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Physical review. B, Condensed matter

Quantum version of a spherical model: Crossover from quantum to classical critical behavior

Thomas Vojta

Published Jan 1, 1996• 102 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.53.710

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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 consider the influence of a quenched random field on the critical behavior of a quantum spherical model at the zero-temperature quantum phase transition. We find a complete solution of the model for arbitrary translationally invariant pair interactions. It turns out that the critical behavior for zero as well as finite temperatures is dominated by static random field fluctuations rather than by quantum or thermal fluctuations. Therefore the critical behavior close to the zero-temperature quantum phase transition is identical to that close to a finite-temperature transition. The system does not show a crossover from quantum to classical behavior. \textcopyright{} 1996 The American Physical Society.
Abstract The spherical Berlin-Katz model is considered in the framework of the epsilon expansion in one-dimensional and two-dimensional space. For the two-dimensional and threedimensional cases in this model, an exact solution was previously obtained in the presence of a field, and for the two-dimensional case the critical temperature is zero, that is, a “quantum” phase transition is observed. On the other hand, the epsilon expansion of critical exponents with an arbitrary number of order parameter components is known. This approach is consistent with the scaling paradigm. Some critical exponents are found for the spherical model in one-and twodimensional space in accordance with the generalized scaling paradigm and the ideas of quantum phase transitions. A new formula is proposed for the critical heat capacity exponent, which depends on the dynamic index z, at a critical temperature equal to zero. An expression is proposed for the order of phase transition with a change in temperature (developing the approach of R. Baxter), which also depends on the z index. An interpolation formula is presented for the effective dimension of space, which is valid for both a positive critical temperature and a critical temperature equal to zero. This formula is general. Transitions with a change in the field in a spherical model at absolute zero are also considered.
Abstract This work is dedicated to the study of a supersymmetric quantum spherical spin system with short-range interactions. We examine the critical properties both a zero and finite temperature. The model undergoes a quantum phase transition at zero temperature without breaking supersymmetry. At finite temperature the supersymmetry is broken and the system exhibits a thermal phase transition. We determine the critical dimensions and compute critical exponents. In particular, we find that the model is characterized by a dynamical critical exponent z = 2. We also investigate properties of correlations in the one-dimensional lattice. Finally, we explore the connection with a nonrelativistic version of the supersymmetric nonlinear sigma model and show that it is equivalent to the system of spherical spins in the large N limit.
Abstract The entropy, the internal energy and the specific heat of a d-dimensional quantum spherical model with a long-range interaction, the decreasing of which is controlled by a parameter σ (0 < σ ≤ 2), are studied close to its quantum critical point in the context of the finite-size scaling (FSS) theory for d = σ. The obtained specific heat critical exponent σ does not depend on σ and satisfies the quantum hyperscaling relation. Based on the derived FSS forms and asymptotes of the universal scaling functions of the considered quantities, the leading temperature dependencies of the entropy, the internal energy and the specific heat are obtained in both the renormalized classical and the quantum disordered regions of the phase diagram. It has been shown that when the temperature T → +0, the entropy and the specific heat in the renormalized classical region tend to zero as powers of T, while in the quantum disordered region they tend to zero exponentially.
A d-dimensional quantum spherical model with a long-range interaction (decreasing at long distances r as r−d−σ, 0 <σ ≤ 2) is considered close to its quantum critical point. The low-temperature behaviors of the entropy, the internal energy and the specific heat capacity are studied for σ/2 < d < 3σ/2 in the context of the finite-size scaling (FSS) theory. For a σ-dimensional system and a d-dimensional system with short-range interaction (σ = 2) the results for the corresponding critical amplitudes as functions of the dimensionality of the system are presented.
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 .