Collective communication in a transparent world: Phase transitions in a many-body Potts model and social-quantum duality

TL;DR

The paper analyzes a $q$-state Potts model with up to $r$-body interactions on a complete graph to model collective decision-making in a digitally connected, transparent society. It derives an exact occupation-number formalism, leading to a thermodynamic free-energy $F_ extinfty$ that combines energetic terms $- frac{}{}\sum_p\sum_k \frac{J_k}{k!} c_p^k$ with entropic mixing $T\sum_p c_p\log c_p$, and identifies three principal phases: symmetric, reduced-symmetry, and consensus, with a special entropy-dominated regime for $q=2$ when $J_3=-2J_2$. A one-dimensional reduction via permutation symmetry clarifies the phase structure and connects to the mean-field $p$-star model; Monte Carlo simulations corroborate the phase diagram and reveal metastable switching in finite systems. Importantly, the work uncovers an exact social–quantum duality to mean-field $SU(N)$ spin systems with quadratic and cubic Casimir terms, mapping occupation fractions to Young diagram rows and reinterpreting symmetry breaking as opinion stratification. Taken together, the results provide a quantitative framework for sociological phase transitions in highly connected networks and establish a bridge to quantum many-body physics through representation theory and Casimir-based energetics.

Abstract

Digitally connected societies approach a \enquote{transparent} regime where all agents can interact without geographic or social barriers -- a limit realized by complete graph topologies. We solve exactly a $q$-state Potts model with many-body interactions on this geometry, modeling agents from $q$ distinct communities. Analyzing the illustrative case of competing pairwise and three-body couplings, we identify three key phases in the thermodynamic limit: democratic (all communities equal), marginalized ($q-1$ communities surviving), and consensus (one dominant group). For two-community systems, we identify a special coupling regime where interaction energies cancel, yielding purely entropy-driven dynamics -- a statistical physics representation of atomized societies without structured influence. Monte Carlo simulations confirm these phases and reveal metastable switching dynamics in finite systems. Furthermore, we establish an exact correspondence between this social model and mean-field $SU(N)$ quantum spin systems with quadratic and cubic Casimir interactions, revealing a \enquote{social-quantum} duality. This duality enables quantitative classification of social structures via Young diagrams and reinterprets quantum symmetry breaking as opinion stratification, bridging statistical sociology and quantum many-body physics.

Figures (7)

• Figure 1: The system of $q=2$ colors with two- and three-body interactions. (a) Phase diagram in the $(J_2, J_3)$ plane at temperature $T=1$. Grey regions indicate multiple degenerate global minima. (b) Effective potential from \ref{['eq:f-infty-q2']} for various interaction coefficients. Red crosses denote minima. Grey dashed curve is $c \log c + (1-c) \log(1-c)$, negative entropy. All potentials are vertically shifted so that $f_{\infty, q=2}(0) = 0$.
• Figure 2: The effective potential parametrized in two forms: \ref{['eq:reduced-potential-finf']} and \ref{['eq:reduced-potential-poly']}. The former (black curve) is formed by $q \approx 3.05019$, $J_2 \approx 6.93398$, $J_3 \approx -11.9041$ at the unit temperature $T=1$. The latter (red dashed curve) is formed by $t_1=-1.342$, $t_2=1.871$, $t_3=-0.756$. The values of $t_1, t_2, t_3$ are taken from biondiniPstarModelsMeanfield2022 for comparison.
• Figure 3: Phase diagram in $(J_2, J_3)$ plane at temperature $T=1$ based on the reduced potential given in \ref{['eq:reduced-potential-finf']}. For illustration, $q=3$. Permutation symmetry allows us to choose any color to be central. $c_{\min}$ is the concentration of the chosen color in the minimum configuration; other colors are equally populated. Grey regions indicate multiple degenerate global minima. The full potential landscape for each set of parameters is depicted below the phase diagram. Red crosses denote global minima.
• Figure 4: Minima of the potential \ref{['eq:reduced-potential-finf']} of the system with two- and three-body couplings and various values of $q$. Grey color is used when there is more than one value of $c$ where the global minimum is attained.
• Figure 5: Discrete effects in phases classified in \ref{['fig:configurations-and-landscapes']}. (a) Projected potential \ref{['eq:reduced-potential-fn']} on the critical subspace. (b) Finite $n$ effects on the position $c_{\min}$ of the minimum of \ref{['eq:reduced-potential-fn']}.