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

arXiv:1801.04452

An exact alternative solution method of 1D Ising model with Block-spin transformation at $H=0$

Tuncer Kaya

Published Jan 13, 2018• 0 citations
arXivHTMLPDF

Abstract

An alternative exact explicit solution of 1D Ising chain is presented without using any boundary conditions (or free boundary condition) by the help of applying successively block-spin transformation. Exact relation are obtained between spin-spin correlation functions in the absence of external field. To evaluate average magnetization (or the order parameter), it is assumed that the average magnetization can be related to infinitely apart two spin correlation function as $^{2}=$. A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that $=\tanh{K}$, which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of $\{σ_{i}σ_{i+1}\}=\{s_{i}\}$ and the result of transfer matrix method in the absence of external field. By applying further block-spin transformation, it is obtained that $\{σ_{0}σ_{j}\}=(\tanh{K})^{j}$, here $j$ assumes the values of $j=2^n$, here $n$ is an integer numbers. We believe that this result is really important in that it is the only exact unique treatment of the 1D Ising chain beside with the transfer matrix method. It is also pointed out the irrelevances of some of the alternative derivation appearing in graduate level text books. The obtained unique correlation relation in this this study leads in the limit of $N\rightarrow\infty$ to $=(\tanh{K})^{N/2}$, indicating that the second order phase transition is only possible in the limit of $K\rightarrow\infty$. The obtained results in this work are exactly the same as those of obtained by the transfer matrix method which considers periodic boundary condition.

DOI
10.48550/arxiv.1801.04452

Related papers

Exact Analytical Results for the 1D Ising Chain with Periodic Impurity Fields

Telfah, Malek, Obeidat, Abdalla

We present an exact analytical solution for the one-dimensional Ising model in the presence of an external magnetic field applied periodically to every $k$-th site. The problem is handled using the symmetrized transfer matrix approach, we derive a compact closed-form expression for the system's eigenvalues for arbitrary period $k$. From the resulting free energy, we obtain exact expressions for the magnetization and zero-field susceptibility. Explicit results are presented for $k = 1$, $k = 2$, and $k = 3$ which is considered a novel result. We further analyze the spin-spin correlation functions, deriving the correlation length and the set of position-dependent correlation strength prefactors, $A_{ij}$. The framework highlights how impurity spacing suppresses thermodynamic responses, with susceptibility scaling as $χ\sim β/ k$ for large $k$, offering insights into diluted magnetic systems and serving as a benchmark for quasiperiodic modulations. The correlation strengths exhibit a strong anisotropy, revealing a complex, non-local structure of spin fluctuations. These results provide a complete and rigorous benchmark for understanding the effects of periodic modulation in 1D systems.

arXiv:2511.15399•Nov 19, 2025•0 citations

Difference Equation Solutions to the One-Dimensional Ising Model with General Spin

Howard Silver, Norman E. Frankel

The one-dimensional Ising model with general spin S and either open or periodic boundary conditions has been formulated as a difference equation of order 2S+1 with non-zero magnetic field B. This difference equation has the characteristic equation of the transfer matrix, V, as its indicial equation. When B=0, for the open chain, the order of the difference equation is reduced to [S+1]. It is shown that the correlation functions for the open chain satisfy similar difference equations. This result explicitly displays the lack of translational invariance of the open chain. For several cases it is shown that the partition function can be represented as the difference of two Chebyshev polynomials of the second kind.

Progress of Theoretical Physics•Sep 1, 1971•2 citations

Chebyshev Polynomials in the Physics of the One-Dimensional Finite-Size Ising Model: An Alternative View and Some New Results

N. S. Tonchev, Daniel Dantchev

For studying of the finite-size behavior of the Ising model under different boundary conditions we propose an alternative to the standard transfer matrix technique approach based on Abel\`{e}s theorem and Chebyshev polynomials. Using it we easily reproduce the known for periodic boundary conditions results for Lee-Yang zeros, the exact position space renormalization group transformation, etc., and extend them deriving new results for antiperiodic boundary conditions. Note that in the latter case one has a nontrivial order-parameter profile, which we also calculate, where the average value of a given spin depends on the distance from the seam with opposite bond in the system. It is interesting to stress that under both boundary conditions, the one-dimensional case exhibits Schottky anomaly.

Condensed Matter•Dec 2, 2024•1 citation

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

Exact solution of Ising model of an alternating delta spin chain

Elías Ríos

The Ising model on an alternating triangular lattice with the nearest-neighbor interaction in a magnetic field is presented. Exact solution of this model is found. The thermodynamic quantities, like free energy, specific heat a finite temperature and function correlaction, are calculated by treatment based on the transfer matrix method. Our results show that the model proposed does not have phase transition to a finite temperature, but the curve of specific heat presented a Schottky anomaly.

arXiv:1409.2127•Sep 7, 2014•0 citations

A New Exact Method for Solving the Two-Dimensional Ising Model

S. Ranjbar, Gholamabbas Parsafar

We have used the two-dimensional Ising model with a limited number of rows, but with the coordination number of four for each site, to set up the transfer matrix for the model. From the solution of such a matrix, the exact thermodynamic properties have been obtained for the model with a definite number of rows, n . We have solved the matrix for n ≤ 7 and n ≤ 10 in the presence and absence of a magnetic field, respectively. On the basis of such solutions, we have proposed an analytical expression for the partition function of the model with any number of rows in the absence of a magnetic field. The proposed expression becomes more accurate when n is larger, in such a way that it becomes very accurate for n ≥ 8 and is exact for n → ∞. Our results show that the singularity of the specific heat occurs only for the model with infinite number of rows. In the presence of a magnetic field, the solution to the matrixes is too complicated to propose a general analytical expression for the partition function of the model with any number of rows. However, our exact solution for the model with n = 4, 5, 6, and 7 reveals an important point that such results are independent of n when the field−spin interaction energy is almost equal to or larger than that of the spin−spin interaction.

