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 , it perfectly works for nonrelativistic systems in . 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 , 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. 1. Introduction While the methods used in particle physics tend to be rather different from the microscopic approaches taken by condensed matter physicists, there is though one fully systematic analytic method that can be applied to both sectors. The effective Lagrangian method, based on a symmetry analysis of the underlying theory, makes use of the fact that the low-energy dynamics is dominated by Goldstone bosons which emerge from the spontaneously broken symmetry: chiral symmetry in quantum chromodynamics (QCD) and spin rotation symmetry in the context of ferromagnets. The method thus connects systems as disparate as QCD and ferromagnets from a universal point of view based on symmetry. The low-energy properties of the system are an immediate consequence of the spontaneously broken symmetry, while the specific microscopic details only manifest themselves in the values of a few effective constants. Still, as we are dealing with nonrelativistic kinematics in the case of the ferromagnet, apart from analogies, there are important differences: most remarkably, the effective Lagrangian method, unlike for systems with relativistic kinematics, perfectly works for ferromagnets in one spatial dimension ( ). While the low-temperature behavior of QCD was discussed more than two decades ago within effective field theory [1–3], the
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