Physical Review Letters

Renormalization-Group Approach to the Percolation Properties of the Triangular Ising Model

W. Klein, Eugene Stanley, PETER J. REYNOLDS, Antonio Coniglio A

Published Oct 23, 1978• 78 citations

Abstract

We present a renormalization-group approach for treating the percolation properties of the nearest-neighbor triangular Ising model. We obtain exponents for the line of percolation transitions ${T}_{c}<~T<~\ensuremath{\infty}$. In particular, we find a possibly exact result for the connectedness-length exponent ${\ensuremath{\nu}}_{p}=\frac{\mathrm{ln}\sqrt{3}}{\mathrm{ln}(\frac{3}{2})}$ in the "pure" percolation ($T=\ensuremath{\infty}$) limit, which holds for ${T}_{c}<T<~\ensuremath{\infty}$. At $T={T}_{c}$ we find the connectedness-length exponents of the percolation problem to be identical to the correlation-length exponents for the thermal problem.

DOI: 10.1103/physrevlett.41.1145

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Physical Review Letters

Renormalization-Group Approach to the Percolation Properties of the Triangular Ising Model

W. Klein, Eugene Stanley, PETER J. REYNOLDS, Antonio Coniglio A

Published Oct 23, 1978• 78 citations

Publisher

Abstract

We present a renormalization-group approach for treating the percolation properties of the nearest-neighbor triangular Ising model. We obtain exponents for the line of percolation transitions

{T}_{c}<~T<~\ensuremath{\infty}

. In particular, we find a possibly exact result for the connectedness-length exponent

{\ensuremath{\nu}}_{p}=\frac{\mathrm{ln}\sqrt{3}}{\mathrm{ln}(\frac{3}{2})}

in the "pure" percolation (

T=\ensuremath{\infty}

) limit, which holds for

{T}_{c}<T<~\ensuremath{\infty}

. At

T={T}_{c}

we find the connectedness-length exponents of the percolation problem to be identical to the correlation-length exponents for the thermal problem.

DOI: 10.1103/physrevlett.41.1145

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Percolation renormalization-group approach to the hard-square model

Chin‐Kun Hu, Chi-Ning Chen

Physical review. B, Condensed matterMar 1, 199111 citations

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Physical review. B, Condensed matterAug 1, 19840 citations

We present a renormalization-group approach for treating the percolation properties of the nearest-neighbor triangular Ising model. We obtain exponents for the line of percolation transitions

{T}_{c}<~T<~\ensuremath{\infty}

. In particular, we find a possibly exact result for the connectedness-length exponent

{\ensuremath{\nu}}_{p}=\frac{\mathrm{ln}\sqrt{3}}{\mathrm{ln}(\frac{3}{2})}

in the "pure" percolation (

T=\ensuremath{\infty}

) limit, which holds for

{T}_{c}<T<~\ensuremath{\infty}

. At

T={T}_{c}

we find the connectedness-length exponents of the percolation problem to be identical to the correlation-length exponents for the thermal problem.

p

depending on the NN coupling constant

J

and the temperature

T

, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.

{T}_{c}

are derived for the susceptibility, specific heat, structure factor, and correlation length. The magnetization is found for

T<{T}_{c}

. The critical exponents and amplitudes for these quantities are calculated. The agreement between the known static properties and the renormalization-group results is good to excellent, and shows that this renormalization-group method can accurately calculate nonuniversal, as well as universal, quantities on different lattices. The computed dynamic structure factor, however, exhibits nonmonotonic behavior as a function of temperature. This nonmonotonic behavior is conjectured to be due to approximations in determining the expansion parameters.