Comparison of an improved self-consistent lower bound theory with Lehmann\u2019s method for low-lying eigenvalues

Miklos Ronto,Eli Pollak,Rocco Martinazzo

doi:10.1038/s41598-021-02473-y

Miklos Ronto, Eli Pollak + Show 1 more

Open Access

https://doi.org/10.1038/s41598-021-02473-y

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Abstract

Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper, we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT method. Using two lattice Hamiltonians, we compared the improved SCLBT (iSCLBT) with its previous implementation as well as with Lehmann’s lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and iSCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the iSCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the iSCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds

Highlights

Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues
The lower bounds from the previous implementation of the Self-Consistent Lower Bound Theory (SCLBT) are compared with those from the improved SCLBT (iSCLBT) method presented in this paper for the ground-state energy levels of the lattice Hamiltonians
The lower bounds for the Ritz eigenvalues obtained from the iSCLBT were compared with those from the SCLBT and from the Lehmann methods

Summary

Introduction

Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The lower bounds from the previous implementation of the SCLBT are compared with those from the iSCLBT method presented in this paper for the ground-state energy levels of the lattice Hamiltonians.

Results

Conclusion