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arXiv:cond-mat/0304150 · Physical Review E

Quantum spherical spin models

Rubèn Serral Gracià, Theo M. Nieuwenhuizen

Published May 28, 2004• 22 citations
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Abstract

A recently introduced class of quantum spherical spin models is considered in detail. Since the spherical constraint already contains a kinetic part, the Hamiltonian need not have kinetic term. As a consequence, situations with or without momenta in the Hamiltonian can be described, which may lead to different symmetry classes. Two models that show this difference are analyzed. Both models are exactly solvable and their phase diagram is analyzed. A transversal external field leads to a phase transition line that ends in a quantum critical point. The two considered symmetries of the Hamiltonian considered give different critical phenomena in the quantum critical region. The model with momenta is argued to be analog to the large-N limit of an SU(N) Heisenberg ferromagnet, and the model without momenta shares the critical phenomena of an SU(N) Heisenberg antiferromagnet.

DOI
10.1103/physreve.69.056119

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A recently introduced class of quantum spherical spin models is considered in detail. Since the spherical constraint already contains a kinetic part, the Hamiltonian need not have kinetic term. As a consequence, situations with or without momenta in the Hamiltonian can be described, which may lead to different symmetry classes. Two models that show this difference are analyzed. Both models are exactly solvable and their phase diagram is analyzed. A transversal external field leads to a phase transition line that ends in a quantum critical point. The two considered symmetries of the Hamiltonian considered give different critical phenomena in the quantum critical region. The model with momenta is argued to be analog to the large-N limit of an SU(N) Heisenberg ferromagnet, and the model without momenta shares the critical phenomena of an SU(N) Heisenberg antiferromagnet.
Motivated by an analogy with the spin anisotropies in the quantum XY chain and its reformulation in terms of spin-less Majorana fermions, its bosonic analogue, the spin-anisotropic quantum spherical model, is introduced. The exact solution of the model permits to analyse the influence of the spin-anisotropy on the phase diagram and the universality of the critical behaviour in a new way, since the interactions of the quantum spins and their conjugate momenta create new effects. At zero temperature, a quantum critical line is found, which is in the same universality class as the thermal phase transition in the classical spherical model in d+1d+1d+1 dimensions. The location of this quantum critical line shows a re-entrant quantum phase transition for dimensions $1
We study d-dimensional ferromagnetic spherical models with power-law, J(R)\ensuremath{\sim}1/${\mathit{R}}^{\mathit{d}+\mathrm{\ensuremath{\sigma}}}$ (\ensuremath{\sigma}2), or short-range (\ensuremath{\sigma}\ensuremath{\simeq}2) interactions and on-site terms of the form Ds2{\mathit{Ds}}^{2}Ds2-\ensuremath{\Vert}U\ensuremath{\Vert}s4{\mathit{s}}^{4}s4+Vs6{\mathit{Vs}}^{6}Vs6, and establish that the phase diagrams for all d and \ensuremath{\sigma} with \ensuremath{\sigma}d2\ensuremath{\sigma}, may exhibit critical end points terminating nonclassical critical lines; tricritical points may arise only for 3/2\ensuremath{\sigma}\ensuremath{\le}d. The phase boundary ${\mathit{D}}_{\mathrm{\ensuremath{\sigma}}}$(T,h), where h is the magnetic field, between the noncritical, spectator phase and the ordering and critical phase is shown to exhibit singularities at the end point with amplitudes obeying the universal ratio laws advanced on phenomenological grounds in Part I.
The quantum-rotors model can be regarded as an effective model for the low-temperature behavior of the quantum Heisenberg antiferromagnets. Here, we consider a ddd-dimensional model in the spherical approximation confined to a general geometry of the form ${L}^{d\ensuremath{-}{d}^{\ensuremath{'}}}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{{d}^{\ensuremath{'}}}\ifmmode\times\else\texttimes\fi{}{L}_{\ensuremath{\tau}}^{z}$ (L(L(L-linear space size and ${L}_{\ensuremath{\tau}}$-temporal size) and subjected to periodic boundary conditions. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions 1<d<31<d<31<d<3 and $0<~{d}^{\ensuremath{'}}<~d$ a detailed analysis, in terms of the special functions of classical mathematics, for the susceptibility and the equation of state is given. Particular attention is paid to the two-dimensional case.
Part II analyzed the properties of quantum Ising and rotor models in some detail at T = 0. We related the quantum phase transitions in these models to the N -component relativistic field theory (2.11), and used it to understand the critical properties. The purpose of Part III is to extend this understanding to T > 0. We will demonstrate that the T = 0 quantum critical point controls a wide “quantum critical” region at T > 0, as illustrated in Fig. 1.2. We are especially interested in dynamic properties in this region: an interesting feature is that many “friction” coefficients are universal and depend only on fundamental constants of nature. We also explore the other regions of the phase diagrams in Fig. 1.2, including behavior in the vicinity of the phase transition at T > 0. We begin this chapter by extending results of the d = 1 quantum Ising model of Chapter 5 to T > 0. This model does not have any phase transition at any T > 0, and so the crossover structure of the phase diagram is in the class in Fig. 1.2a. Phase transitions at T > 0 appear in models to be studied in the following chapter.
A variational approach based on Bogoliubov inequality for the free energy is employed in order to treat the quantum spin-1 anisotropic ferromagnetic Heisenberg model in the presence of a crystal field. Within the Bogoliubov scheme an improved pair approximation has been used. The temperature-dependent thermodynamic functions have been obtained and provide much better results than the previous simple mean-field scheme. In one dimension, which is still nonintegrable for quantum spin-1, we get the exact results in the classical limit, or near-exact results in the quantum case, for the free energy, magnetization, and quadrupole moment, as well for the transition temperature. In two and three dimensions the corresponding global phase diagrams have been obtained as a function of the parameters of the Hamiltonian. First-order transition lines, second-order transition lines, tricritical and tetracritical points, and critical endpoints have been located through the analysis of the minimum of the Helmholtz free energy and a Landau-like expansion in the approximated free energy. Only first-order quantum transitions have been found at zero temperature. Limiting cases, such as isotropic Heisenberg, Blume-Capel, and Ising models, have been analyzed and compared to previous results obtained from other analytical approaches as well as from Monte Carlo simulations.
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.