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

Canadian Journal of Physics

Isothermal and Frequency Dependent Perpendicular Susceptibilities of the Ising Model on the Cayley Tree

T. Horiguchi, T. Morita

Published Nov 1, 1975• 9 citations
Publisher

Abstract

The isothermal and frequency dependent perpendicular susceptibilities under zero external field are obtained for the Ising model on the Cayley tree. The difference between the isothermal and isolated susceptibilities is discussed. It is shown that two simple identities about the spin operators facilitate the derivation of the basic formulas for the perpendicular susceptibilities of the Ising model on a regular or irregular lattice.

DOI
10.1139/p75-288

Related papers

Isothermal and Frequency‐Dependent Perpendicular Susceptibilities of the One‐Dimensional Ising Model with Nearest and Next‐Nearest Neighbour Interactions

L.L. Gonçalves

Abstract Exact expressions for the isothermal and frequency‐dependent perpendicular susceptibilities are obtained for the Ising model (s = 1/2) with nearest and next‐nearest neighbour interactions under zero external field. The static correlations are obtained by mapping the model in a pseudo Ising model with nearest‐neighbour interactions under an external field. The difference between the isothermal and isolated susceptibilities is discussed.

physica status solidi (b)•Apr 1, 1981•0 citations

The frequency-dependent initial perpendicular susceptibility of the Ising model

G. A. T. Allan, Donald D. Betts

The Kubo linear response theory is applied to the quantum mechanical Ising model to obtain a general expression which is valid for all regular lattices for the perpendicular response function in terms of spin correlations. For the honeycomb and the square lattices, exact expressions are found for the frequency-dependent initial perpendicular susceptibility, [Formula: see text] valid for all temperatures. For the honeycomb lattice, the present result for zero frequency agrees with Fisher's static calculation, whereas there is disagreement in the case of the square lattice. The discrepancy is resolved by using an extended calculation which explicitly illustrates the difference between isothermal and adiabatic susceptibilities.

Canadian Journal of Physics•Jan 1, 1968•20 citations

Calculation of the isothermal susceptibility by the Kubo formula

T. Morita, T. Horiguchi

The relation between the zero frequency limit of the frequency-dependent susceptibility and the isothermal susceptibility is made clearer by expressing them in terms of the spectral density function. The general formulas are illustrated for the perpendicular susceptibilities of the Ising model.

Canadian Journal of Physics•Jul 15, 1976•1 citation

Modified Admittance, Perpendicular Susceptibilities, and Transformation of Correlations of the Ising Model

Shoon Kyung Kim

A modified admittance is introduced to give Kubo's admittance at nonzero frequencies and to give the isothermal static admittance at zero frequency within the scope of the Kubo linear-response theory. The method is demonstrated by exact calculations of the frequency-dependent perpendicular susceptibility at zero field and its modified susceptibility of the regular Ising model. The results appear as linear combinations of the equilibrium spin-spin correlation functions of the lattice. The results are valid for all dimensions and all frequencies and temperatures. A (q + 1) × (q + 1) matrix a(q) describes the linear combinations explicitly, where q is the coordination number of the lattice. The properties of this matrix are extensively discussed as a special case of a matrix A(q)(ξ), which satisfies a simple quadratic equation of the form [A(q)(ξ)]2 = (1 + ξ)q for arbitrary values of ξ. Fisher's algebraic transformation of the spin-spin correlation functions for the regular Ising lattice is derived from the linear relation which holds between the perpendicular susceptibility and the corresponding modified susceptibility. By means of the product rules of the matrix A(q)(ξ), the higher-order spin-spin correlation functions are expressed in terms of the lower-order ones in complete generality.

Journal of Mathematical Physics•Oct 1, 1968•5 citations

Perpendicular Susceptibility of the Ising Model

Michael Ellis Fisher

A high-temperature expansion of the partition function for a lattice of N spins with Hamiltonian H=−J ∑ (ij)σizσjz−mHz ∑ iσiz−mHx ∑ iσixis derived and thence an expansion for the zero-field perpendicular susceptibility is found. By perturbation theory, χ⊥(T) is also expanded at low temperatures and seen, in general, to increase with T from the value χ⊥(O) = Nm2/q | J |, where q is the coordination number of the lattice. The perpendicular susceptibility is re-expressed in terms of near neighbor pair and higher-order spin correlation functions in zero field. This yields exact closed formulas for the linear chain, the Bethe pseudolattices, and for the plane square and honeycomb lattices. The behavior of χ⊥(T) in the critical region is investigated for these lattices and for the plane superexchange lattice.

