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Progress of Theoretical Physics

Divergence of the Transverse Nonlinear Susceptibility along the GT Line in Heisenberg Spin Glasses

Masuo Suzuki

Published Mar 1, 1985• 3 citations
Publisher

Abstract

It it shown phenomenologically that the transverse nonlinear susceptibility x2 along the Gabay-Toulouse line in m-component spin glasses, behaves as x2∝(T-Tsg(Hz))-γ near the transition point Tsg(Hz), and that Tsg(Hz)-Tsg(0) ∝Hz4 β/(β+ γ) with a magnetic field Hz and the critical exponent β of the spin glass order parameter.
DOI
10.1143/ptp.73.830

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A general phenomenological theory of spin-glasses is presented, which predicts the weak singularities (like cusps) of susceptibilities and specific heat near the transition point TSG and also predicts generally the divergence of the second derivative of the nonlinear susceptibility with respect to a magnetic field in zero field at TSG. A similar second derivative of specific heat is also shown to diverge at TSG except for the molecular field theory. Scaling equations of state for magnetization m and spin-glass order-parameter q are derived, which yield new scaling relations of critical exponents for m(T), q(T), χ0(T), C0(T) and the corresponding nonlinear quantities. These general results explain qualitatively very well the magnetic field dependence of magnetization and specific heat observed experimentally. A non-uniform phenomenological Hamiltonian is also proposed for the renormalization group approach, which yields the critical dimensionality dc = 6, as usual. Intermediate random statistics and real replica method are proposed. There is also presented an exactly soluble model showing a spin-glass like phase transition.
We examine the quantum phase transition at zero temperature between paramagnetic and spin-glass-ordered phases as the strength of a uniform transverse field, \ensuremath{\Gamma}, is varied. At the critical point, ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$, the spin autocorrelation function decays with time, t, as ${\mathit{t}}^{\mathrm{\ensuremath{-}}2}$. As the transition is approached from the paramagnetic phase the nonlinear susceptibility diverges as {\ensuremath{\Vert}ln(\ensuremath{\Gamma}-${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$)\ensuremath{\Vert}/(\ensuremath{\Gamma}-${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$)${\mathrm{}}}^{1/2}$.
We present a scaling theory for the low-temperature phase transition of the Ising spin glass in a transverse magnetic field (\ensuremath{\Gamma}). The theory provides relations and bounds on some critical exponents, which are supported by results from actual calculations. A renormalization-group analysis of the zero-temperature transition in the presence of a longitudinal symmetry-breaking field enables us to estimate the critical exponents \ensuremath{\beta} and \ensuremath{\gamma}, associated with the Edwards-Anderson order parameter and the nonlinear susceptibility, respectively. We have found no indications of the transition being of first order.
We examine three-dimensional ± J Heisenberg models with and without random anisotropies in a magnetic field. We calculate both the stiffness exponent θ s at zero temperature and spin-glass correlation lengths for the longitudinal and transverse spin components at finite temperatures. We suggest that, contrary to a chirality scenario predicted by Kawamura and his co-workers (Kawamura 1992 Phys. Rev. Lett. 68 3785, 1998 Phys. Rev. Lett. 80 5421, Hukushima and Kawamura 2000 Phys. Rev. E 61 R1008), a Gabay–Toulouse phase transition might occur when the anisotropies are absent, although the results, except for the correlation length of the chirality for larger sizes, suggest no phase transition when they are present.