# Order parameter, symmetry breaking Landau style

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• paralleltransport

#### paralleltransport

TL;DR Summary
I'd like to understand what people mean by an order parameter
Hi all,

I am somewhat familiar the Landau Ginzburg paradigm for phase transition. My understanding is that it is a phenomological model of 2nd order phase transitions by "guessing" that the free energy can be expanded a configuration integral (path integral) of a functional of a local order parameter.

In math the guess would look like this:
exp(F)∝∫exp(∫ddxL(ϕ))exp⁡(F)∝∫Dϕexp⁡(∫ddxL(ϕ)) for some order scalar order parameter ϕϕ.

Once that is done, one can apply all the field theory technologies that studies such (path) integral (RG flow, fixed point etc...).

However when I read papers, I notice there is no clear definition of what an order parameter must satisfy beyond pheonomological consideration.

I'd like to ask if the following must be true:

1) it must be 0 in some phase and non-zero in the phase that breaks the symmetry. Therefore defining an order parameter must be with respect to some broken symmetry.

2) it can be local or non-local although landau symmetry breaking theory is concerned with local order parameter (for example, Wilson loop expectation value and Chern Numbers are non-local order parameters).

3) it must be a physical quantity (for gauge theories, one cannot use charged operators as order parameter since they transform non-trivially under the gauge group)... this seems a bit like a tautology but people often cite elitzur's theorem to justify this, I'm not sure why it's a big deal it seems a bit obvious to me.

4) it has to be a macroscopic variable and can be measured on macroscopic scale (its fluctuations diverge at the phase transition)
* one has to measure it on scales >> lattice spacing so that it can be relatively UV insensitive. I have in mind averaging magnetic spins over mesoscopic scale for the O(N) model.

I feel like order parameters are ultimately very subjective. There doesn't seem to be a "canonical" way to define it, but maybe I'm very naive.

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Have a look at how L&L first introduce and then use an order parameter in book 5.

it can be local or non-local although landau symmetry breaking theory is concerned with local order parameter (for example, Wilson loop expectation value and Chern Numbers are non-local order parameters).
Topological phase transitions are an other beast, and they are not necessary related to any symmetry breaking. You can not define an order parameter and a LG theory to describe such transitions, and the Chern Number is thus referred as a topological invariant, that is, a constant quantity that characterize a single topological phase. I don't know how you would work in TQFT, but do not mix LG symmetry breaking and topological transitions.