Time-dependent quantum transport: causal superfermions, exact fermion-parity protected decay mode, and Pauli exclusion principle for mixed quantum states
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Time-dependent quantum transport: causal superfermions, exact fermion-parityprotected decay mode, and Pauli exclusion principle for mixed quantum states
R. B. Saptsov (1 , and M. R. Wegewijs (1 , , (1) Peter Gr¨unberg Institut, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany(2) JARA- Fundamentals of Future Information Technology(3) Institute for Theory of Statistical Physics, RWTH Aachen, 52056 Aachen, Germany (Dated: November 4, 2018)We extend the recently developed causal superfermion approach to the real-time diagrammatictransport theory to time-dependent decay problems. Its usefulness is illustrated for the Andersonmodel of a quantum dot with tunneling rates depending on spin due to ferromagnetic electrodes and/ or spin polarization of the tunnel junction. This approach naturally leads to an exact result for oneof the time-dependent decay modes for any value of the Coulomb interaction compatible with thewideband limit. We generalize these results to multilevel Anderson models and indicate constraintsthey impose on renormalization-group schemes in order to recover the exact noninteracting limit.(i) We first set up a second quantization scheme in the space of density operators constructing“causal” field superoperators using the fundamental physical principles of causality / probabilityconservation and fermion-parity superselection (univalence). The time-dependent perturbationseries for the time-evolution is renormalized by explicitly performing the wideband limit on the su-peroperator level. As a result, the occurrence of destruction and creation superoperators are shownto be tightly linked to the physical short- and long-time reservoir correlations, respectively. Thiseffective theory takes as a reference a damped local system, which may also provide an interestingstarting point for numerical calculations of memory kernels in real time.(ii) A remarkable feature of this approach is the natural appearance of a fermion-parity protected decay mode which can be measured using a setup proposed earlier [Phys. Rev. B 85, 075301(2012)]. This mode can be calculated exactly in the fully Markovian, infinite-temperature limit byleading order perturbation theory, but surprisingly persists unaltered for finite temperature, for anyinteraction and tunneling spin polarization.(iii) Finally, we show how a Liouville-space analog of the Pauli principle directly leads to an exactexpression in the noninteracting limit for the time evolution, extending previous works by startingfrom an arbitrary initial mixed state including spin- and pairing coherences and two-particle correla-tions stored on the quantum dot. This exact result is obtained already in finite-order renormalizedperturbation theory, which surprisingly is not quadratic but quartic in the field superoperators,despite the absence of Coulomb interaction. The latter fact we relate to the time-evolution of thetwo-particle component of the mixed state, which is just the fermion-parity operator, a cornerstoneof the formalism. We illustrate how the super-Pauli principle also simplifies problems with nonzeroCoulomb interaction.
PACS numbers: 73.63.Kv, 03.65.Yz, 05.60.Gg, 71.10.-w
I. INTRODUCTIONA. Experimental motivation
Quantum dynamics of open systems is of interest invarious research fields, ranging from transport throughmeso- and nanoscopic systems, quantum informationprocessing, and quantum optics to physical chemistryand biology. Typically, the object of investigation is somesmaller part of a larger system, e.g., a single molecule at-tached to macroscopically large contacts, which act asreservoirs and impose strong nonequilibrium boundaryconditions. In the field of quantum transport a high de-gree of control has been achieved over fermionic subsys-tems, such as few-electron quantum dots coupled to var-ious kinds of electrodes (e.g, metals, ferromagnets, or su-perconductors). This control relies mostly on the strongelectrostatic effects, which for very small systems makesthe theoretical description challenging. This progresshas enabled detailed studies of not only stationary but also of time-dependent transport phenomena down tothe scale of atomic quantum dots.
Interaction ef-fects in the time domain have been investigated early on,such as the SET oscillations in the weak tunnel couplingregime, and continue to be of interest. Quantum fluc-tuations between such a strongly correlated dot and theelectrodes, lead to additional effects, such as level renor-malization, inelastic tunneling effects, and Kondo physicsin stationary transport and their nontrivial signatures inthe time-domain have also attracted interest. A problemthat received quite some attention, is the time-dependentresponse of a quantum dot in the Kondo regime.
Theoretically, it has been studied using various modelsand methods.
