By Abhay Ashtekar
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Extra info for 2D Quantum Gravity and SC at high Tc
38) where the continuous angular variables i , run from 0 to 2 . 40) 0 i=1 2 We again assume periodic boundary conditions with M +1 1 . Notice that in zero eld, H remains invariant if all the spins are rotated by the same angle , i ! i + , and so our results will not depend upon the particular orientation chosen for h. The partition function can be evaluated by transfer matrix methods 156, 255] quite similar to those used for the Ising chain. Although we will not use such a method to obtain our results, we nevertheless describe the main steps for completeness.
In particular, we will introduce the central ideas of the scaling limit and universality in this very simple context. We will then go on to map the classical models to equivalent zero-dimensional quantum models and demonstrate that this mapping becomes exact in the scaling limit. The following sections will actually carry out the quantum-to-classical mapping in reverse. With the bene t of hindsight, we will begin by examining certain D = 1 classical statistical mechanics model and show that they are intimately related to single-site quantum Ising and rotor models.
We have chosen the de nition of by anticipating a later computation in which will be seen to be the correlation length. 45) for N = 2. 68) in this form. 45) we had a simple angular parameterization in which H became purely quadratic in the angular variable: one could parameterize the 3-component n using spherical co-ordinates, but the resulting H is not simply quadratic. Further progress towards the evaluation of Zc can however be made after the quantum-classical mapping. 68) can be interpreted as the imaginarytime Feynman path integral for a particle moving in a 3-dimensional space with co-ordinate n.
2D Quantum Gravity and SC at high Tc by Abhay Ashtekar