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Applications
2. Applications Introduction Finite-size scaling really became an important tool with the development by Nightingale of the so-called phenomenological renormalization method. Although in principle it can be used in any number of dimensions for which finite-size scaling is valid, for practical purposes its usefulness has largely been restricted to two dimensions. Consider then some two-dimensional lattice model with a single adjustable parameter which we can think of as the temperature Γ, defined on an infinite strip of width L with periodic boundary conditions. The transfer matrix t for this strip is usually a finite matrix, and we can find its largest eigenvalues λ 0 , λ ΐ9 As is well known, — In λ 0 gives the free energy per unit length, \Ώ.(\0/\Ύ) gives the inverse correlation length ξ(Τ, L ) _ 1 , and so on. The simplest way to calculate these eigenvalues is often to calculate in acting on a suitable state for n large. Finite-size scaling predicts that ξ(Τ, L) = L((T-Tc)1/PL),
(2.1)
which suggests that we define a mapping T-*T' by L~li{T, L) = (L- 1) _1 ί(Γ', L- 1).
(2.2)
From (2.1) we see that this mapping has a fixed point at T= Tc, and that
In practice, then, one numerically calculates ξ(Τ, L) and defines an effective TC(L) by L-^(TC(L), L) = {L-\yli{Tc{L),
L-l).
(2.4)
Equation (2.3), with Tc replaced by TC(L), then defines an approximant v(L). To get rehable values of Tc and v, it is necessary to perform an extrapolation for large L. It is possible to also calculate other exponents by adding suitable terms of the Hamiltonian. That these ideas work for the Ising model, where, thanks to Onsager, we know exact results in finite-width strips, was shown by Nightingale (paper 2.1). In paper 2.2 we see these methods applied to the Potts model. One should remark that, despite its name, phenomenological renormalization is not a true RG transformation, in that it does not preserve a whole probability distribution, or set of correlation functions, but rather just one quantity (L _ 1 £ in the above example). Barber [1] has shown how, by considering more quantities, one can gain access to the projection of the scaling variables into the space of physical parameters, but this process becomes increasingly complicated as more information is required. 183
184
J.L. Cardy
The methods described above are limited by the size of the matrices with which one can calculate. Larger values of L can be treated by going to the so-called quantum Hamiltonian limit. This involves considering anisotropic couplings, parallel to and across the strip. Call them l£|l and K± . Then in the limit in which K^ -> oo and K± -> 0, in such a way that we stay near the critical curve, the transfer matrix simplifies: it can be written ? « 1 — K±H, where H is a sum of local matrices, and can be thought of as the Hamiltonian of a quantum system in one less dimension. For many models, this limit of extreme anisotropy does not modify the universahty class, and the advantage is that powerful methods are available to calculate the lowest few eigenvalues of H. Paper 2.3, by Hamer and Barber, illustrates the method. For some systems, however, anisotropy is an essential feature. For the model of directed percolation, for example, it is known that there is a preferred direction, and that the correlation lengths parallel and perpendicular to this direction, £y and ξ±9 diverge with different exponents v^ and v± in the infinite system. In the finite-width strip, finite-size scaling is thus modified. If we align the preferred direction along the strip, we would expect finite-size effects to occur when ξ± ~ L. At this point £ü - Lv"/p±. Thus the factors of L _ 1 and (L— 1 ) _ 1 in (2.2) should be replaced by L~v"/p± and (L - \)~v\\/p± respectively. Kinzel and Yeomans (paper 2.4) showed that v^ and p± can then be extracted by comparing the values of £y on strips of three different widths. Many modern problems of critical behavior in statistical physics are of a geometrical nature, for example percolation and self-avoiding random walks. In that case there are no Boltzmann weights, and the concept of a transfer matrix requires rethinking. Nevertheless, it does make sense, and finite-size scaling has had some spectacular successes for these types of model, as shown by papers 2.5-2.7, by Derrida et al. The question of the rate of convergence of all the above methods is interesting. RG ideas would suggest that corrections should be of the form L~y, where —y is the leading irrelevant RG eigenvalue. Numerically this does not seem to be the case. Privman and Fisher (paper 2.8) show that such terms can however be seen by looking at very large values of L. Why these terms have such small amplitudes that, for practical purposes, they are swamped by L~2 terms is not understood. We continue the selection of papers in this section with a different subject: how do finite-size effects enter a massive, asymptotically free theory (papers 2.9, 2.10)? This question is important for lattice gauge theories. Finally we include a study (paper 2.11) of the zeroes of the partition function of a finite Ising model. Such studies are important because they may give us clues to the eventual solution of such models.
