Journal of Applied Physics

Transfer matrix studies of d≥3 Ising models

M. A. Novotny

Published May 1, 1990• 27 citations

Abstract

A novel numerical transfer matrix approach to obtain the critical behavior of Ising models in dimensions higher than two is described. The approach augments the physical transfer matrices with other matrices which do not necessarily have a straightforward physical interpretation. These additional matrices have approximately the same form as the physical transfer matrices, and for high dimensions may provide the only way of performing finite size scaling using transfer matrix calculations. Results are presented for the Ising model in three, four, and five dimensions.

DOI: 10.1063/1.345839

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Journal of Applied Physics

Transfer matrix studies of d≥3 Ising models

M. A. Novotny

Published May 1, 1990• 27 citations

Publisher

Abstract

DOI: 10.1063/1.345839

Related papers

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Finite-size scaling of corner transfer matrices for the two-dimensional Ising model

Ingo Peschel

Journal of Physics A Mathematical and GeneralFeb 7, 19889 citations

{\mathit{y}}_{\mathit{T}}

and

{\mathit{y}}_{\mathit{H}}

as a function of dimension. All results are compared with series-expansion results in 1d\ensuremath{\le}2.

Finite-size scaling allows the extraction of a dimension from scaling

{\mathit{d}}_{\mathit{s}}

, as well as critical exponents for systems with a second-order phase transition. The calculation of

{\mathit{d}}_{\mathit{s}}

allows comparisons to be made with other expansions in dimension, in which

{\mathit{d}}_{\mathit{s}}

plays the role of the physical dimension. This is demonstrated by a numerical transfer-matrix study of the pure ferromagnetic Ising model in two and three dimensions, as well as for quenched random bonds in the two-dimensional Ising ferromagnet.

The

d

-dimensional Ising transfer matrix is expressed exactly in terms of Fermi-type operators. In two dimensions this leads to the standard solution of the model. In three or more dimensions the transfer matrix is no longer a quadratic form in the Fermi variables, but rather contains quartic and higher-order terms. However, it can be expressed as a well-defined expansion about a quadratic form in Fermi operators. The zeroth-order transfer-matrix-called herein the "fermion model"---can be exactly diagonalized, and is well approximated in the critical region by ${V}^{(0)}\ensuremath{\approx}{[2sinh(2K)]}^{\frac{N}{2}}\mathrm{exp}\left[\ensuremath{-}N\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}{\left({\ensuremath{\kappa}}^{2}+\ensuremath{\Sigma}{n=1}^{d\ensuremath{-}1}\ensuremath{\Sigma}{m=1}^{d\ensuremath{-}1}{q}_{n}{q}_{m}\right)}^{\frac{1}{2}}t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})\right]$where

N

is the number of sites per layer, $\ensuremath{\mu}$ is the inverse correlation length, $t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ takes values \ifmmode\pm\else\textpm\fi{}1/2,

K

is the reduced inverse temperature, and the integral extends over a Brillouin zone of the ($d\ensuremath{-}1$)-dimensional layer. The critical point of the model is located by the equation ${x}_{c}^{d}+{x}_{c}^{d\ensuremath{-}1}+{x}_{c}=1$, where ${x}_{c}={e}^{\ensuremath{-}2{K}_{c}}$. The critical exponent $\ensuremath{\nu}$ describing the divergence of the correlation length at

{K}_{c}

is found to take on its two-dimensional value, $\ensuremath{\nu}=1$, in all dimensions

d

. The possibility of using the fermion model as the basis for an expansion of the Ising critical exponents as a function of $\ensuremath{\delta}(=d\ensuremath{-}2)$ is pointed out.