Finite-Size Effects in the φ⁴ Field and Lattice Theory Above the Upper Critical Dimension

Abstract

We demonstrate that the standard O(n) symmetric ϕ 4 field theory does not correctly describe the leading finite-size effects near the critical point of spin systems on a d-dimensional lattice with d > 4. We show that these finite-size effects require a description in terms of a lattice Hamiltonian. For n → ∞ and n = 1 explicit results are given for the susceptibility and for the Binder cumulant. They imply that recent analyses of Monte-Carlo results for the five-dimensional Ising model are not conclusive.PACS numbers: 05.70. Jk, 64.60.Ak, 75.40.Mg The effect of a finite geometry on systems near phase transitions is of basic interest to statistical physics and elementary particle physics. In both areas the ϕ 4 Hamiltonianfor an n-component field ϕ(x) in a finite volume V plays a fundamental role [1]. For simplicity we considerThe summation runs over discrete k vectors with componentsIt is generally believed that the leading finite-size effects near the critical point of d-dimensional systems can be described by H both for d ≤ d u and for d > d u where d u = 4 is the upper critical dimension. Since for d > 4 the bulk critical behavior is mean-field like, it is plausible that the leading finite-size effects for d > 4 appear to be describable in terms of a simplified Hamiltonian [2,3]involving only the homogeneous fluctuations of the low- In this Letter we shall demonstrate that the lowestmode approach fails for the Hamiltonian H in Eq.(1) for d > 4 and that the leading finite-size effects of spin systems on a d-dimensional lattice with d > 4 are not correctly described by H. We show that this defect of H is due to the (▽ϕ) 2 term. These unexpected findings shed new light on the role of lattice effects for d > d u and imply that recent analyses of the MC data [4][5][6][7] in terms of the continuum ϕ 4 theory are not conclusive. We shall prove our claims first in the large-n limit where a saddle point approach [1] can be employed. Our proof is not based on the renormalization group. We have extended the saddle point approach to the finite system to derive the order-parameter correlation functionwith the statistical weight exp(−H). In the limit n → ∞ at fixed u 0 n we have found the exact resultWe shall denote the bulk critical temperature by T c . For T ≥ T c , χ can be interpreted as the susceptibility (per component) of the finite system. In the bulk limit the standard equation [8] for the bulk susceptibility χ b for T ≥ T c is recovered from Eq.(4) aswhere k stands for (2π)It is convenient to rewrite Eq.(4) in terms of r 0 − r 0c = a 0 t where r 0c = −4u 0 n k k −2 is the bulk critical value of r 0 as determined from Eq. (5) δr 0 = a 0 t − ∆,

The Finite-Size Scaling Study of the Specific Heat and the Binder Parameter for the Five-Dimensional Ising Model

“…9 Our data for lattice sizes also indicate a critical cumulant g L (T c ) = −1.049. This result is in good agreement with the Monte-Carlo simulations results of g L (T c ) ∼ = −1.00 for sizes between L = 3 and 7, 6 and g L (T c ) = −0.958 for lattice sizes up to 17,15,16 while in disagreement with the analytic d = 5 prediction of The finite-size scaling relation for the Binder parameter has the following form:…”

“…15,16 Therefore, these works were criticized 8 as being in contradiction to MC data. According to Chen and Dohm (CD) 17 for larger systems the behavior in five dimensions, above the upper critical dimension of four, is far from trivial: some infinite-size field theories are invalid, and finite-size scaling is not of the usual one-parameter type.…”

Monte Carlo investigations of phase transitions: status and perspectives

“…Chen and Dohm [59,60] have recently criticized the whole approach sketched above and maintained that one must return to a finite-size scaling description in which both variables t(L/ξ) 2 and (L/ℓ 0 ) 4−d are kept separate, as in Eqs. (11) and (12).…”

Test of renormalization predictions for universal finite-size scaling functions

“…However, apart from the validity of this suggestion, it is difficult to envisage how this would lead to the (dis)appearance of a square-root contribution in the ε-expansion. Furthermore, it is an open question to what extent the breakdown of the field-theoretic description of finite-size scaling for d ≥ 4 [19] influences the nature of the ε-expansion. We feel that an understanding of these problems is of some significance for the understanding of finite-size scaling of critical phenomena in general.…”