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.

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.

Equations

$$H[\{s\}]=-\sum _{k=2}^r\frac{J_k}{n^{k-1}}\sum _{i_1<\cdots <i_k}\delta (s_{i_1},\dots ,s_{i_k}),$$

$$P[\{s\}]=\frac{e^{-H[\{s\}]/T}}{Z_n},Z_n=\sum _{\{s\}}{e}^{-H[\{s\}]/T},$$

$$\begin{gathered} \underset{\text{nr terms}}{\underbrace{\sum _{\text{all poss. ind.}}}}=\underset{\text{r!S(r,r)(rn) terms}}{\underbrace{\sum _{\text{r dist. ind.}}}} \\ +\underset{\text{(r−1)!S(r,r−1)(r−1n) terms}}{\underbrace{\sum _{\text{r−1 dist. ind.}}}}+\dots \\ +\underset{\text{1!S(r,1)(1n)=n terms}}{\underbrace{\sum _{\text{1 dist. ind.}}}}, \end{gathered}$$

$$\sum _{i_1<\dots <i_r}\delta (s_{i_1},\dots ,s_{i_r})$$

$$=\frac{1}{r!}\sum _{i_1,\dots ,i_r}\delta (s_{i_1},\dots ,s_{i_r})$$

$$-\frac{S(r,r-1)}{(r{)}_1}\sum _{i_1<\dots <i_{r-1}}\delta (s_{i_1},\dots ,s_{i_{r-1}})$$

$$-\dots$$

$$-\frac{S(r,2)}{(r{)}_{r-2}}\sum _{i_1<i_2}\delta (s_{i_1},s_{i_2})$$

$$-\frac{n}{r!},$$

$$\sum _{i_1,\dots ,i_r}\delta (s_{i_1},\dots ,s_{i_r})=\sum _{p=1}^q{n}_p^r,$$

$$H[\{s\}]$$

$$=\mathrm{const}-\sum _{p=1}^q\sum _{k=2}^r{n}_p^k\left(\frac{J_k}{k!n^{k-1}}+\sum _{m=k+1}^r\frac{J_m}{m!n^{m-1}}\sum _{\begin{array}{c}\text{sequences}\\ a_1=k<\cdots <a_l=m\end{array}}(-1{)}^{l+1}S(a_l,a_{l-1})\cdots S(a_2,a_1)\right),$$

$$H_n(c_1,.,c_q)$$

$$=-n\sum _{p=1}^q\sum _{k=2}^r\frac{J_k}{k!}{c}_p^k+O(1),$$

$$Z_n$$

$$=\sum _{n_1+\cdots +n_q=n}{e}^{-nF(n_1,...,n_q)/T},$$

$$\begin{gathered} F(n_1,...,n_q)=\frac{H(n_1,...,n_q)}{n}-T\frac{\ln \Gamma (1+n)}{n} \\ +\sum _{p=1}^{q}T\frac{\ln \Gamma (1+n_p)}{n}, \end{gathered}$$

$$F_n(c_1,.,c_q)$$

$$=-\sum _{p=1}^q\sum _{k=2}^r\frac{J_k}{k!}{c}_p^k+T\sum _{p=1}^q{c}_p\log c_p$$

$$+O\left(\frac{\log n}{n}\right).$$

$$U_{\infty }(c_1,...,c_q)=-\sum _{p=1}^q\sum _{k=2}^r\frac{J_k}{k!}{c}_p^k,c_1+...+c_q=1,$$

$$S_{\infty }(c_1,...,c_q)=-\sum _{p=1}^q{c}_p\log c_p,c_1+...+c_q=1.$$

$$J(x)=\sum _{k=0}^{\infty }{J}_k\frac{x^k}{k!},J_0=J_1=J_{r+1}=...=0,$$

$$F_{\infty }(c_1,.,c_q)$$

$$=-\sum _{p=1}^q\sum _{k=2}^r\frac{J_k}{k!}{c}_p^k+T\sum _{p=1}^q{c}_p\log c_p.$$

$$-\sum _{p=1}^q{h}_p\sum _{i=1}^n\delta (s_i,p)=-\sum _{p=1}^q{h}_p{n}_p=-n\sum _{p=1}^q{h}_p{c}_p.$$

$$\begin{gathered} f_{\infty ,q=2}(c)\equiv F_{\infty }(c,1-c)=-\sum _{k=2}^r\frac{J_k}{k!}\left[(1-c{)}^k+c^k\right] \\ +T\left[c\log c+(1-c)\log (1-c)\right]. \end{gathered}$$

$$\begin{gathered} U_{\infty }(c,1-c){|}_{r\equiv 1(\mathrm{mod}2)}=\dots \\ +\frac{2J_{r-1}+J_r}{2(r-2)!}{c}^{r-2}-\frac{2J_{r-1}+J_r}{(r-1)!}{c}^{r-1}. \end{gathered}$$