The Journal of Physical Chemistry B•Aug 12, 1999•9 citations
An exact alternative solution method of 1D Ising model with Block-spin transformation at H=0H=0H=0
An alternative exact explicit solution of 1D Ising chain is presented without using any boundary conditions (or free boundary condition) by the help of applying successively block-spin transformation. Exact relation are obtained between spin-spin correlation functions in the absence of external field. To evaluate average magnetization (or the order parameter), it is assumed that the average magnetization can be related to infinitely apart two spin correlation function as 2=^{2}=2=. A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that =tanh⁡K=\tanh{K}=tanhK, which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of {σiσi+1}={si}\{σ_{i}σ_{i+1}\}=\{s_{i}\}{σi​σi+1​}={si​} and the result of transfer matrix method in the absence of external field. By applying further block-spin transformation, it is obtained that {σ0σj}=(tanh⁡K)j\{σ_{0}σ_{j}\}=(\tanh{K})^{j}{σ0​σj​}=(tanhK)j, here jjj assumes the values of j=2nj=2^nj=2n, here nnn is an integer numbers. We believe that this result is really important in that it is the only exact unique treatment of the 1D Ising chain beside with the transfer matrix method. It is also pointed out the irrelevances of some of the alternative derivation appearing in graduate level text books. The obtained unique correlation relation in this this study leads in the limit of N→∞N\rightarrow\inftyN→∞ to =(tanh⁡K)N/2=(\tanh{K})^{N/2}=(tanhK)N/2, indicating that the second order phase transition is only possible in the limit of K→∞K\rightarrow\inftyK→∞. The obtained results in this work are exactly the same as those of obtained by the transfer matrix method which considers periodic boundary condition.
We present an exact analytical solution for the one-dimensional Ising model in the presence of an external magnetic field applied periodically to every kkk-th site. The problem is handled using the symmetrized transfer matrix approach, we derive a compact closed-form expression for the system's eigenvalues for arbitrary period kkk. From the resulting free energy, we obtain exact expressions for the magnetization and zero-field susceptibility. Explicit results are presented for k=1k = 1k=1, k=2k = 2k=2, and k=3k = 3k=3 which is considered a novel result. We further analyze the spin-spin correlation functions, deriving the correlation length and the set of position-dependent correlation strength prefactors, AijA_{ij}Aij​. The framework highlights how impurity spacing suppresses thermodynamic responses, with susceptibility scaling as χ∼β/kχ\sim β/ kχ∼β/k for large kkk, offering insights into diluted magnetic systems and serving as a benchmark for quasiperiodic modulations. The correlation strengths exhibit a strong anisotropy, revealing a complex, non-local structure of spin fluctuations. These results provide a complete and rigorous benchmark for understanding the effects of periodic modulation in 1D systems.
The one-dimensional Ising model with general spin S and either open or periodic boundary conditions has been formulated as a difference equation of order 2S+1 with non-zero magnetic field B. This difference equation has the characteristic equation of the transfer matrix, V, as its indicial equation. When B=0, for the open chain, the order of the difference equation is reduced to [S+1]. It is shown that the correlation functions for the open chain satisfy similar difference equations. This result explicitly displays the lack of translational invariance of the open chain. For several cases it is shown that the partition function can be represented as the difference of two Chebyshev polynomials of the second kind.
For studying of the finite-size behavior of the Ising model under different boundary conditions we propose an alternative to the standard transfer matrix technique approach based on Abel\`{e}s theorem and Chebyshev polynomials. Using it we easily reproduce the known for periodic boundary conditions results for Lee-Yang zeros, the exact position space renormalization group transformation, etc., and extend them deriving new results for antiperiodic boundary conditions. Note that in the latter case one has a nontrivial order-parameter profile, which we also calculate, where the average value of a given spin depends on the distance from the seam with opposite bond in the system. It is interesting to stress that under both boundary conditions, the one-dimensional case exhibits Schottky anomaly.
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.
We have used the two-dimensional Ising model with a limited number of rows, but with the coordination number of four for each site, to set up the transfer matrix for the model. From the solution of such a matrix, the exact thermodynamic properties have been obtained for the model with a definite number of rows, n . We have solved the matrix for n ≤ 7 and n ≤ 10 in the presence and absence of a magnetic field, respectively. On the basis of such solutions, we have proposed an analytical expression for the partition function of the model with any number of rows in the absence of a magnetic field. The proposed expression becomes more accurate when n is larger, in such a way that it becomes very accurate for n ≥ 8 and is exact for n → ∞. Our results show that the singularity of the specific heat occurs only for the model with infinite number of rows. In the presence of a magnetic field, the solution to the matrixes is too complicated to propose a general analytical expression for the partition function of the model with any number of rows. However, our exact solution for the model with n = 4, 5, 6, and 7 reveals an important point that such results are independent of n when the field−spin interaction energy is almost equal to or larger than that of the spin−spin interaction.