For instance, when starting from aKondo model description, the real-time diagram-matic approach, which is at the focus of this paper,provides deep analytical insight into the renormalizationof exchange interactions as well as the renormalizationof the various dissipative effects that ultimately destroythe Kondo effect. On the other hand, recent numericalstudies starting from an Anderson model haveinvestigated the development of the Kondo effect in time,in particular, the much debated splitting of the Kondopeaks. Application of the real-time diagrammatic ap-proach to the Anderson model at T = 0 is of high in-terest as it can provide analytical insight, especially re-garding the time-evolution towards stationarity. Outsidethe Kondo regime we recently reported some progressin this direction in the stationary limit and noted aninteresting relation to the time-evolution decay modesthat were studied before in the weak / moderate tunnelcoupling limit. In Ref. 31, motivated by experimentalprogress on single-electron sources , new measurementsetups were suggested to probe the relaxation rates of aquantum dot using a quantum point contact (QPC).This study and a more recent one focused on the ef-fect of the Coulomb interaction and surprisingly foundthat certain multiparticle correlators show a remarkablerobustness with respect to most details of the setup (seebelow), in particular to the interaction. As argued there,this absence of interaction corrections is really an effectthat can be measured. A key result of the present paper,expressed by Eq. (111), is that this conclusion holds be-yond various of the approximations made in Ref. 31 and33. This result can be written as follows: h ( n ↑ − )( n ↓ − ) i ( t ) = e − Γ t h ( n ↑ − )( n ↓ − ) i (0) + . . . (1)Here, n σ , σ = ↓ , ↑ are the spin-resolved occupations ofthe dot. The (equal-time) two-particle correlation func-tion (1) contains a term that decays strictly Markovianwith rate Γ. This function appears as a coefficient inthe expansion of the mixed state of the quantum dot. Ithas been shown that the experimental observationof the decay of the mixed state is possible: one can op-timally choose the parameters that determine the initialand final state of the time-dependent decay such thaton a well-separated time scale the current through a de-tector coupled to the quantum dot directly probes thispart of the decay (1). This has been worked out in de-tail for a quantum-point contact detector in Ref. 31 andwas recently extended to a quantum-dot detector . Itis therefore of interest to calculate the full mixed statedynamics, and not just focus on the current through thequantum dot itself which does not reveal this effect. Thisis undertaken here: motivated by the above experimentalconnection, we investigate the mixed-state dynamics andour conclusions strengthen the experimental importanceof this effect: First, the decay is exactly exponential inthe wideband limit, i.e., the Markovian assumption madein Refs. 31 and 33 continues to hold, but only for thisspecial mode. Second, this form of the decay is valid forany tunnel coupling, including possible spin-dependence:still, Γ in Eq. (1) is simply given by the sum of the GoldenRule expressions for the spin-resolved tunnel rates of thevarious junctions r = L, R : Γ = P rσ Γ rσ , cf. Eq. (23).Finally, we show that any more realistic quantum dotmodel taking into account multiple orbitals labeled by l has such a “protected” mode, the decay rate beingΓ = P rlσ Γ rlσ . Notably, this is independent of the exper-imental details of the quantum dot as long as its energyscales are much below the electrode band width. Thiscan include more complex forms of the Coulomb inter-action – including all local two-particle matrix elements,not just the charging part – or spin-orbit interaction, etc.The only crucial assumption is that the tunnel couplingis bilinear in the electron operators, a basic starting pointof virtually all modeling of quantum transport throughstrongly interacting systems. In fact, even the simplify-ing assumption of collinear spin dependence of the tun-neling made in this work, turns out not to be crucial. The interesting question is raised as to which physicalprinciple can be responsible for this remarkable effect.The theoretical importance of the key result (1) lies inthe fact that it arises naturally in the real-time frame-work – by mere formulation, without real calculation –when using a particular kind of superfermion approach.This particular approach arose in the context of station-ary state transport problems and further below we givean overview of other superfermion constructions. The ex-perimental relevance of the striking result (1) thus phys-ically motivates a reformulation of the general real-timeframework. Perhaps, the impact of this should be com-pared with that of second quantization in standard quan-tum mechanics and field theory, which by itself presentsno new physical theory or prediction. That approach,however, had a big impact by making the general frame-work more intuitively accessible (e.g., by introducing fieldoperators to represent quasi-particles), simplifying cal-culations to such an extent that their results becomeintuitively clear and often revealing their physical ori-gin (e.g, particle exchange). Such a second quantizationscheme is well-established for closed quantum systemsbut still under active study for open systems (see be-low). Only recently, this idea has been combined withthe real-time diagrammatic theory targeting stationarytransport. By itself, the real-time diagrammatic theoryis already a very successful framework for the calculationof transport properties of nanoscale, strongly interact-ing systems, allowing various levels of approximationsto be systematically formulated and worked out, bothanalytically and numerically, which have foundapplication to transport experiments . Any generalprogress in simplifying or clarifying the general structureof this theory is therefore ultimately of experimental rel-evance since more accurate approximations come withinreach. For example, as mentioned above, the Andersonmodel and its generalizations present technical obstaclesfor gaining analytical insight into the low T nonequi-librium physics. By combining it with a superfermiontechnique, we were able to make detailed predictions at T = 0 for measurable stationary dI/dV maps, cover-ing large parameter regimes for strong interactions. Thisincludes level renormalization effects, energy-dependentbroadening, charge-fluctuation renormalization of cotun-neling peaks. The restriction of the approximations (onlyone plus two-loop renormalization-group diagrams), how-ever, precluded a study of the Kondo regime for the An-derson model. Clearly, addressing the time-dependentproblem for this model presents an even greater chal-lenge. Therefore the superfermion technique deservesfurther attention and development before such attemptsare to be made.Besides the aforementioned general indirect impor-tance to experiments and the concrete nonperturbativepredictions (1) the present paper also reports an exten-sive discussion of the time-dependence in the effectivelynoninteracting limit U ≪ Γ. In contrast to previousworks, we include spin coherence, electron-pair coherence(superconducting correlations ) and two-particle corre-lations in the initial state of the quantum dot. In ad-dition to various theoretical motivations mentioned inthe following, this limit is also of experimental relevance.For example, the mentioned highly controllable single-electron sources can be understood very well in such apicture. B. Theoretical motivation
The above mentioned experimental progress thus mo-tivates theoretical developments and in this paper weinvest in a reexamination of the fundamental startingpoints of transport theory and show that they can be ex-ploited more explicitly. As we now outline, this leads tothe key physical principle underlying Eq. (1). To describea quantum dot in the presence of the reservoirs one uses amixed-state theoretical description. The mixed quantumstate is described by the reduced density operator andcan be conveniently considered as an element of a linearspace of operators, referred to as Liouville space. Thetime evolution of the state is quite generally describedby a kinetic or quantum master equation, whose time-nonlocal kernel or self-energy is a superoperator on thisLiouville space. This picture is formally quite analogousto that of quantum mechanics of closed systems describedin a Hilbert space. However, the Liouville-space self-energy describes dissipative / non-Hamiltonian dynam-ics, including non-Markovian memory effects.Technically, the dynamics in Liouville space is morecomplicated because one needs to keep track of the evo-lution of state vectors (kets) as well as their adjoints(bras): in the language of Green’s functions, the evo-lution on two Keldysh contours must be described. Asa result, the usual concepts of quasiparticles correspond-ing to quantum field operators breaks down. For open fermion systems the anticommutation sign presents ad-ditional problems in Liouville space.
