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arXiv:cond-mat/9501013 · Journal of the Physical Society of Japan

Universal Finite-Size Scaling Function of the Ferromagnetic Heisenberg Chain in a Magnetic Field

Hiroaki Nakamura, Naomichi Hatano, Minoru Takahashi

Published Jun 15, 1995• 7 citations
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Abstract

The finite-size scaling function of the magnetization of the ferromagnetic Heisenberg chain is argued to be universal with respect to the magnitude of the spin. The finite-size scaling function is given explicitly by an analytical calculation in the classical limit $S=\infty.$ The universality is checked for $S=1/2$ and $1$ by means of numerical calculations. Critical exponents are obtained as well. It is concluded that this universal scaling function originates in the universal behavior of the correlation function.

DOI
10.1143/jpsj.64.1955

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The finite-size scaling function of the magnetization of the ferromagnetic Heisenberg chain is argued to be universal with respect to the magnitude of the spin. The finite-size scaling function is given explicitly by an analytical calculation in the classical limit S=∞.S=\infty.S=∞. The universality is checked for S=1/2S=1/2S=1/2 and 111 by means of numerical calculations. Critical exponents are obtained as well. It is concluded that this universal scaling function originates in the universal behavior of the correlation function.
The finite-size scaling function of the nonlinear susceptibility of the ferromagnetic Heisenberg chain is given explicitly. It is conjectured that the scaling function is universal for any values of SSS. The conjecture is based on the exact solution of the nonlinear susceptibility for S=∞S=\inftyS=∞, and numerical calculations for S=1/2S=1/2S=1/2 and S=1S=1S=1.
A noncritical scaling model is reported to describe the behavior of low-dimensional magnetic systems. Unlike the classical phase transition approach, nonsingular solutions are deduced that are worthwhile when correlations exist, but which are not large enough to trigger a long range order at $T\ensuremath{\ne}0$. The notion of ``universality class'' is extended to these systems that stand at or below a lower critical dimensionality, and illustrated for the quantum Heisenberg ferromagnetic chain. In this system, the TTT dependence of $\ensuremath{\chi}T$ exhibits a power law with a negative critical exponent, $\ensuremath{-}1.25S$, and a negative critical temperature.
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 scaling limit of the Heisenberg XXZ spin chain at zero magnetic field is studied in the gapped antiferromagnetic phase. For a spin-chain ring having N_{x} sites, the universal Casimir scaling function, which characterizes the leading finite-size correction term in the large-N_{x} expansion of the ground-state energy, is calculated by numerical solution of the nonlinear integral equation of the convolution type. It is shown that the same scaling function describes the temperature dependence of the free energy of the infinite XXZ chain at low enough temperatures in the gapped scaling regime.