By Igor Herbut
Serious phenomena is likely one of the most enjoyable components of contemporary physics. This 2007 publication offers an intensive yet financial creation into the foundations and strategies of the idea of severe phenomena and the renormalization crew, from the point of view of contemporary condensed subject physics. Assuming easy wisdom of quantum and statistical mechanics, the publication discusses part transitions in magnets, superfluids, superconductors, and gauge box theories. specific awareness is given to themes comparable to gauge box fluctuations in superconductors, the Kosterlitz-Thouless transition, duality differences, and quantum part transitions - all of that are on the vanguard of physics examine. This ebook comprises various difficulties of various levels of trouble, with ideas. those difficulties supply readers with a wealth of fabric to check their figuring out of the topic. it's perfect for graduate scholars and more matured researchers within the fields of condensed subject physics, statistical physics, and many-body physics.
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Extra resources for A Modern Approach to Critical Phenomena
In the first approximation we may set λ = 0, so that the action becomes quadratic in the fluctuating fields. Z [ j] can then be straightforwardly computed by simply completing the square in the exponent. 44) where the constant is independent of the source field. The mean-field susceptibility in the normal phase, which equals the susceptibility for the 38 Ginzburg–Landau–Wilson theory non-interacting system, is then dk (2π )d χ0 (r ) = kB T eik·r 2k2 2m −μ . 47) √ and the correlation length ξ = / 2m|μ|.
Since μ ∝ Tc − T near Tc , the correlation length exponent is ν = 1/2, and the anomalous dimension η = 0. In the Fourier space χ(k = 0) ∼ |μ|−1 , and thus the exponent γ = 1. The saddle-point approximation to our functional integral therefore leads to the already familiar mean-field values of the critical exponents. In fact, the approximation we employed is nothing else but the general mean-field theory of phase transitions introduced first phenomenologically by L. Landau. The critical exponents are the same as in the Curie–Weiss theory of magnetism and in the van der Waals–Maxwell theory of the liquid–gas transition.
X|→∞ Note also that the result at T > TBEC could have been obtained by neglecting all but ωn = 0 contribution to the sum over Matsubara frequencies. 4 Landau’s mean-field theory Let us consider the finite temperature transition. In accordance with the previous discussion we omit the regular (analytic) piece of the free energy coming from the modes with ωn > 0, and retain only the ω0 = 0 modes in the partition function. 4 Landau’s mean-field theory the partition function becomes Z = S[ ] = 1 kB T 2 dr 2m D ∗ (r )D (r )e−S with the action |∇ (r )|2 − μ| (r )|2 + λ| (r )|4 .