To address such problems, Schmutz introduced su-perfermions , i.e., analogs of quantum field operators thatact on the many-particle Liouville space and obey a simi-lar algebra. It was shown that these, in fact, generate theLiouville space starting from some vacuum supervectorand can thus be used to construct mixed state density operators. Following this analogy, insights from quan-tum field theory in Hilbert space could then also be ap-plied to density-operator approaches to nonequilibriumsystems. In these works superfermions were appliedmostly to Markovian quantum dynamics as describedby a given self-energy or time-evolution kernel and werefound to simplify the diagonalization of Lindblad time-evolution generators, in particular, finding their station-ary eigenvectors. In a recent work, Ref. 30, we have extended the ap-plication of superfermion techniques to the derivation ofthe reduced dynamics from a system - bath approachwithin the framework of the general real-time trans-port theory formulated in Liouville-space. Thisdoes not rely on Born and/or Markov approximations.Moreover, in contrast to the previous superfermion ap-proaches, the special superfermions that areinvolved simultaneously incorporate the structure im-posed by causality, related to probability conserva-tion, as well as the fermion-parity superselection ruleof quantum mechanics. The fermion-parity was alreadyincluded by Schmutz, but turns out to play a far moreprominent role. This causal structure furthermore ex-ploits a Liouville-space analog of the “Keldysh rota-tion” , well-known from Green’s function approaches.Although these particular causal superoperators were in-troduced earlier, their role as quantum fields in Li-ouville space was not recognized or taken advantageof. In this formulation of the real-time approach, theunit operator plays the special role of the vacuum statein Liouville-Fock space of the reduced system. How-ever, physically this operator describes the infinite-temperature mixed quantum state of the reduced systemwith maximal von Neumann entropy. It was realizedthat there is a corresponding natural decomposition ofthe self-energy into an infinite-temperature part and non-trivial finite-temperature corrections. The causal super-fermion operators are constructed in such a way to max-imally simplify and emphasize this fundamental struc-ture of the perturbation theory for self-energy kernels,for time-evolution, and for arbitrary observables. Thisis a general feature of open fermionic quantum systemswhich other superfermion formulations do not explicitlyreveal. The causal superfermions, furthermore, translateother fundamental properties of the underlying Hilbert-Fock space fields in a particularly clear way, such astheir irreducible transformation under spin and particle-hole symmetry transformations, as well as fluctuation-dissipation relations (related to the Liouville-space Wicktheorem ).One of the aims of this paper is to highlight simple ap-plications of causal superfermions and illustrate the phys-ical insight they convey into nonequilibrium transportthrough an Anderson quantum dot, sketched in Fig. 1.These particular quantum-field superoperators were in-troduced in the admittedly rather complicated context ofRef. 30, which constituted one of its major technical ap-plications: Only by exploiting the properties of the causal FIG. 1. Anderson model with spin-dependent tunneling ofelectrons from reservoirs r = L, R into orbitals with energy ǫ σ and local Coulomb interaction U . The dependence of thetunneling on the spin σ = ↑ , ↓ can arise either from the tunnelbarriers or from a spin-polarization of the density of statesin the electrodes (e.g. ferromagnets) or from both. For sim-plicity the magnetic field B = B e z that causes the Zeemansplitting ǫ ↑ − ǫ ↓ = B and the axes of spin-polarization ofthe tunneling are assumed to coincide. The sum of all tunnelrates Γ = P r,σ Γ rσ that connect the quantum dot to the elec-trodes turns out to be an exact decay rate in the interacting( U = 0), nonequilibrium Anderson model. The correspond-ing decay mode is the fermion-parity operator ( − n , a centralquantity in the construction of the causal superfermions. superfermions the two-loop real-time renormalization-group (RG) calculation of the T = 0 transport couldbe kept tractable, even when using the minimalistic An-derson model for the quantum dot. This may conveythe incorrect impression that superfermions are not use-ful in simpler calculations or that they even rely on theadvanced RG machinery. Indeed, in Ref. 30 we alreadyoutlined how various aspects of the Liouville-space real-time approach are further clarified, independent of theRG formulated “on top” of it. These more formal in-sights have already found useful applications to real-timecalculations in several works dealing with simplerproblems and / or approximations. The superfermionapproach also allowed an exact result to be found thatis more specific to the Anderson model: two complex-valued eigenvalues of the exact self-energy superoperatorlie symmetric with respect to an average value dependingon known, bare parameters. This implies a nonpertur-bative sum rule for the level positions and broadeningsof nonequilibrium excitations of the quantum dot in thepresence of coupling to the reservoirs and interaction. Itwas indeed noted earlier in real-time perturbation the-ory and more recently in a Liouville-space Green func-tion study. Time-dependence and fermion-parity.
