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Progress of Theoretical Physics

Da Costa-Nobre-Yokoi Model with Spin S

Katsuki Katayama, T. Horiguchi

Published May 1, 2000• 0 citations
Publisher

Abstract

An infinite-ranged Ising spin glass model with spin S, including S=∞, with uniform biquadratic interactions is investigated using replica symmetry (RS) solutions, de Almeida and Thouless (AT) stability analyses, and one-step replica symmetry breaking (RSB) solutions. We find that the model with integral spin S and the model with half-integral spin S exhibit different thermodynamic behavior. Namely, we show that for the model with integral spin S, certain physical quantities display two-sublattice structure, but that for the model with half-integral spin S, including S=∞, this is not the case.

DOI
10.1143/ptp.103.881

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An infinite-ranged Ising spin glass model with spin S, including S=∞, with uniform biquadratic interactions is investigated using replica symmetry (RS) solutions, de Almeida and Thouless (AT) stability analyses, and one-step replica symmetry breaking (RSB) solutions. We find that the model with integral spin S and the model with half-integral spin S exhibit different thermodynamic behavior. Namely, we show that for the model with integral spin S, certain physical quantities display two-sublattice structure, but that for the model with half-integral spin S, including S=∞, this is not the case.
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.
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.
The one-step replica symmetry breaking (RSB) is used to study a two-sublattice fermionic infinite-range Ising spin-glass (SG) model in a transverse field $\ensuremath{\Gamma}$. The problem is formulated in a Grassmann path integral formalism within the static approximation. In this model, a parallel magnetic field HHH breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase (SG+AF)(\mathrm{SG}+\mathrm{AF})(SG+AF) characterizing an asymmetric RSB region. In this region, intrasublattice disordered interactions VVV increase the difference between the RSB solutions of each sublattice. The freezing temperature shows a higher increase with HHH when VVV enhances. A discontinuous phase transition from the replica symmetry (RS) solution to the RSB solution can appear with the presence of an intrasublattice ferromagnetic average coupling. The $\ensuremath{\Gamma}$ 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. However, in the asymmetric RSB region, $\ensuremath{\Gamma}$ 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 intrasublattice spins couple with disordered interactions.
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 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.
We determine the low-energy spectrum and Parisi replica-symmetry-breaking function for the spin-glass phase of the quantum Ising model with infinite-range random exchange interactions and transverse and longitudinal ( h ) fields. We show that, for all h , the spin-glass state has full replica symmetry breaking and the local spin spectrum is gapless, with a spectral density that vanishes linearly with frequency. These results are obtained using an action functional—argued to yield exact results at low frequencies—that expands in powers of a spin-glass order parameter, which is bilocal in time, and a matrix in replica space. We also present the exact solution of the infinite-range spherical quantum p -rotor model at nonzero h : here, the spin-glass state has one-step replica symmetry breaking and gaplessness only appears after imposition of an additional marginal stability condition. Possible connections to experiments on random arrays of trapped Rydberg atoms are noted. Published by the American Physical Society 2024