From Laplacian-to-Adjacency Matrix for Continuous Spins on Graphs

TL;DR

This work analyzes the large $n$ limit of the ferromagnetic classical $O(n)$ model on general graphs, revealing a fundamental Laplacian–Adjacency interplay: the Laplacian spectrum governs the low-temperature regime, while the Adjacency spectrum dominates at high temperature. The authors derive site-dependent saddle-point equations for the Lagrange multipliers and obtain exact results for trees, decorated lattices, strips, and a complete bipartite graph, illustrating how graph topology selects between $L$ and $A$-driven physics. The Gaussian reduction in the large $n$ limit enables analytical access to equilibrium properties and sets the stage for $1/n$ expansions and quantum generalizations. These insights highlight how non-translational invariance and network structure shape critical behavior and spectral properties in complex graphs.

Abstract

The study of spins and particles on graphs has many applications across different fields, from time dynamics on networks to the solution of combinatorial problems. Here, we study the large n limit of the $O(n)$ model on general graphs, which is considerably more difficult than on regular lattices. Indeed, the loss of translational invariance gives rise to an infinite set of saddle point constraints in the thermodynamic limit. We show that the free energy at low and high temperature $T$ is determined by the spectrum of two crucial objects from graph theory: the Laplacian matrix at low $T$ and the Adjacency matrix at high $T$. Their interplay is studied in several classes of graphs. For regular lattices the two coincide. We obtain an exact solution on trees, where the Lagrange multipliers interestingly depend solely on the number of nearest neighbors. For decorated lattices, the singular part of the free energy is governed by the Laplacian spectrum, whereas this is true for the full free energy only in the zero temperature limit. Finally, we discuss a bipartite fully connected graph to highlight the importance of a finite coordination number in these results. Results for quantum spin models on a loopless graph are also presented.

Figures (3)

• Figure 1: Appearance of the Laplacian (denoted in the figure as "$L$") and Adjacency ("$A$") matrices for the graphs considered in the text. A) For trees there is no critical point such that one only has the high and low temperature limit. B) In the decorated lattice the singular part of the free energy is determined by the Laplacian matrix (denoting this as "$L_{sing}$"). C) For the strip the Laplacian matrix is retrieved in the whole low temperature regime. D) The complete bipartite graph has diverging coordination numbers and therefore does not have to obey the result in burioni2000limit, instead the transition is governed by the Adjacency matrix.
• Figure 2: Saddle point values of the Lagrange multipliers $\lambda_i$ corresponding to sites with $i$ neighbors of the Quantum Rotor model at $T=0$ in the large $n$ limit on a $Y$ graph (solid lines) as a function of the coupling $g$ (defined as in vojta1996quantum). The solution depends on the position inside the leg of the junction. In contrast the classical model only depends on the number of neighbors (dashed lines). Temperature $T$ in the classical model can identified with $g^2$ of the quantum model by direct comparison.
• Figure 3: Saddle point values of the Lagrange multipliers on a decorated lattice in three dimensions. Lagrange multipliers corresponding to decorating sites are denoted by $\lambda_d$, while the ones associated to the sites of the underlying lattice by $\lambda_l$. Above $T_c$, Eqs. \ref{['decor_constraint1']} and \ref{['decor_constraint2']} must be used, while below $T_c$ one has to minimize \ref{['decorated_below_tc']}.