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The following situation is ubiquitous in mathematical physics. Let $\Lambda_N$ be a finite-size lattice with linear size $N$. An typical example would be the subset of $\mathbb{Z}\times\mathbb{Z}$ given by those pairs of integers $(j,k)$ such that $j,k \in$ { $0,\ldots,N-1$}. On each vertex $j$ of the lattice place a copy of the vector space $\mathbb{C}^d$. The total space will be the tensor product of all of these spaces. Then define a Hamiltonian acting on this total space as follows: $$H = \sum_{k \in \Lambda_N} h_k$$ for some Hermitian matrices $h_k$ which act like the identity everywhere except on the vector spaces located on site $k$ and in the neighborhood surrounding $k$. Typically, one is interested in the case where there is a translational symmetry (except at the boundary) in the definition of the $h_k$. Denote the eigenvalues of $H$ in increasing order by $\lambda_1 \le \lambda_2 \le \ldots \le \lambda_M$.
For an arbitrary fixed family of Hamiltonians $H$, what proof techniques exist for computing an upper and a lower bound on $\Delta = \lambda_2 - \lambda_1$ as a function of $N$? In particular, we want to know if $\Delta$ decays to zero as a function of $N$, or if it is lower-bounded by some constant independent of $N$.
The gap $\Delta$ is the energy gap between the ground state and the first excited state of an interacting quantum system. Understanding this quantity tremendously impacts our understanding of the different phases of matter, but it is extremely difficult to compute or even bound for all but the simplest cases (like when all the $h_k$ commute). This difficulty persists even when there is significant additional (physically motivated) structure in the problem, such as considering only $h_k$ which are projectors, and where there is a unique zero-energy eigenstate (all others having positive energy for any finite $N$).