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arXiv:cond-mat/9506087 · Journal of the Physical Society of Japan

Universal Finite-Size Scaling Function of the Ferromagnetic Heisenberg Chain in a Magnetic Field. II –Nonlinear Susceptibility–

Hiroaki Nakamura, Naomichi Hatano, Minoru Takahashi

Published Nov 15, 1995• 7 citations
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Abstract

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 $S$. The conjecture is based on the exact solution of the nonlinear susceptibility for $S=\infty$, and numerical calculations for $S=1/2$ and $S=1$.

DOI
10.1143/jpsj.64.4142

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The phenomenological expression $\ensuremath{\chi}T∕(N{g}^{2}{\ensuremath{\mu}}_{B}^{2}∕k)={C}_{1n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{1n}∕T)+{C}_{2n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{2n}∕T)$ describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins $S=1$, whose size $n$ is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For $n$ large, the coefficients of the expression converge towards ${C}_{1}=0.125$, ${W}_{1}=0.451J$, ${C}_{2}=0.564$, ${W}_{2}=1.793J$ ($J$ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at $T=0$ and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of ${\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}$ and of $\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}$ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.

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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.
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.
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.
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.
Finite-size scaling (phenomenological renormalization) techniques are trusted and widely applied in low-dimensional magnetism and, particularly, in lattice gauge field theory. Recently, investigations have begun which subject the theoretical basis to systematic and intensive scrutiny to determine the validity of finite-size scaling in a variety of situations. The 2D ANNNI model is an example of a situation where finite-size scaling methods encounter difficulty, related to the occurrence of a disorder line (one-dimensional line). A second example concerns the behavior of the spin-1/2 antiferromagnetic XXZ model where the T=0 critical behavior is exactly known and features an essential singularity at the isotropic Heisenberg point. Standard finite-size scaling techniques do not convincingly reproduce the exact phase behavior and this is attributable to the essential singularity. The point is relevant in connection with a finite-size scaling analysis of a spin-one antiferromagnetic XXZ model, which claims to support a conjecture by Haldane that the T=0 phase behavior of integer-spin Heisenberg chains is significantly different from that of half-integer-spin Heisenberg chains.
The susceptibility of simple-cubic-lattice ferromagnetic Ising films of nnn two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for n=3,4,…,14n=3,4,\dots{},14n=3,4,…,14 layers with periodic boundary conditions, and n=3,4,…,10n=3,4,\dots{},10n=3,4,…,10 layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.
The phenomenological expression $\ensuremath{\chi}T∕(N{g}^{2}{\ensuremath{\mu}}_{B}^{2}∕k)={C}_{1n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{1n}∕T)+{C}_{2n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{2n}∕T)$ describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins S=1S=1S=1, whose size nnn is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For nnn large, the coefficients of the expression converge towards C1=0.125{C}_{1}=0.125C1​=0.125, W1=0.451J{W}_{1}=0.451JW1​=0.451J, C2=0.564{C}_{2}=0.564C2​=0.564, W2=1.793J{W}_{2}=1.793JW2​=1.793J (JJJ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at T=0T=0T=0 and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of Y2BaNiO5{\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}Y2​BaNiO5​ and of Ni(C2H8N2)2NO2ClO4\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}Ni(C2​H8​N2​)2​NO2​ClO4​ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.