Iterated star–triangle transformation on inhomogeneous 2D Ising lattices. (30th June 2021)

• Record Type: Journal Article
• Authors: Hilhorst, H J
• Is Part Of: Journal of statistical mechanics. (2021:Jun.)
• Publication Date: 2021-06-30
• Subjects: exact results -- interfaces in random media -- spin glasses -- surface effects; Statistical mechanics -- Periodicals; Mechanics -- Statistical methods -- Periodicals; 530.1305
• DOI: 10.1088/1742-5468/ac06bf
• Languages: English
• ISSNs: 1742-5468

Abstract

Abstract: We consider infinite or periodic 2D triangular Ising lattices with arbitrary positive or negative nearest-neighbor couplings K i ( r ⃗ ), where r ⃗ and i indicate the bond position and orientation, respectively. Iterative application of the star–triangle transformation to an initial lattice T ( 0 ) with a set of couplings { K i ( 0 ) ( r ⃗ ) } generates a sequence of lattices T ( n ), for n = 1, 2, …, with couplings { K i ( n ) ( r ⃗ ) } . When T ( 0 ) includes sufficiently strongly frustrated plaquettes, complex couplings will appear. We show that, nevertheless, the variables 1 / sinh 2 K i ( n ) ( r ⃗ ) remain confined to the union R ∪ i R of the real and the imaginary axis. The same holds for a lattice with free boundaries, provided we distinguish between 'receding' and 'advancing' boundaries, the latter having degrees of freedom that must be fixed by an appropriately chosen protocol. This study establishes a framework for future analytic and numerical work on such frustrated Ising lattices.