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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 .

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