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

## TL;DR

This paper models societal interactions using a Potts model on complete graphs, revealing phase transitions between democratic, marginalized, and consensus states, and establishes a duality with quantum spin systems to connect social and quantum physics.

## Contribution

It provides an exact solution for a many-body Potts model on complete graphs and uncovers a social-quantum duality linking societal phases to quantum symmetry breaking.

## Key findings

- Identifies three key societal phases: democratic, marginalized, and consensus.
- Discovers a purely entropy-driven regime with no structured influence.
- Establishes a duality connecting social models to quantum spin systems.

## 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.