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

Ising Chain with a Spin Impurity

Harold Falk

Published Nov 4, 1966• 9 citations
Publisher

Abstract

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 $S=\frac{3}{2},\frac{5}{2}$.

DOI
10.1103/physrev.151.304

Related papers

II. Exact solutions of disordered Ising spin chains in a magnetic field

Stephen B. Haley

The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.

Journal of Mathematical Physics•May 1, 1978•3 citations

Magnetic properties of the frustrated Ising chain

D. V. Laptiev, O. O. Kryvchikov, Yu. V. Savin +1 more

Using the Kramers–Wannier transfer matrix method, we studied several decorated Ising chains. The exact expressions for thermodynamic characteristics, including the ground state characteristics, were obtained. We considered several modeling chains with different signs and absolute values of exchange constants for the nearest- and the next-nearest neighbors. For these models, we calculated the magnetization curves. The critical values of magnetic fields and corresponding magnetization plateau parameters were obtained. Analytic expressions for the ground state entropy were obtained for the chains with different interaction constants. The dependencies of the number of states with minimum energy (the degeneration of the ground state) as the function of the number of particles were found. It was shown that these dependencies are expressed in terms of well-known numerical sequences, namely Lucas numbers and Pell numbers, which, in the limit of a large number of particles, are proportional to the powers of the golden and silver sections. Therefore, the ground state entropy (per particle) of the systems under consideration can be described in terms of these sections and, therefore, is nonzero.

Low Temperature Physics•Feb 1, 2024•2 citations

Free energy, entropy, and magnetization of a one-dimensional Ising model of a diluted magnet

S. V. Semkin, Виктор Павлович Смагин

We consider a one-dimensional Ising model (chain) with the the nearest-neighbor interaction and with a random nonmagnetic dilution. We find the exact free energy of such a chain as a function of the impurity concentration, temperature, and the external magnetic field. In the case of antiferromagnetic interaction in the chain, we find the specific magnetization, the mean value of the product of neighboring spins, and the entropy as functions of these parameters. We study the residual system entropy.

Theoretical and Mathematical Physics•Nov 1, 2023•2 citations

Magnetic properties of the frustrated decorated Ising chain

D. V. Laptiev, O. O. Krivchikov, Savin, Yu. V. +1 more

Using the Kramers-Wannier transfer matrix method we studied several decorated Ising chains. The exact expressions for thermodynamic characteristics, including the ground state characteristics, were obtained. We considered a number of modeling chains with different signs and absolute values of exchange constants for the nearest- and the next-nearest neighbors. For these models we calculated the magnetization curves. The critical values of magnetic fields and corresponding magnetization plateau parameters were obtained. Analytic expressions for the ground state entropy were obtained for the chains with different interaction constants. The dependencies of the number of states with minimum energy (the degeneration of the ground state) as the function of the number of particles were found. It was shown that these dependencies are expressed in terms of well-known numerical sequences - Lucas numbers and Pell numbers, which, in the limit of a large number of particles, are proportional to the powers of the golden and silver sections. Therefore, the ground state entropy (per particle) of the systems under consideration can be described in terms of these sections and, therefore, is nonzero.

arXiv:2506.12512•Jun 14, 2025•0 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

Exact Solution of 1D Ising Model on Linear Chain with Arbitrary Spin

Alexey I. Proshkin, Felix A. Kassan-Ogly

We investigated the Ising model on a linear chain with arbitrary spin including interactions between nearest and next-nearest neighbors in an applied magnetic field. A series of exact solutions and formulas for frustration fields, magnetizations and entropies at these fields at T→0 are found.

Materials science forum•Mar 1, 2016•9 citations
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→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞, 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​.
The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.
Using the Kramers–Wannier transfer matrix method, we studied several decorated Ising chains. The exact expressions for thermodynamic characteristics, including the ground state characteristics, were obtained. We considered several modeling chains with different signs and absolute values of exchange constants for the nearest- and the next-nearest neighbors. For these models, we calculated the magnetization curves. The critical values of magnetic fields and corresponding magnetization plateau parameters were obtained. Analytic expressions for the ground state entropy were obtained for the chains with different interaction constants. The dependencies of the number of states with minimum energy (the degeneration of the ground state) as the function of the number of particles were found. It was shown that these dependencies are expressed in terms of well-known numerical sequences, namely Lucas numbers and Pell numbers, which, in the limit of a large number of particles, are proportional to the powers of the golden and silver sections. Therefore, the ground state entropy (per particle) of the systems under consideration can be described in terms of these sections and, therefore, is nonzero.
We consider a one-dimensional Ising model (chain) with the the nearest-neighbor interaction and with a random nonmagnetic dilution. We find the exact free energy of such a chain as a function of the impurity concentration, temperature, and the external magnetic field. In the case of antiferromagnetic interaction in the chain, we find the specific magnetization, the mean value of the product of neighboring spins, and the entropy as functions of these parameters. We study the residual system entropy.
Using the Kramers-Wannier transfer matrix method we studied several decorated Ising chains. The exact expressions for thermodynamic characteristics, including the ground state characteristics, were obtained. We considered a number of modeling chains with different signs and absolute values of exchange constants for the nearest- and the next-nearest neighbors. For these models we calculated the magnetization curves. The critical values of magnetic fields and corresponding magnetization plateau parameters were obtained. Analytic expressions for the ground state entropy were obtained for the chains with different interaction constants. The dependencies of the number of states with minimum energy (the degeneration of the ground state) as the function of the number of particles were found. It was shown that these dependencies are expressed in terms of well-known numerical sequences - Lucas numbers and Pell numbers, which, in the limit of a large number of particles, are proportional to the powers of the golden and silver sections. Therefore, the ground state entropy (per particle) of the systems under consideration can be described in terms of these sections and, therefore, is nonzero.
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.