Another exactresult obtained using the causal superfermions providedmore concrete physical insights into another previous work: we showed that generically the exact effective self-energy has an eigenvalue that is protected by the fermion-parity superselection rule of quantum mechanics. Thiseigenvalue corresponds to the experimentally measurabledecay mode, the key result (1) mentioned in the previoussection. Surprisingly, this decay mode depends only onthe sum of all tunnel rate constants but not on any ofthe remaining parameters, including the Coulomb inter-action U . Using the superfermion approach this result,first obtained perturbatively in Ref. 31, could be shown to hold nonperturbatively in the tunnel coupling. How-ever, our study, Ref. 30, did not consider spin-dependenttunneling, in contrast to Ref. 31, a restriction that welift in this paper. Moreover, this result can be easily gen-eralized for an arbitrary number of spin orbitals and, infact, is independent of the details of the interaction onthe quantum dot (i.e., the concrete form of the quan-tum dot model Hamiltonian): only the wideband limitand the bilinear form of the tunnel Hamiltonian mat-ter. This striking result motivates another aim of thispaper, namely, to illustrate the usefulness of causal su-perfermions for time-dependent decay problems, ratherthan the stationary state problems at the focus of Ref. 30.The first part of the paper is concerned with formulat-ing the general time-dependent perturbation theory us-ing causal superfermions, and discusses several insightsoffered into interacting problems. For instance, we showthat the exact time-evolution superoperator has an effec-tive expansion involving only causal “ creation superop-erators“ with intermediate propagators which are expo-nentially damped in time, i.e., dissipative. This physicalpicture emerges when we integrate out Markovian corre-lations, leaving only that part of the bath dynamics thatleads to the nontrivial, low-temperature phenomena inthe Anderson model. This applies generally, i.e., also tointeracting systems, and may be an interesting startingpoint for direct numerical simulation schemes of reduceddynamics, since it eliminates some “Markovianoverhead” from the start. Throughout the paper we high-light such connections of admittedly formal expressionsto physical insights, which is important for effectively ap-plying the technique to more complex problems that havemotivated this work. Noninteracting limit and super-Pauli principle.
Tomost clearly highlight applications of causal super-fermions, the second part of this paper focuses on thesimplest case, the noninteracting limit ( U = 0) of theAnderson model. Importantly, we include the spin – dis-tinguishing it from the spinless noninteracting resonantlevel model (NRLM) – and we also include a magneticfield and spin dependence of the tunneling.First, our study provides a clear illustration of howthe causal field superoperators allow one to deal withlarge fermionic Liouville spaces encountered in trans-port problems. (Already for the very simplistic single-level Anderson model the dimension of this space equals16.) We show by direct calculation that in the nonin-teracting limit the perturbation theory – after a sim-ple skeleton resummation that exploits the widebandlimit (a discrete renormalization step ) – naturally ter-minates at the second loop order. Moreover, the lackof interactions on the quantum dot results in an ad-ditional simplification, namely that the two-loop partof the time-evolution superoperator factorizes. As weshow, this is less clear when considering the two-loopself-energy in Laplace space as is often done when fo-cusing on stationary state properties. Most impor-tantly, we show that this termination and factorizationdirectly follow from the fundamental anticommutationrelations of the causal fermionic superoperators. Thetermination is in fact a consequence of the super-Pauliprinciple , the Liouville-space analog of the correspond-ing principle in the Hilbert-space of the quantum me-chanics of closed systems, which additionally relies onthe independent principle of fermion-parity superselec-tion (this principle is discussed in Sec. II A). Interactioneffects bring additional complications, and also here thecausal superfermions bring about simplifications, some ofwhich were not yet noted in Ref. 30 and are pointed outhere.Second, the analysis of the noninteracting limit providesan important benchmark for studies employing the real-time transport theory. This approach is tailored to dealwith strongly interacting problems, but it has provendifficult to see on a general level how the exact solu-tion of the noninteracting limit is recovered, in particu-lar, when including the effect of multiple spin-orbitals.Without spin, the exact solutions were checked to bereproduced explicitly by nonperturbative diagrammaticsummation but the simplifications due to the vanish-ing of interactions arise only after a detailed analysis ofcancellations. In the presence of spin- and orbital degen-eracies, this is even less obvious when working with ex-plicit, model-dependent matrix representations of super-operators in large Liouville spaces required for interact-ing systems. Our application of the causal superfermionapproach to the real-time formalism shows immediatelyhow the noninteracting limit is correctly reached on thegeneral superoperator level. This illustrates that it mayactually pay off to understand the noninteracting limitin the best possible way, when one is interested in ad-dressing interacting problems and even when one is notexpanding around the noninteracting limit. So far thisaspect has not been given much attention within the real-time framework.Third, the causal superfermion formulation also facili-tates comparison of the real-time transport theory withother approaches, which usually take the noninteractinglimit as a reference (using, e.g., path-integrals or Green’sfunctions) and therefore always contain its solution ex-plicitly on a general level.Our previous study, Ref. 30, provides an example forwhich the stationary, noninteracting limit of the Ander-son model – exhaustively analyzed here – functions asa benchmark. The one- plus two-loop real-time renor-malization group approach worked out there has the im- portant property that it includes the exact solution forthe noninteracting limit U = 0, while for large U it stillprovides a good approximation that is nonperturbativein Γ. It was found that even for the noninteracting limitone still needs a full one- plus two -loop RG to obtainthe exact density operator. Reformulated, the exact ef-fective Liouvillian of the quantum dot turns out to be quartic (instead of quadratic) in the field superoperators.This may seem to be surprising at first in view of the ab-sence of interactions. As mentioned above, in the presentpaper we show that although the renormalized pertur-bation theory terminates, it does so only at the second loop order (i.e., terms quartic in the fields), finding ex-plicit agreement with the stationary RG results for thefull density matrix, self-energy and charge current. Thetime-dependent solution for the current in the spinlessnoninteracting Anderson model was also used as bench-mark for another real-time RG study , dealing with theinteracting resonant level model, in the limit where thenonlocal interaction coupling between the quantum dotand the reservoirs vanishes. The benchmark result ofthe present paper provides the complete time-dependentpropagator, density operator as well as the current andalso includes the spin.Combined with the second quantization tools of causalsuperfermions, the real-time Liouville-space approach be-comes a more accessible tool for dealing with noninteract-ing problems. Such problems continue to attract atten-tion, especially regarding non-Markovian effects thatarise beyond the wideband limit, for which no general an-alytic solution seems to be known. In the wideband limit,noninteracting problems can be solved by means of vari-ous other techniques, for both the stationary limit andfor the transient approach. However, we demonstratehow in the real-time approach the full time-evolutioncan be calculated quite straightforwardly in this limit,i.e., avoiding a self-energy calculation and without trans-forming to Laplace space and back. In the descriptionof the noninteracting decay we include the effects of spinand pairing coherence and of two-particle correlations inthe initial quantum dot state that have been ignored sofar. Solving such problems with the real-time approachhas the additional advantage of directly allowing one togauge the effect of interactions, and to include them, ineither a perturbative or nonperturbative way. A case inpoint is the key result for fermion-parity protected decaymode, Eq. (1). Finally, our formulation also provides aframework for calculating corrections beyond the wide-band limit.
