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

TL;DR

This study numerically investigates the finite-size scaling behavior of the five-dimensional Ising model in cylindrical geometry, revealing deviations from conventional scaling due to hyperscaling violation and supporting new geometric-dependent formulas.

Contribution

It provides the first numerical analysis of FSS in d=5 Ising model with cylindrical boundary conditions, demonstrating geometry-dependent scaling indices and validating new theoretical formulas.

Findings

Scaling indices differ from naive expectations

Data supports formulas =(d-1)/3, y^*_t=2(d-1)/3, y^*_h=d-1

Method employs transfer-matrix with Novotny's technique

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 \xi=L^\Omega f(\epsilon L^{y^*_t}, H L^{y^*_h}) with system size L, reduced temperature \epsilon, 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, under postulating \Omega=4/3, 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 the cylindrical geometry in d \ge 4.