Green ’ s function technique for nonequilibrium steady state

Nonperturbative dynamic theory has a particular advantage in studying the transport in a quantum impurity system in nonequilibrium steady state. Therefore, a method for obtaining the retarded Green’s function is studied here. For that purpose, we use the resolvent Green’s function in the Heisenberg picture and find all linearly independent basis vectors spanning the Liouville space. We propose a new systematic method of selecting the basis vectors, which is the most crucial step in obtaining the resolvent Green’s function. We demonstrate the utility of this method by applying it to the single-impurity Anderson models with one and two reservoirs. The latter is an appropriate model for studying nonequilibrium Kondo phenomenon, which remains an unexplored area of theoretical condensed matter physics. PACS numbers: 71.10.-w, 72.15.Qm, 73.23.-b, 73.63.Kv, 71.27.+a Nonperturbative Green’s function technique for nonequilibrium steady state 2 The single-particle Green’s function is a basic tool for studying the correlation effects in many-body systems. However, calculation of this function for a strongly correlated system is not usually successful because nonperturbative treatment is required, in general. Moreover, the situation becomes even worse for strongly correlated systems under nonequilibrium conditions, such as that in the case of peak splitting in differential conductance [1, 2, 3, 4], a phenomenon of recent interest. The nonequilibrium Kondo phenomenon remains unexplored because no successful theory is available that can handle the nonequilibrium steady state for a quantum impurity system with strong electron correlation. A simple formula for a steady current through a small interacting region connected to charge reservoirs having different chemical potentials has been presented more than a decade ago [5, 6]. The formula is given by J = − e h̄ ∫ dω 2π Γ(ω)Γ(ω) ΓL(ω) + ΓR(ω) [fL(ω)− fR(ω)] ImG + ddσ(ω) (1) for the proportional lead functions, i.e., Γ(ω) ∝ Γ(ω), where fL(ω) denotes the Fermi distribution of the left reservoir and G+ddσ(ω) is the retarded Green’s function of the small interacting region. However, the nonequilibrium Green’s function appearing in Eq. (1) has never been determined for a system with strong interaction. The purpose of this study is to present a new method of determining the nonequilibrium retarded Green’s function appearing in Eq. (1). The method must be nonperturbative to treat a quantum impurity system with strong correlation. For this purpose, we adopt the resolvent Green’s function in the Heisenberg picture [7], which is given by iGijσ(z) = 〈ê1|(zI + iL) |êj〉, (2) where z = −iω + η, ê1 = ciσ, êj = cjσ indicating a fermion of spin σ annihilating at a state or site j, and L is the Liouville operator defined by LÔ = [H, Ô], where H and Ô are the Hamiltonian and an arbitrary operator, respectively. Although we are familiar with the resolvent form in the Schrödinger picture, Gij(ω) = 〈φi|(ω + iη −H) |φj〉, (3) on the basis of the static basis {φj|j = 0, 1, · · · ,∞}, we employ Eq. (2) using the dynamic basis in this study because it is more appropriate for handling nonequilibrium situations. Calculating the retarded Green’s function in resolvent form starts from ensuring a complete set of basis vectors. A notable point is that those resolvent forms do not provide information about the basis vectors. This is the reason why the Lanczos algorithm [8, 9] and the projection operator technique [10, 11] are needed to obtain the static and dynamic basis vectors, respectively. These techniques, however, are not significantly useful for a nontrivial system because they employ the Gram-Schmidt orthogonalization procedure for the linearly independent vectors H|φ0〉 or L ê1, where n = 0, 1, · · · ,∞, in which great complications are involved. Because restrictions are inevitable in obtaining all the members of a basis for a nontrivial system when using the old methods, we need a new methodology. Nonperturbative Green’s function technique for nonequilibrium steady state 3 Once a complete dynamic basis {êl|l = 1, · · · ,∞} is given, the Green’s function of Eq. (2) can be obtained by calculating the matrix inverse (M)1j , where M is composed of the elements Mij = zδij − 〈iLêj|êi〉 = zδij + 〈êj|iLêi〉. (4) The inner product in the Liouville space is defined by 〈êk|êl〉 ≡ 〈{êk, ê † l}〉, where the angular and curly brackets denote statistical average and anticommutator, respectively. In the previous report [12], we successfully explained some parts of the nonequilibrium Kondo phenomenon [2] in terms of the basis vectors that were chosen intuitively. Although the explanation of experimental data was perfect, the intuitive manner of selecting the basis vectors cannot have universal validity and extensive applicability. The new method presented in this study, however, is systematic and, therefore, universally valid and generally applicable. Another important advantage over the old projection operator technique is that the physical meaning of the basis vectors obtained by this method is so clear that one can identify negligible basis vectors in describing the dynamics of a system at a particular parameter regime. Thus, one can construct a reduced Liouville space in which calculation is much simpler. The most crucial step for obtaining the basis vectors in systematic manner is to express the resolvent Green’s function in a matrix form such as iGijσ(z) = (〈Â| 〈B̂|) ( ĜA 0 0 ĜB )( |Â〉 |B̂〉 ) . (5) Then, one may be sure that the Liouville or Hilbert space of Gijσ(z) will be spanned completely by the linearly independent components of vectors |Â〉 and |B̂〉. The formula yielding static basis can be obtained by replacing the operator êj by the state vector φj and the Liouville operator L by the Hamiltonians H, respectively in the formula for dynamic basis that will be derived below. Therefore, we concentrate on the expression of Eq. (2) from now on. To obtain the matrix form of Eq. (5), we rewrite the Green’s function operator Ĝ = (zI + iL) as Ĝ = (zI + iLI + iLC) , where LI and LC represent the Liouville operator using the isolated part HI and the connecting part HC of the Hamiltonian, respectively. We then expand the operator in powers of LC , using the operator identity (Â+ B̂) = Â − ÂB̂(Â+ B̂). Then, the retarded Green’s function of Eq. (2) is written as iGijσ(z) = 〈ciσ|ĜI |cjσ〉 − 〈ciσ|ĜI |iLCĜIcjσ〉+ 〈ciσĜIiLC |Ĝ|iLCĜIcjσ〉, (6) where ĜI ≡ (zI + iLI) , which can be rewritten in matrix form as follows: iGijσ(z) = (〈ciσ| 〈Φi|)G (|cjσ〉 |Φj〉) T , (7) where G = ( ĜI − ĜI 0 Ĝ ) , |Φj〉 = |iLCĜIcjσ〉, and the superscript T denotes the transpose. After a similarity transformation that diagonalizes G, the retarded Green’s function is expressed as iGijσ(z) = (〈c̃iσ| 〈Φi|)GD (|c̃jσ〉 |Φj〉) T , (8) Nonperturbative Green’s function technique for nonequilibrium steady state 4 where GD = ( ĜI 0 0 Ĝ ) and c̃jσ = cjσ + (ĜI − Ĝ) ĜIΦj = cjσ + L −1 C (−izI + L)Φj . (9) Equation (8) is the required form and the linearly independent components of vectors |c̃j↑〉 and |Φj〉 completely span the Liouville space of G + ijσ(ω). In conclusion, the systematic method for selecting the dynamic basis vectors, which are given in operator form, involves finding all linearly independent components comprising the operator c̃jσ because Φj is contained in it. We will subsequently demonstrate the determination of the basis vectors with an example. The system we are interested in is a single electron transistor with a quantum dot, which can be described by a single-impurity Anderson model with two metallic reservoirs, whose Hamiltonian is written as