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This is because Landau theory is a mean field theory, and does not include long-range correlations. This theory does not explain non-analyticity at the critical point, but when applied to superfluid and superconductor phase transition, Landau's theory provided inspiration for another theory, the Ginzburg–Landau theory of superconductivity.
The classical Landau theory (also known as mean field theory) values of the critical exponents for a scalar field (of which the Ising model is the prototypical example) are given by = ′ =, =, = ′ =, =
In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final calculation of a statistic that are free to vary).
In Landau mean-field theory, at temperatures near the superconducting critical temperature , () (/). Up to a factor of 2 {\displaystyle {\sqrt {2}}} , it is equivalent to the characteristic exponent describing a recovery of the order parameter away from a perturbation in the theory of the second order phase transitions.
Based on Landau's previously established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy density of a superconductor near the superconducting transition can be expressed in terms of a complex order parameter field () = | | (), where the quantity | | is a measure of the local density of superconducting electrons () analogous to a quantum mechanical wave ...
Using this in the Landau theory, which is identical to the mean field theory for the Ising model, the value of the upper critical dimension comes out to be 4. If the dimension of the space is greater than 4, the mean-field results are good and self-consistent. But for dimensions less than 4, the predictions are less accurate.
The Ising model can be solved analytically in one and two dimensions, numerically in higher dimensions, or using the mean-field approximation in any dimensionality. Additionally, the ferromagnet to paramagnet phase transition is a second-order phase transition and so can be modeled using the Landau theory of phase transitions. [1] [6]
The idea of quasiparticles originated in Lev Landau's theory of Fermi liquids, which was originally invented for studying liquid helium-3. For these systems a strong similarity exists between the notion of quasiparticle and dressed particles in quantum field theory. The dynamics of Landau's theory is defined by a kinetic equation of the mean ...