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 Theoretical

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

Kazuhiko Minami

Published Nov 26, 2013• 2 citations
Publisher

Abstract

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 .

DOI
10.1088/1751-8113/46/50/505005

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

Kazuhiko Minami

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.

Journal of Physics A Mathematical and General•Oct 7, 1996•11 citations

Critical behavior of random transverse-field Ising spin chains

Daniel S. Fisher

A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.

Physical review. B, Condensed matter•Mar 1, 1995•828 citations

Phase transitions in randomly diluted transverse Ising model

V. K. Saxena

Recently, there has been interest in the problems of disorder in the Ising model in a transverse field, which may apply to DKDP-KDP mixed systems and diluted vanadates. The site and bond diluted transverse Ising models have already been studied using Green's function techniques along with the coherent potential approximation. However, the situation as regards to the existence of a second order phase transition and the divergence of the static susceptibility is not very clear. In the present paper, a spin ½ Ising model in a transverse field and random bond (site) dilution is studied by variational cluster and perturbation expansion method. The critical temperature for the existence of a second order phase transition is calculated as a function of concentration of bonds (spins) and the transverse field. Variation of the critical concentration, for the existence of phase transition, is studied as a function of the transverse field. A phase diagram in the temperature-concentration-transverse field space is constructed. The static susceptibility is calculated and is found to diverge at the critical temperature, as is expected for the existence of a second order phase transition.

Ferroelectrics•Oct 1, 1981•0 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

Effect of dilution on phase transitions in a transverse Ising model

V. K. Saxena

The effect of random bond (site) dilution on the phase transition of a three-dimensional transverse Ising model is studied by a cluster variation, along with perturbation expansion method. Complete phase diagrams in the temperature-concentration-transverse-field space are constructed for both the cases. The value of critical transverse field as a function of concentration, at zero temperature, exhibits a discontinuity at the critical value of concentration for the existence of phase transition, as conjectured by Harris. As is expected for the existence of a second-order phase transition, the static susceptibility diverges on the critical surface.

Physical review. B, Condensed matter•Jun 1, 1983•46 citations
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 .
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
Recently, there has been interest in the problems of disorder in the Ising model in a transverse field, which may apply to DKDP-KDP mixed systems and diluted vanadates. The site and bond diluted transverse Ising models have already been studied using Green's function techniques along with the coherent potential approximation. However, the situation as regards to the existence of a second order phase transition and the divergence of the static susceptibility is not very clear. In the present paper, a spin ½ Ising model in a transverse field and random bond (site) dilution is studied by variational cluster and perturbation expansion method. The critical temperature for the existence of a second order phase transition is calculated as a function of concentration of bonds (spins) and the transverse field. Variation of the critical concentration, for the existence of phase transition, is studied as a function of the transverse field. A phase diagram in the temperature-concentration-transverse field space is constructed. The static susceptibility is calculated and is found to diverge at the critical temperature, as is expected for the existence of a second order phase transition.
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.
The effect of random bond (site) dilution on the phase transition of a three-dimensional transverse Ising model is studied by a cluster variation, along with perturbation expansion method. Complete phase diagrams in the temperature-concentration-transverse-field space are constructed for both the cases. The value of critical transverse field as a function of concentration, at zero temperature, exhibits a discontinuity at the critical value of concentration for the existence of phase transition, as conjectured by Harris. As is expected for the existence of a second-order phase transition, the static susceptibility diverges on the critical surface.