Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 2 / g

Table of contents
- Solution g

Solution

For a damped ferromagnetic spin wave, shifting both poles into the lower half-plane preserves causality:

G_{R} (ω, k) = \frac{ω _{k} / s}{( ω + i Γ _{k} ) ^{2} - ω _{k}^{2}} .

Its spectral function is

ϱ (ω, k) = \frac{4 ( ω _{k} / s ) ω Γ _{k}}{( ω ^{2} - ω _{k}^{2} - Γ _{k}^{2} ) ^{2} + 4 ω ^{2} Γ _{k}^{2}} .

The classical Fluctuation-dissipation theorem then gives

S (ω, k) = \frac{4 T ( ω _{k} / s ) Γ _{k}}{( ω ^{2} - ω _{k}^{2} - Γ _{k}^{2} ) ^{2} + 4 ω ^{2} Γ _{k}^{2}} .

Hence the zero-frequency static structure factor is

S (k) = \frac{4 T ω _{k} Γ _{k}}{s ( ω _{k}^{2} + Γ _{k}^{2} ) ^{2}},

which has the stated proportionality after absorbing the normalization

4/ s

.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version