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

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 ξ= LΩ f ( L yt*, H L yh*) with system size L, reduced temperature , and magnetic field H; the indices y t,h * and Ω 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 (= Ld-1) =16,81,256,.... As a result, we estimate the scaling indices as Ω=1.40 (15), yt* =2.8 (2), and yh* =4.3 (1). Additionally, postulating Ω=4 3, we arrive at yt* =2.67 (10) and yh* =4.0 (2). These indices differ from the naively expected ones Ω=1, yt* =2 and yh* =3. Rather, our data support the generic formulas Ω= (d-1) 3, yt* =2 (d-1) 3, and yh* =d-1, advocated for a cylindrical geometry in d≥4.