Comparison of two approaches for the treatment of Gutzwiller variational wave functions

Abstract

In this work, we analyse the variational problem emerging from the Gutzwiller approach to strongly correlated systems. This problem comprises two main steps: evaluation and minimization of the ground state energy for the postulated Gutzwiller Wave Function. We discuss the available methods for evaluating , in particular the recently proposed diagrammatic expansion method. We compare the two existing approaches to minimize : the standard approach based on the effective single-particle Hamiltonian and the so-called Statistically-consistent Gutzwiller Approximation (SGA). On the example of the superconducting phase analysis, we show that these approaches lead to the same minimum as it should be. However, the calculations within the SGA method are easier to perform and the two approaches allow for a simple cross-check of the obtained results. Finally, we show two ways of solving the equations resulting from the variational procedure, as well as how to incorporate the condition for a fixed number of particles.

Keywords:

• high-temperature superconductivity
• Gutzwiller wave function
• Hubbard model
• unconventional superconductivity

Acknowledgements

I would like to thank J. Spałek, J. Bünemann, and M. Wysokiński for discussions and comments on the manuscript. The work was supported by the Foundation for Polish Science (FNP) under the ‘TEAM’ program. I also acknowledge the hospitality of the Leibniz Universität in Hannover where a large part of the work was performed.

Notes

1 Note that for superconducting states the correlated and non-correlated numbers of particles ( and ) may differ, and hence we minimize the functional at a constant chemical potential , and not the energy at a constant number of particles .

2 Note that there exists an alternative derivation of the effective Hamiltonian, see e.g. [29], Appendix C.

3 The non-zero temperature is introduced for technical reasons. In the following, we will take the zero-temperature limit.

4 For example, in a system with coexistence of superconductivity and magnetism [30, 37] with both ferro- and antiferromagnetic order.

5 The damping procedure can also be introduced for the effective parameters and .