Journal of Mathematical Physics•Jan 1, 1963•308 citations

Exact solutions for perpendicular susceptibilities of kagomé and decorated-kagomé Ising models

Jeremiah H. Barry, Mahfuza Khatun

For more than two decades, exact and explicit solutions for the initial isothermal transverse susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) of the quantum-mechanical Ising model ferromagnet have been known on the honeycomb, square, and triangular lattices but not as yet on the kagom\'e lattice which is the only other regular lattice in two dimensions. Recently, however, exact solutions have been found for localized even-number correlations of the ferromagnetic kagom\'e Ising model, rendering its ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) exactly calculable. Similar to the other three regular lattices, the resulting continuous curve for ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T)/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0) shows both a weak (energy-type) singularity at the critical temperature ${T}_{c}$ where ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(${T}_{c}$)/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0)=1.1426. . . and a rounded maximum at ${T}_{\mathrm{max}>{T}_{c}}$ with the parameters of this maximum given by ${\ensuremath{\chi}}_{\ensuremath{\perp}}^{\mathrm{max}}$/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0)=1.1945. . . at ${T}_{\mathrm{max}}$/${T}_{c}$=1.0994. . . . The exact solution for ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) of the decorated-kagom\'e Ising-model ferromagnet is also obtained having modified features illustrating the effects of the decoration spins (singly decorated bonds) upon ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T).

Physical review. B, Condensed matter•Jun 1, 1987•16 citations
The isothermal and frequency dependent perpendicular susceptibilities under zero external field are obtained for the Ising model on the Cayley tree. The difference between the isothermal and isolated susceptibilities is discussed. It is shown that two simple identities about the spin operators facilitate the derivation of the basic formulas for the perpendicular susceptibilities of the Ising model on a regular or irregular lattice.
The Kubo linear response theory is applied to the quantum mechanical Ising model to obtain a general expression which is valid for all regular lattices for the perpendicular response function in terms of spin correlations. For the honeycomb and the square lattices, exact expressions are found for the frequency-dependent initial perpendicular susceptibility, [Formula: see text] valid for all temperatures. For the honeycomb lattice, the present result for zero frequency agrees with Fisher's static calculation, whereas there is disagreement in the case of the square lattice. The discrepancy is resolved by using an extended calculation which explicitly illustrates the difference between isothermal and adiabatic susceptibilities.
A modified admittance is introduced to give Kubo's admittance at nonzero frequencies and to give the isothermal static admittance at zero frequency within the scope of the Kubo linear-response theory. The method is demonstrated by exact calculations of the frequency-dependent perpendicular susceptibility at zero field and its modified susceptibility of the regular Ising model. The results appear as linear combinations of the equilibrium spin-spin correlation functions of the lattice. The results are valid for all dimensions and all frequencies and temperatures. A (q + 1) × (q + 1) matrix a(q) describes the linear combinations explicitly, where q is the coordination number of the lattice. The properties of this matrix are extensively discussed as a special case of a matrix A(q)(ξ), which satisfies a simple quadratic equation of the form [A(q)(ξ)]2 = (1 + ξ)q for arbitrary values of ξ. Fisher's algebraic transformation of the spin-spin correlation functions for the regular Ising lattice is derived from the linear relation which holds between the perpendicular susceptibility and the corresponding modified susceptibility. By means of the product rules of the matrix A(q)(ξ), the higher-order spin-spin correlation functions are expressed in terms of the lower-order ones in complete generality.
A high-temperature expansion of the partition function for a lattice of N spins with Hamiltonian H=−J ∑ (ij)σizσjz−mHz ∑ iσiz−mHx ∑ iσixis derived and thence an expansion for the zero-field perpendicular susceptibility is found. By perturbation theory, χ⊥(T) is also expanded at low temperatures and seen, in general, to increase with T from the value χ⊥(O) = Nm2/q | J |, where q is the coordination number of the lattice. The perpendicular susceptibility is re-expressed in terms of near neighbor pair and higher-order spin correlation functions in zero field. This yields exact closed formulas for the linear chain, the Bethe pseudolattices, and for the plane square and honeycomb lattices. The behavior of χ⊥(T) in the critical region is investigated for these lattices and for the plane superexchange lattice.
For more than two decades, exact and explicit solutions for the initial isothermal transverse susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) of the quantum-mechanical Ising model ferromagnet have been known on the honeycomb, square, and triangular lattices but not as yet on the kagom\'e lattice which is the only other regular lattice in two dimensions. Recently, however, exact solutions have been found for localized even-number correlations of the ferromagnetic kagom\'e Ising model, rendering its ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) exactly calculable. Similar to the other three regular lattices, the resulting continuous curve for ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T)/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0) shows both a weak (energy-type) singularity at the critical temperature Tc{T}_{c}Tc​ where ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(Tc{T}_{c}Tc​)/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0)=1.1426. . . and a rounded maximum at Tmax>Tc{T}_{\mathrm{max}>{T}_{c}}Tmax>Tc​​ with the parameters of this maximum given by ${\ensuremath{\chi}}_{\ensuremath{\perp}}^{\mathrm{max}}$/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0)=1.1945. . . at Tmax{T}_{\mathrm{max}}Tmax​/Tc{T}_{c}Tc​=1.0994. . . . The exact solution for ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) of the decorated-kagom\'e Ising-model ferromagnet is also obtained having modified features illustrating the effects of the decoration spins (singly decorated bonds) upon ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T).