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

Journal of Physics A Mathematical and General

The zero-field susceptibility of the transverse Ising chain with arbitrary spin

Kazuhiko Minami

Published Oct 7, 1996• 11 citations
Publisher

Abstract

The zero-field susceptibility of the transverse Ising chain with arbitrary spin-S is expressed in terms of the eigenvector for the maximum eigenvalue of its transfer matrix. As a result, the exact susceptibility is explicitly obtained for , 1, and can be obtained at least for . The numerical calculations of the susceptibility for arbitrary spin are possible and those for S = 6, 12 and 24 are given. It is also derived that the zero-temperature limit of the susceptibility is independent on spin-S.

DOI
10.1088/0305-4470/29/19/023

Related papers

The Exact Susceptibility of the Spin-S Transverse Ising Chain with Next-nearest-neighbor Interactions

Kazuhiko Minami

The zero-field susceptibility of the spin-S transverse Ising chain with next-nearest-neighbor interactions is obtained exactly. The susceptibility is given in an explicit form for S=1/2, and expressed in terms of the eigenvectors of the transfer matrix for general spin S. It is found that the low-temperature limit is independent of spin S, and is divergent at the transition point.

arXiv:2212.12693•Journal of the Physical Society of Japan•Apr 14, 2023•1 citation

The Susceptibility in Arbitrary Directions and the Specific Heat in General Ising-Type Chains of Uniform, Periodic and Random Structures

Kazuhiko Minami

The susceptibility in arbitrary directions, the specific heat, the energy and the magnetization of general Ising-type chains are exactly expressed in terms of the eigenvectors of the transfer matrix corresponding to the Ising-type interaction. These quantities, as a result, can be explicitly obtained for small spin values and generally obtained from the solution of corresponding eigenvalue problem of finite degree. Numerical estimations are easy for arbitrary spin values. This formula includes the spin- S transverse Ising model with vanishing or non-vanishing parallel external field, the Blume-Emery-Griffiths model and the Blume-Capel model, mixed spin and mixed bond periodic systems such as alternating Ising chains or Ising ferrimagnets and also includes Ising models with random structures. Low temperature behaviors of these systems and the crossover to infinite spin systems are also investigated.

Journal of the Physical Society of Japan•Jul 15, 1998•7 citations

An Exact Calculation of the Transverse Susceptibility for an Antiferromagnetic Ising Chain

N. Kunisada, Y. Fukumoto

We study the transverse susceptibility of the fully frustrated antiferromagnetic Ising Δ-chain, extending Minami's transfer-matrix method to the transverse susceptibility of general-type Ising linear-chains [J. Phys. Soc. Jpn. 67 (1998), 2255]. For the transverse fields Γ1 in tip spin sites and Γ2 in bottom spin sites, we calculate zero-field transverse susceptibilities χxtip = lim Γ1 →0Mxtip/Γ1 and χxbottom = lim Γ2 →0Mxbottom/Γ2 while keeping the ratio Γ1/Γ2 fixed, where Mxtip (bottom) denotes the magnetization for tip (bottom) spin sites. Both the transverse susceptibilities follow Curie's law at low temperatures. We also calculate χxbottom(Γ1 > 0), the transverse susceptibility of the bottom spin chain under a finite tip-spin transverse field, to understand the Curie-type behavior in zero-field susceptibility. Using the second-order perturbation theory, we also discuss the Γ1 dependence of χxbottom (Γ1) at zero temperature.

arXiv:0712.4306•Progress of Theoretical Physics•Jun 1, 2008•1 citation

Transverse susceptibility in spin S Ising chains

Michael F. Thorpe, M. F. Thomsen

It is shown that the transverse susceptibility of a one-dimensional spin S Ising system can be calculated by combining linear response theory with the transfer matrix technique. Closed form expressions for infinite chains are obtained for S= 1 / 2 and S=1, and numerical results are displayed for S= 1 / 2 to S= 5 / 2 .

Journal of Physics C Solid State Physics•Mar 20, 1983•13 citations

Exact transverse susceptibility of one-dimensional random bond Ising model with alternating spin

Kazuhiko Minami

The zero-field transverse susceptibility of a one-dimensional random bond Ising model with alternating spin 1/2 and S is exactly obtained. The integral form of the susceptibility is written. The thermal curve of the susceptibility becomes irregular when weak interactions appear with a small probability. The leading-order terms of an expansion, in terms of the moment of the distribution of a random variable, are obtained. The first-order term vanishes and the second-order term is written explicitly. The expansion up to infinite order is exactly obtained in the case of uniform spin S = 1/2. The classical limit S → ∞ is also considered. It is found that the low-temperature limit T → 0 of the susceptibility is independent of S .

