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arXiv:cond-mat/0010479 · The European Physical Journal B

Finite-size correlation length and violations of finite-size scaling

Sergio Caracciolo A, Andrea Gambassi, Massimiliano Gubinelli, Andrea Pelissetto C

Published Mar 1, 2001• 23 citations
arXivHTMLPDF
DOI
10.1007/bf01352587

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Finite-size scaling of the d=5 Ising model embedded in a cylindrical geometry: The influence of hyperscaling violation

Yoshihiro Nishiyama

Finite-size scaling (FSS) of the five-dimensional (d=5) Ising model is investigated numerically. Because of the hyperscaling violation in d>4 , FSS of the d=5 Ising model no longer obeys the conventional scaling relation. Rather, it is expected that the FSS behavior depends on the geometry of the embedding space (boundary condition). In this paper, we consider a cylindrical geometry and explore its influence on the correlation length xi=L;{Omega}f(L;{y_{t};{*}},HL;{y_{h};{*}}) with system size L , reduced temperature , and magnetic field H ; the indices y_{t,h};{*} and Omega characterize FSS. For that purpose, we employed the transfer-matrix method with Novotny's technique, which enables us to treat an arbitrary (integral) number of spins, N=8,10,...,28 ; note that, conventionally, N is restricted in N(=L;{d-1})=16,81,256,... . As a result, we estimate the scaling indices as Omega=1.40(15) , y_{t};{*}=2.8(2) , and y_{h};{*}=4.3(1) . Additionally, postulating Omega=43 , we arrive at y_{t};{*}=2.67(10) and y_{h};{*}=4.0(2) . These indices differ from the naively expected ones Omega=1 , y_{t};{*}=2 and y_{h};{*}=3 . Rather, our data support the generic formulas Omega=(d-1)3 , y_{t};{*}=2(d-1)3 , and y_{h};{*}=d-1 , advocated for a cylindrical geometry in d4 .

arXiv:cond-mat/0611767•Physical Review E•Jan 10, 2007•7 citations
We address the problem of the definition of the finite-volume correlation length. First, we study the large-N limit of the N-vector model, and we show the existence of several constraints on the definition if regularity of the finite-size scaling functions and correct anomalous behaviour above the upper critical dimension are required. Then, we study in detail a model in which the zero mode is prohibited. Such. a model is a generalization of the fixed-magnetization Ising model which is equivalent to the lattice gas. Also in this case, we find that the finite-volume correlation length must satisfy appropriate constraints in order to obtain regular finite-size scaling functions, and, above the upper critical dimension, an anomalous scaling behaviour. The large-N results are confirmed by a one-loop calculation in the lattice phi (4) theory.
We study the large-distance behavior of the bulk order-parameter correlation function for T > T c within the lattice version of the theory including lattice effects. We also study the large-L behavior of the susceptibility for T > T c of the confined lattice system of linear size L with periodic boundary conditions. We find that the structure of the large-L behavior of of the confined system is closely related to the structure of the large-distance behavior of of the bulk system. Explicit results are derived in the spherical (large-n) limit and in one-loop order for general dimensions d > 2. For the lattice model with cubic symmetry we find that finite-size scaling must be formulated in terms of the anisotropic bulk correlation length (exponential correlation length) that governs the exponential decay of for large r rather than in terms of the ordinary isotropic bulk correlation length defined via the second moment of . We show that it is the exponential bulk correlation length in the direction of the cubic axes that determines the exponential finite-size scaling behavior of lattice systems in a rectangular geometry. This result modifies a recent interpretation concerning an apparent violation of finite-size scaling in terms of the second-moment correlation length . Exact results for the one-dimensional Ising model illustrate our conclusions. Furthermore we show for general d>2 that a description of finite-size effects for finite n in the entire region requires different perturbative approaches that are applicable either to the region or , respectively. In particular we show that the exponential finite-size behavior for above T c is not captured by the standard perturbation approach that separates the homogeneous lowest mode from the inhomogeneous higher modes. Consequences for the theory of finite-size effects above four dimensions are discussed. We show that the two-variable finite-size scaling form predicts an exponential approach to the bulk critical behavior above T c whereas the reduction to a single-variable scaling form implies a power-law approach .
Finite-size scaling (FSS) of the five-dimensional (d=5) Ising model is investigated numerically. Because of the hyperscaling violation in d>4 , FSS of the d=5 Ising model no longer obeys the conventional scaling relation. Rather, it is expected that the FSS behavior depends on the geometry of the embedding space (boundary condition). In this paper, we consider a cylindrical geometry and explore its influence on the correlation length xi=L;{Omega}f(L;{y_{t};{*}},HL;{y_{h};{*}}) with system size L , reduced temperature , and magnetic field H ; the indices y_{t,h};{*} and Omega characterize FSS. For that purpose, we employed the transfer-matrix method with Novotny's technique, which enables us to treat an arbitrary (integral) number of spins, N=8,10,...,28 ; note that, conventionally, N is restricted in N(=L;{d-1})=16,81,256,... . As a result, we estimate the scaling indices as Omega=1.40(15) , y_{t};{*}=2.8(2) , and y_{h};{*}=4.3(1) . Additionally, postulating Omega=43 , we arrive at y_{t};{*}=2.67(10) and y_{h};{*}=4.0(2) . These indices differ from the naively expected ones Omega=1 , y_{t};{*}=2 and y_{h};{*}=3 . Rather, our data support the generic formulas Omega=(d-1)3 , y_{t};{*}=2(d-1)3 , and y_{h};{*}=d-1 , advocated for a cylindrical geometry in d4 .