# Graph dimension and phase transitions

In this question I am mainly concerned with phase transitions associated with symmetry breaking. For phase transitions of this type, there are a number of results describing the relationship between the existence of a phase transition and the dimension of the system.

For example, roughly speaking, the Mermin-Wagner theorem states that there cannot be spontaneous breaking of a continuous symmetry at finite temperature in dimensions $$d \leq 2$$. This has been proven rigorously for a number of models, including for example the quantum spin-$$S$$ Heisenberg model with isotropic and finite-range interactions on a $$d$$-dimensional hypercubic lattice (c.f. Scholarpedia).

Broadly I am interested in whether it is possible to obtain results of this type (i.e. relating the existence of a phase transition to the dimension of the system) for models defined on more general graphs than regular lattices. For a graph $$G$$, one notion of dimension is the graph dimension, which roughly corresponds to the minimum dimension $$d$$ for which there is an embedding of the graph in $$d$$-dimensional Euclidean space such that all edges are of unit length.

To make this question more answerable, here is a specific question:

Is there a version of the Mermin-Wagner theorem for the quantum spin-$$S$$ Heisenberg model defined on a general graph $$G$$ (say with nearest-neighbour interactions), relating the existence of a phase transition to the graph dimension of $$G$$?

I would also be interested in answers which provide counterexamples to this idea.