Spin 1 ising model on the cubic lattices : critical temperature

We have extended the series for the zero-field susceptibility of the spin- $S$ Ising model to eighth order in the reduced temperature $K$ , on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series ${h}_{n}(S)$ are expressed as simple polynomials in $X=S(S+1)$ . For the face-centered-cubic lattice, an accurate polynomial fit to the critical point ${K}_{c}(S)$ is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all $S$ . However, the large- $S$ series seems to exhibit more rapid apparent convergence.We have extended the series for the zero-field susceptibility of the spin-$S$ Ising model to eighth order in the reduced temperature $K$, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series ${h}_{n}(S)$ are expressed as simple polynomials in $X=S(S+1)$. For the face-centered-cubic lattice, an accurate polynomial fit to the critical point ${K}_{c}(S)$ is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all $S$. However, the large-$S$ series seems to exhibit more rapid apparent convergence.

Physical review. B, Solid stateApr 1, 197442 citations

The Critical Temperature of a Spin‐One Ising Model on a Rectangular Lattice

Viktor Urumov

physica status solidi (b)Jan 1, 19882 citations

Extension to order β23 of the high-temperature expansions for the spin- 12 Ising model on simple cubic and body-centered cubic lattices

P. Butera, M. Comi

Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$

arXiv:hep-lat/0006009Physical review. B, Condensed matterDec 1, 200040 citations

The phase transition in the simple cubic ising lattice

M. Helm

PhysicaNov 1, 19720 citations

Phase Diagram and Ground State of a Decorated Ising Model on a Cubic Lattice

Vadim A. Mutailamov, Akai Kurbanovich Murtazaev

Abstract The static critical behavior of a three-dimensional decorated Ising model on a cubic lattice is investigated by the methods of computational physics. The case is considered when the exchange interaction Jn = –1 between nodal spins is antiferromagnetic and the interaction Jd between nodal spins and decorated spins varies within [–3.0; 3.0]. For various values of Jd, the ground state of the model is determined, the critical temperatures and critical exponents are calculated, and the phase diagram is constructed. It is shown that, for any ratios of exchange interactions in the model, a second-order phase transition is observed from an ordered to a paramagnetic state, and the critical exponents are close to the values for the standard Ising model.Abstract The static critical behavior of a three-dimensional decorated Ising model on a cubic lattice is investigated by the methods of computational physics. The case is considered when the exchange interaction Jn = –1 between nodal spins is antiferromagnetic and the interaction Jd between nodal spins and decorated spins varies within [–3.0; 3.0]. For various values of Jd, the ground state of the model is determined, the critical temperatures and critical exponents are calculated, and the phase diagram is constructed. It is shown that, for any ratios of exchange interactions in the model, a second-order phase transition is observed from an ordered to a paramagnetic state, and the critical exponents are close to the values for the standard Ising model.

Journal of Experimental and Theoretical PhysicsJul 1, 20216 citations

We have extended the series for the zero-field susceptibility of the spin-

S

Ising model to eighth order in the reduced temperature

K

, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series

{h}_{n}(S)

are expressed as simple polynomials in

X=S(S+1)

. For the face-centered-cubic lattice, an accurate polynomial fit to the critical point

{K}_{c}(S)

is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all

S

. However, the large-

S

series seems to exhibit more rapid apparent convergence.