Heider balance of a square lattice in an external field

TL;DR

This work studies Heider balance on two-dimensional lattices under an external field, introducing a balance energy $U = -\varepsilon P - h M$ with loop parity $P$ and edge magnetization $M$, and explores how the external field breaks symmetry between friendly and hostile ties under thermal fluctuations. Using Monte Carlo simulations, exact enumeration for small systems, and analytic solutions in limiting cases, the authors map the limiting case $\varepsilon'\to\infty$ to the nearest-neighbor Ising model and derive exact edge-magnetization susceptibilities, revealing a second-order phase transition with a logarithmic divergence at a critical field. The study highlights distinct symmetry properties between square (quadruples) and triangular (triads) lattices and shows close agreement between finite-size results and Ising/Onsager solutions, underscoring a deep link between social balance dynamics and Ising universality. These results provide quantitative tools to characterize balance ($p$) and field sensitivity ($m$, $\chi_m$, $\chi_p$) under noise, with potential implications for understanding how external social fields influence interpersonal relations in lattice-structured networks.

Abstract

We discuss the Heider model in the presence of an external social field. This field was introduced to break the symmetry between the probabilities of hostile and friendly relationships. We consider the system in the presence of fluctuations generated by thermal noise and present the results of a comparative study of two-dimensional triangular and square networks with periodic boundary conditions. The results were obtained using three different methods: exact calculations for small systems, Monte Carlo simulations of medium-sized systems, and exact calculations in the thermodynamic limit (corresponding to infinite size) of certain limiting cases for which analytical solutions are possible. In particular, we exploit the recently discovered equivalence between structurally balanced systems and the Ising model to derive an exact form of the edge magnetization susceptibility for systems in Heider equilibrium.

Figures (6)

• Figure 1: A $5 \times 5$ square lattice with periodic boundary conditions. Periodic copies of edges are shown as dotted lines. Spins of edges belonging to the spanning tree (in red) are fixed $s_{ab}=1$. Remaining spins (black edges) are effective degrees of freedom in Eq. \ref{['Uh=0']} under this gauge fixing. Due to the gauge fixing, interactions in each column take the form of 1D Ising interactions, for instance in the first column they are $s_{a_2} + s_{a_2} s_{a_3} + s_{a_3}s_{a_4} + s_{a_4}s_{a_5}$, in the fifth $s_{e_1} s_{e_2} + s_{e_2} s_{e_3} + s_{e_3}s_{e_4} + s_{e_4}s_{e_5}$. The only exception is the top row, where interactions involve products of three or four spins per plaquette, for example the contribution of the top left plaquette is $s_{\alpha_5} s_{a_5} s_{\beta_5}$
• Figure 2: Average density $m$\ref{['eq:m']} vs. $\beta h/n_e$ for different values of the parameter $\beta \varepsilon/n_p$ for (a) square lattice, (b) triangular lattice. Monte Carlo results for $16 \times 16$ lattices are shown as dashed lines in colors corresponding to different values of $\beta \varepsilon/n_p$. The results for small systems ($N_p=9$ squares; $N_p=8$ triangles) are displayed as thin lines in the same colors as the corresponding Monte Carlo results. Transparent yellow line represent analytic solution for $\varepsilon=0$ and transparent green analytic solution for $\varepsilon\to\infty$. For all other figures, we will use the same method of presenting results and will not repeat details in the captions. The tangent of all curves at the zero point is one. For $\varepsilon \to \infty$ and $N\rightarrow \infty$, the tangent at the critical points $\beta h/n_e = \pm \ln (\sqrt{2}+1)/2 \approx \pm 0.44069$ for the square lattice and $\beta h/n_e = \ln (3)/4 \approx 0.27465$ for the triangular lattice is infinite
• Figure 3: Average density $p$\ref{['eq:p']} for (a) square lattice, (b) triangular lattice
• Figure 4: Susceptibility $\chi_m n_e / \beta$\ref{['eq:chim']} for (a) square lattice, (b) triangular lattice. All curves take the value one at the zero point. For $\varepsilon\to \infty$ and $N\to \infty$, the curves diverge at the critical points $\beta h/n_e = \pm \ln (\sqrt{2}+1)/2\approx \pm 0.44069$ for the square lattice and $\beta h/n_e = \ln (3)/4 \approx 0.27465$ for the triangular lattice
• Figure 5: Susceptibility $\chi_p n_p/ \beta$\ref{['eq:chim']} for (a) square lattice, (b) triangular lattice