Outline
The paper is organized as follows. In Sec. II,we directly formulate the model in Liouville-space nota-tion, introduce the causal superfermion fields, constructthe Liouville-Fock space and formulate the super-Pauliprinciple. In Sec. III, we formulate the time-dependentLiouville-space perturbation theory and derive a renor-malized series that explicitly incorporates the widebandlimit on the superoperator level. We give general rulesfor the simplifications that arise in the noninteracting( U = 0) limit. This critically relies on the causal struc-ture which is made explicit by the causal superfermions.Importantly, in this limit the renormalized series nat-urally terminates at the second loop, and higher-ordercorrections are identically equal to zero. This analysisalso reveals the special importance of the physical infinitetemperature limit T → ∞ , serving as a reference pointfor both the construction of Liouville-Fock space and therenormalized perturbation theory. In Sec. IV, we per-form the explicit one- and two-loop calculations givingthe exact, full time-evolution propagator, the density op-erator and the charge current in the noninteracting limit( U = 0). To better understand the stationary limits ofthese results and to directly compare with the real-timeRG-results of Ref. 30, we additionally perform the calcu-lation directly in Laplace space. We summarize our re-sults in Sec. V and discuss their generalization to multipleorbitals, the implications for real-time renormalization-group schemes, and possible further application of thedeveloped ideas. II. ANDERSON MODEL IN LIOUVILLE-FOCKSPACEA. Fermion-parity superselection rule
In this paper, we make explicit use of the postulate offermion-parity (or univalence) superselection in quantummechanics and quantum field theory.
It is a part ofquantum kinematics and can therefore be discussed be-fore any model of the dynamics is formulated. Here, webriefly illustrate the main substance of this postulate anddiscuss one of its aspects, which is crucial for starting upthe formulation of our approach in the following. For ex-ample, for a single level quantum dot with field operators d σ and d † σ , where σ = ± corresponds to spin up ( ↑ ) anddown ( ↓ ) along the z -axis, the fermion-parity operator,recurring at many crucial steps in the paper, is definedas ( − n := e iπn = Y σ (1 − n σ ) , (2)where n = P σ n σ is the fermion number operator. Forthis simple case, the fermion-parity superselection rulecan be phrased as follows: the density operator andthe operator of any physical observable A must commutewith the total fermion parity operator of the system:[( − n , A ] − = [( − n , ρ ] − = 0 . (3)This excludes the possibility of interference (superposi-tions of) states with even and odd number of fermions.The operator ( − n has been applied in Ref. 82 to makefield operators for different fermion species commute(rather than anticommute), and it plays a key role insetting up the second quantization in Liouville space. Here and in the following, it is convenient to introducean additional particle-hole index η : d ησ = (cid:26) d † σ , η = + d σ , η = − . (4)Throughout the paper we will denote the inverse valueof a two-valued index with a bar, e.g.,¯ η = − η (5)We combine all indices into a multi-index variable writtenas a number: 1 = η, σ ; ¯1 = ¯ η, σ, (6)where, by way of exception, the bar denotes inver-sion of the particle-hole index only. If we have morethan one multi-index, we distinguish their componentsby using the multi-index number as a subscript: 1 = η , σ , r , ω , η , σ , r , ω and use a multi-indexKronecker symbol δ = δ η ,η δ σ ,σ . (7)For clarity, we usually omit these subscripts if there isonly one multi-index as in Eq. (6). Then d = d ησ and d ¯1 = d − ησ , and we can summarize all fundamental rela-tions simply by ( d ) † = d ¯1 and [ d , d ] + = δ . Through-out the paper, we denoted the (anti)commutators by[ A, B ] − = AB − BA and [ A, B ] + = AB + BA .A first application of the fermion-parity arises when weconnect the quantum dot to reservoirs with field opera-tors a σrk , where σ is the spin index, k the orbital index,and the reservoir index r = ± corresponds to left/right.We have to make a choice for commutation relations of a ησrk relative to d η ′ σ ′ : in setting up the second quanti-zation, one is free to choose either commutation or anti-commutation relations for fermions of different states /particles, whereas one must have anticommutation rela-tions for fermions in the same state. Both choices pro-duce identical, correctly antisymmetrized multi-particlestates. Usually, the most elegant choice, indicated hereby a prime on the field operators, is to let them all anti-commute: [ a ′ , d ′ ] + = 0 , (8a)[ d ′ , d ′ ] + = δ , (8b)[ a ′ , a ′ ] + = δ . (8c)The fields are pairwise Hermitian adjoints, ( d ′ ) † = d ′ ¯1 and ( a ′ ) † = a ′ ¯1 . The