$$\begin{gathered} F_{\infty }(c,1-c){|}_{r=3}=\mathrm{const}+T\left[c\log c+(1-c)\log (1-c)\right], \\ \text{when}{J}_3=-2J_2,J_4=J_5=...=0. \end{gathered}$$

$$c_1=c,c_2=\dots =c_{q-1}=\frac{1-c}{q-1},c\in [0,1].$$

$$f_n(c)=F_n\left(c,\frac{1-c}{q-1},\dots ,\frac{1-c}{q-1}\right)$$

$$\begin{gathered} f_{\infty }(c)=\underset{\text{polynomial in c}}{\underbrace{-\sum _{k=2}^r\frac{J_k}{k!}\left[\frac{(1-c{)}^k}{(q-1{)}^{k-1}}+c^k\right]+cT\log (q-1)}} \\ +T\underset{-\text{entropy in c}}{\underbrace{\left[c\log c+(1-c)\log (1-c)\right]}},c\in [0,1], \end{gathered}$$

$$c^r\frac{J_r}{r!}\left(\frac{(-1{)}^r}{(q-1{)}^{r-1}}+1\right).$$

$$f_{\infty }(c){|}_{q\in \mathbb{R}}=-2\sum _{k=1}^r{t}_k{c}^k+T\left[c\log c+(1-c)\log (1-c)\right],$$

$$H=-\frac{J_2}{n}\sum _{i<j}\delta (s_i,s_j)-\frac{J_3}{n^2}\sum _{i<j<k}\delta (s_i,s_j,s_k)$$

$$\mathcal{H}$$

Full text

I Introduction

The Schelling model [schellingDynamicModelsSegregation1971] is a widely studied framework for understanding spontaneous segregation in social systems, illustrating how collective behavior emerges from individual preferences within communities. In the original formulation, agents on a lattice relocate if their neighbors differ too much in type, eventually reaching a dynamic equilibrium distribution. At a critical satisfaction threshold, the equilibrium configuration transitions from dispersed agents to monochromatic clusters with fluctuating boundaries. The model demonstrates how local preferences for similar neighbors, combined with agent mobility, can drive large-scale structural changes. For mathematical details and generalizations, see, for example, [dallastaStatisticalPhysicsSchelling2008, vinkovicPhysicalAnalogueSchelling2006, hatnaCombiningSegregationIntegration2015, singhCreatingAdaptiveNetwork2006, fagioloSegregationNetworks2007, henryEmergenceSegregationEvolving2011].

Beyond spatial lattices, recent work has explored Schelling dynamics on general networks. Avetisov et al. [avetisovPhaseTransitionsSocial2018a] studied a variant on graphs, where a society is modeled as a social network with different colors indicating distinct social categories. They showed that competition between two-body interactions among individuals from different groups and three-body interactions within connected triads can reproduce a wide range of social behaviors beyond simple spatial segregation. Numerical analysis demonstrated phase transitions between structures of various topologies, including bipartite graphs and clustered graphs linked via “ambassadors”.

Our work extends these ideas in two key directions. First, we formulate the problem as a $q$-state Potts system with many-body couplings on a complete graph. The complete graph topology reflects the limiting case of contemporary “transparent societies” where digital technologies enable individuals to observe and influence each other across traditional boundaries. Exact methods for mean-field Potts models with pairwise interactions have been developed using diffusion equations [lorenzoniExactAnalysisPhase2019a], but our combinatorial approach provides a systematic framework for general many-body couplings. Our model also connects to the mean-field theory of $p$-star exponential random graph models [biondiniPstarModelsMeanfield2022], though we defer this technical connection to Section III.2. Second, unlike the original Schelling model where community sizes are fixed, our framework determines equilibrium community sizes $n_{1},\ldots,n_{q}$ by free energy minimization, allowing spontaneous emergence of majority and minority groups.

We solve the model exactly for finite $n$ and analyze the thermodynamic limit $n\to\infty$, deriving phase diagrams for systems with up to three-body interactions. Three distinct phases emerge: a symmetric phase (all $q$ communities equally populated), a reduced-symmetry phase ($q-1$ equal communities), and a consensus phase (one dominant group with symmetric minorities). These phases represent different patterns of spontaneous symmetry breaking of the Hamiltonian’s permutation symmetry under color relabeling. We complement analytical results with Monte Carlo simulations that confirm the phases and reveal metastable switching dynamics between degenerate configurations. Notably, we establish an exact correspondence with mean-field $SU(N)$ quantum spin systems, enabling classification of social stratification via Young diagrams and reinterpreting quantum phase transitions as opinion formation dynamics.

