Coulomb gas methods

Date: 2015-10-07; view: 549.

Many important results concerning the O (n) model can be derived in a non-rigorous fashion using so-called Coulomb gas methods. For the purposes of comparison with later results from SLE, we now summarise these methods and collect a few relevant formulae. A much more complete discussion may be found in the review by Nienhuis [3].

We assume that the boundary conditions on the O (n) spins are free, so that the partition function is a sum over closed loops only. First orient each loop at random. Rather than giving clockwise and anti-clockwise orientations the same weight n/2, give them complex weights e^±6iχ, where n = e^6iχ + e^−6iχ = 2 cos 6χ. These may be taken into account, on the honeycomb lattice, by assigning a weight e^±iχ at each vertex where an oriented loop turns R (respectively, L). This transforms the non-local factors of n into local (albeit complex) weights depending only on the local configuration at each vertex.

Next transform to the height variables described above. By convention, the heights are taken to be integer multiples of π. The local weights at each vertex now depend only on the differences of the three adjacent heights. The crucial assumption of the Coulomb gas approach is that, under the RG, this model flows to one in which the lattice can be replaced by a continuum, and the heights go over into a gaussian free field, with partition function Z = ∫ e^{−S [h]} [dh], where

(8)

As it stands, this is a simple free field theory. The height fluctuations grow logarithmically: (h (r₁) − h (r₂))² (2/g) ln|r₁ − r₂|, and the correlators of exponentials of the height decay with power laws

eiqh(r1)e-iqh(r2) |r1-r2|-2xq,

(9)

where x_q = q²/2g. All the subtleties come from the combined effects of the phase factors and the boundaries or the topology. This is particularly easy to see if we consider the model on a cylinder of circumference ℓ and length L ℓ. In the simple gaussian model (8) the correlation function between two points a distance L apart along the cylinder decays as exp (−2πx_qL/ℓ). However, if χ ≠ 0, loops which wrap around the cylinder are not counted correctly by the above prescription, because the total number of left turns minus right turns is then zero. We may arrange the correct factors by inserting e^±6iχh/π at either end of the cylinder. This has the effect of modifying the partition function: one finds ln Z (πc/6) (L/ℓ) with

(10)

This dependence of the partition function is one way of determining the so-called central charge of the corresponding CFT (Section 5). The charges at each end of the cylinder also modify the scaling dimension x_q to (1/2g)((q − 6iχ/π)² − (6iχ/π)²).

The value of g may be fixed [17] in terms of the original discreteness of the height variables as follows: adding a term −λ∫ cos 2hd²r to S in (8) ensures that, in the limit λ → ∞, h will be an integer multiple of π. For this deformation not to affect the critical behaviour, it must be marginal in the RG sense, which means that it must have scaling dimension x₂ = 2. This condition then determines g = 1 − 6χ/π.