The $O(n\to\infty)$ Rotor Model and the Quantum Spherical Model on Graphs

TL;DR

The paper investigates the large-$n$ limit of the $O(n)$ quantum rotor model on general graphs and proves that its critical behavior matches that of the quantum spherical model, with critical exponents determined solely by the graph's spectral dimension $d_s$. Using a classical-to-quantum mapping, the authors relate the rotor to the spherical model and analyze the interplay between the Laplacian $L$ and Adjacency $A$ matrices via site-dependent Lagrange multipliers. They show that near criticality the singular part of the free energy is governed by the Laplacian, while away from criticality the Adjacency matrix dominates, yielding a comprehensive phase-diagram in the $g$-$T$ plane. The critical exponents are shown to coincide with those of a $(d_s+1)$-dimensional classical model, providing a precise, topology-driven characterization of quantum criticality on graphs and suggesting avenues for $1/n$ expansions and dynamical studies on networks.

Abstract

We show that the large $n$ limit of the $O(n)$ quantum rotor model defined on a general graph has the same critical behavior as the corresponding quantum spherical model and that the critical exponents depend solely on the spectral dimension $d_s$ of the graph. To this end, we employ a classical to quantum mapping and use known results for the large $n$ limit of the classical $O(n)$ model on graphs. Away from the critical point, we discuss the interplay between the Laplacian and the Adjacency matrix in the whole parameter plane of the quantum Hamiltonian. These results allow us to paint the full picture of the $O(n)$ quantum rotor model on graphs in the large $n$ limit.

Figures (2)

• Figure 1: Qualitative phase diagram of the $O(n\to\infty)$ Quantum Rotor model on graphs with dimension $d_s>2$. The singular part of the free energy on the whole line is determined by the Laplacian matrix $L$, which we denote as $L_{\text{sing}}$. The quantum phase transition at $T=0$ has the dynamical critical exponent $z=1$. In the $T\to 0$ and $g\to 0$ limit the full free energy is governed by the Laplacian while for large values of $g$ and $T$ the Adjacency matrix $A$ controls the free energy.
• Figure 2: Qualitative phase diagram of the $O(n\to\infty)$ Quantum Rotor model on graphs with dimension $2\geq d_s>1$ (right) and $d_s\leq1$ (left). The darker region around the critical point $g_c$ only indicates that there the singular part is determined by the Laplacian matrix.