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Europhysics Letters (EPL)

Mean-field treatment of nonlinear susceptibilities for a ferromagnet of arbitrary spin

Yugandhar Bitla, Sharika Nandan Kaul

Published Oct 20, 2011• 23 citations
Publisher

Abstract

The intrinsic linear and nonlinear magnetic susceptibilities, χ n , for a ferromagnet of arbitrary spin, calculated using the mean-field approximation, are shown to diverge in the asymptotic critical region (ACR) with the exponent γ n = n γ+( n − 1)β and n =1, 2, ... . This behaviour of χ n in the ACR is consistent with the scaling equation of state. With increasing spin, the divergence in χ n ( T ), as the ferromagnetic-paramagnetic phase transition temperature, T C , is approached from below or above, progressively slows down with the result that the width of the ACR increases. For a given spin, the higher the order of nonlinear susceptibility, the narrower the ACR. These results are in qualitative agreement with the critical behaviour of χ n ( T ) observed in an archetypal ferromagnet.

DOI
10.1209/0295-5075/96/37012

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The intrinsic linear and nonlinear magnetic susceptibilities, χ n , for a ferromagnet of arbitrary spin, calculated using the mean-field approximation, are shown to diverge in the asymptotic critical region (ACR) with the exponent γ n = n γ+( n − 1)β and n =1, 2, ... . This behaviour of χ n in the ACR is consistent with the scaling equation of state. With increasing spin, the divergence in χ n ( T ), as the ferromagnetic-paramagnetic phase transition temperature, T C , is approached from below or above, progressively slows down with the result that the width of the ACR increases. For a given spin, the higher the order of nonlinear susceptibility, the narrower the ACR. These results are in qualitative agreement with the critical behaviour of χ n ( T ) observed in an archetypal ferromagnet.
In the asymptotic critical region of a ferromagnetic spin system, the magnetic susceptibility χ and the correlation length ξ are related by the power law: χ∼ξ2−η, where η is the correlation function exponent. Somewhat farther from the critical point, but still within the critical region, we expect the behavior to be mean-field in character: χ∼ξ2. We give an exact relationship between χ and ξ in terms of a differential renormalization group approach. The leading behavior in the disordered phase is explicitly calculated for the n-component Wilson-Fisher model to O(ε2), ε≡4−d. The result includes both limiting power laws in a smooth crossover function. For the Ising case, n=1, the relationship is extended to the entire critical region, including the ordered phase. In this case, even in the asymptotic limit, the susceptibility differs from ξ2−η by a factor which depends on the magnetization M. This function is calculated to leading order and correctly gives the associated amplitude ratios to O(ε). Applications to the general p-point vertex function are discussed.
The temperature dependence of the paramagnetic zero-field susceptibility chi is calculated for the second-neighbour S=1/2 Heisenberg model and for the Baxter-Wu model, from Pade approximants to existing high-temperature series expansions and by Monte Carlo simulations. The data are well represented by a power law chi T=(t')-y using the critical susceptibility exponent gamma over the whole paramagnetic temperature range when a nonlinear scaling variable t' is used; namely t'=1-Tc/T for the second-neighbour models and t'=1-(Tc/T)2 for the Baxter-Wu model.
A nonlinear susceptibilities (the third derivative of a magnetization mSm_SmS​ by a magnetic field hhh ) of the SSS=1/2 ferromagnetic Heisenberg chain and the classical Heisenberg chain are calculated at low temperatures T.T.T. In both chains the nonlinear susceptibilities diverge as T−6T^{-6}T−6 and a linear susceptibilities diverge as T−2.T^{-2}.T−2. The arbitrary spin SSS Heisenberg ferromagnet [[[ H=∑i=1N{−JSiSi+1−(h/S)Siz}{\cal H} = \sum_{i=1}^{N} \{ - J{\bf S}_{i} {\bf S}_{i+1} - (h/S) S_{i}^{z} \}H=∑i=1N​{−JSi​Si+1​−(h/S)Siz​} (J>0),(J>0),(J>0), ]]] has a scaling relation between mS,m_S,mS​, hhh and T:T:T: mS(T,h)=F(S2Jh/T2).m_S(T,h) = F( S^2 Jh/T^2).mS​(T,h)=F(S2Jh/T2). The scaling function F(x)F(x)F(x)=(2xxx/3)-(44x3x^{3}x3/135) + O(x5x^{5}x5) is common to all values of spin S.S.S.
The analysis of a previous study (Allen and Pathria. Can. J. Phys. 67, 952 (1989)) on finite-size effects in systems with O(n) symmetry [Formula: see text], confined to geometry Ld−d′ × ∞d′ (where d and d′ are continuous variables such that 2 < d′ < d < 4) and subjected to periodic boundary conditions, is extended (i) to include the region of first-order phase transition (T < Tc) as well as the region of second-order phase transition [Formula: see text] and (ii) to allow the presence of an external field H > 0. Predictions, involving both amplitudes and exponents, are made on the magnetization m and susceptibility χ in different regimes of the variables T, H, and L. Analytical verification of the predicted results is carried out in the case of the spherical model of ferromagnetism (n = ∞), and complete agreement is found.
The critical behaviour along the coexistence curve of isotropic ferromagnets is characterized by two diverging correlation lengths since longitudinal and transverse fluctuations of the order parameter become critical. Therefore, a critical theory valid for the whole phase diagram has to incorporate the crossover between a region characterized by one correlation length to another region where two different correlation lengths dominate. Treating this crossover problem entirely within the framework of the trajectory integral method. The author has calculated the equation of state and the susceptibilities. It is shown that coexistence behaviour is governed by a coexistence fixed point which is characterized by the vanishing interaction among the Goldstone modes. Taking the coupling of the critical transverse modes to the longitudinal modes into account, it is shown that coexistence behaviour is characterized by a Fisher renormalization. This central result sheds new light on the symmetry-broken phase in isotropic systems. The asymptotic forms of the equation of state and the susceptibilities at the coexistence curve are expressed by the specific heat exponent alpha t =1/2 for d>or=3 and for all spin dimensions. These results are in perfect agreement with the nonlinear sigma -model. The crossover from coexistence curve behaviour to ordinary critical behaviour is calculated for the equation of state and for the susceptibilities to O( in )( in =4-d).