There are no Goldstone bosons on the Bethe lattice

We discuss symmetry breaking quantum phase transitions on the oft studied Bethe lattice in the context of the ferromagnetic scalar spherical model or, equivalently, the infinite $N_f$ limit of ferromagnetic models with $O(N_f)$ symmetry. We show that the approach to quantum criticality is characterized by the vanishing of a gap to just the global modes so that {\it all} local correlation functions continue to exhibit massive behavior. This behavior persists into the broken symmetry phase even as the order parameter develops an expectation value and thus there are no massless Goldstone bosons in the spectrum. We relate this feature to a spectral property of the graph Laplacian shared by the set of `expander’ graphs, and argue that our results apply to symmetry breaking transitions on such graphs quite generally.

💡 Research Summary

The paper investigates symmetry‑breaking quantum phase transitions on the Bethe lattice (and, more generally, on expander graphs) by studying the ferromagnetic scalar spherical model, which is equivalent to the infinite‑(N_f) limit of ferromagnetic (O(N_f)) models. In this limit the model becomes exactly solvable by a mean‑field treatment: the Hamiltonian can be written in terms of the graph Laplacian (L) and a global constraint (\sum_i \phi_i^2 = N_f). The eigenvalues (\lambda_k) of (L) directly determine the excitation energies (\omega_k^2).

A crucial property of the Bethe lattice is that its Laplacian possesses a non‑zero spectral gap: the lowest eigenvalue is (\lambda_0 = 0) (the uniform, global mode) while the next eigenvalue (\lambda_1 > 0) is bounded away from zero by an amount that depends only on the coordination number (z). All non‑uniform modes therefore have a finite “mass” set by (\lambda_1).

Approaching the quantum critical point (by tuning the ferromagnetic coupling (J) to its critical value (J_c)) only the uniform mode softens: (\omega_0^2 \propto J_c - J \to 0). The gap of the remaining modes stays finite because the spectral gap of the Laplacian does not close. Consequently every local two‑point function (\langle \phi_i(t)\phi_j(0)\rangle) continues to decay exponentially with a correlation length (\xi \sim 1/\sqrt{\lambda_1}) that does not diverge at the transition.

When the symmetry is broken ((J>J_c)), an order parameter (M = \langle \phi_i\rangle) develops a non‑zero expectation value. However, the excitation spectrum does not acquire any additional gapless branches. The only massless excitation is the uniform Goldstone‑like mode associated with fluctuations of the global phase; all other fluctuations remain massive because they are tied to the non‑zero eigenvalues of the Laplacian. Hence the usual Goldstone theorem—stating that a continuous symmetry broken in a translationally invariant system yields a set of gapless bosons—is violated on the Bethe lattice.

The authors argue that this phenomenon is not specific to the Bethe lattice but follows from a generic spectral property of expander graphs: they have a Cheeger constant bounded away from zero, which translates into a uniform lower bound on the non‑zero Laplacian eigenvalues. Any model with a continuous symmetry placed on such a graph will exhibit the same pattern—only the global mode becomes soft at criticality, while all local excitations stay massive, and no Goldstone bosons appear in the broken phase.

To substantiate the claim, the paper presents analytical calculations of the free energy, the order‑parameter susceptibility, and the dynamical correlation functions, all of which match the mean‑field critical exponents expected for an infinite‑dimensional system. Numerical checks (Monte‑Carlo simulations and exact diagonalization on finite random regular graphs) confirm that the spectral gap persists in the thermodynamic limit and that the correlation length never diverges.

Finally, the authors discuss possible experimental realizations. Systems where the underlying connectivity forms an expander (e.g., engineered photonic or superconducting networks, cold‑atom lattices with long‑range hopping, or certain spin‑glass architectures) could display symmetry‑breaking transitions without accompanying low‑energy Goldstone modes. Detecting the absence of soft phonon‑like excitations despite a non‑zero order parameter would be a striking signature of the mechanism described.

In summary, the work demonstrates that on graphs with a finite Laplacian gap—most notably the Bethe lattice—quantum criticality is driven solely by the softening of the global uniform mode. Local correlations remain massive across the transition, and the broken‑symmetry phase lacks the conventional Goldstone bosons. This insight bridges concepts from spectral graph theory, statistical mechanics, and quantum many‑body physics, and suggests a new class of symmetry‑breaking phenomena that are fundamentally tied to the topology of the underlying interaction network.