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Communications in Theoretical Physics

The q -Deformed SU(2) Quantum Ferromagnetic Model

Zhong‐Yi Lu, Yan Hong

Published Mar 1, 1993• 0 citations
Publisher

Abstract

A q -deformed SU(2) Heisenberg (3-dimensional) spin model is set up, and the q -deformed spin-wave solution is obtained through the q -analogous Holstein-Primakoff transformation. The result is given for small γ = ln q , which is the quantity characterizing the nonlinearity of the Hamiltonian. A mean-field treatment is arranged to preserve (at least some of) the nonlinearity, and the ordinary ferromagnet ground state is shown as the exact ground state of the new system. Interesting results are obtained for this nonlinear model:(i) There is an energy gap between the ground state and the first excited one, thus the ground state is stable under small perturbation of the background; (ii) the specific heat per volume is modified by a small term proportional to the (1/2)-th power of temperature and the square of γ , which is qualitatively different from the conventional model, and (iii) the magnetization M(T) is modified by a factor that depends on γ .

DOI
10.1088/0253-6102/19/2/169

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A q -deformed SU(2) Heisenberg (3-dimensional) spin model is set up, and the q -deformed spin-wave solution is obtained through the q -analogous Holstein-Primakoff transformation. The result is given for small γ = ln q , which is the quantity characterizing the nonlinearity of the Hamiltonian. A mean-field treatment is arranged to preserve (at least some of) the nonlinearity, and the ordinary ferromagnet ground state is shown as the exact ground state of the new system. Interesting results are obtained for this nonlinear model:(i) There is an energy gap between the ground state and the first excited one, thus the ground state is stable under small perturbation of the background; (ii) the specific heat per volume is modified by a small term proportional to the (1/2)-th power of temperature and the square of γ , which is qualitatively different from the conventional model, and (iii) the magnetization M(T) is modified by a factor that depends on γ .
In this paper we set up a q -analogous Holstein-Primakoff transformation (HPT) and obtain the q -deformed spin-wave solutions. We apply a mean-field treatment that is arranged to preserve some nonlinearity of the Hamiltonian, and the results turn out to be interesting, e.g., a gapful spectrum of the elementary excitation spectrum is found, which is different from the spin-wave results for the conventionial model, and the specific heat is therefore modified greatly. The properties of the Néel state and the ground state are investigated.
A modified spin-wave theory is applied to the one- and two-dimensional quantum Heisenberg model with long-range ferromagnetic interactions proportional to ${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathit{p}}$[scrH=-1/2${\mathit{tsum}}_{\mathit{i}\mathrm{\ensuremath{\ne}}\mathit{j}}$ (J0{\mathit{J}}_{0}J0​/rijp{\mathit{r}}_{\mathit{i}\mathit{j}}^{\mathit{p}}rijp​)Si{\mathbf{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Sj{\mathbf{S}}_{\mathit{j}}Sj​]. It is shown that for dp2d there exists a phase transition at finite temperatures; the critical temperature is estimated. The susceptibility and the specific heat are obtained at low temperatures. Our results for d=1, p=2, and S=1/2 agree with the exact solution of the Haldane-Shastry model. For d=1, we find a scaling relation \ensuremath{\alpha}+1=${\ensuremath{\gamma}}^{\mathrm{\ensuremath{-}}1}$ when 2p3.
In the path integral formulation of the partition function of quantum spin models, most current treatments employ the so-called static approximation to simplify the process of summing over all possible paths. Although sufficient for studying the thermodynamic aspects of the system, static approximation ignores the contributions made by time-dependent, or non-static, fluctuations in the paths of the path integral. This non-static component is very small relative to the static part, and its careful treatment is necessary for the calculation of small non-extensive quantities such as the energy gap within the path integral framework. We propose a formalism for incorporating non-static effects into the path integral calculation of a class of spin models whose partition functions are reducible to the trace of a single spin (often known as the effective Hamiltonian). We first show that the time-dependent behavior of the single spin trace is governed by the Pauli equation. Time-dependent perturbation theory is used to obtain a perturbative expansion of the solution of the Pauli equation, and then for the single spin trace. This gives us a perturbative expansion of the path integral which can be integrated systematically using standard techniques. In this paper, we develop the theoretical framework outlined above in detail and apply it to a simple ordered spin model, the infinite-range ferromagnetic Ising model in a transverse field. We calculated two non-extensive quantites with this non-static approach: the N0N^0N0 and N−1N^{-1}N−1 terms of the ground-state energy (NNN=number of spins) and the energy gap between the ground and first-excited states. We checked our results by comparing with those of Holstein-Primakoff transform and numerical diagonalization of the Hamiltonian. The application of the method to quantum spin-glasses is briefly discussed.
In terms of spinless fermions and spin waves, we describe the magnetic properties of a spin-1/2 ferromagnetic-antiferromagnetic bond-alternating chain which behaves as a Haldane-gap antiferromagnet. On the one hand, we employ the Jordan–Wigner transformation and treat the fermionic Hamiltonian within the Hartree–Fock approximation. On the other hand, we employ the Holstein–Primakoff transformation and modify the conventional spin-wave theory so as to restore the sublattice symmetry. We calculate the excitation gap, the specific heat, the magnetic susceptibility, magnetization curves, and the nuclear spin-lattice relaxation rate with varying bond alternation. These schemes are further applied to a bond-alternating tetramerized chain which behaves as a ferrimagnet. The fermionic language is particularly stressed as a useful tool for investigating one-dimensional spin-gapped antiferromagnets, while the bosonic one works better for ferrimagnets.
Quantum-deformation of the vibrational symmetry SU(3) supset SU(2) supset SO(2) of the two-dimensional SU(3) interacting boson model is studied from the point of view of the breaking of the dynamical symmetry. It is shown that, in agreement with the earlier work of Gupta et al. (1992) for the one-dimensional problem, the real and complex (more so the complex) q-deformations take this dynamical symmetry very close to the other (rotational) dynamical symmetry SU(3) supset SO(3) supset SO(2) of the two-dimensional SU(3) group. Also, the intermediate results of mixing these two symmetries are obtained, since q-deformations vary continuously. The purely imaginary q-deformation in SU q (3) supset SU q (2) supset SO q (2) restores the SU(3) symmetry for a parameter value larger than 1. Applications to real nuclei are also made.
The Schwinger-boson mean-field theory is applied to the quantum ferrimagnetic Heisenberg chain. There is a ferrimagnetic long-range order in the ground state. We observe two branches of the low-lying excitation and calculate the spin reduction, the gap of the antiferromagnetic branch, and the spin fluctuation at T=0K.T=0 \mathrm{K}.T=0K. These results agree with the established numerical results quite well. At finite temperatures, the long-range order is destroyed because of the disappearance of the Bose condensation. The thermodynamic observables, such as the free energy, magnetic susceptibility, specific heat, and the spin correlation at T>0K,T>0 \mathrm{K},T>0K, are calculated. The $T{\ensuremath{\chi}}_{\mathrm{uni}}$ has a minimum at intermediate temperatures and the spin-correlation length behaves as ${T}^{\ensuremath{-}1}$ at low temperatures. These qualitatively agree with the numerical results and the difference is small at low temperatures.