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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Green's-function method for the O(N) spin system in 1+1 dimensions

Tetsuo Matsui

Published Apr 15, 1984• 2 citations
Publisher

Abstract

The Green's-function method is employed to study the two-dimensional system of classical $\mathrm{O}(N)$-vector spins. The second-order double-time retarded Green's functions of the corresponding quantum Hamiltonian system on a one-dimensional lattice are decoupled approximately in a selfconsistent manner. The expressions for the mass gap and the spin-correlation function thus obtained reproduce the known exact results in the large-$N$ limit and in the strong-coupling region. Some directions are discussed to improve the present method in the region of weak coupling with $N$ not large.

DOI
10.1103/physrevd.29.1757

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The Green's-function method is employed to study the two-dimensional system of classical O(N)\mathrm{O}(N)O(N)-vector spins. The second-order double-time retarded Green's functions of the corresponding quantum Hamiltonian system on a one-dimensional lattice are decoupled approximately in a selfconsistent manner. The expressions for the mass gap and the spin-correlation function thus obtained reproduce the known exact results in the large-NNN limit and in the strong-coupling region. Some directions are discussed to improve the present method in the region of weak coupling with NNN not large.
The one-dimensional spin-1 Heisenberg antiferromagnetic chains are studied by a generalized two-time Green's-function method at finite temperature. 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 is set to zero at each site to guarantee the lack of the long-range order in the one-dimensional Heisenberg system. The Green's functions are decoupled in terms of the correlation functions of spin operators. A set of self-consistent equations of the correlation functions are derived and solved numerically. Correlation functions of spin operators, thermodynamic properties such as the internal energy, and specific heat in the entire temperature region are obtained. The Haldane gap in the excitation spectrum appears naturally in the analytical result. When k is zero, there is a gap 2\ensuremath{\Delta}, which is 1.0J and close to the theoretical result of White and Huse. And there is a broad maximum in the temperature dependence of the specific heat. The properties of the thermodynamic quantities are consistent with the numerical results on the finite chains. \textcopyright{} 1996 The American Physical Society.
Recent lattice Hamiltonian strong-coupling calculations for the (1+1)-dimensional Ising, O(2), O(3), and O(4) models are extended to d+1d+1d+1 dimensions and to general O(n)\mathrm{O}(n)O(n) symmetry. This allows a nontrivial numerical check on the rate of convergence of the strong-coupling expansions as the multidimensional models all have standard second-order phase transitions which have been well studied by other methods. Dlog-Pad\'e analysis of the fourth-order mass gap series for the (2+1)-dimensional Ising model show the critical exponent $\ensuremath{\nu}$ to be calculated to about the 10% level of accuracy. The behavior of $\ensuremath{\nu}$ as a function of ddd and nnn is exhibited and compared against known results. Mean-field behavior is recovered exactly in the $d\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, while the Berlin-Kac spherical model is obtained in the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
In the usual theory of the two-time Green's function, decoupling techniques are introduced in order to obtain a self-closed equation for the lowest order Green's function. These decouplings give neither the correct temperature dependence (the absence of a T3{T}^{3}T3 term) of magnetization nor the correct correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$, which should be zero by definition. In this paper the equation of motion for the higher order Green's function is solved in the approximation which will lead to correct results up to terms of order T3{T}^{3}T3 in these quantities. Employing this higher order Green's function, it is found that the correction to the magnetization consists of a term of order T4{T}^{4}T4 and a divergent term and that the correlation function consists of a term of order T3{T}^{3}T3 and a divergent term. A method extracting a significant part from this divergent term is found by analyzing a system of two interacting \textonehalf{} spins. By this method, it is shown that the T3{T}^{3}T3 term in the magnetization is completely eliminated and that the correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$ vanishes identically.
(by translator) A perturbation theory is developed for normal many-particle systems by expanding the inverse of the advanced and retarded Green functions. Both systems with direct interactions and systems with interactions through a quantum field are considered. For the former, the infinite set of coupled equations is written down and the various methods of decoupling discussed. Systems with weak, hard sphere type, or coulomb interactions are considered and the connections with other methods discussed. For systems with quantum field interactions, the chain of equations involves both fermion and boson Green functions and the first approximation can be much better than in earlier cases. This latter approach seems to be more convenient when collective oscillations are important.
We calculate the spin-spin correlations and the magnetic susceptibility of the quantum one- and two-dimensional XY\mathrm{XY}XY models with S=1,S=1,S=1, using the two-time Green's function method and performing a decoupling proposed by Kondo and Yamaji for cases with no long-range order. A set of self-consistent equations of the correlation functions are derived and solved numerically. We present results for the correlation functions, susceptibilities, and specific heat for the whole range of temperature.