Automata and the susceptibility of the square lattice Ising model modulo powers of primes

TL;DR

This paper investigates the susceptibility of the Ising model on a square lattice modulo powers of primes, revealing functional equations linked to automata theory and algebraic functions, with implications for understanding symmetries and transformations in statistical physics.

Contribution

It demonstrates that the susceptibility modulo prime powers can be described by automata-related algebraic functions and connects these results to symmetries like elliptic curve isogenies.

Findings

Exact functional equations for susceptibility modulo primes

Susceptibility reduces to algebraic functions modulo prime powers

Links between automata, algebraic functions, and physical symmetries

Abstract

We study the full susceptibility of the Ising model modulo powers of primes. We find exact functional equations for the full susceptibility modulo these primes. Revisiting some lesser-known results on discrete finite automata, we show that these results can be seen as a consequence of the fact that, modulo 2^r, one cannot distinguish the full susceptibility from some simple diagonals of rational functions which reduce to algebraic functions modulo 2^r, and, consequently, satisfy exact functional equations modulo 2^r. We sketch a possible physical interpretation of these functional equations modulo 2^r as reductions of a master functional equation corresponding to infinite order symmetries such as the isogenies of elliptic curves. One relevant example is the Landen transformation which can be seen as an exact generator of the Ising model renormalization group. We underline the importance of studying a new class of functions corresponding to ratios of diagonals of rational functions: they reduce to algebraic functions modulo powers of primes and they may have solutions with natural boundaries.