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

American Journal of Physics

Green functions and impurities in harmonic chains

D. Prato, Carlos Alberto Condat

Published Feb 1, 1983• 2 citations
Publisher

Abstract

The density of vibrational states of harmonic chains is calculated with the Green function formalism. Chains with impurities are considered, and it is explicitly shown how the localized states are built up from the contribution of every lattice site to the density of states. The method can be used, with slight changes, to study the dynamics of a linear ferromagnetic lattice with nearest-neighbor interactions. The procedure employed in this article shows clearly the power and simplicity of the Green function method.

DOI
10.1119/1.13469

Related papers

The Green’s Function in the Problem of Charge Dynamics on A One-Dimensional Lattice with An Impurity Center

Vladimir Nikolaevich Likhachev, George A Vinogradov

In the “tight-binding” approximation (the Hückel model), we consider the evolution of the charge wave function on a semi-infinite one-dimensional lattice with an additional energy U at a single impurity site. In the case of the continuous spectrum (for |U| < 1) where there is no localized state, we construct the Green’s function using the expansion in terms of eigenfunctions of the continuous spectrum and obtain an expression for the time Green’s function in the form of a power series in U. It unexpectedly turns out that this series converges absolutely even in the case where the localized state is added to the continuous spectrum. We can therefore say that the Green’s function constructed using the states of the continuous spectrum also contains an implicit contribution from the localized state.

Theoretical and Mathematical Physics•Jul 1, 2018•1 citation

Green's function and excitation spectrum of finite lattices

S. Cojocaru, Victor Bârsan, Arnout Jozef Ceulemans

Abstract New analytical results are obtained for the Green's function of a finite one‐dimensional lattice with nearest neighbor interaction using discrete Fourier transform. The method offers an alternative to the Bethe Ansatz, but does not require any a priori assumption on the form of the wavefunction. This makes it suitable for extensions to nano‐ferromagnets of higher dimensions. Solutions of the Heisenberg spin chain with periodic and open boundary conditions are considered as examples. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

physica status solidi (b)•Jun 1, 2006•1 citation

Ising Chain with a Spin Impurity

Harold Falk

Consider a closed, linear chain of $N$ localized spins (each $s=\frac{1}{2}$) with nearest-neighbor Ising interaction. Such a chain, exposed to a uniform external magnetic field, will be called the pure host. Replace a single spin in the pure host by an impurity spin which is also subjected to nearest-neighbor Ising interaction and to the external field. The magnitude of the impurity spin, the magnitude of its magnetic moment, and the magnitude and sign of its interaction with the host are allowed to differ from the corresponding values characterizing the host. For the bulk system ($N\ensuremath{\rightarrow}\ensuremath{\infty}$, for constant linear density of spins), the thermodynamic properties, such as the impurity magnetization, the (position-dependent) magnetization of the impurity-host system and the impurity-host spin correlation functions, are obtained exactly in terms of conventional, tabulated functions. Numerical results are presented for impurity spins of magnitude $S=\frac{3}{2},\frac{5}{2}$.

Physical Review•Nov 4, 1966•9 citations

Magnetic impurity coupled to a Heisenberg chain: Density-matrix renormalization-group study

W. Zhang, J. Igarashi, Peter Fulde

By making use of the Lanczos and the density-matrix renormalization-group methods, we study the zero-temperature properties of a spin-1/2 antiferromagnetic Heisenberg chain system with a spin-1/2 magnetic impurity symmetrically coupled to its two nearest-neighbor spins. A comparison is made with the results for a magnetic impurity coupled only to the nearest-neighbor spin in the chain. The binding energy and the spin correlations are calculated. The matrix elements of the impurity spin operators are also calculated between several low-energy states. By carefully analyzing finite-size effects, we show that the dynamical structure factor of the impurity contains a \ensuremath{\delta}-function contribution at zero frequency which does not vanish in the thermodynamic limit for the case of symmetrical coupling. This indicates a partial decoupling of the impurity spin from the spins in the chain, and therefore the impurity susceptibility is predicted to show a Curie behavior. This contradicts the two-channel Kondo behavior suggested from analytic considerations.

