Ferromagnets and antiferromagnets in the effective Lagrangian perspective

Nonrelativistic systems exhibiting collective magnetic behavior are analyzed within the framework of effective Lagrangians. The method, which formulates the dynamics of the system in terms of Goldstone bosons, allows to investigate the consequences of spontaneous symmetry breaking from a unified point of view. Analogies and differences with respect to the Lorentz-invariant situation (chiral perturbation theory) are pointed out. We then consider the low-temperature expansion of the partition function both for ferro- and antiferromagnets, where the spin waves or magnons represent the Goldstone bosons of the spontaneously broken symmetry $O(3) \to O(2)$. In particular, the low-temperature series of the staggered magnetization for antiferromagnets and the spontaneous magnetization for ferromagnets are compared with the condensed matter literature.Comment: Contributed Talk at 8th Mexican Workshop on Particles and Fields, Zacatecas, Mexico, 14-20 Nov 200

Antiferromagnets at low Temperatures

The low-temperature properties of the Heisenberg antiferromagnet in 2+1 space-time dimensions are analyzed within the framework of effective Lagrangians. It is shown that the magnon-magnon interaction is very weak and repulsive, manifesting itself through a term proportional to five powers of the temperature in the pressure. The structure of the low-temperature series for antiferromagnets in 2+1 dimensions is compared with the structure of the analogous series for antiferromagnets in 3+1 dimensions. The model-independent and systematic effective field theory approach clearly proves to be superior to conventional condensed matter methods such as spin-wave theory.Comment: Presented at 12th Mexican Workshop on Particles and Fields, Mazatlan, Sinaloa, Mexico, 9-14 Nov 200

Goldstone Boson Interaction in D=2+1 (Pseudo-)Lorentz-Invariant Systems with a Spontaneously Broken Internal Rotation Symmetry

The low-temperature properties of systems characterized by a spontaneously broken internal rotation symmetry, O($N$) $\to$ O($N$-1), are governed by Goldstone bosons and can be derived systematically within effective Lagrangian field theory. In the present study we consider systems living in two spatial dimensions, and evaluate their partition function at low temperatures up to three-loop order. Although our results are valid for any such system, here we use magnetic terminology, i.e., we refer to quantum spin systems. We discuss the sign of the Goldstone boson interaction in the pressure, staggered magnetization, and susceptibility as a function of an external staggered field for general $N$. As it turns out, the $d$=2+1 quantum XY model ($N$=2) and the $d$=2+1 Heisenberg antiferromagnet ($N$=3), are rather special, as they represent the only cases where the spin-wave interaction in the pressure is repulsive in the whole parameter regime where the effective expansion applies. Remarkably, the $d$=2+1 XY model is the only system where the interaction contribution in the staggered magnetization (susceptibility) tends to positive (negative) values at low temperatures and weak external field.Comment: 31 pages, 12 figure

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

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.Comment: 16 pages, 3 figures; minor typos corrected in v

Thermodynamics of Ferromagnetic Spin Chains in a Magnetic Field: Impact of the Spin-Wave Interaction

The thermodynamic properties of ferromagnetic spin chains have been the subject of many publications. Still, the problem of how the spin-wave interaction manifest itself in these low-temperature series has been neglected. Using the method of effective Lagrangians, we explicitly evaluate the partition function of ferromagnetic spin chains at low temperatures and in the presence of a magnetic field up to three loops in the perturbative expansion where the spin-wave interaction sets in. We discuss in detail the renormalization and numerical evaluation of a particular three-loop graph and derive the low-temperature series for the free energy density, energy density, heat capacity, entropy density, as well as the magnetization and the susceptibility. In the low-temperature expansion for the free energy density, the spin-wave interaction starts manifesting itself at order $T^{5/2}$. In the pressure, the coefficient of the $T^{5/2}$-term is positive, indicating that the spin-wave interaction is repulsive. While it is straightforward to go up to three-loop order in the effective loop expansion, the analogous calculation on the basis of conventional condensed matter methods, such as spin-wave theory, appears to be beyond reach.Comment: 24 pages, 3 figure