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

TL;DR

This paper demonstrates that the large n limit of the quantum rotor model on graphs shares critical behavior with the quantum spherical model, with critical exponents depending only on the graph's spectral dimension.

Contribution

It establishes the equivalence of the large n limit of the quantum rotor model and the quantum spherical model on graphs, highlighting the role of spectral dimension.

Findings

Critical exponents depend solely on spectral dimension d_s.

Large n limit of quantum rotor model matches quantum spherical model behavior.

Analysis of Laplacian and adjacency matrix interplay across parameter space.

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.