3. The Matsubara-Voltage Formalism 13
3.2. Hershfield’s Operator Y
A natural starting point to stationary currents of fermions is thus the quantum field theory which emerges from taking an infinite lattice of fermions.
Mathematical Implications
While an ultraviolet cutoff is inherently imposed by the lattice, still the infinite number of degrees of freedom poses a fundamental problem when included a-priori.
It has been pointed out that in principle one might even face certain conceptual limitations of the Fock-space formulation of quantum field theory.3 In particular, the existence of a (thermal) density operator is not certain, because one may find counter-examples in similar systems [80]. One has to keep these principal precautions in mind when proceeding with the formalism.
When performing calculations with otherwise ill-defined density operators, one can use the fact that the typical length scale associated to certain recurrence phenomena may be identified. It may be given by vFt, where vF is a Fermi velocity. If then the system size L vFt, the recurrence phenomenon will not appear. In order to consider certain convergence processes such as the t → ±∞ limits of scattering theory, even more care has to be taken, in order to correctly model the interplay of the remaining relevant energy scales of the system.
3.2. Hershfield’s Operator Y
For the finite-size equilibrium Anderson impurity model one can formally write down the thermal density matrix in the grand canonical ensemble as
ρeq := e−β(H−µN), (3.28)
whereN is the particle-number operator andµ the chemical potential. Observables may be expressed as hAi = Tr (ρeqA)/Trρeq, and the thermodynamic limit may be taken at the end of the calculation.
In the steady-state nonequilibrium setup, the situation is not as clear. Hershfield postulated4 that the density matrix of the steady-state system effectively may be similarly coped in the Gibbsian form
ρneq := e−β(H−ΦY), (3.29)
where Φ is the bias voltage µL−µR and β = (kBT)−1 is the temperature of the two leads [29]. The newly appearing operatorY is Hermitian and can be constructed by
3Consider, e.g., the paradoxa emerging within the field-theoretic limit of BCS theory as discussed in Ref. [79].
4 Other nonequilibrium Gibbsian approaches to quantum transport may already be found in Zubarev’s textbook [28].
means of certain scattering states. More generally, having not the bias, but maybe a current as defining parameter of non-equilibrium, Hershfield claims that
ρneq := e−β(H−Y˜), (3.30) for some Hermitian operator ˜Y.
3.2.1. Outline of the formalism
The formal details, such as specifying the appropriate limiting procedures in order to make the involvement of ρneq well-defined, with Hershfield’s approach are due to the field-theoretic limit somewhat delicate and still subject to intensive ongoing investigations. Hence, we will only discuss the central ideas of the formalism. Details of the appropriate limiting procedure may be found in the literature [81; 82]. In this work, we will concentrate on the aspects of Hershfield’s approach which are also relevant for its imaginary-voltage extension by Han and Heary.
As in Keldysh perturbation theory, one typically defines the non-equilibrium sys-tem starting with an equilibrium syssys-tem at t = −∞. In our case, one can start with zero hopping from the leads into the nanostructure, with the two leads being in thermal equilibrium at different chemical potentials µL and µR, namely with the initial density matrix for the leads,
ρ(t=−∞) := e−β(HL−µLNL)⊗e−β(HR−µRNR). (3.31) Using the time evolution with the full Hamiltonian of the system, a steady state of the system is approached. For this purpose, the existence of a physical relaxation process is assumed by Hershfield. Doyon and Andrei argue that a formal similarity to Caldeira-Leggett models of quantum decoherence ensures the necessary relaxations automatically within the model [81; 83]. Nevertheless, the actual approach to and even the mere existence of a steady-state may only be proven for the interacting Anderson impurity model under certain assumptions [75]. Another setup consider a steady-state non-interacting system which may be constructed explicitly either with respect to a certain voltage drop or with respect to a certain current. As the interaction is turned on, the full non-equilibrium problem emerges.
None of these strategies is chosen by Hershfield, but he merely points out that in all of these cases, the initial density matrix already has the form
ρ0 = e−β(H0−Y˜0). (3.32) Under the assumption that due to the presence of a physical relaxation process the mathematical details work out well, he was able to show that the interacting stationary state may also be written in the above form. In the particular example of the first switching process, namely the two equilibrium reservoirs with chemical
3.2. Hershfield’s Operator Y
potentials µL, µR, he finds that Y˜ =µRX
k
ψ†R,kψR,k+µLX
k
ψ†L,kψL,k. (3.33)
Here,ψα,k(†) operators are the scattering states evolving from thek-th band electronc(†)α,k of the leadαvia Eq. (3.26). It should again be emphasized thatψα,k† operators create complicated many-body states involving processes such as particle-hole excitations, etc., and the commutator relations of theψ-operators are a priori unclear.
However, it has recently been shown by Han that
{ψα†0,k0, ψα,k}=δα,α0δk,k0, (3.34) as long as the many-body interaction acts only locally on the nanostructure, so that LVc†ασk ∝d†σ, whereLV is the Liouvillian with respect to the hopping and interaction part of the Hamiltonian [84]. Hence, for a large class of mesoscopic systems, including the single-impurity Anderson model, the scattering states are fermions. Han also showed that these fermions provide a complete basis set for the underlying many-body Hilbert space [84].
Note that when we use the common convention of setting the zero of energy to the mean of the two potentials,µL= +Φ/2, µR =−Φ/2, we have
Y =X
αk
α
2ψαk† ψαk. (3.35)
TheY quantum numbers are±1/2 for a single-electron stateψ±,k† |0iand distinguish the source and drain leads to the nanostructure. In the many-body case the Y quantum number is a measure for the balance of the number of electrons scattering into the nanostructure from source or drain leads, respectively.
3.2.2. Application of the approach
The Zubarev-Hershfield approach has the appealing feature of yielding a quasi-Gibbsian density operator. Nevertheless, it took the scientific community some time to develop the idea further for the following reasons.
First, the benamed mathematical difficulties coming with the introduction of the density operator, have raised concern. Recently, the work by Doyon and Andrei appeared to provide some insight to this topic. One of their results is that one can probably only consider expectation values of “local” observables within Hershfield’s formalism, such as the current operator and correlators within the nanostructure.
Second, the definition of theY-operator via interacting scattering states is a highly challenging issue for practical computations. Besides the Lippmann-Schwinger
equa-tion there is basically no other starting point for a systematic buildup of the scat-tering states. The scatscat-tering-states numerical renormalization group approach [22]
imposes a numerically explicit real-time evolution for this purpose. Very recently, Dutt et al. provided a perturbative imaginary-time approach to the Hershfield for-malism [82]. It is however only diagrammatic in a sense that the density operator has to be first computed to the desired order via a hierarchy of differential equations, and then the diagrammatic rules define the Green’s functions of the nanostructure.
Third, these latter complications are very much due to the fact that the statistical operator and the time evolution operator are no longer closely related. In contrast to this, for thermal Green’s functions the Boltzmann factor, using the chemical potential µ = 0 as the reference energy, may be written as an evolution from 0 to −iβ in imaginary time. The same Hamiltonian is used for the time evolution as in real-time dynamics. This coincidence is heavily used in equilibrium quantum statistical mechanics but is not applicable here.