Optimised renormalisation group transformations of lattice spin models

Bennett, Adam

doi:10.1088/0305-4470/20/14/027

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J. Phys. A: Math. Gen.

1987

DOI: 10.1088/0305-4470/20/14/027

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Optimised renormalisation group transformations of lattice spin models

Adam Bennett¹

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1996

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“…The precise agreement of the renormalization results with those obtained by finite-size scaling confirms the validity of hyperscaling [19]. Earlier attempts to accelerate the convergence used optimized transformations [20][21][22][23][24][25]. The present work uses this idea combined with a Hamiltonian with suppressed corrections to scaling.…”

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Monte Carlo Renormalization of the 3D Ising Model: Analyticity and Convergence

et al. 1996

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We review the assumptions on which the Monte Carlo renormalization technique is based, in particular, the analyticity of the block-spin transformations. On this basis, we select an optimized Kadanoff blocking rule in combination with the simulation of a d 3 Ising model with reduced corrections to scaling. This is achieved by including interactions with second and third neighbors. As a consequence of the improved transformation, this Monte Carlo renormalization method yields a fast convergence and a high accuracy. The results for the critical exponents are y H 2.481͑1͒ and y T 1.585͑3͒.PACS numbers: 05.50.+q, 02.70.Lq, 75.10.Hk, 75.40.Mg Applications of the Monte Carlo renormalization group (MCRG) to the three-dimensional Ising model [1][2][3][4] have become increasingly elaborate and complicated, and tend to require considerable computer resources. Nevertheless, there are still uncertainties due to the basic assumptions underlying the renormalization transformations used. In particular, we will focus on the question concerning the analyticity of the transformation, which may be related to the question whether the corrections to scaling vanish at the fixed point of the transformation [5]. In this Letter we first calculate the analytic part of a divergent observable; this demonstrates that gross nonanalyticities are normally absent. However, in general we may expect weak nonanalyticities due to corrections to scaling. Thus, second, we minimize their effects by adjusting the transformation as well as the Hamiltonian that is simulated.The MCRG method has amply been reviewed [6,7], and here we only briefly outline the method. The reduced Hamiltonian is written aswhere S is a spin configuration, the K a are couplings, and the S a are the conjugate lattice sums over spin products, e.g., K 1 is the magnetic field and S 1 P i s i the sum over all spins; K 2 is the nearest-neighbor coupling and S 2 P ͗nn͘ s i s j the sum over all nearest-neighbor pairs ͑s i , s j ͒. A special "coupling" is the background energy density K 0 ; S 0 is the number of spins. Application of a block-spin transformation to Monte Carlo generated configurations S leads to configurations S 0 described by a Hamiltonian H 0 H ͑K 0 0 , K 0 1 , K 0 2 , . . . ; S 0 ͒. The renormalized couplings K 0 a are assumed to be analytic functions of the original ones, even at the infinite system critical point. However, this property remains unproven in general, even for K 0 0 , or the so-called "analytic part" of the transformation.It is straightforward to calculate, using the Monte Carlo method, The dummy index summation rule applies to Greek indices. The matrix T is approximated by solving Eq. (5) after truncation to a finite number of couplings. Under iteration of the block-spin transformation, the K a (a . 0) are assumed to approach a critical fixed point, where the eigenvalues of T determine the critical exponents. Thus, the MCRG method relies on assumptions of (1) analyticity of the transformation, (2) convergence with the dimensionality of the coupling s...

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“…[17] Earlier attempts to accelerate the convergence used optimized transformations. [18][19][20][21][22][23] The present work uses this idea combined with a Hamiltonian with suppressed corrections to scaling. This alleviates the problem with the assumption of analyticity, and leads to a much improved convergence to the fixed point.…”

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confidence: 99%

Monte Carlo Renormalization of the 3D Ising Model: Analyticity and Convergence

et al. 1996

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Optimised renormalisation group transformations of lattice spin models

Cited by 1 publication

References 7 publications

Monte Carlo Renormalization of the 3D Ising Model: Analyticity and Convergence

Monte Carlo Renormalization of the 3D Ising Model: Analyticity and Convergence

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