Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 337 / 2 / c

Table of contents
- Solution c

Solution

At a site with background

η_{i} n

, the linear variation is

δ n_{i} = m

. The supplied variation formula gives

S δ Γ [e t a_{i} n] = S \int d t ma t hb f m \cdot [(η_{i} n) \times (η_{i} \partial_{t} n)] = S \int d t ma t hb f m \cdot (n \times \partial_{t} n) .

Replacing the lattice sum by

a^{- 2} \int d^{2} x

yields

I_{W Z}^{(1)} = \frac{S}{a ^{2}} \int d t d^{2} x m \cdot (n \times \partial_{t} n) .

This Berry-phase term makes the uniform canting field the momentum conjugate to rotations of the Néel order parameter.

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