Throughout this paper, we use standard statistical physics terminology for phase descriptions (symmetric, consensus, ordered, reduced-symmetry), with sociological interpretations provided in parentheses or quotation marks (democratic, hegemonic, totalitarian, etc.).

The paper is organized as follows. Section II derives the Hamiltonian and effective potential for the $q$-state Potts model with many-body interactions on a complete graph. Section III analyzes the phase structure in the thermodynamic limit for systems with two- and three-body interactions. Section IV examines finite-size effects and presents Monte Carlo simulations that confirm the phases. Section V establishes the social-quantum duality. Section VI provides sociological interpretations, discusses limitations, and suggests open questions for future research.

II Model and Methods

II.1 Hamiltonian in terms of occupation numbers

We begin by deriving an exact representation of the Hamiltonian of $n$ nodes with $q\geq 2$ distinct community types in terms of occupation numbers, which reduces the configuration space from exponentially large ($q^{n}$ configurations) to polynomially tractable ($o(n^{q})$ occupation number partitions).

We consider a $q$-state Potts model on a complete graph of $n$ nodes, where each node (agent/spin/particle) has a discrete state $s_{i}\in\{1,\ldots,q\}$ representing community membership. The collective behavior is governed by a Hamiltonian $H[\{s\}]$, which assigns an energy to each configuration $\{s\}$. The Hamiltonian for up to $r$-body interactions reads:

[TABLE]

where $\delta(s_{i_{1}},\ldots,s_{i_{k}})=1$ if all arguments are equal, and [math] otherwise. For clarity, $J_{k}<0$ suppresses alignment (repulsion) among $k$-cliques, while $J_{k}>0$ promotes alignment (attraction).

The scaling $1/n^{k-1}$ ensures extensivity: the total energy scales linearly with the system size $n$ as $n\to\infty$. When $r=2$, the system reduces to the conventional $q$-state Potts model with only pairwise interactions. At $r=2$ and $q=2$, we recover the conventional mean-field Ising model with all-to-all couplings.

At equilibrium at temperature $T$ (in energy units), the configurations follow the Boltzmann distribution:

[TABLE]

where the partition function $Z_{n}$ sums over all $q^{n}$ possible spin configurations.

We aim to represent $H(\{s\})=H(n_{1},\ldots,n_{q})$, where $n_{1},\ldots,n_{q}$ are the occupation numbers that count spins of each kind ($n=n_{1}+\cdots+n_{q}$). This representation is natural because the $\delta$-functions count same-spin groups, thus the Hamiltonian in Eq. 1 on a complete graph depends only on the sizes of spin equivalence classes.

First, consider $r$ (possibly equal) numbers from $1$ to $n$ labeling vertices in the graph. Each combination of these numbers has $r$ or fewer distinct values. Then the sum over all possible $n^{r}$ combinations can be rearranged as follows:

[TABLE]

where $S(r,k)$ is the Stirling number of the second kind, i.e. the number of partitions of $r$ indices into $k$ different equivalence classes. Comparing the number of terms on both sides, we restore a standard relation that the Stirling numbers satisfy.

Second, since $\delta$ is a symmetric function, we can express it as follows:

[TABLE]

where $(r)_{k}=r(r-1)\cdots(r-k+1)$ is the falling factorial. Eq. 4 decouples $r$-body interactions in terms of the occupation numbers:

[TABLE]

since the only non-zero $\delta$-terms correspond to interactions of spins from the same equivalence class. Alternatively, Eq. 5 counts the number of possible words of length $r$ in the $n_{1},\ldots,n_{q}$-letter alphabets.

Now we can proceed recursively with the remaining terms in Eq. 4 involving fewer than $r$ spins. Applying Eq. 3 to interactions of decreasing order from $r-1$ to $2$, we obtain the Hamiltonian Eq. 1 in the desired form:

[TABLE]

where the $\operatorname{const}$-term is a constant depending only on $n=n_{1}+\cdots+n_{q}$. The sequence sum in Eq. 6 reduces to unity when $m=k$.

To study phase transitions, we now take the thermodynamic limit $n\to\infty$. To avoid divergence of occupation numbers, we introduce spin concentrations $c_{p}=n_{p}/n\in[0,1]$. Moreover, we can omit the $\operatorname{const}$-term in Eq. 6 since it does not affect the statistical properties of the Boltzmann distribution Eq. 2. Retaining only the leading-order terms in Eq. 6 (corresponding to $m=k$) yields:

[TABLE]

where the $O(1)$ terms vanish in the thermodynamic limit.

