Spectral Gaps via Imaginary Time

Many open problems in physics from the Haldane conjecture regarding Heisenberg models with integer spins to the existence of the topological spin liquid phase to the Yang-Mills mass gap problem, one of the famed millennium problems, and an explanation of quark confinement are concerned with spectral gaps. The spectral gap is the energy difference between the ground and first excited states of a system and governs many of its behaviors, especially at lower energy. We show that the spectral gap can be calculated as a simple ratio of the two expectation values calculated over the wave function propagated in imaginary time. This method constitutes a significant simplification over the existing methods for spectral gap calculation. We demonstrate the effectiveness of this method on the Fermi-Hubbard and transverse field Ising models. Additionally, we discuss the implementation of the method on a quantum computer. Sandia National Labs is managed and operated by NTESS under DOE NNSA contract DENA0003525. SAND2022-14593 A.