### 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 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.

Original language | English |
---|---|

Article number | 011106 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 75 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Finite-size scaling of the d=5 Ising model embedded in a cylindrical geometry : The influence of hyperscaling violation.** / Nishiyama, Yoshihiro.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Finite-size scaling of the d=5 Ising model embedded in a cylindrical geometry

T2 - The influence of hyperscaling violation

AU - Nishiyama, Yoshihiro

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevE.75.011106

DO - 10.1103/PhysRevE.75.011106

M3 - Article

AN - SCOPUS:33846391930

VL - 75

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 1

M1 - 011106

ER -