fermion number operator of thedot is expressed as n = P σ d ′ σ † d ′ σ and the correspondingfermion-parity operator anticommutes with the dot fields d ′ (using [( − n ] = e iπn = 1),( − n d ′ ( − n = e iπn d ′ e − iπn = − d ′ , (9)but commutes with the reservoir operators a ′ (like anyoperator local to the dot), a ′ ( − n = ( − n a ′ . (10)In approaches where the reservoir degrees of freedomare eliminated by a partial trace operation, it is muchmore convenient to let reservoir and dot fields commute by definition, allowing operators of different subsystemsto be separated easily. By doing this from the start manyunnecessary canceling sign factors can be avoided. Suchfields are used throughout this paper and are indicatedby leaving out the prime. The fields on the different (thesame) systems mutually (anti)commute:[ a , d ] − = 0 , (11a)[ a , a ] + = δ , (11b)[ d , d ] + = δ , ¯2 , (11c)with d † = d ¯1 and a † = a ¯1 . This choice of commutationrelations was used in Refs. 51 and 30, and will be usedhere as well, unless stated otherwise.The fermion-parity operator now appears as the formaldevice relating the above two choices, which is convenientto have at hand for a direct comparison with other ap-proaches, e.g., the Green’s function approach, startingfrom the choice Eq. (8). One way of obtaining the choicein Eq. (11) from the fields satisfying Eq. (8) is the fol-lowing change of variables: a = ( − n a ′ = a ′ ( − n , (12a) d = − η ( − n d ′ = η d ′ ( − n . (12b)Here, ( a ) † = a ¯1 and the η -sign ensures that the ad-joint relation ( d ′ ) † = d ′ ¯1 is also preserved: using Eq. (9)( d ) † = η ( − n ( d ′ ) † = − η ( d ′ ) † ( − n = η ¯1 d ′ ¯1 ( − n = d ¯1 . A key point, needed later, is that when tracingout the reservoirs only averages of products of an evennumber of reservoir fermions can appear, and the quan-tum dot operator ( − n in Eq. (12a) cancels out inTr R a . . . a k = Tr R a ′ . . . a ′ k since ( − kn = 1. Thetransformation (12) is only canonical locally on the quan-tum dot and on the reservoirs. Since it is not globallycanonical we must check how observable operators aretransformed. This is done in Sec. II B once we have spec-ified the dynamics and the physical operators of interest. B. Anderson model
The model that we consider was already sketched inFig. 1. The usual formulation of the single-level Andersonmodel specifies the Hamiltonian H = ǫn + BS z + U n ↑ n ↓ (13)where ǫ denotes the energy of the orbital and n = X σ n σ , n σ = d † σ d σ , (14)is the fermion number operator. Furthermore, S z = P σ σn σ is the z component of the spin vector operator S = P σσ ′ σ σ,σ ′ d † σ d σ ′ along the external magnetic field B = B e z (in units gµ B = 1) and σ is the vector of Paulimatrices. The dot is attached to electrodes which aretreated as free electron reservoirs: H R = X σ,r,k ǫ σrk a † σrk a σrk . (15)The reservoir electron number and spin n R = X r n r , s R = X r s r , (16)respectively, can be decomposed into their contributions n r = P σ,k a † σrk a σrk and s r = P σ,k σ σ,σ ′ a † σrk a σ ′ rk . Be-fore we introduce the coupling, we introduce the nota-tion of Ref. 30 to conveniently deal with the continuumlimit. The reservoirs are described by a density of states ν rσ ( ω ) = P k δ ( ω − ǫ σrk + µ r ) and we go to the energyrepresentation of the fermionic operators, a σr ( ω ) = 1 p ν r ( ω ) X k a σrk δ ( ω − ǫ σrk + µ r ) , (17)with the anticommutation relations:[ a σr ( ω ) , a † σ ′ r ′ ( ω ′ )] + = δ σ,σ ′ δ r,r ′ δ ( ω − ω ′ ) , (18)[ a σr ( ω ) , a σ ′ r ′ ( ω ′ )] + = 0 . (19)Here, we denote (anti)commutators by [ A, B ] ∓ = AB ∓ BA . The continuous reservoir Hamiltonian is thus H R = X σ,r Z dω ( ω + µ r ) a † σr ( ω ) a σr ( ω ) , (20)with the electron energy ω taken relative to µ r for reser-voir r . The junctions connecting the dot and reservoirsare modeled by the tunneling Hamiltonian V = X r V r , (21) V r = X σ Z dω p ν rσ ( ω ) (cid:0) t rσ ( ω ) a † σr ( ω ) d σ + h.c. (cid:1) , (22)with real spin-dependent amplitudes t rσ ( ω ). Using thespectral densityΓ rσ ( ω ) = 2 πν rσ ( ω ) | t rσ ( ω ) | , (23)it is convenient to rescale the field operators: b σr ( ω ) = r Γ rσ ( ω )2 π a σr ( ω ) . (24)We thus incorporate two sources of spin-polarization ofthe tunneling rates: either the attached electrodes areferromagnetic [ ν rσ ( ω )] or the tunnel junctions are mag-netic [ t rσ ], or both. We made the simplifying assumptionthat the magnetizations of the electrodes are collinear (ei-ther parallel or antiparallel), and also collinear with thespin-polarization axes of the tunnel junctions. Moreover,the external magnetic field B is assumed to be collinearwith this axis.As before, we introduce an additional particle-hole in-dex: b ησr ( ω ) = (cid:26) b † σr ( ω ) , η = + b σr ( ω ) , η = − . (25)and combine all indices into a multi-index variable writ-ten as a number, which now