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Progress of Theoretical Physics

Singularity of Nonlinear Response near the Critical Field. I

Masuo Suzuki

Published Nov 1, 1969• 9 citations
Publisher

Abstract

Phenomenological arguments are presented to clarify the singularity of static nonlinear response near the critical magnetic (or electric) field Hc. The singularity of nonlinear susceptibility χ(H) is connected with the expansion of the free energy Φ(m) as a function of a quantity (mc - m), where m is the normalized magnetization and mc its value at the critical field; i.e. if Φ(m) = a0 - a1(mc - m) + a2(mc - m)ψ…, then χ(H) ∼1/(Hc - H)γH; γH = (ψ- 2)/(ψ- 1). That is, for 1 ≪ ψ≪ 2, χ(H) is continuous, for ψ= 2, it is finite but discontinuous at H = Hc, and for ψ> 2, it is divergent. The index γH(or γE) is investigated in the one-dimensional exactly soluble magnetic systems at T = 0 and the two-dimensional ferroelectric (KDP) models at finite temperature. The value of γH is zero for a fermion system with repulsive delta-function interaction. γH is equal to 1/2 for the antiferromagnetic Heisenberg model. In the Hubbard model, γH is equal to 1/2 in the case of half-filled band, and γH is always zero other-wise. In the general two-dimensional ferroelectric models, there exists no critical electric field, though the Wu model has a critical field Ec with γE = 1/2.

DOI
10.1143/ptp.42.1076

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Phenomenological arguments are presented to clarify the singularity of static nonlinear response near the critical magnetic (or electric) field Hc. The singularity of nonlinear susceptibility χ(H) is connected with the expansion of the free energy Φ(m) as a function of a quantity (mc - m), where m is the normalized magnetization and mc its value at the critical field; i.e. if Φ(m) = a0 - a1(mc - m) + a2(mc - m)ψ…, then χ(H) ∼1/(Hc - H)γH; γH = (ψ- 2)/(ψ- 1). That is, for 1 ≪ ψ≪ 2, χ(H) is continuous, for ψ= 2, it is finite but discontinuous at H = Hc, and for ψ> 2, it is divergent. The index γH(or γE) is investigated in the one-dimensional exactly soluble magnetic systems at T = 0 and the two-dimensional ferroelectric (KDP) models at finite temperature. The value of γH is zero for a fermion system with repulsive delta-function interaction. γH is equal to 1/2 for the antiferromagnetic Heisenberg model. In the Hubbard model, γH is equal to 1/2 in the case of half-filled band, and γH is always zero other-wise. In the general two-dimensional ferroelectric models, there exists no critical electric field, though the Wu model has a critical field Ec with γE = 1/2.
Phenomenological arguments on the critical slowing down is presented and “similarity law” is proposed on the indices of the critical slowing down. The similarity law is confirmed in linear spin chains near the critical field and in the kinetic Ising model near the critical temperature. It is exactly shown in the linear spin chains that the critical index of slowing down is different from that of the static susceptibility and that the dynamical susceptibility has a logarithmic singularity with respect to the frequency at the critical field and at zero temperature.
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.
The susceptibility of a quantum X-Y model in a transverse field Gamma is studied in the vicinity of the multicritical point ( Gamma c ,T=0). For dimensions d between two and four we obtain a Hartree type approximation an equation for the corresponding crossover function which is solvable analytically in three dimensions. The critical line Gamma c (T)- Gamma c (0) approximately T psi is asymptotically described by the shift exponent psi =d/2 in accord with self-consistent spin wave theory. For the variable of the crossover scaling function T/( Gamma - Gamma c (T)) phi one obtains the crossover exponent phi =(4-d)/2 which is characteristic for the Gaussian multicritical point. A dimensional-analysis argument suggests that this is valid beyond the approximation.
Previous series expansion studies of the magnetization and susceptibility of the Ising model are extended to the higher derivatives of the free energy with respect to the magnetic field. It is found that just above the critical temperature the 2 n th derivative is given by where in two dimensions γ = 1¾, Δ = 1fraction seven-eighths and in three dimensions γ = 1¼, Δ = 1fraction nine-sixteens. In two dimensions γ is exact and Δ is correct to 0.6%; the three-dimensional indices are correct to 0.2%. Just below the critical temperature where in three dimensions 0.303 <or= β <or= 0.318 and 1·55 <or= Δprime <or= 1.65. If the even derivatives are to have the same index above and below T c , then β = fraction five-sixteens and Δprime = 1fraction nine-sixteens. The latter index lies very close to the lower limit so that, although we cannot rule out Δprime = Δ, Δprime = 1fraction five-eighths is not unlikely. In two dimensions our results are consistent with linearity of the index and, if this is assumed, then β = fraction one-eighths and Δprime = 1fraction seven-eighths exactly and there is symmetry about T c . The amplitudes C 2 n + and C n - are estimated for n = 1 to 6.
Even though the Hubbard model is one of the most fundamental models of highly correlated electrons, analytical and numerical data describing its thermodynamics at nonzero magnetization are relatively scarce. We present a detailed investigation of the thermodynamic properties for the one-dimensional repulsive Hubbard model in the presence of an arbitrary magnetic field for all values of the filling fraction and temperatures as low as $T\ensuremath{\sim}0.005t.$ Our analysis is based on the system of integral equations derived in the quantum transfer matrix framework. We determine the critical exponents of the quantum phase transitions and also provide analytical derivations for some of the universal functions characterizing the thermodynamics in the vicinities of the quantum critical points. Extensive numerical data for the specific heat, susceptibility, compressibility, and entropy are reported. The experimentally relevant double occupancy presents an interesting doubly nonmonotonic temperature dependence at intermediate values of the interaction strength and also at large repulsion and magnetic fields close to the critical value. The susceptibility in zero magnetic field has a logarithmic singularity at low temperatures for all filling factors similar to the behavior of the same quantity in the spin-1/2 isotropic Heisenberg model. We determine the density profiles for a harmonically trapped system and show that while the total density profile seems to depend mainly on the value of chemical potential at the center of the trap the distribution of phases in the inhomogeneous system changes dramatically as we increase the magnetic field.
We study the leading low-temperature dependence of magnetic susceptibility of one-dimensional electrons with a fixed total number of particles at the magnetic fields equal to zero-temperature critical values where the magnetic field induces commensurate-incommensurate quantum phase transitions. For free and repulsively interacting electrons there is only one such critical field corresponding to the transition to the fully polarized state. For attractively interacting electrons besides saturation field there is another critical field equal to the spin gap where zero-temperature magnetization sets in. For all cases, except for the lattice models at half filling, the magnetic susceptibility at critical values of magnetic field has a universal logarithmic temperature dependence, $\ensuremath{\chi}(T)=\ensuremath{\chi}(0)(1+2/ln\phantom{\rule{0.16em}{0ex}}T+\ensuremath{\cdots})$ for $T\ensuremath{\rightarrow}0$.