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"In the 2D Hubbard model (two spatial dimensions) the Mermin-Wagner theorem does not allow a phase transition."

I am quite illiterate concerning this theorem and hearsay. Does the theorem apply to a superconducting transition as well? References that justify its applicability are welcome.

Qmechanic
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Fitzgerald Creen
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    If by 2D you mean two spatial dimensions, then Mermin-Wagner theorem allows spontaneous continuous symmetry breaking at zero temperature, but at finite temperature there is only algebraic long-range order. In one spatial dimension, even at $T=0$ there is no long-range-ordered superconductivity, only algebraic ones. – Meng Cheng Sep 17 '15 at 01:07
  • Yes, i mean 2D. I can educate myself of course but what is algebraic/non-algebraic long range order? Do you have references? I would be interested especially in a comprehensive argument, why the theorem applies to SC as well. So far i war only aware of its application to magnetic phase transitions. – Fitzgerald Creen Sep 17 '15 at 05:01
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    Instead of saying some order parameter has a finite expectation value, a better way to describe spontaneous symmetry breaking is to look at the correlation function of order parameters, and algebraic long-range order just means the correlation function decays as a power-law with distance. – Meng Cheng Sep 17 '15 at 05:30
  • So do i understand correctly that you're saying that the Mermin-Wagner theorem does not exclude superconductivity in the 2D hubbard model as long as the correlation function of the order parameter decays as a power law ,i.e. it does not diverge? Or is it better to call that something like superconducting fluctuations? – Fitzgerald Creen Sep 17 '15 at 05:57
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    This is a generic feature of SSB in 2D. Only at $T=0$ (the ground state), there is true long-range order. When $T>0$ it always becomes algebraic. – Meng Cheng Sep 17 '15 at 06:59
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    See this answer, Wikipedia on Ginzburg–Landau theory, or (if you have access) this article for a discussion of the complex order parameter in superconductors. – Stephen Powell Sep 17 '15 at 12:35
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    Superconductivity is described by a complex order parameter $\Psi$ (sometimes thought of as a macroscopic wavefunction). Complex numbers have both magnitude and phase, and so are equivalent to two-component vectors. The Mermin–Wagner(–Hohenberg) theorem says that, in two spatial dimensions at nonzero temperature, there can be no symmetry-breaking transition where a continuous symmetry is broken. The symmetry broken in a superconductor is phase-rotation symmetry: a nonzero $\Psi$ implies a certain complex phase. – Stephen Powell Sep 17 '15 at 12:42

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