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

Abstract

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 .