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arXiv:cond-mat/9408055 · Physical Review Letters

Quantum Description of Spherical Spins

Theo M. Nieuwenhuizen

Published May 22, 1995• 69 citations
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Abstract

The spherical model for spins describes ferromagnetic phase transitions well, but it fails at low temperatures. A quantum version of the spherical model is proposed. It does not induce qualitative changes near the phase transition. However, it produces a physical low temperature behavior. The entropy is non-negative. Model parameters can be adapted to the description of real quantum spins. Several applications are discussed. Zero-temperature quantum phase transitions are analyzed for a ferromagnet and a spin glass in a transverse field. Their crossover exponents are presented.

DOI
10.1103/physrevlett.74.4293

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The spherical model for spins describes ferromagnetic phase transitions well, but it fails at low temperatures. A quantum version of the spherical model is proposed. It does not induce qualitative changes near the phase transition. However, it produces a physical low temperature behavior. The entropy is non-negative. Model parameters can be adapted to the description of real quantum spins. Several applications are discussed. Zero-temperature quantum phase transitions are analyzed for a ferromagnet and a spin glass in a transverse field. Their crossover exponents are presented.
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A generalization of the usual spherical model of spin-glasses is introduced, and its static and dynamic behavior is fully studied. The generalized model is given the name quantum spherical model to contrast it with the usual model which we call the classical spherical model. The results of the quantum model reduce to those of the classical model in appropriate limits. However, the quantum model shows some new and interesting features. The equilibrium entropy of spin-glasses which is negative at sufficiently low temperatures in the classical model is found positive in the quantum version, and it vanishes at zero temperature. The dynamic properties of quantum and classical spherical models are similar except in the infinite time limit when different equilibrium behavior is obtained as expected. The magnetization, order-parameter, and on-site correlations are found to decay exponentially above the transition temperature, and algebraically at and below it in the long-time limit. A closed-form expression for the dynamic susceptibility valid at all temperatures is also obtained. The algebraic decay predicted by our model may serve as a model form of behavior in the interpretation of experimental relaxation data on spin-glasses.
We investigate a quantum version of the spherical model which is obtained from the classical Berlin-Kac spherical model by a simple canonical quantization scheme. We find a complete solution of the model for short-range as well as for long-range interactions. At finite temperatures the critical behavior is the same as in the classical spherical model whereas at zero temperature we find a quantum phase transition characterized by new critical exponents. Based on a functional-integral representation of the partition function the free energy of the model is shown to be equivalent to that of the nonlinear \ensuremath{\sigma} model in the limit of infinite order-parameter dimensionality. \textcopyright{} 1996 The American Physical Society.
We propose a soluble quantum spherical XY ferromagnet with a random field in the boson space. We obtain a general expression of the critical temperature below which the ordered ferromagnet phase appears. The Imry–Ma result concerning the lower critical dimension is recovered, and the critical exponents near the critical temperature are calculated. We show that the random-field fluctuations rather than the quantum fluctuations dominate the phase transition and critical behavior of the system. The entropy vanishes as at low temperatures, contrary to the classical spherical model.
Abstract In this work we investigate properties of a supersymmetric extension of the quantum spherical model from an off-shell formulation directly in the superspace. This is convenient to safely handle the constraint structure of the model in a way compatible with supersymmetry. The model is parametrized by an interaction energy, , which governs the interactions between the superfields of different sites. We briefly discuss some consequences when corresponds to the case of first-neighbor interactions. After computing the partition function via saddle point method for a generic interaction, , we focus in the mean-field version, which reveals an interesting critical behavior. In fact, the mean-field supersymmetric model exhibits a quantum phase transition without breaking supersymmetry at zero temperature, as well as a phase transition at finite temperature with broken supersymmetry. We compute critical exponents of the usual magnetization and susceptibility in both cases of zero and finite temperature. Concerning the susceptibility, there are two regimes in the case of finite temperature characterized by distinct critical exponents. The entropy is well behaved at low temperature, vanishing as .
A mean-field spherical model with random couplings between pairs, quartets, and possibly higher multiplets of spins is considered. It has the same critical behavior as the Sherrington-Kirkpatrick model. It thus exhibits replica symmetry breaking. The order parameter function is solved exactly in the whole low temperature phase. The zero-field-cooled susceptibility remains finite at low TTT. Next a quantum version of the system is considered. Whereas the magnetic properties are not altered qualitatively, the thermodynamics is now regular at small temperatures.