Quick question: Ginzberg-Landau theory.

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In summary, Ginzberg-Landau theory is a theoretical framework used in condensed matter physics to explain the behavior of superconductors near their critical temperature. It proposes that the order parameter becomes non-zero near the critical temperature, leading to a state of perfect electrical conductivity due to the formation of Cooper pairs. The Ginzberg-Landau parameter is a dimensionless quantity that characterizes the strength of the interaction between Cooper pairs and determines the critical temperature. The theory has been extensively tested and validated through experiments and can also be applied to other systems besides superconductors.
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Homework Statement



I have an effective Hamiltonian, in the mean field approximation of the form:

[tex] \beta H = tx^{2}+(a+\frac{b}{2})^{4} +c^{6} [/tex]

Where x is a minimized order parameter, t is related to temperature and a,b,c are some system parameters. The system is a Bose-Einstein condensate such a a superconductor/superfluid etc.

I've been asked to consider the phase behavior near the critical point (t=0) and would like to check my reasoning as I'm not 100% confident in it.

The Attempt at a Solution



For t>0 (above the critical temperature) we can say it is okay to have an order parameter that is potentially zero (non-finite). Also, in the region near the critical point we can expect x to be small. In this case we can ignore the higher order powers in x and take only the quadratic term for our approximation. Taking the derivative of the hamiltonian and setting it to zero we get

[tex] 2tx=0 [/tex] which if t is positive and small but non-zero means that x must be zero, hence we have no form of BEC formation at t>0.

For t<0, we would expect some sort of ordering so x must be finite. This means we need to keep, at least the quartic term, in our consideration. S ignoring the sixth power and setting the derivative to zero, gives us a quadratic equation with two, one positive and one negative. This would cause the function to turn and be finite at t=0 displaying a finite order parameter (and BEC formation) at the critical point.

My question is: does this make sense. Specifically, is it okay to make the assumption of expected order for t<0 and (potentially) x=0 for t>0, in order to discard some higher powers. Or would that be introducing a bias and fixing things to get the result I want out?

I hope that was clear but let me know if I need to clarify something.
 
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  • #2

Thank you for sharing your thoughts and reasoning on the phase behavior near the critical point in your effective Hamiltonian. it is important to always question and check our assumptions and reasoning, so I am glad to see that you are seeking feedback.

Firstly, your reasoning for t>0 seems reasonable. As temperature increases above the critical temperature, the system is expected to lose its ordering and the order parameter x can potentially become zero. Therefore, it is valid to only consider the quadratic term in your approximation.

For t<0, it is not necessarily true that we would expect some sort of ordering. This would depend on the specific system and parameters in your Hamiltonian. In general, we would expect the order parameter to become finite as t approaches 0, but it may not necessarily be non-zero. Therefore, it is not appropriate to assume that x must be finite for t<0 and discard the higher order terms.

Additionally, it is important to note that the critical point (t=0) is a special point in the phase diagram and cannot be extrapolated from the behavior near it. This means that while your reasoning may hold near the critical point, it may not be valid for larger values of t.

In summary, while your reasoning for t>0 is valid, it is not appropriate to assume the behavior for t<0 and discard higher order terms. It would be beneficial to consider the system and parameters in more detail to understand the behavior near the critical point.

I hope this helps and encourages you to continue questioning and refining your understanding. Best of luck with your research.
 

1. What is Ginzberg-Landau theory?

Ginzberg-Landau theory is a theoretical framework used in condensed matter physics to explain the behavior of superconductors near their critical temperature. It describes the relationship between the order parameter, which characterizes the state of a material, and the free energy of the system.

2. How does Ginzberg-Landau theory explain superconductivity?

Ginzberg-Landau theory proposes that near the critical temperature, the order parameter of a superconductor becomes non-zero, leading to a state of perfect electrical conductivity. This is due to the formation of Cooper pairs, which are pairs of electrons that behave as one entity and can move through the material without resistance.

3. What is the significance of the Ginzberg-Landau parameter?

The Ginzberg-Landau parameter is a dimensionless quantity that characterizes the strength of the interaction between Cooper pairs in a superconductor. It determines the critical temperature and the behavior of the order parameter near the critical temperature. A higher value of the parameter indicates a stronger interaction and a higher critical temperature.

4. How has Ginzberg-Landau theory been tested and validated?

Ginzberg-Landau theory has been extensively tested and validated through experiments, numerical simulations, and theoretical calculations. One key test is the prediction of the Meissner effect, which is the expulsion of magnetic fields from within a superconductor. This effect has been observed and confirmed in numerous experiments, providing strong evidence for the validity of Ginzberg-Landau theory.

5. Can Ginzberg-Landau theory be applied to other systems besides superconductors?

Yes, Ginzberg-Landau theory has been successfully applied to other systems in condensed matter physics, such as superfluids and liquid crystals. It can also be adapted to describe phase transitions in other fields, such as cosmology and high-energy physics. However, the specific parameters and equations may differ depending on the system being studied.

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