includes an additional con-tinuous index ω : η, σ, r, ω ; ¯1 = ¯ η, σ, r, ω. (26)Then b = b ησr ( ω ) and b ¯1 = b − η,σr ( ω ) and the(anti)commutation relations are[ d , b ] − = 0 , (27)[ d , d ] + = δ , (28)[ b , b ] + = Γ π δ , (29)where Γ = Γ rσ ( ω ). In Eq. (29), it is left implicit that themulti-index δ -function δ contains an additional delta-function δ ( ω − ω ) relative to the Kronecker-delta (7) inEq. (28).Since we formulated our model in terms of fields ob-tained by a noncanonical transformation, we should nowcheck the form of the model in terms of the (primed)anticommuting fields [Eq. (8)]. First, any local reser-voir observable has the same form in terms of b opera-tors as in terms of b ′ . This immediately follows fromthe fermion-parity superselection rule: by Eq. (3) lo-cal reservoir observables always contain products of evennumbers of the primed reservoir field operators. Sec-ond, any operator local to the quantum dot also has thesame form due to fermion-parity superselection rule if it conserves the fermion number n . Locally, we canthus express everything in terms of b and d by sim-ply omitting the primes. However, the interaction op-erator V = P σ,r R dω ( b ′† σrω d ′ σ + d ′ σ † b ′ σrω ) = P ηb ′ ¯1 d ′ now changes its form. In fact, it simplifies by loosing its η -sign: V = b ¯1 d , (30)Here, we implicitly sum over all discrete parts of themulti-index 1 (i.e., η, σ, r ) and integrate over its continu-ous part ( ω ). An alternative discussion of the above notexplicitly referring to the fermion parity can be foundin Ref. 51Finally, the reservoirs are assumed to be in thermalequilibrium with temperature T , each described by itsown grand-canonical density operator, ρ R = Y r ρ r , ρ r = 1 Z r e − T ( H r − µ r n r ) , (31) where Z r = Tr r e − T ( H r − µ r n r ) . For the example setupshown in Fig. 1 one can give the electrochemical poten-tials by, e.g., assuming a symmetrically applied bias volt-age, i.e., µ L,R = ± V b /
2. However, most of our resultsapply to any number of electrodes and do not depend onthis.Together with the the Hamiltonian of the total system, H tot = H + H R + V, (32)this specifies the model. The noninteracting resonant-level model (NRLM) is obtained by setting U = 0 in thedot Hamiltonian Eq. (13) in Eq. (32) and discarding thespin. C. Reduced time-evolution propagator
In order to calculate the dynamics of the reduced den-sity operator of the quantum dot, we first need to con-sider the evolution of the total system density operator.It is generated by the Liouville–von Neumann equation: ∂ t ρ tot ( t ) = − i (cid:2) H tot , ρ tot ( t ) (cid:3) − = − iL tot ρ tot ( t ) , (33)with the Liouvillian superoperator L tot • = [ H tot , • ] − .Superoperators are linear transformations of operatorsand throughout the paper (if needed) we let the solidbullet • indicate the operator on which a superoperatoracts. In the following, we will make the common assump-tion that the initial state of the total system at time t is a direct product ρ tot ( t ) = ρ R ρ ( t ) . (34)However, some of the developments reported in the fol-lowing do not depend on this assumption. In a forth-coming work we will show that the causal superfermionapproach is useful also when initial reservoir-dot correla-tions are present. The formal solution of Eq. (33) is: ρ tot ( t ) = e − iL tot ( t − t ) ρ tot ( t ) . (35)The reduced dot density operator is obtained by integrat-ing out of reservoirs degrees of freedom: ρ ( t ) = Tr R ρ tot ( t ) = Tr R (cid:16) e − iL tot ( t − t ) ρ R (cid:17) ρ ( t ) . (36)Equation (36) is the starting point for a perturbationtheory for the propagator superoperatorΠ( t, t ) = Tr R (cid:16) e − iL tot ( t − t ) ρ R (cid:17) • . (37)Decomposing L tot = L + L R + L V , with L = [ H, • ] − , L R = [ H R , • ] − we expand in the tunnel coupling L V =[ V, • ] − ∼ √ Γ. Usually two additional steps are taken, ineither order. First, one derives a Dyson equation for theexact propagator and introduces a self-energy Σ( t, t ′ ).Π( t, t ) = e − iL ( t − t ) − i Z dt dt t ≥ t ≥ t ≥ t e − iL ( t − t ) Σ( t , t )Π( t , t ) . (38)The reduced density operator is then found to satisfyNakajima-Zwanzig / generalized master / kinetic equa-tion ∂ t ρ ( t ) = − i t Z t dt ′ L ( t, t ′ ) ρ ( t ′ ) , (39)where L ( t, t ′ ) = L ¯ δ ( t − t ′ ) + Σ( t, t ′ ) (40)is the so called effective Liouvillian for the quantum dot.We introduced ¯ δ ( t − t ′ ) := 2 δ ( t − t ′ ) such that Z tt dt ′ ¯ δ ( t − t ′ ) = 1 . (41)to absorb the factor 2 that is required to recover theLiouville equation for ∂ t ρ ( t ) = − iLρ ( t ′ ) from Eq. (39)for Σ( t, t ′ ) = 0 (since R tt δ ( t − t ′ ) dt ′ = 1 / Equation (39)in the Laplace representation is then the starting pointfor the calculation of stationary quantities using dif-ferent approximate calculation schemes, e.g., perturba-tive and renormalization group approaches.