II.2 Effective potential in terms of occupation numbers

Now we can easily collect together terms in the partition function Eq. 2 involving the same occupation numbers. $Z_{n}$ becomes the sum over all possible partitions of $n=n_{1}+...+n_{q}$ into $q$ non-negative integer terms:

[TABLE]

where the multinomial coefficients $\binom{n}{n_{1},...,n_{q}}=\frac{n!}{n_{1}!...n_{q}!}$ are the statistical weights of the partitions and $F(n_{1},...,n_{q})$ is an effective per-site potential:

[TABLE]

where $\ln\Gamma$ denotes the log-Gamma function and the Hamiltonian $H$ is given in Eq. 6. To ensure well-defined thermodynamic limits, we again use concentrations (see Eq. 7) when $n\to\infty$ and apply the Stirling approximation:

[TABLE]

The potential function $F_{n}$ admits a direct thermodynamic interpretation. From Eq. 8, we identify the per-site internal energy:

[TABLE]

and the per-site entropy:

[TABLE]

In Eq. 9, the coupling constants define an exponential generating function:

[TABLE]

so that $U_{\infty}(c_{1},...,c_{q})=-\sum_{p=1}^{q}J(c_{p})$.

Finally, the potential $F_{n}$ in the thermodynamic limit takes the standard form of the free energy:

[TABLE]

If we add a magnetic field $\mathbf{h}=(h_{1},...,h_{p})$ acting separately on each spin type, the initial Hamiltonian Eq. 1 and the Hamiltonian in terms of occupation numbers Eq. 6 receive additional terms

[TABLE]

Consequently, the exact per-site potentials for finite $n$ from Eq. 8 and in the thermodynamic limit from Eq. 10 are dressed with $-\sum_{p=1}^{q}h_{p}c_{p}$. The particular case of the resulting system when the interaction is limited to the two-body coupling ($r=2$) is studied in [lorenzoniExactAnalysisPhase2019a].

III Phase Structure in the Thermodynamic Limit

III.1 Two-color system: Critical points and the pure entropic regime

Begin with a $q=2$–color system of $r$-body interacting Ising spins. From Eq. 10 the effective potential reads:

[TABLE]

First, note that Eq. 11 is symmetric under permutation $c\leftrightarrow 1-c$, meaning that $c=1/2$ is a critical point of the potential (see Fig. 1(a)). For two- and three-body interactions, the phase boundary $J_{3}=4T-2J_{2}$ is determined by analyzing the second derivative $f_{\infty,q=2}^{\prime\prime}(c)\big|_{c=1/2}$, which changes sign at the boundary.

Second, coupling $J_{k}$ of an odd order $k$ does not contribute to $c^{k}$ in Eq. 11. Thus, for odd $r=2m+1$, the energy polynomial is a polynomial of order $r-1$:

[TABLE]

The polynomial has $r-2$ relevant coefficients (a constant term in energy does not affect statistical properties of Eq. 2). Thus one can find a $(\geq 1)$-dimensional subspace in the $r-1$-dimensional phase space of couplings $J_{2},...,J_{r}$ which makes $U_{\infty}(c,1-c)$ independent of $c$. For example, in case of only two- and three-body interactions ($r=3$) one has (see Fig. 1(b))

[TABLE]

Proceeding recursively such coupling subspace can be built for any odd $r$. In other words, there is a class of interaction constants which makes Ising spins with the many-body interactions Eq. 1 indistinguishable in the thermodynamic limit from a purely entropic system (when $H[\{s\}]\equiv\operatorname{const}$).

III.2 General $q$-color system: Critical points and the connection to the $p$-star model

The major contribution to the partition function comes from configurations $c_{1},...,c_{q}$ that minimize the potential $F_{n}(c_{1},...,c_{q})$ (see Eq. 8). The permutation symmetry of $F_{n}$ allows us to significantly simplify this optimization problem.

By the Palais’ Principle of Symmetric Criticality [palaisPrincipleSymmetricCriticality1979], all critical points must lie on critical subspaces, i.e. subspaces invariant under the permutation symmetry group. Implementing the constraint through $c_{1}=1-c_{2}-\ldots-c_{q}$ reduces the problem to $q-1$ independent variables in $[0,1/(q-1)]$. This leads to the one-dimensional critical subspace:

[TABLE]

The optimization thus reduces to studying the univariate function:

[TABLE]

where the complete permutation symmetry ensures that this reduction is independent of the specific color chosen to have $c_{1}=c$. In the thermodynamic limit we can substitute Eq. 10 into Eq. 13 and split the resulting expression into two parts:

[TABLE]

where we omitted the non-relevant constant shift. The linear term $cT\log(q-1)$ comes from the $q$-degeneracy of the system and vanishes when $q=2$. The highest order term in Eq. 14 is

[TABLE]

For $q\neq 2$ this term cancels out only if $J_{r}=0$. Thus to cancel out the $c^{r-1}$-term, which depends on $J_{r}$ and $J_{r-1}$, we need to set $J_{r-1}=0$. Proceeding recursively we see that eliminating the $c$-dependence of the energy in Eq. 14 for $q\neq 2$ requires zero coupling constants. Thus the system with non-trivial interactions may become purely entropic only in case of $q=2$ colors (see the discussion around Eq. 12).

