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Physical Review Letters

What is the dimension from scaling of finite systems?

M. A. Novotny

Published Jan 11, 1993• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.70.109

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Finite-size scaling allows the extraction of a dimension from scaling ds{\mathit{d}}_{\mathit{s}}ds​, as well as critical exponents for systems with a second-order phase transition. The calculation of ds{\mathit{d}}_{\mathit{s}}ds​ allows comparisons to be made with other expansions in dimension, in which ds{\mathit{d}}_{\mathit{s}}ds​ 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.
Critical exponents are presented for the ferromagnetic Ising model in dimensions between one and two. The exponents are calculated by applying finite-size-scaling methods to numerical transfer-matrix data. The transfer matrix, which is translationally invariant, is written in such a fashion that interpolation to noninteger dimensions is possible. Results are also obtained for the first derivatives evaluated at d=2 for the critical exponents yT{\mathit{y}}_{\mathit{T}}yT​ and yH{\mathit{y}}_{\mathit{H}}yH​ as a function of dimension. All results are compared with series-expansion results in 1d\ensuremath{\le}2.
The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.
We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density, which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension D=70D=70D=70, the absolute errors of the critical temperature Tc{T}_{c}Tc​ and the critical exponent $\ensuremath{\nu}$, $\ensuremath{\beta}$ are at the order of ${10}^{\ensuremath{-}7},{10}^{\ensuremath{-}5}$, ${10}^{\ensuremath{-}4}$, respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cutoff in bond dimension and elucidate its physical meaning in this context.
INTRODUCTION Finite-size scaling (FSS) is a well established technique for the extraction of critical exponents from finite volume analyses [1]. Such exponents characterise critical phenomena at a second-order phase transition. The simplest model exhibiting such a transition is the Ising model in two dimensions, which, despite a long history and extensive study, still o#ers new results and insights. Here, we study the model under the special boundary conditions of Brascamp and Kunz [2] to extract new information and to help resolve some hitherto puzzling features of FSS. Let CL (#) be the specific heat at inverse temperature # for a system of linear extent L. FSS of the specific heat is characterized by the location of its peak, #L , its height CL (# L ) and its value at the infinite-volume critical point CL (# c ). The peak position, #L , is a pseudocritical point which typically approaches # c as L ## as |# # c L -# , (1) where # is the shift exponent. In two dimensions,
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.