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Extra info for Conformal symmetries and constrained critical phenomena
23) Let us see how this result leads to the familiar phenomenon of Bose–Einstein condensation. 24) where we have included an infinitesimal η to render the frequency sum finite. The sum over Matsubara frequencies may be computed by using the calculus of residues as follows. 25) where the contour C is a large circle in the complex ω-plane as in Fig. 2. In the infinite limit of the circle’s radius the integrand, vanishes exponentially fast so that the integral I = 0. 3 Temperature dependence of the chemical potential in a threedimensional system of non-interacting bosons.
1. Since the atoms of helium are light and interact via weak dipole–dipole interactions, due to quantum zero-point motion helium stays liquid down to the lowest temperatures, at not too high pressures. Instead of solidifying it suffers a continuous normal liquid–superfluid liquid transition at Tc ≈ 2K , also called the λ-transition due to the characteristic form of the specific heat in Fig. 6. The λ-transition represents the best quantitatively understood critical point in nature. 0003, with the power-law behavior being observed over six decades of the reduced temperature!
Although m = 0 for T < Tc , the deviations from the finite magnetization are also uncorrelated at large distances, and the same is true for − (z). The integral in the last equation is therefore finite in both cases. Comparing with the definition of the exponent γ we see that γ = ν(2 − η), which is also known as Fisher’s scaling law. 2 Measured values of critical exponents in different systems, belonging to the Ising, XY, and Heisenberg universality classes. 1 Finally, assuming that the only relevant length scale near Tc is provided by the correlation length ξ , the free energy per unit volume is expected to scale as f ∝ ξ (t)−d .