Correlation trends between the Ising critical temperature and the corresponding spanning tree constant

Abstract In physics, with the advent of topological materials, and in chemistry, within the scope of chemical graph theory, topological invariants and/or indices have been considered to successfully characterize innumerous systems. In particular, strong links have been identified (both numerically and analytically) between properties of the Ising model on a lattice L and features of the so-called spanning trees (STs) of L . Nontheless, studies exploring this connection tend to address only a handful of cases given the demands of the necessary calculations. But examining only a few instances prevents one from looking for general trends across numerous L ’s, which could eventually reveal universal traits. In this contribution, we present the most comprehensive investigation to date, analyzing the Ising-ST relation for all the L ’s belonging to the families <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="fraktur">F</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> </mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mtext>⩽</mml:mtext> <mml:mi>K</mml:mi> <mml:mtext>⩽</mml:mtext> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> –uniform periodic tiling of the plane, in a total of 1248 lattices. With this goal, we develop optimized protocols (taking advantage of a recently proposed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:math> representation for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="fraktur">F</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> </mml:math> ) to compute for each L its ST constant λ and the Kac–Ward matrix. The determinant of the latter yields the Ising model free energy and consequently the critical temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . Then, considering the relatively large sample generated, we use machine learning techniques, which disclose a general correlation between the Ising critical temperature and the ST constant, described by a simple quadratic polynomial function P . As a benchmark, we test P for some arbitrary lattices (outside <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="fraktur">F</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> </mml:math> ), finding rather satisfactory fittings. These results point to a useful classification scheme for Ising <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> in 2D, demonstrating that λ can be a relevant topological concept to investigate lattice models. Finally, as a positive ‘side-effect’ of our computations, for these <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="fraktur">F</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> </mml:math> ’s we confirm (and even improve) a conjectured inequality associating λ and the effective coordinator number κ of a lattice.