Reference [1] M.N. Barber, Phys. Rev. B 27 (1983) 5879; Physica A 130 (1985) 171.
(2.1)
which suggests that we define a mapping T-*T' by L~li{T, L) = (L- 1) _1 ί(Γ', L- 1).
(2.2)
From (2.1) we see that this mapping has a fixed point at T= Tc, and that
In practice, then, one numerically calculates ξ(Τ, L) and defines an effective TC(L) by L-^(TC(L), L) = {L-\yli{Tc{L),
L-l).
(2.4)
Equation (2.3), with Tc replaced by TC(L), then defines an approximant v(L). To get rehable values of Tc and v, it is necessary to perform an extrapolation for large L. It is possible to also calculate other exponents by adding suitable terms of the Hamiltonian. That these ideas work for the Ising model, where, thanks to Onsager, we know exact results in finite-width strips, was shown by Nightingale (paper 2.1). In paper 2.2 we see these methods applied to the Potts model. One should remark that, despite its name, phenomenological renormalization is not a true RG transformation, in that it does not preserve a whole probability distribution, or set of correlation functions, but rather just one quantity (L _ 1 £ in the above example). Barber [1] has shown how, by considering more quantities, one can gain access to the projection of the scaling variables into the space of physical parameters, but this process becomes increasingly complicated as more information is required. 183
184
J.L. Cardy
The methods described above are limited by the size of the matrices with which one can calculate. Larger values of L can be treated by going to the so-called quantum Hamiltonian limit. This involves considering anisotropic couplings, parallel to and across the strip. Call them l£|l and K± . Then in the limit in which K^ -> oo and K± -> 0, in such a way that we stay near the critical curve, the transfer matrix simplifies: it can be written ? « 1 — K±H, where H is a sum of local matrices, and can be thought of as the Hamiltonian of a quantum system in one less dimension. For many models, this limit of extreme anisotropy does not modify the universahty class, and the advantage is that powerful methods are available to calculate the lowest few eigenvalues of H. Paper 2.3, by Hamer and Barber, illustrates the method. For some systems, however, anisotropy is an essential feature. For the model of directed percolation, for example, it is known that there is a preferred direction, and that the correlation lengths parallel and perpendicular to this direction, £y and ξ±9 diverge with different exponents v^ and v± in the infinite system. In the finite-width strip, finite-size scaling is thus modified. If we align the preferred direction along the strip, we would expect finite-size effects to occur when ξ± ~ L. At this point £ü - Lv"/p±. Thus the factors of L _ 1 and (L— 1 ) _ 1 in (2.2) should be replaced by L~v"/p± and (L - \)~v\\/p± respectively. Kinzel and Yeomans (paper 2.4) showed that v^ and p± can then be extracted by comparing the values of £y on strips of three different widths. Many modern problems of critical behavior in statistical physics are of a geometrical nature, for example percolation and self-avoiding random walks. In that case there are no Boltzmann weights, and the concept of a transfer matrix requires rethinking. Nevertheless, it does make sense, and finite-size scaling has had some spectacular successes for these types of model, as shown by papers 2.5-2.7, by Derrida et al. The question of the rate of convergence of all the above methods is interesting. RG ideas would suggest that corrections should be of the form L~y, where —y is the leading irrelevant RG eigenvalue. Numerically this does not seem to be the case. Privman and Fisher (paper 2.8) show that such terms can however be seen by looking at very large values of L. Why these terms have such small amplitudes that, for practical purposes, they are swamped by L~2 terms is not understood. We continue the selection of papers in this section with a different subject: how do finite-size effects enter a massive, asymptotically free theory (papers 2.9, 2.10)? This question is important for lattice gauge theories. Finally we include a study (paper 2.11) of the zeroes of the partition function of a finite Ising model. Such studies are important because they may give us clues to the eventual solution of such models.
Reference [1] M.N. Barber, Phys. Rev. B 27 (1983) 5879; Physica A 130 (1985) 171.