Now consider Eq. 14, treating $q$ as a continuous degree of freedom ($T$ is fixed). Together with $J_{2},...,J_{r}$ we have $r$ free parameters that form a polynomial of order $r$:

[TABLE]

where $t_{1},...,t_{r}$ depend on $q,J_{2},...,J_{r}$. Reverting logic, we can find specific values of $q,J_{2},...,J_{r}$ for a given set of constants $t_{1},...,t_{r}$. Here we connect the reduced potential Eq. 15 with a mean field model of the $p$-star exponential random graph [biondiniPstarModelsMeanfield2022] (in our notation $p=r$). That is, if we replace the concentration $c$ by the mean connectance (mean fraction of the present edges) defined as $\langle L\rangle=\sum_{ij}A_{ij}/(n(n-1))\in[0,1]$ of a random Boolean adjacency matrix $(A_{ij})$, we restore the same expression as in [biondiniPstarModelsMeanfield2022] for the free energy in a random graph model, where all $k\leq r$ tuples of edges interact with the coupling constant $t_{k}$.

Interestingly, to make this mapping quantitatively exact, our setting requires non-integer $q$ (see Fig. 2). Thus, the mapping provides qualitative correspondence for the reduced potential Eq. 15, but not to the initial Hamiltonian Eq. 1 and the direct interpretation of our parameters in terms of the parameters of the $p$-star model is meaningless.

III.3 System with two- and three-body interactions

The global minima of the reduced potential $f_{n}(c)$ and the full potential $F_{n}$ (defined in Eq. 13) depend critically on the interaction coefficients $J_{2},\dots,J_{r}$ and the value of $n$. Specifically, we consider the $q$-color system with two- and three-body couplings at temperature $T=1$. We begin with the thermodynamic limit and discuss finite-size effects in the next section.

The effective potential reaches its global minimum through five distinct scenarios (Fig. 3):

1. (i)

Complete consensus phase ($c_{\min}\to 1$): a fully ordered configuration where all agents share a single opinion (sociologically, a “totalitarian regime”); 2. (ii)

Consensus phase ($1/q<c_{\min}<1$): one dominant opinion coexists with equally populated minority opinions (sociologically, “hegemonic consensus”); 3. (iii)

Symmetric phase ($c_{\min}=1/q$): all $q$ communities equally populated, no dominant opinion (sociologically, “democratic pluralism”); 4. (iv)

Reduced-symmetry phase ($0<c_{\min}<1/q$): $q-1$ equally populated opinions coexist with one emerging minority opinion; this phase vanishes at $T=0$ (sociologically, “emergent marginalization”); 5. (v)

Complete $(q-1)$-symmetry phase ($c_{\min}\to 0$): only $q-1$ communities persist with equal populations (sociologically, “cultural extinction”).

Throughout the remainder of this paper, we use the physical phase names (symmetric, consensus, etc.) with sociological interpretations provided contextually.

Since the energy contribution (Eq. 14) has no singularities in $c$, the minimum of the whole potential $f_{\infty}(c)$ is never reached at $c=0$ or $c=1$. Stepping a little away from the boundary $[0,1]$ might lower the potential $F_{\infty}=U_{\infty}-TS_{\infty}$ due to the increase of the entropy. However, this entropy-driven difference is negligibly small (see Fig. 3) and vanishes in the limit $T\to 0$.

In Fig. 4 we depict phase diagrams which realize the global minimum of the potential, controlled by different values of $q$. A richer phase structure is observed in the quadrant with $J_{2}>0$ (pair attraction, see Eq. 1) and $J_{3}<0$ (triadic repulsion).

IV Finite-Size Effects and MC Simulations

For better understanding the peculiarities of phase behavior in finite systems with competing many-body interactions, we numerically study the $q$-state Potts model on a complete graph. First, we analyze finite-size effects and then we show that analytic results in the thermodynamic limit are consistent with MC simulations.

It is instructive to compare the discrete finite size potential $f_{n}$ defined by Eq. 13 for any $n$ (even for small ones) with the continuous limit $f_{\infty}$ from Eq. 14. The results are presented for $q=3$–Potts model; however, we expect the generic picture to hold for all $q$ based on the symmetry structure of the problem. The discrete and continuous potentials of this model we denote by $f_{n}$ and $f_{\infty}$, respectively. Fig. 5(a) illustrates the potential landscapes for: (i) The discrete system $f_{n}(c)$ (blue vertical bars), representing exact counts of states; (ii) The continuum limit $f_{\infty}(c)$ (green smooth curve), describing the thermodynamic limit.

