The Hubbard model: exact constraints on spectral moments in the strong coupling limit

ABSTRACT

Exact spectral moment relations are derived and presented for the low temperature, strong coupling limit of the Hubbard model, for any dimension, lattice structure and electron density. These results generate an exact, rigorous and quantitative test for proposed solutions to the Hubbard model that claim to be valid in the strong coupling, low temperature region. The test complements analyses for weak coupling and high temperatures, offering insights into the essential physics of the model, specifically the quasiparticle energy (centroid) and lifetime (width) of the spectral weight function in the infinite coupling, finite hopping limit. Numerical results are given for nearest neighbour hopping in 1, 2 and 3D.

KEYWORDS:

• Hubbard model
• strongly correlated electrons
• infinite U
• numerical methods
• spectral moments

Acknowledgements

With pleasure, we acknowledge the inspiration of R.V. Lange. Numerous helpful conversations with B. S. Shastry are also gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For example, Maier et al. [10] states that ‘ … the (Dynamic Cluster Approximation) become(s) exact in … the strong-coupling limit … (where) all the sites in the lattice are decoupled. The effective cluster problem reduces to a single-site problem without coupling to a mean field.’

2 Contrast should be made with low order moments taken across the entire frequency spectrum [25–28]. Six such moments would be needed to replicate the information in the first three moments of the infinite U SWF in the strong coupling (separated peak) region.

3 Equation of motion plus de-coupling approximations continue to be used [35–38], often in conjunction with the DMFT approximation, even though the usual decoupling has long been known to be physically incorrect in the low temperature strong coupling region from a physical point of view (the non-locality discussed in the Atomic Limit section). Indeed, the Appendix lists some exact decoupling results for equal time correlation functions in the low temperature infinite U limit that are functionally quite different from the usual decoupling assumptions.

4 Another peculiar comment is made by Dai et al. [17] that it is ‘well known that deep in the insulating state (here large U), the width of the Hubbard bands as well as the shape of the Hubbard bands has to be the same as the non-interacting density of states.’ This is in clear violation of the low order moments derived here and by Harris and Lange [12] for the centroid (first moment) and width (second central moment) of the lower Hubbard band and is likely another confusion of the high temperature atomic limit with the physically relevant low temperature atomic limit and non-locality.

5 Two errors should be noted in [3]. Eqn. (15a) in [3] is to be corrected for R₁ = R_1’. The correction follows immediately from Equations (A5a) and (A5b) above. That correction is not needed for this analysis. In addition, the line after Eqn. 22 in [3] refers to the ‘left-hand side’ not the ‘right-hand side.’

6 Only relations (A6)–A(7) and (A13) – (A15) are presented in Esterling and Dubin [3].