# Quantum phase transition

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In quantum phase transition, the correlation length diverges at the critical point. How to physically understand it? Means, what exactly happens at the quantum critical point, which leads to the divergence of correlation length.

What happens to time at the quantum critical point?

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OP seems to be seeking an intuitive understanding, here is my understanding, which I think is the common intuitive understanding:

When a phase transition happens, a qualitative change of the system must happen at a macroscopic scale. The system must organize or de-organize itself on a macroscopic scale, which needs different local parts of the system to "cooperate". The correlation length is a quantity that measures the range of influence a local part has, i.e. within this distance, different parts can significantly cooperate. When correlation length diverges, it means the whole system somehow behaves as a whole, which to some degree is the definition of phase transition. This seems to be a common idea lying under both classical and quantum phase transition. I don't know if it can be made rigorous.

answered Aug 17, 2014 by (2,640 points)

I was also thinking that a more rigorous understanding could be made along these lines. If the phases we care about are gapped, then the phase of the system is described by the long range TQFT. After some sharpening of this idea, really the phase is labelled by the path component in the space of TQFTs where the long range limit lives. In other words, if we change some things about the system that amounts to a variation of a continuous parameter of the TQFT then this does not require a phase transition. However, if we want to jump from one path component to another while continuously varying parameters in the original system, there must be some set of parameters that have no long range limit. This can only happen if the gap closes, ie. some modes have diverging correlation lengths.

@RyanThorngren, I have little knowledge on what you talked about. I know what a gapped Hamiltonian is, but what is a gapped phase?

I just mean the phase of a gapped Hamiltonian (or more general system).

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To understand this question, one can make use of the fact that that QM is equivalent to 1D statistical mechanics. The relation is, that the trace over the quantum mechanical time evolution operator taken at imaginary time corresponds to the (canonical) partition function of an 1D statistical mechanics system at finite temperature

$Z(\beta) = \text{trace} \{ U(i\beta)\} = \sum\limits_{\alpha}\langle \alpha | e^{-\frac{H}{k_BT}}|\alpha\rangle = \sum\limits_{\alpha}e^{-\beta\epsilon_{\alpha}}$

In statistical mechanics, a phase transition corresponds to a qualitative change in the behavior of the many particle system. The partition sum diverges and the correlation length becomes infinite.

For the example of an 1D Gaussian model

$Z = \text{Tr}(q)\prod\limits_{j=1}^N\exp(w(q_j,q_{j+1}))$

with

$w(q,q') = -\frac{1}{4} (q^2 + q'^2) + Kqq'$

it can be shown by applying the transfer matrix method that the correlation length is given by

$\xi = \frac{a}{\epsilon_1 -\epsilon_0}$

where $a$ corresponds to the lattice spacing and $\epsilon_0$ and $\epsilon_1$ are the energy of the ground and first excited state of the corresponding quantum system respectively.

From this one can see, that the correlation length becomes infinite as the ground state gets degenerated.

$(\epsilon_1 - \epsilon_0) \rightarrow 0$

Physically, this means in this example that a symmetry of the quantum system gets restored.

answered Aug 17, 2014 by (6,240 points)

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