While the projected potential $f_{n}$ (Fig. 5(a), blue vertical bars) maintains the $S_{q=3}$ symmetry, deviations of the minimum locations from the continuum limit (Fig. 5(b)) reveal non-commensurability effects. Comparing the potential for various values of $n$ shows that different phases respond differently to changes in system size $n$. The phase corresponding to a chosen set of couplings $J$ might find a minimum at finite $n$ corresponding to another phase.

MC simulations visualize the behavior of the analytically derived effective potential. For small systems with small values of $J_{3}$, the concentrations of all three Potts states fluctuate around $\tfrac{1}{3}$, manifesting the symmetric phase, while for moderate values of $J_{3}$, a dominant Potts state emerges, designating the consensus phase — compare Fig. 6a ($J_{3}=1$, symmetric phase) and Fig. 6b ($J_{3}=10$, consensus phase).

Nevertheless, even in the consensus phase, no single state remains dominant throughout the entire simulation — the system jumps among all $q$ states – see Fig. 7 — each maximum of the concentration represents a configuration where one of the Potts states becomes dominant and shallow energy barriers allow the system to switch between degenerate configurations. Comparing Fig. 7(a) and Fig. 7(b), we observe that these jumps become less frequent when the MC temperature lowers. We also observe decrease in jump frequency when we increase the depth of the potential barrier by increasing $J_{3}$.

V Social-Quantum Duality

Having established the phase structure of our social model, we now reveal an exact correspondence with special unitary $SU(N)$ quantum spin systems. This provides a mutual fertilization of ideas from statistical description of social systems and elements of representation theory. We explore in detail the exact correspondence between a Schelling-inspired $q$-state Potts model with many-body interactions and $SU(N)$ quantum spin systems in their mean-field regimes exhaustively studied recently in [polychronakosFerromagneticPhaseTransitions2023, polychronakosPhaseTransitionsDecomposition2024, polychronakosNonabelianFerromagnetsThreebody2025]. The connection emerges through careful analysis of how social interaction patterns map onto the irreducible representations of $SU(N)$ symmetry groups where the number of communities $q$ maps onto the number of internal degrees of freedom $N$. Recall that the $q$-state Potts model Hamiltonian

[TABLE]

describes a $n$-agent system ($n$-atom system in the notation of [polychronakosNonabelianFerromagnetsThreebody2025]) where agents can exist in $q$ distinct states, with $J_{2}$ and $J_{3}$ quantifying the strengths of pairwise and three-body interactions respectively. The complete graph ensures exact mean-field treatment through all-to-all connectivity.

The general form of the $SU(N)$-symmetric quantum spin Hamiltonian with both two-body and three-body interactions can be written as:

[TABLE]

The operators $S_{i}^{a}$ ($a=1,...,N^{2}-1$) are the generators of $SU(N)$ at site $i$, satisfying the Lie algebra:

[TABLE]

where $f^{abc}$ are the structure constants of $SU(N)$. The first term in the Hamiltonian describes pairwise interactions, where $J_{2}$ is the coupling constant (assumed to be positive for ferromagnetic interactions); $N\equiv q$ is the number of colors/states (dimension of $SU(N)$ fundamental representation). This term can be rewritten using the quadratic Casimir operator $C^{(2)}$

[TABLE]

The three-body term introduces higher-order correlations, and $J_{3}$ is the three-body coupling constant; $d_{abc}$ is the symmetric structure constants of $SU(N)$. This term is related to the cubic Casimir operator $C^{(3)}$ as follows:

[TABLE]

In the mean-field approximation the Hamiltonians in Eqs. 16 and 17 derived in [polychronakosNonabelianFerromagnetsThreebody2025] have the same mathematical structure. The key to this correspondence lies in the representation theory of $SU(N)$. For a system of $n$ spins in the fundamental representation, the Hilbert space decomposes into irreducible components labeled by Young diagrams with $n$ boxes and up to $n$ rows. The quadratic and cubic Casimir operators take eigenvalues determined by the row lengths $k_{i}$ of these Young diagrams:

[TABLE]

Constructing the quantum spin Hamiltonian from these operators as is described in detail in [polychronakosNonabelianFerromagnetsThreebody2025],

[TABLE]

we obtain a free energy per site in the thermodynamic limit $n\gg 1$:

[TABLE]

where $x_{i}=k_{i}/n$ are the normalized Young diagram row lengths and the parameters $T,T_{0},\Theta_{0}$ define the effective coupling constants reproduced from [polychronakosNonabelianFerromagnetsThreebody2025]. This establishes a perfect dictionary between the models: the Potts state occupation fractions $c_{p}$ correspond to Young diagram row lengths $x_{i}$, while the relation between couplings is as follows: $J_{2}\leftrightarrow T_{0}$ and $J_{3}\leftrightarrow\Theta_{0}$ (we follow the notations of [polychronakosNonabelianFerromagnetsThreebody2025]).

