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arXiv:1403.0978 · Physical Review E

Spin-glass splitting in the quantum Ghatak-Sherrington model

Dohyun Kim

Published Apr 29, 2013• 1 citation
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Abstract

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.

DOI
10.1103/physreve.87.042127

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•Jun 1, 2007•0 citations
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.
We analyze the infinite range Ising spin glass in a transverse-field $\ensuremath{\Gamma}$ below the critical temperature by a one step replica symmetry breaking theory. The set of nnn replicas is divided in rrr blocks of mmm replicas each. We present results for different values of the block-size parameter mmm. The spin operators are represented by bilinear combinations of fermionic fields and we compare the results of two models: In the four-states (4-S) model the diagonal Siz{S}_{i}^{z}Siz​ operator has two unphysical vanishing eigenvalues, that are suppressed by a restraint in the two-states (2-S) model. In the static approximation we obtain qualitatively similar results for both models. They both exhibit a critical temperature ${T}_{c}(\ensuremath{\Gamma})$ that decreases when $\ensuremath{\Gamma}$ increases, until it reaches a quantum critical point at the same value of ${\ensuremath{\Gamma}}_{c}$ and they are both unstable under replica symmetry breaking in the whole spin glass phase. Below the critical temperature we present results for the order parameters and free energy.
We solve the fermionic version of the Ising spin glass for arbitrary filling µ and temperature T taking into account replica symmetry breaking. Using a simple exact mapping from µ to the anisotropy parameter D , we also obtain the solution of the S = 1 Sherrington-Kirkpatrick model. An analytic expression for T = 0 gives an improved critical value for the first-order phase transition. We revisit the question of stability against replica-diagonal fluctuations and find that the appearance of complex eigenvalues of the Almeida-Thouless matrix is not an artefact of the replica-symmetric approximation.
We investigate the Ghatak-Sherrington model with S ≥1, including S =∞, in a framework of the replica method, namely in terms of the replica symmetry (RS) solutions, the de Almeida and Thouless (AT) stability analysis and the one-step replica symmetry breaking (RSB) solutions. We find that there is a first order transition of van der Waals type between the paramagnetic phase and the spin glass phase and also a first order transition between the RS solution and the RSB solution in the model with integral S in some range of anisotropy parameter, while there is only a continuous transition in the model with half-integral S , including S =∞. We find that the model with integral S has a critical end point under the external magnetic field. We obtain phase diagrams as for the AT stability for the model under the external magnetic field.
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 replica-symmetry-breaking (RSB) solution of the infinite-range Ising spin glass in the presence of a transverse field is obtained. The quenched free energy and the phase boundary of the glass transition temperature versus the transverse field are calculated at first-step RSB without using the static approximation. We demonstrate that replica symmetry (RS) has to be broken in the spin-glass phase by comparing the free energies of the RSB and RS solutions. No evidence is found to support an intermediate spin-glass phase with replica symmetry.
In the present work, a fermionic infinite-range Ising spin glass (SG) model with a transverse field Γ is studied. The model is analyzed in a one-lattice structure (quantum SG) and in a two-sublattice structure (SG/antiferromagnetism (AF) competition). In these two cases, the exchange interactions are random variables that follow Gaussian probability distributions (SK interactions). For the two-sublattice model, spins of distinct sublattices can couple with an SK-type interaction with an AF average. For a general case of this model, the disordered interactions can also occur with spins of same sublattices. The problem is formulated in a Grassmann path integral formalism within the static ansatz. The replica method is applied to treat the disorder, in which the replica symmetry (RS) and one-step replica symmetry breaking (1P-RSB) approximations are used. The one-lattice model results provide evidences that the 1S-RSB can be used to improve the physical description of the ordered quantum SG phase treated by the replica method. In the two-sublattice model, a parallel magnetic field H breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase (SG+AF) characterizing an asymmetric RSB region. In this region, intra-sublattice disordered interactions V increase the difference between the RSB solutions of each sublattice. The freezing temperature (Tf ) shows a higher increase with H when V enhances. A discontinuous phase transition from the replica symmetry (RS) solution to the RSB solution can appear with the presence of an intra-sublattice ferromagnetic average coupling. The Γ field introduces a quantum spin flip mechanism that suppresses the magnetic orders leading them to quantum critical points. Results suggest that the quantum effects are not able to restore the RS solution. Therefore, the presence of both fields, Γ and H, produces effects that are in competition on Tf for instance. However, in the asymmetric RSB region, Γ can produce a stable RS solution at any finite temperature for a particular sublattice while the other sublattice still presents RSB solution for the special case in which only the intra-sublattice spins couple with disordered interactions.