Exact product form for the anisotropic simple cubic lattice Green function

2006 ◽  
Vol 39 (16) ◽  
pp. 4119-4145 ◽  
Author(s):  
R T Delves ◽  
G S Joyce
Keyword(s):  
Green Function ◽  
Product Form ◽  
Cubic Lattice ◽  
Simple Cubic ◽  
Simple Cubic Lattice ◽  
Exact Product

On the simple cubic lattice Green function

1973 ◽  
Vol 273 (1236) ◽  
pp. 583-610 ◽  
Keyword(s):  
Green Function ◽  
Lattice Statistics ◽  
Quadratic Transformation ◽  
Cubic Lattice ◽  
Summation Formulae ◽  
Simple Cubic ◽  
Simple Cubic Lattice ◽  
Heun Function ◽  
Heun Functions ◽  
The Green Function

The analytical properties of the simple cubic lattice Green function G ( t ) = 1 π 3 ∫ ∫ ∫ 0 π [ t − ( cos x 1 + cos x 2 + cos x 3 ) ] − 1 d x 1 d x 2 d x 3 are investigated. In particular, it is shown that tG(t) can be written in the form t G ( t ) = [ F ( 9 , − 3 4 ; 1 4 , 3 4 , 1 , 1 2 ; 9 / t 2 ) ] 2 , where F ( a , b ;α, β, γ, β; z) denotes a Heun function. The standard analytic continuation formulae for Heun functions are then used to derive various expansions for the Green function G − ( s ) ≡ G R ( s ) + i G I ( s ) = lim ∈→ 0 + G ( s − i ∈ ) ( 0 ≤ s < ∞ ) about the points s = 0,1 and 3. From these expansions accurate numerical values of G R ( s ) and G I ( s ) are obtained in the range 0≤ s ≤3, and certain new summation formulae for Heun functions of unit argum ent are deduced. Quadratic transformation formulae for the Green function G(t) are discussed, and a connexion between G(t) and the Lamé-Wangerin differential equation is established. It is also proved that G(t) can be expressed as a product of two complete elliptic integrals of the first kind. Finally, several applications of the results are made in lattice statistics.

scholarly journals Phase Transitions of Ferromagnetic Potts Models on the Simple Cubic Lattice

2014 ◽  
Vol 31 (7) ◽  
pp. 070503 ◽  
Author(s):  
Shun Wang ◽  
Zhi-Yuan Xie ◽  
Jing Chen ◽  
Bruce Normand ◽  
Tao Xiang
Keyword(s):  
Phase Transitions ◽  
Potts Models ◽  
Cubic Lattice ◽  
Simple Cubic ◽  
Simple Cubic Lattice

Finite-size scaling of the three-state Potts model on a simple cubic lattice

1990 ◽  
Vol 59 (5-6) ◽  
pp. 1397-1429 ◽  
Author(s):  
M. Fukugita ◽  
H. Mino ◽  
M. Okawa ◽  
A. Ukawa
Keyword(s):  
Potts Model ◽  
Finite Size ◽  
Finite Size Scaling ◽  
Cubic Lattice ◽  
Size Scaling ◽  
Simple Cubic ◽  
Simple Cubic Lattice
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