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Journal of Physics A Mathematical and General

Optimized lower bounds forN-body Hamiltonians

Kheir-Eddine Boudjemaa, Sami R. Zouzou

Published May 23, 2006• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/39/23/015

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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.
Given a renormalization scheme, we show how to formulate a tractable convex relaxation of the set of feasible local density matrices of a many-body quantum system. The relaxation is obtained by introducing a hierarchy of constraints between the reduced states of ever-growing sets of lattice sites. The coarse-graining maps of the underlying renormalization procedure serve to eliminate a vast number of those constraints, such that the remaining ones can be enforced with reasonable computational means. This process can be used to obtain rigorous lower bounds on the ground-state energy of arbitrary local Hamiltonians by performing a linear optimization over the resulting convex relaxation of reduced quantum states. The quality of the bounds crucially depends on the particular renormalization scheme, which must be tailored to the target Hamiltonian. We apply our method to 1D translation-invariant spin models, obtaining energy bounds comparable to those attained by optimizing over locally translation-invariant states of n≳100 spins. Beyond this demonstration, the general method can be applied to a wide range of other problems, such as spin systems in higher spatial dimensions, electronic structure problems, and various other many-body optimization problems, such as entanglement and nonlocality detection. Published by the American Physical Society 2024
Lower bounds to the ground-state energy of a system composed of N identical particles interacting by two-body forces (and possibly also with an external potential) are derived. They are expressed in terms of the ground-state energies of (simpler) systems composed of a smaller number of particles having masses and/or interactions different from the original ones. This reduction process may be continued all the way down to systems involving one and two particles only. The generalization to systems containing some identical and some distinguishable particles is also discussed.
A system of N identical particles with masses m1 interact with each other via a pair potential V and with an additional particle of mass m0 by the pair potential U, and the system obeys nonrelativistic quantum mechanics. A general lower bound formula is derived for the ground-state energy, which reduces to Coleman’s result in the special case m0=∞. The lower bound is compared with the exact energy for the harmonic oscillator problem V (x) =k21x2 and U (x) =k02x2, which has recently been solved exactly (Hall).
Variation principles for obtaining lower bounds by means of density matrices are applied to a number of many-body problems. This is done by means of restrictions derived from the study of exacly solvable models. Two classes of problems are considered: (a) obtaining a lower bound to the ground-state energy of a system of particles, and (b) obtaining a lower bound to the Helmholtz free energy. For the first case an exact lower bound is obtained for the ground-state energy of the particle-conserving Bogoliubov Hamiltonian. Of particular significance in the nonzero temperature case is a rigorous lower bound to the free energy of the Ising model with small external field at low temperatures.