 Solution

ID: past-exam-of-the-phase-transitions-and-collective-phenomena-course-of-the-university-of-cambridge/2012/paper-1/2/b/solution

 Top articles  Latest articles New article in topic

Past exam of the phase transitions and collective phenomena course of the University of Cambridge / 2012 / Paper 1 / 2 / b / Solution by

Codex  0 2026-10-20

The magnetization correlator is an average of a phase difference, not merely its lowest Taylor expansion. Gaussian phase averaging gives

C_{m} (r) = \overset{m}{ˉ}^{2} ⟨ cos [θ (x) - θ (0)]⟩ = \overset{m}{ˉ}^{2} exp [- \frac{1}{2} D (r)],

(1)

where

D (r) = ⟨[θ (x) - θ (0)]^{2} ⟩ = \frac{2}{K ˉ} \int \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1 - c o s ( q \cdot x )}{q ^{2}} = 2 [G (0) - G (r)] .

(2)

The coincident value is ultraviolet-regularized at a microscopic scale

a

. At large

r

For $d > 2$ , the infrared integral for $G (0)$ converges. Choosing $G (\infty) = 0$ , $D (\infty) = 2 G (0) < \infty$ , so $C_{m} (\infty) = m_{0}^{2} = \overset{m}{ˉ}^{2} e^{- G (0)} > 0$ within the low-temperature spin wave thermodynamic phase.
For $d = 2$ , $D (r) = (π \overset{ˉ}{K})^{- 1} lo g (r / a) + O (1)$ , hence $C_{m} (r) \propto r^{- 1/ (2 π \overset{ˉ}{K})} \to 0$ .
For $0 < d < 2$ , the leading growth is $D (r) = 2 r^{2 - d} / [(2 - d) S_{d} \overset{ˉ}{K}] + O (1)$ , hence $C_{m} (r) \propto exp {- r^{2 - d} / [(2 - d) S_{d} \overset{ˉ}{K}]} \to 0$ . In particular $d = 1$ , $S_{1} = 2$ , gives $C_{m} (r) \propto e^{- r / (2 \overset{ˉ}{K})}$ .

Figure 1.
Infrared spin-wave correlation limits
. Original regulated phase-only model curves for dimensions one, two and three. They illustrate exponential decay, algebraic decay and a nonzero limit respectively; no experimental dataset or phase-transition temperature is fitted. **Details**.

Thus two is the lower critical dimension for conventional positive-temperature continuous-symmetry order under the short-range assumptions. Two dimensions can still have quasi-long-range order and a thermodynamic transition; zero limiting magnetization does not forbid them. The

d > 2

spin wave result describes an admissible low-temperature ordered thermodynamic phase, not order at arbitrarily high temperature. The Mermin-Wagner theorem supplies the more general no-order statement below or at two dimensions.^[ref]

 Read the full article

 New to topics? Read the docs here!