Dual fermion method as a prototype of generic reference-system approach for correlated fermions

Highlights

• Dual fermion theory is accurately derived from a weak-coupling diagrammatic expansion.

• Generalization for an arbitrary reference system is straightforward.

• Dual denominator in the self-energy is essential for any approximation of the theory.

• Simple reference system of an isolated 2x2 cluster provides promising results.

Abstract

We present a purely diagrammatic derivation of the dual fermion scheme (Rubtsov et al., 2008). The derivation makes particularly clear that a similar scheme can be developed for an arbitrary reference system provided it has the same interaction term as the original system. Thereby no restrictions are imposed by the locality of the reference problem or by the nature of the original problem as a lattice one. We present new arguments in favour of keeping the dual denominator in the expression for the lattice self-energy independently of the truncation of the dual interaction. As an example we present the computational results for the half-filled 2D Hubbard model with the choice of a 2 × 2 plaquette with periodic boundary conditions as a reference system. We observe that obtained results are in a good agreement with numerically exact lattice quantum Monte Carlo data.

Introduction

The dynamical mean-field theory (DMFT) [1] has opened new ways in theory of correlated electron systems, in particular, due to its implementation into electronic structure calculations and the related progress in description and understanding the properties of real materials [2], [3], [4]. It originates from the consideration of a rather artificial limit of infinite dimensionality [5] and strictly speaking is exact only in this limit. Viewing the DMFT approximation in terms of the Luttinger–Ward functional [4] sheds light on the question why it works surprisingly well for real systems that are three- or sometimes even two-dimensional. The dual fermion (DF) approach [6] gave a different view on the DMFT. In this approach a change of variables in the path integral over fermionic degrees of freedom is suggested (which can be considered as a functional analog of Fourier transformation) such that in the new variables DMFT Green’s function is just the bare Green’s function, and a regular diagrammatic expansion starting from this new zeroth-order approximation is possible. Specific applications of DF as well as other diagrammatic approaches beyond DMFT are reviewed in Ref. [7]. Here we rederive DF theory by a completely different approach, using topological analysis of Feynman diagrams instead of explicit manipulations with the functional integrals. The advantage of this view is that it allowed for new insights and possible nontrivial generalizations. We consider DF as a particular case of the reference system ideology where one relates the initial system described by the Hamiltonian $H$ to some auxiliary system, easier to treat, described by the Hamiltonian $H^{\ast }$ as was initially proposed in Peierls–Feynman–Bogoliubov variational principle [8], [9], [10] (for recent applications of this method to correlated fermions see [11], [12]). The problem with variational approaches is that generally speaking there is no regular way to improve them systematically. Instead, we develop here a diagrammatic approach to the reference system, with two types of Green’s functions, related to the system with the Hamiltonian $H$ and to the system with the Hamiltonian $H^{\ast }$. We show that DF can be considered as a particular case of this approach when $H^{\ast }$ corresponds to the effective impurity i.e. a lattice site plus a bath [1]. We consider here a generalization of this approach. As a simple numerical test we use the Hubbard model on the square lattice as an example, with a plaquette playing the role of the reference system [13]. The results look promising. The approach developed here can have more applications, for example, it may potentially help to solve the famous sign problem in quantum Monte Carlo (QMC) calculations [14], by mapping of the system with sign problem (e.g., $t-t^{\prime }$ Hubbard model on a bipartite lattice with finite doping) onto the model without sign problem (e.g., the same problem with $t^{\prime }$ and doping being equal to zero).

The paper is organized as follows: in Section 2 we give a brief review of the DF formalism and outline some concepts used further. In Section 3 we give a detailed derivation of the generalized DF from the diagrammatic point view. Section 4 is devoted to a discussion of the truncation of the DF scheme. In Section 5 we show numerical results for a simple test problem and compare it with diagrammatic-QMC results [15]. Finally in Section 6 we give conclusions and a brief outlook.