Theory Seminar: Partial bosonization of the extended Hubbard model

Evgeny Stepanov
Mean-field theory is a simple and transparent method that is suitable for a description of collective fermionic excitations in a broad range of physical problems.

Evgeny Stepanov
Herbert-Walther Lecture Hall G0.25
Wed, 17. July 2019, 11:30 am (MEZ)

Abstract: 

Mean-field theory is a simple and transparent method that is suitable for a description of collective fermionic excitations in a broad range of physical problems. The underlying idea of the method is based on the partial bosonization of collective fermionic fluctuations in leading (charge, spin, and etc.) channels of instability. This allows to solve the initial problem diagrammatically in terms of original fermionic and effective bosonic propagators in the GW fashion.

    Theoretical description of electronic effects in the regime of strong electron-electron interaction requires more advanced approaches than the mean-field solution of the problem. One of them is the (extended) dynamical mean-field theory (EDMFT), which is found to be a good approximation for single-particle quantities, especially when properties of the system are dominated by local correlations. Description of collective electronic fluctuations in correlated materials requires more efforts, since they are essentially nonlocal. Their accurate description is possible only within a certain extension of (E)DMFT.

    In the case when accurate quantum mechanical calculations are challenging, the initial quantum problem can be replaced by an appropriate classical one. Indeed, some collective phenomena, such as magnetism, can be sufficiently described by simple Heisenberg-like models that are formulated in terms of bosonic variables. This fact suggests that other many-body excitations can also be described by simple bosonic models in the spirit of Heisenberg theory. Following the mean-field idea, a partially bosonized description of collective electronic effects in strongly interacting systems can also be performed on the basis of the dynamical mean-field theory. This allows to account for nonlocal fluctuations in a simplified GW way, and resulted in the development of GW+(E)DMFT, TRILEX, and Dual Boson methods.

    Although the GW extension of the dynamical mean-field theory is an efficient and inexpensive numerical approach, it has a significant drawback, which is common for every partially bosonized theory. This severe problem is known as the Fierz ambiguity in decomposition of the local Coulomb interaction into different bosonic channels, which drastically affects the result of the method. Surprisingly, this issue is yet unsolved even for a simple mean-field theory, let alone the GW+DMFT method that became a standard approach for calculation of physical properties of realistic multiband systems and for the solution of interacting time-dependent problems.

    Here, I will show how the partial bosonization of the extended Hubbard model can be performed avoiding the Fierz ambiguity. The resulting theory is formulated in terms of fermionic degrees of freedom and new bosonic fields that describe the interaction in a certain channel. I will also define a physical regime where the obtained model action reduces to an effective Ising and Heisenberg models for charge and spin variables, respectively. This method accounts for nonlocal fluctuations diagrammatically beyond the EDMFT level and allows to include the effect of magnetic fluctuations in the GW scheme. As the result, the proposed method improves already existing GW, GW+(E)DMFT, and TRILEX approaches that are used for description of wide range of physical problems. Although the method is discussed here in the context of the extended Hubbard model, it is not restricted only to this particular model and can be applied to other fermionic problems broadly defined.