Differences and analogies between quantum chromodynamics and ferromagnets

Abstract

Abstract. The low-energy physics of quantum chromodynamics (QCD) and ferromagnets is dominated by Goldstone bosons. While the effective theory of QCD -chiral perturbation theory -is well established in the particle physics community, the systematic studies of ferromagnetic systems within the effective Lagrangian framework are not well-known. We analyze the lowtemperature properties of ferromagnets in one, two and three space dimensions up to three-loop order in the effective expansion, i.e., beyond the accuracy of any previous results obtained with conventional condensed matter methods. In particular, in the nonrelativistic domain, the effective method perfectly works in one space dimension. MotivationThe effective field theory of quantum chromodynamics (QCD), chiral perturbation theory, is very-well established in particle physics. The method was originally devised for zero temperature [1, 2, 3], but has been extended to finite temperature soon after [4,5]. There are many excellent outlines of the method available -the interested reader may want to consult Refs. [6,7,8,9,10]. In the present overview, we are interested in the description of relativistic and nonrelativistic systems at nonzero temperature. In fact, the low-temperature properties of QCD have been systematically analyzed in a series of papers a long time ago [4,5,11]. One important result of these studies is the three-loop formula for the order parameter, i.e., the quark condensate, which takes the following form [11],where 0|qq|0 is the quark condensate at zero temperature. Note that the boldfaced terms are those contributions that result from the interaction among the Goldstone bosons, while the leading temperature-dependent term corresponds to free Goldstone bosons. The effective field theory method, however, is not restricted to the relativistic domain. Indeed, the effective Lagrangian technique has been transferred to nonrelativistic systems in Ref. [12]. In particular, the effective Lagrangian for the ferromagnet was established in that reference. In analogy to the low-temperature expansion of the quark condensate and other thermodynamic quantities, in this talk, we will be interested in the behavior of the (spontaneous) magnetization and other observables of ferromagnets in one, two and three space dimensionsdetailed information can be found in the original papers [13,14,15,16,17,18]. We will present

Cubic ideal ferromagnets at low temperature and weak magnetic field

“…In order to convince the community that we are dealing with a method that indeed deserves attention, we like to enumerate some references where condensed matter problems have been addressed and solved within the effective Lagrangian framework. Systems whose low-energy properties are governed by magnons include ferromagnetic spin chains [46][47][48], as well as ferromagnets in two [49][50][51] and three [1,5,6,[52][53][54][55][56][57][58] spatial dimensions. Antiferromagnetic and XY-type systems in two [59][60][61][62][63][64] or three [65][66][67] spatial dimensions can be analyzed along the same lines.…”