Two-time Green Function Research Articles

Projection Operators in the Theory of Two-Time Green Functions

New projection operators are introduced in the theory of two-time Green functions. The self-energy part of the Green function is expressed in terms of the random force whose time development differes from usual one by a factor (1- P ), P being the projection operator. In the case of the Bose system, we obtained the time-relaxation expression for the width function \(\varGamma\)( k , ω) of the dynamical structure factor, which satisfies the sum rule by Puff.

Temperature dependence of damping of lattice resonance in alkali halides

It is shown that the explicit T 2 dependence of damping due to quartic contributions is small in comparison with the T dependence from cubic anharmonicity in ionic crystals. A two-time Green function method is used.

Thermal Behaviors of Magnon Side-Bands in Antiferromagnets

The absorption coefficient for magnon side-bands is obtained in the whole temperature region by the two-time Green function method in the simple RPA description. The Hamiltonian is composed of excitons and spins, exciton operators being spin-dependent. It is shown that the magnon side-bands have the following behaviors. (1) For T ∼0, only one band exists and its expression agrees with the result obtained by the simple spin wave approximation. (2) For 0< T < T N , additional hot bands appear and the total intensity of absorption increases gradually as T increases. (3) For T ∼ T N , all bands come together to from a single band at frequency ω= E 0 where E 0 is the orbital excitation energy. The total intensity is considerably temperature-dependent near and above T N . (4) For T ≫ T N , the total intensity tends to a constant value.

Decoupling of chains of equations for two-time Green's functions

Decoupling of chains of equations for two-time Green's functions

Quantum theory of correlation functions for liquid metals and normal liquids in the zero sound wave range

The correlation functions of liquid metals and normal liquids in the zero sound wave range are formulated using the quantum-mechanical two-time Green function method, and the classical formula of Bogolyubov-Sadovnikov and Nelkin-Ranganathan for normal liquids is extended to the quantum-mechanical case. The formula obtained clarifies the relation between the coherent-correlation function and the self-correlation function of a free gas, and becomes the quantum-mechanical extension of Kerr's classical formula for the relation. The effective potential obtained classically by Percus and Zwanzig from the direct correlation function is extended to the quantum-mechanical case, and the relation between the effective potential and S( k, E) is obtained. The effective potential between ions of liquid metals is formulated by taking into account the contribution from the conduction electrons. And the formula obtained by March is improved by taking into account the electron correlation effect together with the exchange interaction.

Theory of temperature dependence of spin-wave impurity modes in an antiferromagnetic perovskite

The method of two-time Green function is applied to the study of the behaviour of spin-wave impurity modes at low temperatures. Results are compared with two experiments on Ni 2+ and Eu 3+ impurities in KMnF 3.

Improvement of Approximations in the Method of Two-Time Green's Function

It is shown that the method of the two-time Green's function, in the first order approximation of the binary kernel, leads to the same type of divergence for the many-boson systems, as a divergence noticed for the case of the Heisenberg model. By introducing the “collision kernel” and collecting all the terms of the second order of the collision kernels, a convergent result is achieved. The result gives a justification for the manipulation which Morita and Tanaka proposed in order to obtain the Dyson's T 4 term for the Heisenberg ferromagnet.

Decoupling of two-time Green's functions and diagrammatic analysis

Decoupling of two-time Green's functions and diagrammatic analysis

Tetradics Formulation of the Two-Time Green's Function Method and Its Application to the Heisenberg Ferromagnet

It is shown that the “tetradic” interpretation of Kubo's H × operator is convenient in writing the equation of motion for two-time Green's function. In the formulation, the tetradics are used to express an anti-commutator of an operator as well as the commutator. For the Green's function defined in terms of a commutator, the resulting equation of motion is equivalent to the one recently derived by Shimizu. A new approximation scheme is proposed on the basis of the obtained expression. Its lowest approximation is applied to the Heisenberg ferromagnet. It is shown how the results of the decouplings due to Tyablikov and others are derived as special cases in the present approximation scheme.

Diagrammatical Method for the Two-Time Green's Function for Many-Body Problems at Finite Temperatures

The diagram technique, which was developed to solve a hierarchy of equations of motion for the reduced density matrices, is applied in order to solve a hierarchy of the equations of motion for the two-time Green's functions. The diagrammatical representations of the two-time Green's functions are obtained either in terms of the equilibrium one- and many-particle reduced density matrices or in terms only of the equilibrium one-particle reduced density matrix. The latter representation for the one-particle two-time Green's function and the formula by which the reduced density matrix is calculated from the Green's function give us a set of coupled equations which determines both the reduced density matrix and the two-time Green's function. The calculation with the aid of the present technique is illustrated for an example of Hubbard's calculation for the “Hubbard Hamiltonian”.

Decoupling a retarded green function defined in a non-equilibrium state

It is shown how to decouple and calculate the two-time Green function defined so that it is easy to represent a strongly disturbed system.

Longitudinal spin waves as secondary excitations of the spin system of a ferromagnet

The method of two-time Green's functions is used to calculate the natural oscillations of the longitudinal component of the spontaneous magnetization of a ferromagnet. The spectrum of these oscillations or spin waves has no gap, even when magnetic interactions are taken into account. For smallk the spectrum of longitudinal spin waves has the same properties as secondary excitations of the spin system of a ferromagnet In which the primary excitations are transverse (ordinary) spin waves.

