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arXiv:cond-mat/9605019 · Journal of Physics A Mathematical and General

Computing the roughening transition of Ising and solid-on-solid models by BCSOS model matching

Martin Hasenbusch B, K. Pinn

Published Jan 7, 1997• 89 citations

Abstract

We study the roughening transition of the dual of the two-dimensional (2D) XY model, of the discrete Gaussian model, of the absolute value solid-on-solid model and of the interface in an Ising model on a three-dimensional (3D) simple cubic lattice. The investigation relies on a renormalization group finite size scaling method that was proposed and successfully tested a few years ago. The basic idea is to match the renormalization group flow of the interface observables with that of the exactly solvable body-centred solid-on-solid (BCSOS) model. Our estimates for the critical couplings are , and for the XY model, the discrete Gaussian model and the absolute value solid-on-solid model, respectively. For the inverse roughening temperature of the Ising interface we find . To the best of our knowledge, these are the most precise estimates for these parameters published so far.

DOI: 10.1088/0305-4470/30/1/006

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Physical review. B, Condensed matter•Apr 1, 1984•318 citations

We study the roughening transition of the dual of the two-dimensional (2D) XY model, of the discrete Gaussian model, of the absolute value solid-on-solid model and of the interface in an Ising model on a three-dimensional (3D) simple cubic lattice. The investigation relies on a renormalization group finite size scaling method that was proposed and successfully tested a few years ago. The basic idea is to match the renormalization group flow of the interface observables with that of the exactly solvable body-centred solid-on-solid (BCSOS) model. Our estimates for the critical couplings are , and for the XY model, the discrete Gaussian model and the absolute value solid-on-solid model, respectively. For the inverse roughening temperature of the Ising interface we find . To the best of our knowledge, these are the most precise estimates for these parameters published so far.

Space renormalization group calculations are performed for a class of spin-

S

Ising systems, using a modification of Migdal's recursion relation. Interface-roughening models are obtained when $S\ensuremath{\rightarrow}\ensuremath{\infty}$ and tricritical points and critical lines become the roughening temperature and rough phase. The latter corresponds to the low-temperature phases of the classical Coulomb gas and two-dimensional

\mathrm{XY}

model and implications for the

\mathrm{XY}

-model exponent $\ensuremath{\eta}$ are discussed.

Ising models on the square and simple cubic lattices are studied by a slightly modified version of the Niemeijer-van Leeuwen renormalization-group transformation. For the simple cubic lattice, we find the transition temperature Kc = 0.226 with critical exponents α= -0.420 and δ= 5.04 from the third order cumulant expansion. Ising models are also treated by the two-, three- and four-step decimation transformations among the simple cubic, body-centered cubic and face-centered cubic lattices. It is found from the three-step transformations that three systems have the same critical exponents. In contrast to the results from the cumulant expansion method, this method yields the positive specific heat exponent except one case. These results are compared with the known series values.

Using recently developed histogram techniques and an ultrafast multispin coding simulation algorithm, we have investigated the critical behavior of the d=3 simple-cubic Ising model. We have studied lattice sizes ranging from L=8 to 96 using between 3\ifmmode\times\else\texttimes\fi{}

{10}^{6}

and 12\ifmmode\times\else\texttimes\fi{}

{10}^{6}

Monte Carlo steps (complete lattice updates). By accurately measuring the finite-size behavior of several different thermodynamic quantities, we are able to determine the critical properties with a precision comparable to that obtained with Monte Carlo renormalization-group and sophisticated series-expansion techniques. The best estimate of the inverse critical temperature from our analysis is

{\mathit{K}}_{\mathit{c}}

=0.221 659 5\ifmmode\pm\else\textpm\fi{}0.000 002 6. The advantages of the histogram technique are discussed, as are the potential problems that can arise at this level of resolution.

The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with

{8}^{3}

,

{16}^{3}

,

{32}^{3}

, and

{64}^{3}

sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling

{K}_{1}^{c}=0.221654(6)

. After allowing for (i) interpolation to this best estimate for

{K}_{1}^{c}

, (ii) an apparent finite-size effect in the renormalization-group transformation due to the measurement of correlation functions of too few (seven) operators, and (iii) the extrapolation for the effect of a slow transient towards the fixed point, the values $\ensuremath{\nu}=0.629(4)$ and $\ensuremath{\eta}=0.031(5)$ are obtained for the thermal and magnetic exponents. The correction-to-scaling exponent $\ensuremath{\omega}$ is estimated to be around 1; to obtain an accuracy competitive with other methods requires measurements with more than seven operators. We briefly review the problem of redundant operators and indicate the future prospects for this kind of calculation.