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 |
Fingerprint
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.
In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 75, No. 1, 011106, 2007.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.
UR - http://www.scopus.com/inward/record.url?scp=33846391930&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=33846391930&partnerID=8YFLogxK
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 -