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Journal of Physics Condensed Matter

The tricritical point in the quantum IsingS= 1 spin glass with biaxial crystal-field effects

Zbigniew Domański, T. K. Kopeć

Published Jun 13, 2000• 6 citations
Publisher

Abstract

We study a quantum generalization of the infinite-range Sherrington-Kirkpatrick spin-glass model with biaxial crystal-field effects described by two uniaxial anisotropy parameters D x and D y . For spin dimensionality S = 1 we report an analytical and numerical analysis in the ( T , D x , D y ) parameter space (with T being the temperature). For D ≡ D x = D y the model effectively becomes classical and identical with the crystal-field-split spin-glass Ising model (introduced by Ghatak and Sherrington) showing a discontinuous phase transition to the spin-glass phase on a portion of the T - D line.

DOI
10.1088/0953-8984/12/26/318

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We study a quantum generalization of the infinite-range Sherrington-Kirkpatrick spin-glass model with biaxial crystal-field effects described by two uniaxial anisotropy parameters D x and D y . For spin dimensionality S = 1 we report an analytical and numerical analysis in the ( T , D x , D y ) parameter space (with T being the temperature). For D ≡ D x = D y the model effectively becomes classical and identical with the crystal-field-split spin-glass Ising model (introduced by Ghatak and Sherrington) showing a discontinuous phase transition to the spin-glass phase on a portion of the T - D line.
The phase diagram of the spin 1 quantum van Hemmen spin glass model with biaxial crystal-field anisotropy and a transverse magnetic field is studied. We investigate the effects of the two transverse crystal-field parameters (Dx{D}_{x}Dx​ and Dy{D}_{y}Dy​) and transverse field $(\ensuremath{\Omega})$ on the phase diagrams in the $T\ensuremath{-}{D}_{x}$ plane for various values of Dy{D}_{y}Dy​ and $\ensuremath{\Omega}$. Tricritical and reentrant behavior can be seen for appropriate ranges of these parameters.
A generalisation of the Sherrington-Kirkpatrick Ising spin glass model to spins S z =0,+or-1, . . . +or-S with exchange J ij S i z S j z , where the J ij are randomly distributed, and crystal field D(S i z ) 2 , is examined with regard to the character of its transitions and the stability of its phases, in order to clarify and correct earlier work. Both continuous and discontinuous transitions are found. Conventional Parisi theory with renormalised parameters suffices near the continuous transition away from the tricritical point, but near the tricritical point and the first-order transitions, new effects are predicted.
We propose an expanded spin-glass model, called the quantum Ghatak-Sherrington model, which considers spin-1 quantum spin operators in a crystal field and in a transverse field. The analytic solutions and phase diagrams of this model are obtained by using the one-step replica symmetry-breaking ansatz under the static approximation. Our results represent the splitting within one spin-glass (SG) phase depending on the values of crystal and transverse fields. The two separated SG phases, characterized by a density of filled states, show certain differences in their shapes and phase boundaries. Such SG splitting becomes more distinctive when the degeneracy of the empty states of spins is larger than one of their filled states.
I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).
The p-component quantum Potts spin-glass model is considered in the mean-field approximation via the thermofield-dynamics (TFD) method. Quantum effects appear as the action of a transverse field \ensuremath{\Gamma} which represents the tunneling frequency between different discrete Potts states. Applying the Sherrington-Kirkpatrick infinite-interaction-range scheme in the TFD formalism, we find a continuous spin-glass transition for p\ensuremath{\le}6 and discontinuous for p>6 at the field-dependent critical line Tc{\mathit{T}}_{\mathit{c}}Tc​(\ensuremath{\Gamma}), as demonstrated in the jump of the temperature dependence of the spin-glass order parameter. Interestingly, in contrast to the statics the dynamical theory leads to a continuous slowing down as Tc{\mathit{T}}_{\mathit{c}}Tc​(\ensuremath{\Gamma}) is approached from the ergodic side, indicating the existence of the infinitely-long-lived metastable states.