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Few-Body Systems

Optimized Lower Bound for Four-Body Hamiltonians

Achour Benslama, Abboud Metatla, A. Bachkhaznadji, Sami R. Zouzou, Ali Krikeb, Jean-Louis Basdevant, J. M. Richard, Tai Tsun Wu

Published May 24, 1998• 12 citations
Publisher
DOI
10.1007/s006010050075

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Generalizing a method used previously for three-body, four-body and very recently for five-body systems, we derive a lower bound for the ground state energy of an N-body Hamiltonian, with arbitrary N. Because the expression of the lower bound obtained in this way depends on a number of parameters, we obtain in fact a family of lower bounds, a lower bound for each set of values of these parameters. The best of these is of course obtained by maximizing over these parameters and is correspondingly named optimized lower bound. The set of values of the parameters corresponding to the optimized lower bound satisfy a number of relations, named universal dynamical constraints, which result from the application of a dynamical principle and are independent of the particular form of the potential. For N = 3, 4, 5, they can be worked out in the most general case. For N = 6 up, they can be worked out only for particular mass configurations. Furthermore, the optimized lower bound proves to be saturated in the harmonic oscillator case.
We begin by deriving explicit formulae for the energy levels of a system of N harmonic oscillators for two special mass configurations but for arbitrary N . Under the same conditions, we can perform analytically all the calculational procedure leading to an optimized lower bound for the ground state energy of an N -body system. The lower bound obtained in this way proves to be identical to the exact result. It is the first time, to our knowledge, that an explicit analytical proof of saturability has been worked out.
Estimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad applicability, ranging from many-body physics to quantum computing and classical sampling techniques. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate – a proof of a lower bound on the gap – is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe's bound and its subsequent improvements, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria in both the accuracy of the lower bound on the gap, as well as the range of parameters in which a gap is detected.