Finite-size scaling of the d=5 Ising model embedded in the cylindrical geometry: An influence of the 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 the cylindrical geometry, and explore its influence on the correlation length ξ=L^Ωf(εL^{y^*_t}, H L^{y^*_h}) with system size L, reduced temperature ε, 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(=L^{d-1})=16,81,256,.... As a result, we estimate the scaling indices as Ω=1.40(15), y^*_t=2.8(2), and y^*_h=4.3(1). Additionally, under postulating Ω=4/3, we arrive at y^*_t=2.67(10) and y^*_h=4.0(2). These indices differ from the naively expected ones, Ω=1, y^*_t=2 and y^*_h=3. Rather, our data support the generic formulas, Ω=(d-1)/3, y^*_t=2(d-1)/3, and y^*_h=d-1, advocated for the cylindrical geometry in d \ge 4.