Journal of Physics A Mathematical and Theoretical•Nov 26, 2013•2 citations

Susceptibility of the one-dimensional Ising model: is the singularity at T = 0 an essential one?

Taylor, James H.

The zero-field isothermal susceptibility of the one-dimensional Ising model with nearest-neighbor interactions and a finite number of spins is shown to have a relatively simple singularity as the temperature approaches zero, proportional only to the inverse temperature. This is in contrast to what is seen throughout the literature for the inifinite chain: an essential singularity that includes an exponential dependence on the inverse temperature. Assuming an arbitrary (but finite) number of spins and retaining terms that are usually considered ignorable in the thermodynamic limit, the analysis involves nothing beyond straightforward series expansions, starting either from the partition function for a closed chain in a magnetic field, obtained using the transfer-matrix approach; or from the expression for the zero-field susceptibility found via the fluctuation-dissipation theorem. In both cases, the exponential singularity is exactly removed. In addition, the susceptibility per spin is found to increase with the number of spins (except in the case of noninteracting spins), a result which is also at variance with what is normally reported for an infinite chain.

arXiv:2001.01693•Jan 6, 2020•0 citations
The zero-field susceptibility of the transverse Ising chain with arbitrary spin-S is expressed in terms of the eigenvector for the maximum eigenvalue of its transfer matrix. As a result, the exact susceptibility is explicitly obtained for , 1, and can be obtained at least for . The numerical calculations of the susceptibility for arbitrary spin are possible and those for S = 6, 12 and 24 are given. It is also derived that the zero-temperature limit of the susceptibility is independent on spin-S.
The susceptibility in arbitrary directions, the specific heat, the energy and the magnetization of general Ising-type chains are exactly expressed in terms of the eigenvectors of the transfer matrix corresponding to the Ising-type interaction. These quantities, as a result, can be explicitly obtained for small spin values and generally obtained from the solution of corresponding eigenvalue problem of finite degree. Numerical estimations are easy for arbitrary spin values. This formula includes the spin- S transverse Ising model with vanishing or non-vanishing parallel external field, the Blume-Emery-Griffiths model and the Blume-Capel model, mixed spin and mixed bond periodic systems such as alternating Ising chains or Ising ferrimagnets and also includes Ising models with random structures. Low temperature behaviors of these systems and the crossover to infinite spin systems are also investigated.
We study the transverse susceptibility of the fully frustrated antiferromagnetic Ising Δ-chain, extending Minami's transfer-matrix method to the transverse susceptibility of general-type Ising linear-chains [J. Phys. Soc. Jpn. 67 (1998), 2255]. For the transverse fields Γ1 in tip spin sites and Γ2 in bottom spin sites, we calculate zero-field transverse susceptibilities χxtip = lim Γ1 →0Mxtip/Γ1 and χxbottom = lim Γ2 →0Mxbottom/Γ2 while keeping the ratio Γ1/Γ2 fixed, where Mxtip (bottom) denotes the magnetization for tip (bottom) spin sites. Both the transverse susceptibilities follow Curie's law at low temperatures. We also calculate χxbottom(Γ1 > 0), the transverse susceptibility of the bottom spin chain under a finite tip-spin transverse field, to understand the Curie-type behavior in zero-field susceptibility. Using the second-order perturbation theory, we also discuss the Γ1 dependence of χxbottom (Γ1) at zero temperature.
The zero-field transverse susceptibility of a one-dimensional random bond Ising model with alternating spin 1/2 and S is exactly obtained. The integral form of the susceptibility is written. The thermal curve of the susceptibility becomes irregular when weak interactions appear with a small probability. The leading-order terms of an expansion, in terms of the moment of the distribution of a random variable, are obtained. The first-order term vanishes and the second-order term is written explicitly. The expansion up to infinite order is exactly obtained in the case of uniform spin S = 1/2. The classical limit S → ∞ is also considered. It is found that the low-temperature limit T → 0 of the susceptibility is independent of S .
The zero-field isothermal susceptibility of the one-dimensional Ising model with nearest-neighbor interactions and a finite number of spins is shown to have a relatively simple singularity as the temperature approaches zero, proportional only to the inverse temperature. This is in contrast to what is seen throughout the literature for the inifinite chain: an essential singularity that includes an exponential dependence on the inverse temperature. Assuming an arbitrary (but finite) number of spins and retaining terms that are usually considered ignorable in the thermodynamic limit, the analysis involves nothing beyond straightforward series expansions, starting either from the partition function for a closed chain in a magnetic field, obtained using the transfer-matrix approach; or from the expression for the zero-field susceptibility found via the fluctuation-dissipation theorem. In both cases, the exponential singularity is exactly removed. In addition, the susceptibility per spin is found to increase with the number of spins (except in the case of noninteracting spins), a result which is also at variance with what is normally reported for an infinite chain.