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International Journal of Modern Physics E

LOWER BOUNDS FOR THE ENERGY OF A MANY-BODY MODEL

Amalio F. Pacheco, Jaime Sañudo A

Published Jun 1, 2008• 0 citations
Publisher

Abstract

The well known and analytically solvable Lipkin–Meshkov–Glick model was devised to test several many-body approximation approaches used in nuclear physics. Here, we use this model to illustrate a method that sets lower bounds for the ground state energy of a quantum Hamiltonian. The idea of the method consists of dividing the Hamiltonian into identical groups of terms that represent subdivisions in the particle population of the system, and then to make an appropriate use of the ground energy of one of these groups. The bounds clearly improve as the size of these groups increases.

DOI
10.1142/s0218301308010350

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The well known and analytically solvable Lipkin–Meshkov–Glick model was devised to test several many-body approximation approaches used in nuclear physics. Here, we use this model to illustrate a method that sets lower bounds for the ground state energy of a quantum Hamiltonian. The idea of the method consists of dividing the Hamiltonian into identical groups of terms that represent subdivisions in the particle population of the system, and then to make an appropriate use of the ground energy of one of these groups. The bounds clearly improve as the size of these groups increases.
Two methods which give the lower bound for the ground-state energy of a system with N interacting particles are described. The methods exploit the solution of the Schrodinger equation for the comparison system with K<N particles. An illustrative example of fermions interacting with the harmonic force is discussed. More realistic interactions used in nuclear physics are treated numerically with K=2, 3 and 4, showing that the lower bound tends to the exact value of the energy when K increases. The lower bound is strongly dependent on the choice of the interaction.
Upper bounds, lower bounds, and estimates based on moments are compared for the ground state energy of a soluble Hamiltonian due to Lipkin. Several bounding methods based on variational approaches and one based on the method of moments are employed. We show that lower bounds are of comparable quality with upper bounds and are no more difficult to obtain. No single bounding method is found superior over the entire range of parameters we study for the soluble Hamiltonian. However, in one region we find an upper bound from one method which coincides with the lower bound of another method, thus yielding the exact ground state energy. We also show that estimation methods based on the easily calculated first and second moments of the eigenvalue distribution are surprisingly accurate for a wide range of coupling parameters and number of active particles.NUCLEAR STRUCTURE Upper and lower bounds to ground state energy with soluble model; spectral properties of Lipkin model; variational methods; moment methods.
The importance of ground-state correlations in the spectroscopy of a finite many-particle system, such as the nucleus, is investigated for the model Hamiltonian of Lipkin, Meshkov, and Glick. In particular, we have investigated their effect on the solution of the equations of motion for excitation energies, transition matrix elements, and the correlation energy of the ground state. Methods are studied for calculating the correlations needed to make the necessary corrections to the equations of motion. Also studied is the behavior, with increase of interaction strength, of the phase transition from the spherical to the strongly deformed intrinsic states, in the projected Hartree-Fock scheme.
A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all that variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin- 1 / 2 Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way. Published by the American Physical Society 2024
A variational upper bound on the ground state energy E gs of a quantum system, E gs ⩽ 〈Ψ| H |Ψ〉, is well-known (here H is the Hamiltonian of the system and Ψ is an arbitrary wave function). Much less known are variational lower bounds on the ground state. We consider one such bound which is valid for a many-body translation-invariant lattice system. Such a lattice can be divided into clusters which are identical up to translations. The Hamiltonian of such a system can be written as , where a term H i is supported on the i 'th cluster. The bound reads , where is some wisely chosen set of reduced density matrices of a single cluster. The implementation of this latter variational principle can be hampered by the difficulty of parameterizing the set , which is a necessary prerequisite for a variational procedure. The root cause of this difficulty is the nonlinear positivity constraint ρ > 0 which is to be satisfied by a density matrix. The squaring parametrization of the density matrix, ρ = τ 2 /tr τ 2 , where τ is an arbitrary (not necessarily positive) Hermitian operator, accounts for positivity automatically. We discuss how the squaring parametrization can be utilized to find variational lower bounds on ground states of translation-invariant many-body systems. As an example, we consider a one-dimensional Heisenberg antiferromagnet.