Landau-Ginzburg Theory

Onwards we continue in our quest for understanding complicated interacting systems! Last week, we built up our understanding of mean field theory, and this week, we'll extend the theory a bit further, thinking about how magnetization can vary throughout space, and working towards the concept of an order parameter field.

Overview

In my notes, we're going to take a slightly different approach than in class. Rather than jumping right into the Landau-Ginzburg functional F[phi(vec{r})]

, we're going to go step by step to properly build up our intuition. Here's the game plan:

Game Plan

We want to understand what F[phi(vec{r})] really means, and why it's such a useful object to consider. We'll start with F(m) , then generalize to $F({m_j})$ , and then finally talk about .

To start off, we will introduce a fictitious ‘‘probe field’’ as a cute trick to easily calculate the magnetization . After a Legendre transform, we will end up with the free energy as a function of magnetization – and as we'll find, F(m) is a fantastic little object.

Note: In class we jumped directly into a spatially-varying magnetization density rather than beginning with a uniform . I'm taking a different approach here.

If time permits, we'll take a little detour to understand free energy, and why we care so much about it. Spoiler alert: the minima of the free energy tell you where equilibrium lies.

Armed with intuition about free energy, we'll explore the behavior of F(m) (Landau theory), which will teach us many fabulous things:

a beautiful intuition for the behavior of second-order phase transitions
how you can find the magnetization density at any temperature by looking at the minima of
a direct picture for how symmetry is broken when
insight into how you can deduce the shape of merely with symmetry arguments
and far more….

Once we've solidified our understanding of F(m) , we'll move on and generalize to a spatially-varying magnetization density m to m_j , which lets us account for textures in m_j at different sites . Also, we'll discuss how to generalize from to an order parameter phi to describe all sorts of phase transitions beyond ferromagnetic ordering.

Then we'll take the continuum limit $phi_j to phi(vec{r})$ and pretend that the order parameter is defined everywhere in space vec{r} rather than just sites . This approach allows us to construct an order parameter field and to think about the free energy as a function(al) of the ‘‘texture’’ of the field (Landau-Ginzburg theory). If time permits, we'll also explain how to arrive at phi(vec{r}) by coarse-graining. (Depending on how we cover the renormalization group later, this may or may not be useful!)

Finally, we'll discuss how we could have arrived at the shape of F[phi(vec{r})] just by symmetry arguments. This sort of top-down or phenomenological approach is the sort of stuff that theorists fantasize about. The argument is simple, general, and powerful.

Motivation: Lingering Questions about Mean Field Approaches

We spent the last few weeks learning about the variational principle and mean field theory, but these appraoches still leave us with a few lingering questions.

As we'll see, the more sophisticated methods that we'll learn in class over the next few weeks will let us address some of these nagging questions which keep us up at night and gnaw at our sanity.

Landau-Ginzburg Theory

Overview

Motivation: Lingering Questions about Mean Field Approaches

Outline

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