The phase structure of both systems exhibits remarkably similar behavior characterized by symmetry breaking patterns. At high temperatures $T>T_{c}$, the quantum spin system remains in a symmetric $SU(N)$-invariant phase where all states are equally probable, represented by Young diagrams with equal row lengths. In social interpretation this corresponds to the symmetric (democratic) phase. As temperature decreases below the critical value $T_{c}$, the symmetry spontaneously breaks according to the pattern $SU(N)\rightarrow SU(N-1)\times U(1)$ (see [polychronakosNonabelianFerromagnetsThreebody2025]). This transition has a counterpart in social interpretation shown in Fig. 3, corresponding to the transition from the $q$-symmetric phase to the $(q-1)$-symmetric reduced-symmetry phase.

The mathematical foundation of this correspondence becomes particularly clear when examining the partition functions. The quantum spin system can be expressed as sums over Young diagrams:

[TABLE]

where the dimensions are given by the Weyl dimension formula: $\dim(V_{\lambda})$ is the dimension of the irreducible representation and $\dim(\mathcal{M}_{\lambda})$ is its multiplicity (see [knappLieGroupsIntroduction2005] and [polychronakosPhaseTransitionsDecomposition2024, polychronakosNonabelianFerromagnetsThreebody2025] – for specific case related to the model under consideration):

[TABLE]

In the thermodynamic limit ($n\to\infty$), application of Stirling’s approximation reveals that the entropy term $\ln\dim(\mathcal{M}_{\lambda})$ exactly reproduces the Potts model entropy contribution $\sum c_{p}\ln c_{p}$ in Eq. 10.

VI Discussion

VI.1 Sociological interpretation of phase transitions in the $q$-state Potts model on a complete graph

We have analyzed a $q$-state Potts model with many-body interactions on a complete graph, motivated by Avetisov et al.’s work [avetisovPhaseTransitionsSocial2018a] on Schelling dynamics in networks. The complete graph topology serves dual purposes. Mathematically, it makes mean-field arguments exact. Sociologically, it represents the limiting case of “transparent societies” where digital technologies allow all agents to observe and influence each other across traditional boundaries.

As shown in Section III.3, the system exhibits three principal phases with two extreme variants (see Fig. 3). These phases offer insights into how social diversity and ideological consensus spontaneously emerge in structured societies forming a transparent world.

The symmetric phase (iii) corresponds to a “democratic” configuration where no single identity dominates. Cultural or ideological pluralism is preserved through balanced coexistence, with social power distributed equally across $q$ distinct communities (or viewpoints). Such a phase reflects a political landscape where no group monopolizes influence.

In phases (iv) and (v), only $q-1$ communities persist despite the system supporting $q$ states, forming a reduced-symmetry phase. Sociologically, this represents a society where one group becomes culturally disengaged, marginalized, or assimilated. Though the structure appears pluralistic, underlying asymmetry hints at latent tensions or loss of voice for one population segment. For the mean-field $SU(N)$ quantum spin system [polychronakosNonabelianFerromagnetsThreebody2025], this corresponds to symmetry breaking $SU(N)\to SU(N-1)$.

The consensus phase (ii) describes a society where one ideology or cultural identity becomes hegemonic. A dominant viewpoint attracts many followers while minority groups, though equally represented, exert diminished influence. This phase is characteristic of political or ideological consolidation. As coupling strengths increase, the consensus phase sharpens into a fully ordered or “totalitarian” phase (i) where complete consensus suppresses alternative viewpoints.

These transitions can be understood as “sociopolitical phase transitions”, exhibiting spontaneous symmetry breaking where the system’s inherent $S_{q}$ permutation symmetry is broken by the equilibrium configuration. This generates centralized norms even from unbiased initial conditions. The transition from symmetric (democratic) to consensus phases suggests democracy persists only to a critical threshold. Beyond this point, the system undergoes a sharp first-order phase transition to a regime of ideological dominance and suppressed diversity.

The complete graph assumption warrants reflection. With advancing surveillance, control, and communication technologies, individual behavior becomes increasingly observable and analyzable, producing qualitative changes in social interactions. Our analysis reveals the equilibrium patterns of opinion clustering that emerge under such conditions: symmetric, reduced-symmetry, or consensus.

VI.2 The entropy-driven regime as an atomized society

Section III.1 identifies a special regime for $q=2$ when $J_{3}=-2J_{2}$. At this critical balance, the effective potential becomes independent of interactions, and collective behavior is governed solely by entropy of mixing. In physical terms, this corresponds to a purely disordered state, free of energetic preference. Sociologically, it provides a compelling image for modern societies where structured social influence disappears and individual opinion is driven largely by randomness.