Physical review. B, Condensed matter•Jul 1, 1997•24 citations

GREEN'S FUNCTION OF A FINITE CHAIN AND THE DISCRETE FOURIER TRANSFORM

S. Cojocaru

Unlike the Fourier series expansion, the discrete Fourier transform is defined on a finite basis set of harmonic functions. The first approach is widely used in condensed matter to describe the thermodynamic limit of various lattice models, while the latter did not receive sufficient development that would allow to address finite lattices. In the present paper a general expression for the Green's function of a finite one-dimensional lattice with nearest neighbor interaction is derived for the first time via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Although the final expressions are completely equivalent to Bethe ansatz, the examples allow us to clarify the differences between the two approaches. On the other hand, it is explained why the well known results obtained by Fourier series expansion were incomplete and thus provides a deeper understanding of the approach.

arXiv:0704.2898•International Journal of Modern Physics B•Feb 20, 2006•10 citations

Theory of the Ising Ferromagnet Using the Green Function Method

Takehiko Oguchi, Ikuo Ono

The Green function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.

Progress of Theoretical Physics•Jun 1, 1966•15 citations
The density of vibrational states of harmonic chains is calculated with the Green function formalism. Chains with impurities are considered, and it is explicitly shown how the localized states are built up from the contribution of every lattice site to the density of states. The method can be used, with slight changes, to study the dynamics of a linear ferromagnetic lattice with nearest-neighbor interactions. The procedure employed in this article shows clearly the power and simplicity of the Green function method.
In the “tight-binding” approximation (the Hückel model), we consider the evolution of the charge wave function on a semi-infinite one-dimensional lattice with an additional energy U at a single impurity site. In the case of the continuous spectrum (for |U| < 1) where there is no localized state, we construct the Green’s function using the expansion in terms of eigenfunctions of the continuous spectrum and obtain an expression for the time Green’s function in the form of a power series in U. It unexpectedly turns out that this series converges absolutely even in the case where the localized state is added to the continuous spectrum. We can therefore say that the Green’s function constructed using the states of the continuous spectrum also contains an implicit contribution from the localized state.
Abstract New analytical results are obtained for the Green's function of a finite one‐dimensional lattice with nearest neighbor interaction using discrete Fourier transform. The method offers an alternative to the Bethe Ansatz, but does not require any a priori assumption on the form of the wavefunction. This makes it suitable for extensions to nano‐ferromagnets of higher dimensions. Solutions of the Heisenberg spin chain with periodic and open boundary conditions are considered as examples. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Consider a closed, linear chain of NNN localized spins (each s=12s=\frac{1}{2}s=21​) with nearest-neighbor Ising interaction. Such a chain, exposed to a uniform external magnetic field, will be called the pure host. Replace a single spin in the pure host by an impurity spin which is also subjected to nearest-neighbor Ising interaction and to the external field. The magnitude of the impurity spin, the magnitude of its magnetic moment, and the magnitude and sign of its interaction with the host are allowed to differ from the corresponding values characterizing the host. For the bulk system ($N\ensuremath{\rightarrow}\ensuremath{\infty}$, for constant linear density of spins), the thermodynamic properties, such as the impurity magnetization, the (position-dependent) magnetization of the impurity-host system and the impurity-host spin correlation functions, are obtained exactly in terms of conventional, tabulated functions. Numerical results are presented for impurity spins of magnitude S=32,52S=\frac{3}{2},\frac{5}{2}S=23​,25​.
By making use of the Lanczos and the density-matrix renormalization-group methods, we study the zero-temperature properties of a spin-1/2 antiferromagnetic Heisenberg chain system with a spin-1/2 magnetic impurity symmetrically coupled to its two nearest-neighbor spins. A comparison is made with the results for a magnetic impurity coupled only to the nearest-neighbor spin in the chain. The binding energy and the spin correlations are calculated. The matrix elements of the impurity spin operators are also calculated between several low-energy states. By carefully analyzing finite-size effects, we show that the dynamical structure factor of the impurity contains a \ensuremath{\delta}-function contribution at zero frequency which does not vanish in the thermodynamic limit for the case of symmetrical coupling. This indicates a partial decoupling of the impurity spin from the spins in the chain, and therefore the impurity susceptibility is predicted to show a Curie behavior. This contradicts the two-channel Kondo behavior suggested from analytic considerations.
Unlike the Fourier series expansion, the discrete Fourier transform is defined on a finite basis set of harmonic functions. The first approach is widely used in condensed matter to describe the thermodynamic limit of various lattice models, while the latter did not receive sufficient development that would allow to address finite lattices. In the present paper a general expression for the Green's function of a finite one-dimensional lattice with nearest neighbor interaction is derived for the first time via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Although the final expressions are completely equivalent to Bethe ansatz, the examples allow us to clarify the differences between the two approaches. On the other hand, it is explained why the well known results obtained by Fourier series expansion were incomplete and thus provides a deeper understanding of the approach.
The Green function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.