It is shown that there is a class of graphs of the Ising model (or (Heisenberg model for $T>{T}_{\mathrm{Curie}}$ ) which is comprised of cycle graphs plus some excluded volume effects which sum to the spherical model. The spherical model, suitably generalized for $T<{T}_{\mathrm{Curie}}$ , was conjectured in a previous work to be the high-density limit of the Ising model, correct to $\frac{1}{z}$ , where $z$ is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). ${z}^{\frac{1}{3}}$ measures the range of the exchange potential. This is now proved by examining the omitted graphs. The error is shown to be $O(\frac{1}{{z}^{2}})$ .It is shown that there is a class of graphs of the Ising model (or (Heisenberg model for $T>{T}_{\mathrm{Curie}}$) which is comprised of cycle graphs plus some excluded volume effects which sum to the spherical model. The spherical model, suitably generalized for $T<{T}_{\mathrm{Curie}}$, was conjectured in a previous work to be the high-density limit of the Ising model, correct to $\frac{1}{z}$, where $z$ is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). ${z}^{\frac{1}{3}}$ measures the range of the exchange potential. This is now proved by examining the omitted graphs. The error is shown to be $O(\frac{1}{{z}^{2}})$.

Physical ReviewApr 15, 196168 citations

Note on the Critical Behavior of Ising Ferromagnets

Ryuzo Abe

Progress of Theoretical PhysicsJul 1, 196776 citations

Anisotropic Heisenberg interface between Ising bulk ferromagnets: A renormalization-group approach

Roberto J. V. dos Santos, Enaldo F. Sarmento, Constantino Tsallis +1 more

We consider two semi-infinite spin-1/2 Ising media (respectively characterized by the ferromagnetic coupling constants ${J}_{1}$ and ${J}_{2}$ ) separated by a spin-1/2 anisotropic Heisenberg interface (characterized by a ferromagnetic coupling constant ${J}_{S}$ and by an anisotropy \ensuremath{\eta}\ensuremath{\in}[0,1] in spin space; \ensuremath{\eta}=1 and \ensuremath{\eta}=0, respectively, correspond to the Ising and isotropic Heisenberg limits). Within a real-space renormalization-group framework, we discuss the full phase diagram; this diagram exhibits, besides the paramagnetic phase, three physically different ordered phases, namely the double-bulk, single-bulk, and surface ferromagnetic ones. We also analyze the various universality classes appearing in this problem, and determine, in particular, the existence of a new high-order multicritical point associated with the ${J}_{1}$ = ${J}_{2}$ case.We consider two semi-infinite spin-1/2 Ising media (respectively characterized by the ferromagnetic coupling constants ${J}_{1}$ and ${J}_{2}$) separated by a spin-1/2 anisotropic Heisenberg interface (characterized by a ferromagnetic coupling constant ${J}_{S}$ and by an anisotropy \ensuremath{\eta}\ensuremath{\in}[0,1] in spin space; \ensuremath{\eta}=1 and \ensuremath{\eta}=0, respectively, correspond to the Ising and isotropic Heisenberg limits). Within a real-space renormalization-group framework, we discuss the full phase diagram; this diagram exhibits, besides the paramagnetic phase, three physically different ordered phases, namely the double-bulk, single-bulk, and surface ferromagnetic ones. We also analyze the various universality classes appearing in this problem, and determine, in particular, the existence of a new high-order multicritical point associated with the ${J}_{1}$=${J}_{2}$ case.

Physical review. B, Condensed matterFeb 1, 19866 citations

Bounds on the critical temperature of Ising ferromagnets

George V. Waldo, J. R. Dorfman

Physics Letters AJul 1, 19722 citations

Ising limit of a Heisenberg XXZ magnet and some temperature correlation functions

N. M. Bogoliubov, Cyril Leonidovich Malyshev

arXiv:1106.5455Theoretical and Mathematical PhysicsNov 1, 201110 citations

AXIAL AND OFF-AXIAL DYNAMIC TRANSITIONS IN UNIAXIALLY ANISOTROPIC HEISENBERG FERROMAGNET: A COMPARISON

Muktish Acharyya

Uniaxially anisotropic Heisenberg ferromagnet, in the presence of a magnetic field varying sinusoidally in time, is studied by Monte Carlo simulation. The axial (field applied only along the direction of anisotropy) and off-axial (field applied only along the direction which is perpendicular to the direction of anisotropy) dynamic transitions are studied. By studying the distribution of the dynamic order parameter component, it is observed that the axial transition is discontinuous for low anisotropy and becomes continuous in high anisotropy. The off-axial transition is found to be continuous for all values of anisotropy. In the infinite anisotropy limit, both types of transitions are compared with that observed in an Ising ferromagnet for the same value of the field and frequency. The infinitely anisotropic axial transition and dynamic transition in the Ising ferromagnet occur at different temperatures, whereas the infinitely anisotropic off-axial transition and the equilibrium ferro-para transition in the Ising model occur at the same temperature.Uniaxially anisotropic Heisenberg ferromagnet, in the presence of a magnetic field varying sinusoidally in time, is studied by Monte Carlo simulation. The axial (field applied only along the direction of anisotropy) and off-axial (field applied only along the direction which is perpendicular to the direction of anisotropy) dynamic transitions are studied. By studying the distribution of the dynamic order parameter component, it is observed that the axial transition is discontinuous for low anisotropy and becomes continuous in high anisotropy. The off-axial transition is found to be continuous for all values of anisotropy. In the infinite anisotropy limit, both types of transitions are compared with that observed in an Ising ferromagnet for the same value of the field and frequency. The infinitely anisotropic axial transition and dynamic transition in the Ising ferromagnet occur at different temperatures, whereas the infinitely anisotropic off-axial transition and the equilibrium ferro-para transition in the Ising model occur at the same temperature.

arXiv:cond-mat/0307225International Journal of Modern Physics CJan 1, 200315 citations

It is shown that there is a class of graphs of the Ising model (or (Heisenberg model for

T>{T}_{\mathrm{Curie}}

) which is comprised of cycle graphs plus some excluded volume effects which sum to the spherical model. The spherical model, suitably generalized for

T<{T}_{\mathrm{Curie}}

, was conjectured in a previous work to be the high-density limit of the Ising model, correct to

\frac{1}{z}

, where

z

is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions).

{z}^{\frac{1}{3}}

measures the range of the exchange potential. This is now proved by examining the omitted graphs. The error is shown to be

O(\frac{1}{{z}^{2}})

We consider two semi-infinite spin-1/2 Ising media (respectively characterized by the ferromagnetic coupling constants

{J}_{1}

and

{J}_{2}

) separated by a spin-1/2 anisotropic Heisenberg interface (characterized by a ferromagnetic coupling constant

{J}_{S}

and by an anisotropy \ensuremath{\eta}\ensuremath{\in}[0,1] in spin space; \ensuremath{\eta}=1 and \ensuremath{\eta}=0, respectively, correspond to the Ising and isotropic Heisenberg limits). Within a real-space renormalization-group framework, we discuss the full phase diagram; this diagram exhibits, besides the paramagnetic phase, three physically different ordered phases, namely the double-bulk, single-bulk, and surface ferromagnetic ones. We also analyze the various universality classes appearing in this problem, and determine, in particular, the existence of a new high-order multicritical point associated with the

{J}_{1}

{J}_{2}

case.