Effects of Spin Fluctuations on Magnetic Properties
In this chapter, we show a number of interesting magnetic properties are derived based on the idea that the local spin fluctuation amplitude is conserved, independent of temperature and irrespective of the presence of externally applied magnetic field. As
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Effects of Spin Fluctuations on Magnetic Properties
3.1 Basic Idea of the Spin Fluctuation Theory In our treatment of spin fluctuations of large amplitudes, it is inappropriate to employ a kind of expansion method with respect to their amplitudes. Instead it is better to rely on some general ideas justified independent of the magnitude of amplitudes. We propose the following ideas [1, 2] as the basis of our following discussions. 1. Total spin amplitude conservation (TAC) The total spin amplitude on each magnetic site of the crystal is conserved independent of temperature. It is also unaffected by external magnetic field. 2. Global consistency in the effect of magnetic field (GC) We mean by this that the magnetic isotherm, i.e., the functional relation between the external magnetic field H and the induced magnetization M, is globally consistent with the first condition. Specifically, the condition of TAC is explicitly written in the form
2 Sloc
tot
2 2 = δSloc (y, yz , T ) + δSloc (y, yz ) + σ 2 . T
Z
(3.1)
The left-hand side represents a conserved constant amplitude. The first two terms on the right-hand side are the thermal and the zero-point amplitudes, respectively, while the last term is a mean static local magnetic moment squared. In the presence of external magnetic field, both these amplitudes are determined by the reduced inverse of magnetic susceptibilities y(σ, t) and yz (σ, t) as functions of variables σ and t. These functions are related to each other by yz (σ, t) = y(σ, t) + σ
∂ y(σ, t) . ∂σ
Y. Takahashi, Spin Fluctuation Theory of Itinerant Electron Magnetism, Springer Tracts in Modern Physics 253, DOI: 10.1007/978-3-642-36666-6_3, © Springer-Verlag Berlin Heidelberg 2013
(3.2)
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3 Effects of Spin Fluctuations
Our GC requirement imposes on y(σ, t) the condition that its σ dependence has always to be determined to satisfy (3.1). It means that (3.1) is regarded as an ordinary differential equation for y(σ, t). With use of the thermal amplitude A(y, t) in (2.83), the total amplitude conservation is written as follows: TA 2 TA 2 σ = Δ Sloc tot 2 A(y, t) + A(yz , t) − c(2y + yz ) + 3T0 3T0 2 2 2 Δ Sloc tot ≡ Sloc tot − Sloc Z (0, 0)
(3.3)
At the critical temperature in the absence of the external magnetic field, y = yz = σ = 0 is satisfied for ferromagnets. The right-hand side of (3.3) is then given by 3A(0, tc ) on the left hand side. For paramagnets in the ground state (T = 0) with no thermal amplitude it is given by −3cy0 because both y and yz have the same finite value y0 . To summarize, the right-hand side of (3.3) is given by TA 2 Δ Sloc = tot 3T0
3A(0, tc ), for ferromagnets −3cy0 , for paramagnets
(3.4)
In the following sections we will show that various magnetic properties are derived by solving the differential equation (3.3).
3.2 Magnetic Isotherm in the Ground State To begin with, for convenience in later sections, we first show how the magnetic isotherm in the ground state is derived as a solution of Eq. (3.3). Since no thermal spin
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