Question. Are there any references or techniques which can be applied to obtain energy gaps for ferromagnetic XXX spin-1/2 Hamiltonians, on general interaction graphs, or tree-graphs?
I'm interested in frustration-free Hamiltonians such as the XXX ferromagnet on spin-1/2 particles, i.e. $$ H \;=\; -\sum_{j=1}^{n} \Bigl( \sigma^{(x)}_j \sigma^{(x)}_{j+1} \;+\; \sigma^{(y)}_j \sigma^{(y)}_{j+1} \;+\; \sigma^{(z)}_j \sigma^{(z)}_{j+1} \Bigr) $$ acting on $n$ spins (where we identify the spins labelled $1$ and $n+1$ to obtain periodic boundary conditions). The ground-state manifold here is the entire symmetric subspace on $n$ spin-1/2 particles, which has dimension $n+1$. In the usual manner of speaking, it has no "eigenvalue gap". What I am interested in, however, is the gap between the two lowest distinct eigenvalues $E_0$ and $E_1$ for such systems. However: I'm interested in such systems where the pair-wise interaction may occur over an arbitrary undirected graph — such as a tree graph — and not just the line.
In my searches, I have only found articles addressing the problem of 1d spin chains or rectangular lattices in 2,3 dimensions. The references I have found are:
Koma, Nachtergaele (1997). The spectral gap of the ferromagnetic XXZ chain.
Lett. Math. Phys., 40, pp. 1–16. [arXiv:cond-mat/9512120]Starr (2001, PhD thesis).
Some properties for the low-lying spectrum of the ferromagnetic, quantum XXZ spin system.
[arxiv.org:math-ph/0106024]
where the first reference formally shows that for spin-chains of length $n$, the spectral gap is precisely $$(E_1 - E_0) \;=\; 1 - \cos(\pi/n) \;>\; \tfrac{\pi^2\!\!}{2} n^{-2} -\, \tfrac{\pi^4\!\!}{24} n^{-4}\, $$ and the second reference discusses that result and others on rectangular lattices for spin-1/2 particles (for instance in its preliminary chapters), as well as for higher spins.
These references represent the simplest cases of the problem that I would like to consider. To consider arbitrary graphs, it would be sufficient for me to know what results apply for trees, which are a reasonable generalization from spin chains topologically, and which I can use to obtain a lower bound for all graphs. However, I'm only interested in spin-1/2 systems.
