The relation between the 2d Ising partition function and spin network evaluations, reflecting a bulk-boundary duality between the 2d Ising model and 3d quantum gravity, promises an exchange of results and methods between statistical physics and quantum geometry. We apply this relation to the case of the tetrahedral graph. First, we find that the high/low temperature duality of the 2d Ising model translates into a new self-duality formula for Wigner’s -symbol from the theory of spin recoupling. Second, we focus on the duality between the large spin asymptotics of the -symbol and Fisher zeros. Using the Ponzano–Regge formula for the asymptotics for the -symbol at large spins in terms of the tetrahedron geometry, we obtain a geometric formula for the zeros of the (inhomogeneous) Ising partition function in terms of triangle angles and dihedral angles in the tetrahedron. While it is well known that the 2d intrinsic geometry can be used to parametrize the critical point of the Ising model, e.g., on isoradial graphs, it is the first time to our knowledge that the extrinsic geometry is found to also be relevant. This outlines a method towards a more general geometric parametrization of the Fisher zeros for the 2d Ising model on arbitrary graphs.
Publié le :
DOI : 10.4171/aihpd/114
Mots-clés : Ising model, Fisher zeros, duality, quantum gravity, $6j$-symbol, spin network
@article{AIHPD_2022__9_1_73_0, author = {Bonzom, Valentin and Livine, Etera R.}, title = {Self-duality of the $6j$-symbol and {Fisher} zeros for the tetrahedron}, journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D}, pages = {73--119}, volume = {9}, number = {1}, year = {2022}, doi = {10.4171/aihpd/114}, mrnumber = {4407999}, zbl = {1492.82007}, language = {en}, url = {http://archive.numdam.org/articles/10.4171/aihpd/114/} }
TY - JOUR AU - Bonzom, Valentin AU - Livine, Etera R. TI - Self-duality of the $6j$-symbol and Fisher zeros for the tetrahedron JO - Annales de l’Institut Henri Poincaré D PY - 2022 SP - 73 EP - 119 VL - 9 IS - 1 UR - http://archive.numdam.org/articles/10.4171/aihpd/114/ DO - 10.4171/aihpd/114 LA - en ID - AIHPD_2022__9_1_73_0 ER -
%0 Journal Article %A Bonzom, Valentin %A Livine, Etera R. %T Self-duality of the $6j$-symbol and Fisher zeros for the tetrahedron %J Annales de l’Institut Henri Poincaré D %D 2022 %P 73-119 %V 9 %N 1 %U http://archive.numdam.org/articles/10.4171/aihpd/114/ %R 10.4171/aihpd/114 %G en %F AIHPD_2022__9_1_73_0
Bonzom, Valentin; Livine, Etera R. Self-duality of the $6j$-symbol and Fisher zeros for the tetrahedron. Annales de l’Institut Henri Poincaré D, Tome 9 (2022) no. 1, pp. 73-119. doi : 10.4171/aihpd/114. http://archive.numdam.org/articles/10.4171/aihpd/114/
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