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

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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 languageEnglish
Article number011106
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume75
Issue number1
DOIs
Publication statusPublished - 2007

Fingerprint

Finite-size Scaling
Ising model
Ising Model
scaling
geometry
Transfer Matrix Method
Scaling Relations
Correlation Length
Scaling Behavior
Temperature Field
Magnetic Field
Scaling
Boundary conditions
matrix methods
embedding
Influence
Arbitrary
temperature distribution
Estimate
boundary conditions

ASJC Scopus subject areas

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

Cite this

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title = "Finite-size scaling of the d=5 Ising model embedded in a cylindrical geometry: The influence of 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 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.",
author = "Yoshihiro Nishiyama",
year = "2007",
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language = "English",
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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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