datomerki8a5yj

2022-05-18

I am trying to get the relation between developing a Ginzburg-Landau theory, let's say for a ferromagnet with magnetization field $\overrightarrow{m}=\overrightarrow{m}(\overrightarrow{r})$, and the formal expansion of the free energy density $\mathcal{F}=\mathcal{F}(\overrightarrow{m})$ in terms of a Taylor series.

Considering an isotropic ferromagnet, the lowest-order terms in our Ginzburg-Landau theory should be given by

$\mathcal{F}=\frac{r}{2}{\overrightarrow{m}}^{2}+\frac{U}{4}{\left({\overrightarrow{m}}^{2}\right)}^{2}+\frac{J}{2}[{\left({\mathrm{\partial}}_{x}\overrightarrow{m}\right)}^{2}+{\left({\mathrm{\partial}}_{y}\overrightarrow{m}\right)}^{2}+{\left({\mathrm{\partial}}_{z}\overrightarrow{m}\right)}^{2}]$

with r<0 and U,J>0.

However, when I think of a Taylor expansion of $\mathcal{F}(\overrightarrow{m})$ around the origin

$\mathcal{F}={\mathcal{F}}_{0}+{\overrightarrow{m}}^{T}\cdot D\mathcal{F}+\frac{1}{2}{\overrightarrow{m}}^{T}\cdot {D}^{2}\mathcal{F}\cdot \overrightarrow{m}+\dots $

this is giving me terms of all the individual powers in $\overrightarrow{m}$ , which are either zero or identified with the r- and U-term, but no gradient terms for the J-term? How to motivate these through a Taylor expansion?

Considering an isotropic ferromagnet, the lowest-order terms in our Ginzburg-Landau theory should be given by

$\mathcal{F}=\frac{r}{2}{\overrightarrow{m}}^{2}+\frac{U}{4}{\left({\overrightarrow{m}}^{2}\right)}^{2}+\frac{J}{2}[{\left({\mathrm{\partial}}_{x}\overrightarrow{m}\right)}^{2}+{\left({\mathrm{\partial}}_{y}\overrightarrow{m}\right)}^{2}+{\left({\mathrm{\partial}}_{z}\overrightarrow{m}\right)}^{2}]$

with r<0 and U,J>0.

However, when I think of a Taylor expansion of $\mathcal{F}(\overrightarrow{m})$ around the origin

$\mathcal{F}={\mathcal{F}}_{0}+{\overrightarrow{m}}^{T}\cdot D\mathcal{F}+\frac{1}{2}{\overrightarrow{m}}^{T}\cdot {D}^{2}\mathcal{F}\cdot \overrightarrow{m}+\dots $

this is giving me terms of all the individual powers in $\overrightarrow{m}$ , which are either zero or identified with the r- and U-term, but no gradient terms for the J-term? How to motivate these through a Taylor expansion?

Cortez Hughes

Beginner2022-05-19Added 23 answers

Note that this is rather an opinion than a fully rigorous statement:

For vanishing gradients, i.e. for uniform systems (the case originally considered by Landau), the expansion of the free energy is indeed a Taylor expansion (in even powers of m) near the transition. However the addition of gradient terms in the case of non-uniform systems removes this interpretation as is rather a phenomenological modification. However, you could still see this an expression for a classical field theory.

For vanishing gradients, i.e. for uniform systems (the case originally considered by Landau), the expansion of the free energy is indeed a Taylor expansion (in even powers of m) near the transition. However the addition of gradient terms in the case of non-uniform systems removes this interpretation as is rather a phenomenological modification. However, you could still see this an expression for a classical field theory.

Raphael Mccullough

Beginner2022-05-20Added 1 answers

It looks to me like you are Taylor expanding $\mathcal{F}$ as a function of $\overrightarrow{m}$ without considering that $\overrightarrow{m}$ is a function of space. In particular, your $D$ and ${D}^{2}$ involve things like $\frac{\mathrm{\partial}}{\mathrm{\partial}{m}_{x}}$ but not $\frac{\mathrm{\partial}}{\mathrm{\partial}x}$. You are implicitly assuming that $\mathcal{F}(\overrightarrow{x})$ only depends on $\overrightarrow{m}(\overrightarrow{x})$ and not also for example $\overrightarrow{m}(\overrightarrow{x}+d\overrightarrow{x})$ for small $d\overrightarrow{x}$

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