Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Green's-Function Theory of a Damped Boson System (Application to an Interacting Magnon System)

U. Balucani, F. Barocchi, V. Tognetti

Published Jan 1, 1972• 16 citations
Publisher

Abstract

The many-time Green's-function theory for a damped boson system subjected to an external time-dependent excitation is presented. The damping is properly taken into account by means of a generalized form of the Senitzky-Haken theory of a boson system interacting with a heat bath. The theory leads to a hierarchy of equations for the Green's functions of the system, which are averaged over the heat-bath variables and which take into account the damping of the various boson modes. Two examples for a ferromagnetic magnon system coupled by an anharmonic interaction are given.

DOI
10.1103/physreva.5.442

Related papers

Green's-Function Analysis of the Ising Ferromagnet

F. Burr Anderson

The method of the two-time temperature-dependent Green's function has been used to analyze the Ising model of a ferromagnet in an external magnetic field. The selection of a particular Green's function enables us to write an exact expression for the equation of motion. We are then led to a differential difference equation for the correlation function corresponding to the Green's function. No decoupling assumptions have been made, so the equation is exact for both arbitrary spin and range of interaction. It is shown how various approximate theories may be extracted from our formalism. The exact differential difference equation may be reduced to a partial differential equation. The latter form allows us to generate relations among the magnetization and spin-spin correlation functions. These relations are given in detail for the case of spin \textonehalf{} and $z$ nearest neighbors.

Physical review. B, Solid state•Jun 1, 1971•9 citations

Dynamics of quantum dissipation systems interacting with bosonic canonical bath: Hierarchical equations of motion approach

Rui‐Xue Xu, YiJing Yan

A nonperturbative theory is developed, aiming at an exact and efficient evaluation of a general quantum system interacting with arbitrary bath environment at any temperature and in the presence of arbitrary time-dependent external fields. An exact hierarchical equations of motion formalism is constructed on the basis of a calculus-on-path-integral algorithm, via the auxiliary influence generating functionals related to the interaction bath correlation functions in a parametrization expansion form. The corresponding continued-fraction Green's functions formalism for quantum dissipation is also presented. Proposed further is the principle of residue correction, not just for truncating the infinite hierarchy, but also for incorporating the small residue dissipation that may arise from the practical difference between the true and parametrized bath correlation functions. The final residue-corrected hierarchical equations of motion can therefore be used practically for the evaluation of arbitrary dissipative quantum systems.

arXiv:cond-mat/0612649•Physical Review E•Mar 15, 2007•177 citations

Equation-of-motion approach to the problem of damped motion in quantum mechanics

Mohsen Razavy

The motion of a particle subject to an external force and coupled to a large number of harmonic oscillators is reducible to a nonlinear integro-differential equation for the position operator. This is achieved by eliminating the degrees of freedom corresponding to the motion of the oscillators. Upon this elimination the commutator of the position and the mechanical momentum of the particle becomes time dependent and a c number for a quadratic potential, but in general a q number for other fields of force. For a harmonically bound particle interacting with the heat bath the time evolution of the position and momentum operators are given in terms of the solutions of an integro-differential equation. Certain forms of the coupling of the motion of the particle to the heat bath allow for further reduction of these equations to quartic differential equations with constant coefficients. For these cases the time dependence of the position-momentum commutator, the energy of the particle as well as the heat bath are studied.

Physical Review A•Feb 1, 1990•7 citations

Green's-Function Formalism of the One-Dimensional Heisenberg Spin System

Jun Kondo, Yamaji K.

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.

Progress of Theoretical Physics•Mar 1, 1972•246 citations

Perturbation theory for double-time temperature-dependent Green functions

Sergei Vladimirovich Tyablikov, В. Л. Бонч-Бруевич

(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.

Advances In Physics•Oct 1, 1962•56 citations

Statistical diagonalization of the Hamiltonian with interaction in the Green function theory

充 福地, 一夫 椎木

A new method to obtain the knowledge for the ground state energy as well as one for the elementary excitation energy in an interacting many particle system is considered from the intuitive approach in the double time Green function theory.Using this method of approach the statistical diagonalization which may include properly the effects of temperature and other external fields can be performed when the Green functions are solved by an appropriate approximation, and the physical meaning of the results is considered.The application to the Heisenberg antiferromagnet is shown to lead the temperature and external magnetic field dependent results in the approximation of Tyablikov decoupling.

Proceedings of the Fujihara Memorial Faculty of Engineering Keio University•Jan 1, 1970•0 citations
The many-time Green's-function theory for a damped boson system subjected to an external time-dependent excitation is presented. The damping is properly taken into account by means of a generalized form of the Senitzky-Haken theory of a boson system interacting with a heat bath. The theory leads to a hierarchy of equations for the Green's functions of the system, which are averaged over the heat-bath variables and which take into account the damping of the various boson modes. Two examples for a ferromagnetic magnon system coupled by an anharmonic interaction are given.
The method of the two-time temperature-dependent Green's function has been used to analyze the Ising model of a ferromagnet in an external magnetic field. The selection of a particular Green's function enables us to write an exact expression for the equation of motion. We are then led to a differential difference equation for the correlation function corresponding to the Green's function. No decoupling assumptions have been made, so the equation is exact for both arbitrary spin and range of interaction. It is shown how various approximate theories may be extracted from our formalism. The exact differential difference equation may be reduced to a partial differential equation. The latter form allows us to generate relations among the magnetization and spin-spin correlation functions. These relations are given in detail for the case of spin \textonehalf{} and zzz nearest neighbors.
A nonperturbative theory is developed, aiming at an exact and efficient evaluation of a general quantum system interacting with arbitrary bath environment at any temperature and in the presence of arbitrary time-dependent external fields. An exact hierarchical equations of motion formalism is constructed on the basis of a calculus-on-path-integral algorithm, via the auxiliary influence generating functionals related to the interaction bath correlation functions in a parametrization expansion form. The corresponding continued-fraction Green's functions formalism for quantum dissipation is also presented. Proposed further is the principle of residue correction, not just for truncating the infinite hierarchy, but also for incorporating the small residue dissipation that may arise from the practical difference between the true and parametrized bath correlation functions. The final residue-corrected hierarchical equations of motion can therefore be used practically for the evaluation of arbitrary dissipative quantum systems.
The motion of a particle subject to an external force and coupled to a large number of harmonic oscillators is reducible to a nonlinear integro-differential equation for the position operator. This is achieved by eliminating the degrees of freedom corresponding to the motion of the oscillators. Upon this elimination the commutator of the position and the mechanical momentum of the particle becomes time dependent and a c number for a quadratic potential, but in general a q number for other fields of force. For a harmonically bound particle interacting with the heat bath the time evolution of the position and momentum operators are given in terms of the solutions of an integro-differential equation. Certain forms of the coupling of the motion of the particle to the heat bath allow for further reduction of these equations to quartic differential equations with constant coefficients. For these cases the time dependence of the position-momentum commutator, the energy of the particle as well as the heat bath are studied.
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.
(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.
A new method to obtain the knowledge for the ground state energy as well as one for the elementary excitation energy in an interacting many particle system is considered from the intuitive approach in the double time Green function theory.Using this method of approach the statistical diagonalization which may include properly the effects of temperature and other external fields can be performed when the Green functions are solved by an appropriate approximation, and the physical meaning of the results is considered.The application to the Heisenberg antiferromagnet is shown to lead the temperature and external magnetic field dependent results in the approximation of Tyablikov decoupling.