A convergent hierarchy of spectral gap certificates for qubit Hamiltonians
Sujit Rao
TL;DR
The paper develops a convergent SDP certificate hierarchy to bound the spectral gap of local qubit Hamiltonians from below. It combines a noncommutative SOS-based lower-bound hierarchy defined on a generator-relations system for $\mathfrak{su}(2^{n})$ with an upper-bound hierarchy based on a noncommutative Lasserre framework (and entropy-energy bounds) to obtain a bound $\\gamma \ge A - 2B$, where $A \le \lambda_1(H)+\lambda_2(H)$ and $B \ge \lambda_1(H)$. Convergence is proven by showing all allowed representations align with the antisymmetric subspace $\\wedge^{2}(\\mathbb{C}^{2^{n}})$, which encodes the sum of the two smallest eigenvalues, and the level-$n$ bound is exact. The framework yields polynomial-size certificates at fixed degree and includes an explicit example certifying a nontrivial gap bound for a simple commuting 1-local Hamiltonian, highlighting the method's potential for frustrated and nonfrustrated systems alike.
Abstract
We give a convergent hierarchy of SDP certificates for bounding the spectral gap of local qubit Hamiltonians from below. Our approach is based on the NPA hierarchy applied to a polynomially-sized system of constraints defining the universal enveloping algebra of the Lie algebra $\mathfrak{su}(2^{n})$, as well as additional constraints which put restrictions on the corresponding representations of the algebra. We also use as input an upper bound on the ground state energy, either using a hierarchy introduced by Fawzi, Fawzi, and Scalet, or an analog for qubit Hamiltonians of the Lasserre hierarchy of upper bounds introduced by Klep, Magron, Massé, and Volčič. The convergence of the certificates does not require that the Hamiltonian be frustration-free. We prove that the resulting certificates have polynomial size at fixed degree and converge asymptotically (in fact, at level $n$), by showing that all allowed representations of the algebra correspond to the second exterior power $\wedge^2(\mathbb{C}^{2^n})$, which encodes the sum of the two smallest eigenvalues of the original Hamiltonian. We also give an example showing that for a commuting 1-local Hamiltonian, the hierarchy certifies a nontrivial lower bound on the spectral gap.