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 the Physical Society of Japan

Correlation Index of the Ising Model with a Transverse Field

Akira Yanase

Published Jun 1, 1977• 11 citations
Publisher

Abstract

The critical behaviour of the correlation function of the Ising model with a transverse field, Δ , is investigated by means of the perturbation expansions. Exact expressions for the coefficients up to seventh order in J / k T ( J the exchange interaction) are given as functions of Δ / k T for the quadratic, triangular, simple cubic, body-centerd cubic and face-centerd cubic lattices. The universality of the critical index for the correlation length is concluded.

DOI
10.1143/jpsj.42.1816

Related papers

Critical Behaviour of the Susceptibility of the Ising Model with a Transverse Field

Akira Yanase, Yukito Takeshige, Masaaki Suzuki

The critical behaviour of the susceptibility of the Ising model with a transverse field is investigated by means of the perturbation expansions. Exact expressions for the coefficients up to seventh order in J / k T are given as functions of Δ/ k T for the quadratic, triangular, simple cubic, b.c.c. and f.c.c. lattices. The universality of the critical index is concluded from the analysis of the coefficients and the critical temperature is obtained as function of the transverse field for the five kinds of lattices.

Journal of the Physical Society of Japan•Oct 1, 1976•35 citations

Series expansions for the 3D transverse Ising model at T=0

Zheng Weihong, J. Oitmaa, Chris J. Hamer

Both weak-coupling and strong-coupling series expansions are calculated for the Ising model in a transverse field at zero temperature in three dimensions. Series are obtained for the ground-state energy, the magnetization, the susceptibility, and the energy gaps, on the simple cubic, the body-centred cubic and the face-centred cubic lattices. The analysis of the critical behaviour is consistent with the behaviour predicted by renormalization group theory for the four-dimensional simple Ising model. There is a remarkable degree of universality between all three lattices.

Journal of Physics A Mathematical and General•Aug 21, 1994•29 citations

The Ising model with a transverse field. I. High temperature expansion

Roger James Elliott, C Wood

The Ising model (exchange constant J) with a transverse field Gamma can be regarded as a simple model of hydrogen bonded ferroelectrics and of certain magnetic systems. The transition temperature T C decreases as Gamma increases and falls to zero at a critical field Gamma C . This transition is studied in the quadratic and simple cubic lattices with nearest neighbour interactions by a power series expansion of the susceptibility at high T up to fifth order in (J/kT) to all orders in ( Gamma /kT). The coefficients are calculated by diagram counting and time ordering perturbation methods. The critical Gamma against T curve is derived and the critical exponent gamma of the susceptibility. This appears to vary smoothly with Gamma but it is argued that this is an artifact due to the use of a short series and that the critical exponents are probably independent of Gamma as long as T C remains finite but change discontinuity at T C =0.

Journal of Physics C Solid State Physics•Oct 28, 1971•121 citations

The Ising model with a transverse field. II. Ground state properties

Pierre M. Pfeuty, Roger James Elliott

For Pt. I see ibid., vol. 4, 2359 (1971). The properties of the Ising model with a transverse field, with the Hamiltonian written H=- Gamma Sigma i S i x - 1 / 2 Sigma i,j J ij S i z S j z , are studied by perturbation expansions at T=0 in the ground state, both below and above the critical field Gamma C , for an hypercubic lattice and short range interactions. The order parameter (S z ) and the correlation function G=) Sigma i S i z Sigma j S j z ) are calculated respectively below and above Gamma C . The gap in the excitation spectrum is also studied above the critical field. The critical behaviour obtained from these expansions at T=0 is different from that obtained at T finite and is shown to be analogous to the critical behaviour of the Ising model for a lattice with one more dimension.

Journal of Physics C Solid State Physics•Oct 28, 1971•214 citations

Series expansions for the zero-temperature transverse Ising model

L.G. Marland

Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .

Journal of Physics A Mathematical and General•Aug 1, 1981•50 citations

General transverse Ising paramagnet

G. Kamieniarz

An Ising model in a transverse field with general spin S is considered within RPA Green-function theory in the paramagnetic region. Expressions for the temperature renormalised collective excitation spectrum with a gap decreasing as the exchange interaction increases, the transverse correlation function, and the critical exchange-field ratio for magnetic ordering, are derived. Zero-temperature expansions are given in the limit of weak interaction. Critical values at T=0K are calculated for several values of S and an asymptotical expression for S>>1 is given.

Journal of Physics C Solid State Physics•Nov 14, 1976•7 citations
The critical behaviour of the correlation function of the Ising model with a transverse field, Δ , is investigated by means of the perturbation expansions. Exact expressions for the coefficients up to seventh order in J / k T ( J the exchange interaction) are given as functions of Δ / k T for the quadratic, triangular, simple cubic, body-centerd cubic and face-centerd cubic lattices. The universality of the critical index for the correlation length is concluded.
The critical behaviour of the susceptibility of the Ising model with a transverse field is investigated by means of the perturbation expansions. Exact expressions for the coefficients up to seventh order in J / k T are given as functions of Δ/ k T for the quadratic, triangular, simple cubic, b.c.c. and f.c.c. lattices. The universality of the critical index is concluded from the analysis of the coefficients and the critical temperature is obtained as function of the transverse field for the five kinds of lattices.
Both weak-coupling and strong-coupling series expansions are calculated for the Ising model in a transverse field at zero temperature in three dimensions. Series are obtained for the ground-state energy, the magnetization, the susceptibility, and the energy gaps, on the simple cubic, the body-centred cubic and the face-centred cubic lattices. The analysis of the critical behaviour is consistent with the behaviour predicted by renormalization group theory for the four-dimensional simple Ising model. There is a remarkable degree of universality between all three lattices.
The Ising model (exchange constant J) with a transverse field Gamma can be regarded as a simple model of hydrogen bonded ferroelectrics and of certain magnetic systems. The transition temperature T C decreases as Gamma increases and falls to zero at a critical field Gamma C . This transition is studied in the quadratic and simple cubic lattices with nearest neighbour interactions by a power series expansion of the susceptibility at high T up to fifth order in (J/kT) to all orders in ( Gamma /kT). The coefficients are calculated by diagram counting and time ordering perturbation methods. The critical Gamma against T curve is derived and the critical exponent gamma of the susceptibility. This appears to vary smoothly with Gamma but it is argued that this is an artifact due to the use of a short series and that the critical exponents are probably independent of Gamma as long as T C remains finite but change discontinuity at T C =0.
For Pt. I see ibid., vol. 4, 2359 (1971). The properties of the Ising model with a transverse field, with the Hamiltonian written H=- Gamma Sigma i S i x - 1 / 2 Sigma i,j J ij S i z S j z , are studied by perturbation expansions at T=0 in the ground state, both below and above the critical field Gamma C , for an hypercubic lattice and short range interactions. The order parameter (S z ) and the correlation function G=) Sigma i S i z Sigma j S j z ) are calculated respectively below and above Gamma C . The gap in the excitation spectrum is also studied above the critical field. The critical behaviour obtained from these expansions at T=0 is different from that obtained at T finite and is shown to be analogous to the critical behaviour of the Ising model for a lattice with one more dimension.
Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .
An Ising model in a transverse field with general spin S is considered within RPA Green-function theory in the paramagnetic region. Expressions for the temperature renormalised collective excitation spectrum with a gap decreasing as the exchange interaction increases, the transverse correlation function, and the critical exchange-field ratio for magnetic ordering, are derived. Zero-temperature expansions are given in the limit of weak interaction. Critical values at T=0K are calculated for several values of S and an asymptotical expression for S>>1 is given.