Partition Function in One, Two and Three Spatial Dimensions from Effective Lagrangian Field Theory

Abstract: The systematic effective Lagrangian method was first formulated in the context of the strong interaction: chiral perturbation theory (CHPT) is the effective theory of Quantum Chromodynamics (QCD). It was then pointed out that the method can be transferred to the nonrelativistic domain -- in particular, to describe the low-energy properties of ferromagnets. Interestingly, whereas for Lorentz-invariant systems the effective Lagrangian method fails in one spatial dimension ($d_s$=1), it perfectly works for nonrelativistic systems in $d_s$=1. In the present brief review, we give an outline of the method and then focus on the partition function for ferromagnetic spin chains, ferromagnetic films and ferromagnetic crystals up to three loops in the perturbative expansion -- an accuracy never achieved by conventional condensed matter methods. We then compare ferromagnets in $d_s$=1,2,3 with the behavior of QCD at low temperatures by considering the pressure and the order parameter. The two apparently very different systems (ferromagnets and QCD) are related from a universal point of view based on the spontaneously broken symmetry. In either case, the low-energy dynamics is described by an effective theory containing Goldstone bosons as basic degrees of freedom.