This regime closely resembles an “atomized” society, a concept rooted in classical sociology. In such societies, individuals are no longer integrated into strong collective identities or institutions. The weakening of social connections — typically maintained by religion, stratification, or tradition — leads to what Durkheim termed “anomie” [durkheimSuicideStudySociology1951]. This is a state of normlessness where individuals behave as isolated agents, forming opinions independently of shared norms. This produces fragmented opinions driven not by democracy or consensus, but by pure randomness, much like the entropy-driven distributions in our model.

Contemporary digital societies exhibit clear parallels. Social media platforms provide individuals with highly diverse, randomized, and provocative content, weakening stable group influence and amplifying opinion heterogeneity. In such environments, opinion formation becomes stochastic rather than driven by traditional social institutions such as political ideologies, national identities, or religious affiliations. This entropy-driven phase also echoes, in a certain sense, postmodernity, where identities are fluid, social ties are optional, and meaning is increasingly self-constructed. Thus, the entropy-dominated regime in our model captures a crucial sociological insight: the screening of structured interactions.

VI.3 Social-quantum duality

We have established a mathematical correspondence between our model and mean-field $SU(N)$ spin systems with two- and three-body couplings [polychronakosNonabelianFerromagnetsThreebody2025]. We believe this exact correspondence between Schelling-type social models on complete graphs and $SU(N)$ quantum spin systems represents more than a mathematical curiosity — it establishes a bridge between statistical physics and sociology, enabling transfer of quantitative methods and conceptual frameworks between disciplines.

The Young diagram formulation provides a powerful language for describing social stratification patterns, while the social interpretations offer new perspectives on quantum many-body phenomena. This interdisciplinary connection promises to yield insights in both fields. Moreover, this correspondence provides tools for analyzing finite-size effects. While the exact equivalence holds in the thermodynamic limit, real social and quantum systems have finite sizes. The spin occupation language, merged with the Young diagram approach, could enable systematic $1/N$ corrections, potentially revealing how small cluster sizes affect segregation thresholds.

This “social-quantum” duality enables significant cross-disciplinary insights:

• •

For social modeling, it provides: (i) quantitative measures of segregation through Casimir eigenvalues; (ii) classification schemes for social structures via Young diagrams.

• •

Conversely, for quantum physics, the social analogies offer: (i) new interpretations of symmetry breaking as “opinion stratification”; (ii) interpretation of quantum fluctuations as individual opinions shared by social agents in the fully connected network; (iii) interpretation and visualization of quantum phenomena in terms of cross-cultural mental behavior.

VI.4 Limitations and Future Directions

While our model provides exact results in the mean-field limit of complete graphs, we identify several limitations.

First, real social networks exhibit complex topologies with clustering, degree heterogeneity, and community structure that deviate significantly from all-to-all connectivity. Extending our framework to sparse graphs would require approximation schemes beyond mean-field theory.

Second, our equilibrium analysis assumes ergodic sampling of configuration space, but real social systems may exhibit path- or history-dependent behavior not captured by Boltzmann statistics. The metastable state switching observed in our MC simulations (Section IV) hints at such non-equilibrium effects.

The “social-quantum” correspondence may extend beyond equilibrium statistics. The framework naturally suggests dynamical generalizations where social mobility corresponds to quantum fluctuations between different Young diagram configurations. The time evolution of social structures could be modeled using master equations for transitions between different symmetry-broken states.

Third, our model treats agents as distinguishable only by community membership, neglecting individual heterogeneity in influence, persuadability, or opinion formation mechanisms.

Despite these limitations, the “social-quantum” duality suggests novel research directions in both fields:

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For social physics: (i) what is the role of higher-order ($k$-body) interactions in community formation; (ii) how to account for network topology effects beyond complete graphs; (iii) what is the non-equilibrium dynamics of opinion formation.

• •

For quantum magnetism: (i) investigation of new phases stabilized by competing Casimir terms; (ii) development of social interpretations of topological excitations; (iii) mapping of social influence networks onto spin coupling patterns including higher-order couplings.

Acknowledgements.

Inspired by the results of work [avetisovPhaseTransitionsSocial2018a] we thank A. Gorsky, M. Tamm, and O. Valba for fruitful discussions. We express our special gratitude to O. Evnin for critical remarks and important suggestions at different stages of the work. We are grateful to A. Polychronakos for providing us with results of his works on many-body quantum spin chains prior to publication which permitted us to speculate about the “social-quantum” duality. SN thanks the hospitality of Beijing Institute for Mathematical Sciences and Applications (BIMSA) where part of this work was done.