Dynamical properties of the isotropic XY model

The dynamical longitudinal properties of the isotropic one-dimensional XY model are obtained by the two-time Green function method, which involves 《a + λ a μ |a + ν a σ 》 (7minus;) , 《a + λ a μ |a + ν a σ 》 (+) and 《a μ |a + λ a σ a + λ 》 (+) defined by the anticommulator and the commutator. The frequency-and wave number- dependent susceptibility, the response function and the relaxation function (canonical) correlation function) are obtained and discussed for all temperatures. The imaginary part of the frequency- and wave number-dependent susceptibility diverges at ω=4J sin ( κ 2 ) and vanishes for ω >; 4J sin ( κ 2 ) . The difference between isothermal and zero-frequency isolated susceptibilities is resolved by taking the limit κ →0 χzz ( κ , 0). The contribution of the zero-frequency pole in the two-frequency pole in the two-time Green function is also discussed and clarified.

Temperature dependence of the afr frequencies of a uniaxial antiferromagnet

The method of two-time Green's functions is used to calculate the temperature dependences of the antiferromagnetic-resonance (AFR) frequencies for a uniaxial antiferromagnet in the field range below sublattice-reversal fields. The AFR frequency has a [K(T)/M(T)]1/2 dependence on the anisotropy constant K(T). For the simplest case of single-ion anisotropy, the normalized AFR frequency is proportional to the relative sublattice magnetization: ω(T) /ω(0) ∼ σ(T).

Magnetic Properties of Low Dimensional Spin Systems

Using the method of the two-time Green's function, both static and dynamic properties of low dimensional spin systems with an anisotropic exchange interaction were investigated. It was shown that the magnetic behaviours of one- and two-dimensional spin systems are considerably different from those of the usual three-dimensional systems. That is, the transi­ tion temperature is lower than that expected from the magnitude of the coupling constant when an anisotropy is small enough. The short range order is developed more remarkably in the neighbourhood of the transition temperature. The damping constant increases in both ferro- and antiferromagnet with decreasing temperature owing to the anomalous growth of fluctuation. At the same time a broad shoulder is developed in the line shape, which may be considered as indicating quasi-collective modes of motion which persist in the paramag­ netic phase.

Calculation of the isothermal susceptibility by the Kubo formula

An analysis is made to clarify why Allan and Betts and Katsura and Horiguchi could give the correct isothermal perpendicular susceptibility of the Ising model and the XY model, respectively, with the aid of the Kubo formula and some limiting procedure. A comparison is made with the similar situation in the calculation of the two-time correlation function with the use of the two-time Green function. The discussion includes the calculation of the isothermal susceptibility for the ideal spin-wave approximation of a ferromagnet.

First Order Green Function Theory of Ferromagnetism

A method of two-time Green function for the Heisenberg model of ferromagnet of S =1/2 is developed. The equations of motion of two kinds of Green functions G (-) and G (+) , which are defined using commutator and anticommutator, respectively, are derived. Higher order Green functions are decoupled taking into account the anticommutability of Pauli operators at the same lattice site, giving coupled equations for G (-) and G (+) . The solution from G (+) are analyzed. The effect of the kinematical interaction is considered automatically, and over-all (in the entire temperature region and in the entire magnetic field region) good properties of the magnetization (which are not realized in some previous works using the first-order decoupling) are obtained. Inverse Curie temperature, J β c , is 0.398 and the critical index of the magnetization is 1/2. Several previous theories of the first-order are rederived and analyzed from a unified point of view.

Two-Time Green's Function Theory for Laser

By means of the retarded Green's function (response function) we derive the threshold condition, the oscillation frequency and the time-limited band width of lasers. The statistical property of the set of identical two-level atoms are taken into account on the basis of the Dicke-Haken's model. This property gives rise to atom-field correlation terms in the polarization operator. It is found that this terms influence the imaginary part of the oscillation frequency and the effect becomes large as the photon density increases or the occupancy difference tends to zero. In this respect the relation with the super-radiation is discussed. The customary approximation of considering only the interaction which is conservative for the particle number is tested. Under the resonance condition this approximation is correct up to the order of |ω k -ω 0 |/ω 0 , where ω 0 is the transition frequency of the atoms and ω k is the frequency of the free field.

The ground-state sublattice magnetization in an antiferromagnet

A calculation is given of the additional zero-point sublattice deviation created by a single non-magnetic impurity ion in a Heisenberg antiferromagnet. A knowledge of this quantity is shown to permit a measure of the sublattice magnetization in the pure host to be made from an observation of the cross section for the diffuse elastic scattering of thermal neutrons from the mixed system. The form of the cross section is discussed in detail. It is shown that the pure host sublattice magnetization can be obtained directly from a relative measurement of the intensity of the first Bragg peak and the diffuse elastic scattering intensity at this position. The calculation of the magnetic defect created on the host is based on the linear spin-wave approximation and is accomplished with the use of thermal two-time Green functions. We also discuss how nuclear magnetic resonance and Mossbauer experiments can be used with the mixed system to give an estimate of the sublattice magnetization in the pure host.

On the quasipotential description of the negaton-positon interaction

The negaton-positon interaction is investigated to second order in the coupling constant using the quasipotential approach to the quantum field theory. The quasipotential is constructed with the help of the two-time Green function. Terms depending on the energy are obtained. Their possible influence on the positonium energy levels is briefly discussed.