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 (ds=1), it perfectly works for nonrelativistic systems in ds=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 ds=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.
Highlights
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)
Whereas for Lorentzinvariant systems the effective Lagrangian method fails in one spatial dimension, it perfectly works for nonrelativistic systems in ds = 1
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
Summary
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 SU(3)R × SU(3)L → SU(3)V in quantum chromodynamics (QCD) and spin rotation symmetry O(3) → O(2) in the context of ferromagnets. In the second part we present the low-temperature expansions for the pressure and the order parameters, that is, the quark condensate in QCD and the spontaneous magnetization in the context of ferromagnets. To appreciate the power of the effective method, we mention that—until Dyson’s monumental work on the d = 3 + 1 ferromagnet [10, 11]—it was unclear at which order in the low-temperature expansion of the spontaneous magnetization the spin-wave interaction shows up. The impact of the spin-wave interaction in ferromagnetic films and spin chains was first addressed systematically and conclusively solved with effective Lagrangians in [6,7,8,9]
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