Frustrations on Decorated Square Lattice in Ising Model

TL;DR

This paper analytically investigates frustration phenomena in the Ising model on a decorated square lattice, deriving formulas for residual entropies and analyzing how frustrations influence thermodynamic behavior, revealing complex properties absent in the undecorated lattice.

Contribution

It provides exact formulas for residual entropies of all frustrations in a decorated square lattice Ising model, expanding understanding of frustration effects with multiple exchange interactions.

Findings

Derived rigorous formulas for residual entropies.

Identified rich and intricate frustration properties.

Compared decorated and undecorated lattice behaviors.

Abstract

In this paper, we performed the comprehensive studies of frustration properties in the Ising model on a decorated square lattice in the framework of an exact analytical approach based on the Kramers--Wannier transfer matrix method. The assigned challenge consisted in finding rigorous formulas for residual entropies of all possible frustrations without exception and elucidating the influence of frustrations on the behavior of the thermodynamic functions of the system under discussion. The accomplished results turned out to be very rich and intricate, especially in comparison with the original undecorated square lattice, in which frustration properties are completely absent. Such an abundance of properties is due to the multiparametric problem of decorated lattice with four exchange interactions $J_{x}$, $J_{y}$, $J_{dx}$ and $J_{dy}$ taking on arbitrary continuous values and different signs: positive (ferromagnetic) and negative (antiferromagnetic), as well as with two discrete variables $d_{x}$ and $d_{y}$ (multiplicities of decorating spins in both lattice directions), corresponding to infinite natural series of numbers.