Connection between QFT and statistical physics of phase transitions

I have heard that there is a deep connection between QFT (emphasized by its path-integral formulation) and statistical physics of critical systems and phase transitions. I have only a basic course in QFT and stat mech and they looked like separate disciplines to me, could someone briefly explain or summarize what is the connection?

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Read the first chapter of these notes: maths.dur.ac.uk/users/kasper.peeters/eft.html – Hunter Apr 7 '14 at 14:40
What you are asking is a hard question to answer at one go. Its true. What you need is to read one of three genre of texts. i) Quantum field theory Methods in Statistical Physics ii) Thermal fluctuations and euclidean field theory iii) QFT a statistical field theory approach. The truth about what I just said is its too much stuff to do. As far as you know how partition functions make it into QFT and how wick rotations work in the non-trivial case you should be fine. I will try to put an answer, but it would take a while to prepare. @Hunter's link is good so start with that. – kevin Tah N. Apr 7 '14 at 22:17

I will not directly answer your question, rather I'll try to make plausible the connexion between QFT and statistical physics. To my mind the mathematical details are somehow obscure and confusing, whereas using the theory is worth a deal, and give interesting results, especially in condensed matter and nuclear matter problems. For more details you can have a look on the historical references

A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinsky, Methods of Quantum Field Theory in Statistical Physics (Prentice Hall, 1963).

L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Westview Press, 1962)

and some modern accounts (including the path-integral if it's all you want)

A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge University Press, 2010).

A. Zee, Quantum Field Theory in a Nutshell, (Princeton University Press, 2003).

as well as the intermediary book

J. W. Negele and H. Orland, Quantum Many-particle Systems (Westview Press, 1998)

The basic idea of the connexion between quantum field theory and statistical physics is the ressemblance between the Schrödinger evolution operator $e^{-\mathbf{i}Ht/\hbar}$ and the Gibbs weight $e^{-\beta H}$ with $\beta=\left(k_{B}T\right)^{-1}$ representing the temperature.

In many-body problems at zero-temperature, we are mostly interested by the ground state, so we want to calculate things like the expectation value of an observable $O$, defined as $$\left\langle O\right\rangle =\left\langle \emptyset\right|O\left(t\right)\left|\emptyset\right\rangle =\left\langle \emptyset\right|e^{\mathbf{i}Ht/\hbar}Oe^{-\mathbf{i}Ht/\hbar}\left|\emptyset\right\rangle$$ with $\left|\emptyset\right\rangle$ the ground state. Its time dependency comes from the Heisenberg equation, and for simplicity I supposed a time-independent Hamiltonian $H$. Generalisation are not complicated, and read $$\left\langle O\right\rangle =\left\langle \emptyset\right|U\left(-\infty,t\right)OU^{\dagger}\left(t,-\infty\right)\left|\emptyset\right\rangle$$ with the evolution operator $U$ verifying the Schrödinger equation $\mathbf{i}\hbar dU/dt=HU$ for any time-dependent Hamiltonian $H$. We supposed that at time $t\rightarrow-\infty$ the system was in its ground-state. The above formula is the basics for perturbation theory (see below).

In statistical physics, we are interested in the temperature-dependent ground-state. So we just want to know something like $$\left\langle O\right\rangle =\text{Tr}\left\{ \rho O\right\} \rightarrow\text{Tr}\left\{ e^{-\beta H}O\right\} =\left\langle \emptyset\right|e^{-\beta H}O\left|\emptyset\right\rangle$$ where you recognise the quantum average on the left of the arrow, with $\rho$ the density matrix. On the right of the arrow, you can recognise the statistical average, when the density matrix becomes the Gibbs averaging with $\beta=\left(k_{B}T\right)^{-1}$ ($k_{B}$ is the Boltzmann constant and $T$ is the temperature). The chemical potential is discarded for simplicity.

Now the magic trick: thanks to the resemblance between the zero- and finite-temperature writings, we define an operator $\tilde{O}\left(\tau\right)=e^{H\tau}Oe^{-H\tau}$ which looks the same as the previous $e^{\mathbf{i}Ht/\hbar}Oe^{-\mathbf{i}Ht/\hbar}$ except for the redefinition of the time $\tau=\mathbf{i}t/\hbar$, which in practise represents the temperature. If you wish, you can see that rewriting as a Heisenberg-representation of the operator $O$ (in imaginary time, though). The important point is that one can adapt the perturbation techniques to the temperature problems. For instance suppose as usual that $H=H_{0}+H_{\text{int}}$ with a known unperturbed $H_{0}$ and a perturbative $H_{\text{int}}$. Then we define the $S$ matrix as $e^{-H\tau}=e^{-H_{0}\tau}S\left(\tau\right)$ which gives, after differentiation $$-\frac{dS}{d\tau}=\tilde{H}_{\text{int}}S\left(\tau\right)\Rightarrow S\left(\tau\right)=T_{\tau}\exp\left[-\int\tilde{H}_{\text{int}}d\tau\right]$$ with $\tilde{H}_{\text{int}}=e^{H_{0}\tau}H_{\text{int}}e^{-H_{0}\tau}$ being the interaction-representation of the interaction-Hamiltonian. We recognise the Schrödinger equation verified by the $S$ operator, and $T_{\tau}$ is the chronological order operator in the $\tau$ variable. So in principle everything you know about Schrödinger equation, or evolution operator, or more generally about quantum physics applies straightforwardly to the classical statistical physics.

Using this trick we have $e^{-\beta H}=e^{-H_{0}\beta}S\left(\beta\right)$ in the statistical average of the $O$-observable. Now perturbation theory for $S$ helps you finding $\left\langle O\right\rangle$. That's not more complicated.

Now the connexion with the path-integral: since we want to discuss statistical physics (or many-body systems), one needs a way to quantised the fermions and bosons fields. That's how we use path-integrals.

And finally the connexion to phase transition: a phase transition appears when the ground state of the full Hamiltonian does not verify all the symmetries of $H$. So the perturbation theory can help you finding the new ground-state. You can use for instance Hubbard-Stratonovitch decoupling to get the new ground state.

Final remark: Sometimes you can not get the new ground state perturbatively, so you need more evolved techniques. They all come from the path-integral rewriting of what I said above.

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I started writing an appropriate answer for you then realized life is too short. The answer you are looking for is **Hagen Kleinert: "Path integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets" ** It is only a little over 1500 pages. If you read 200 pages everyday and then think hard about the information, you can find the connection you've been hunting for in just about 10 days.

Godspeed!

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