[Paper Review] Statistically-consistent Gutzwiller approach and its equivalence with the mean-field slave-boson method for correlated systems

Jakub Jȩdrak, Jan Kaczmarczyk+1 more|arXiv (Cornell University)|Jul 30, 2010

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper introduces the Statistically-consistent Gutzwiller Approximation (SGA), a new mean-field approach based on the Maximum Entropy principle with statistical consistency constraints. It proves equivalence between SGA and the mean-field slave-boson method for the single-band Hubbard model, providing a more transparent and controlled derivation without ad hoc auxiliary fields, while enabling systematic improvements beyond standard Gutzwiller approximations.

ABSTRACT

We propose a new method of solving a class of mean-field (MF) models, which is based on the Maximum Entropy (MaxEnt) principle with additional constraints included. Next, we show equivalence of our method when applied to the Gutzwiller approximation (GA), with the mean-field slave-boson (SB) formalism (on the example of the single-band Hubbard model). This equivalence provides thus an alternative justification of the results obtained within the SB approach which, however, contains ad hoc assumptions to position it in agreement with GA. Our approach implies that all predictions of the MF SB method can be obtained in a simpler, transparent, and controllable manner within GA when supplemented with the statistical-consistency conditions. We call the method as the Statistically-consistent Gutzwiller Approximation (SGA). Explicitly, the present formulation does not require introducing the condensed amplitudes of auxiliary Bose fields, which do not have a direct physical meaning and do not appear in the present formulation. Although the results of SGA are in the present case equivalent to SB, one can improve them further by utilizing more advanced schemes of calculating averages beyond the standard GA. To illustrate our approach, as well as to outline its advantages over alternative treatments of GA, we select the case of almost localized Fermi liquid (ALFL) in two dimensions and analyze it in detail within the tight-binding approximation. We also comment on significance of our method for describing correlated fermions. Namely, the reasoning used here can be applied to the corresponding MF treatment of the multiband Hubbard, the periodic Anderson, the $t-J$, and the $t-J-U$ models. In this manner our method may be applied for strongly correlated electron systems, optical lattices, and other related situations.

Motivation & Objective

• To develop a statistically consistent, controlled mean-field approach for strongly correlated electron systems.
• To establish a rigorous connection between the Gutzwiller approximation and the slave-boson formalism without ad hoc assumptions.
• To eliminate the need for unphysical auxiliary Bose fields in the slave-boson method by deriving its results via a more transparent statistical mechanics framework.
• To enable systematic improvements beyond standard Gutzwiller approximations through advanced average calculations.
• To extend the applicability of the method to multiband Hubbard, t-J, and t-J-U models, as well as optical lattices.

Proposed method

• The method employs the Maximum Entropy (MaxEnt) principle with additional statistical consistency constraints to derive a variational mean-field approximation.
• It formulates the ground-state energy functional under constraints on particle number and magnetization, ensuring thermodynamic consistency.
• The approach uses a constrained grand potential functional that incorporates Lagrange multipliers for global constraints on occupation numbers.
• The resulting self-consistent equations are derived by extremizing the grand potential with respect to variational parameters, including chemical potential and magnetic field terms.
• The method avoids introducing auxiliary Bose fields by directly enforcing physical constraints on the Gutzwiller wave function's expectation values.
• It allows for systematic extensions beyond the standard Gutzwiller approximation by incorporating higher-order corrections or improved average calculations.

Experimental results

Research questions

• RQ1 Can a statistically consistent mean-field approach be formulated for the Gutzwiller approximation that avoids unphysical auxiliary fields?
• RQ2 Is the mean-field slave-boson method equivalent to a properly constrained Gutzwiller approach in the context of the single-band Hubbard model?
• RQ3 Can the results of the slave-boson method be derived in a more transparent and controllable way using statistical mechanics principles?
• RQ4 Does the proposed method allow for systematic improvements beyond the standard Gutzwiller approximation?
• RQ5 Can this framework be generalized to multiband and other correlated electron models such as t-J and t-J-U?

Key findings

• The Statistically-consistent Gutzwiller Approximation (SGA) is formally equivalent to the mean-field slave-boson method for the single-band Hubbard model, providing a new justification for the latter.
• The SGA method reproduces all predictions of the slave-boson approach without introducing auxiliary Bose fields with no direct physical meaning.
• The method ensures statistical consistency by enforcing global constraints on particle number and magnetization, leading to a well-defined variational principle.
• The SGA framework allows for systematic improvements beyond the standard Gutzwiller approximation through more advanced average calculations.
• The approach is applicable to a wide class of correlated fermion systems, including multiband Hubbard, periodic Anderson, t-J, and t-J-U models.
• The method is demonstrated to be effective in analyzing the almost localized Fermi liquid in two dimensions within the tight-binding approximation.