} = 0$ which contains the stress-energy tensor, the spin-density $\sigma = \varphi _{\left( {1,2} \right)} = \varphi _{\left( {2,2} \right)} \left( {\bar \Delta _{1,2} = \bar \Delta _{1,2} = {1 \mathord{\left/ {\vphantom {1 {16}}} \right. \kern-\nulldelimiterspace} {16}}} \right)$, and the energy-density $\varepsilon = \varphi _{\left( {2,1} \right)} = \varphi _{\left( {1,3} \right)} \left( {\Delta _{2,1} = \bar \Delta _{2,1} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \right)$. The central charge is c = 1/2 and the model is labeled by m = 3 in the Kac notation (eqs. (4.48) and (4.49)). We will calculate the most important four-point functions and determine exactly the monodromy group and the operator product coefficients. The fermionic operators will appear when we reduce the constraint of locality to a subgroup of the monodromy group. We stress however, that the particular virtue of these techniques is that the exact solution of a lattice system is not required. This allows to discuss more complicated systems along exactly the same lines.