The time evolution can be obtained by calculating thefull Laplace image of the reduced density operator bymeans of perturbation theory or renormalization groupapproaches, and then performing the inverseLaplace transformation. However, a direct approach in the time representationis of interest. For instance, even if one is interested in sta-tionary properties in the end, some manipulations maybe easier or clearer in the time representation, for in-stances, in problems of noise and counting statistics,
Markovian approximations and adiabatic driving cor-rections or simplifications for higher-order tunnelrates in the stationary limit. Whereas the above citedworks mostly deal with strongly interacting (Anderson)quantum dots, here, the noninteracting limit of the An-derson model ( U = 0) has our interest. We show that forthis case it is convenient to work directly with the propa-gator Π( t, t ) in the time-representation. The self-energyis only used in an intermediate renormalization step ofthe perturbation series for Π( t, t ) to deal with the wide-band limit. For now, however, we make no assumptionon U unless stated otherwise.Although we only calculate Schr¨odinger picture quanti-ties, it is useful to extend the standard interaction repre-sentation to the Liouville space. The solution of thevon Neumann equation (35) for the total density operatorhas a form familiar from the Hamiltonian time-evolutionof the state vector in Hilbert-space quantum mechanics: ρ tot ( t ) = e − i ( L + L R ) ( t − t ) ˆ T e − i t R t L V ( τ ) dτ ρ tot ( t ) , (42)where ˆ T denotes the time-ordering of superoperators and L V ( τ ) are the tunnel Liouvillians in the interaction pic-ture: L V ( τ ) = e i ( L + L R ) ( τ − t ) L V e − i ( L + L R ) ( τ − t ) . (43)Expanding Eq. (42) in L V ( t ), one obtains the time-dependent perturbation expansion ρ tot ( t ) = e − i ( L + L R ) ( t − t ) − i t Z t dt L V ( t ) (44)+( − i ) t Z t dt t Z t dt L V ( t ) L V ( t ) + ... ρ tot ( t ) . Making use of Tr R L R = 0, the perturbation expansionfor the time-dependent reduced density operator in pow-ers of L V reads as ρ ( t ) = e − iL ( t − t ) Tr R [1 + ∞ X m =0 ( − i ) m Z dt . . . dt mt ≥ t m ... ≥ t ≥ t L V ( t m ) . . . L V ( t )] ρ tot ( t ) (45a)= e − iL ( t − t ) ρ ( t ) + ∞ X m =0 ( − i ) m Z dt . . . dt mt ≥ t m ... ≥ t ≥ t e − iL ( t − t m ) Tr R h L V e − i ( L + L R )( t m − t m − ) . . . L V e − i ( L + L R )( t − t ) ρ tot ( t ) i . (45b)0A direct analysis of the series Eq. (45a) is cumbersome,even for the noninteracting case. To derive a man-ageable series, we now introduce the causal fermionic-superoperators in the next section. D. Second quantization in Liouville Fock space
The above Liouville space formulation of quan-tum mechanics is well-known and has found manyapplications.
However, when applied to quantummany particle systems, it becomes more powerful ifanalogs of field-theoretical techniques are introduced,in particular, field super operators and Liouville-Fockspace.
1. Causal superfermions
Special causal field superoperators may beconstructed: for the quantum dot they read as G q • = 1 √ n d • + q ( − n • ( − n d o , (46)and for the reservoir J q • = 1 √ n b • − q ( − n R • ( − n R b o . (47)Here, q = ± labels the components obtained after a“Keldysh rotation” of simpler superoperators defined byleft and right multiplication with a Hilbert-Fock spacefield. Moreover, ( − n = e iπn and ( − n R = e iπn R arethe fermion-parity operators of the dot and reservoirs, re-spectively [cf. Eqs. (14) and (16)]. The additional minussign in the second term of the definition of J q , relative tothe definition of G q is purely conventional but is advanta-geous later [cf. Eqs. (89)-(90)]. The tunneling Liouvillian L V can be written compactly as L V = X q = ± G q J q ¯1 = ¯ G ¯ J ¯1 + ˜ G ˜ J ¯1 , (48)where in the second equality we used “bar-tilde” nota-tion of Refs. 51 and 30 for the q = ± components, re-spectively: ¯ G = G +1 , ¯ J = J +1 , ˜ G = G − , ˜ J = J − , (49)which is sometimes more convenient. We note that in therewriting of L V as Eq. (48) the fermion-parity superse-lection rule is already used, see Ref. 30 for a discussion.The superoperators G q and J q obey fermionic anticom-mutation relations: h ¯ G , ˜ G i + = δ , ¯2 , h ¯ G , ¯ G i + = h ˜ G , ˜ G i + = 0 , (50) h ¯ J , ˜ J i + = Γ π δ , ¯2 , h ¯ J , ¯ J i + = h ˜ J , ˜ J i + = 0 , (51) where [ , ] + now denotes the anticommutator of superop-erators. The superoperators of the dot and the reservoirscommute with each other,[ ˜ J , ˜ G ] − = [ ¯ J , ¯ G ] − = [ ¯ J , ˜ G ] − = [ ˜ J , ¯ G ] − = 0 . (52)This follows from our assumption that the dot and thereservoir operators commute [see Sec. II A]. Furthermore,the field superoperators are pairwise related by the super-Hermitian conjugation (which will be defined shortlyhereafter): ¯ G † = ˜ G ¯1 , ¯ J † = ˜ J ¯1 . (53)In this relation, note the reversal of the causal index (¯ q )as well as the multi-index (¯1): ( G q ) † = G ¯ q ¯1 and ( J q ) † = J ¯ q ¯1 .The crucial properties of the causal fermionic field su-peroperators (46)-(47), distinguishing them from previ-ously introduced field superoperators (see Ref. 30 for adetailed comparison) are Tr D ¯ G • = 0 , Tr R ˜ J • = 0 , (54)¯ G ( − n = 0 , ˜ J ( − n R = 0 , (55)for all values of the multi-index 1. In the following, wewill see that Eq. (54) relates to the fundamental causalstructure of the correlation functions expressed in thesesuperfields and plays a key role in maximally simplify-ing them [cf. Eq. (89)-(90)]. Equation (55) arises sincethe fermion-parity operator is used to ensure that thefield superoperators anticommute. This property leadsto an interesting exact result for the interacting Ander-son model discussed in Sec. III C 2.It is of interest to also give the superoperators G q interms of the anticommuting dot fields d ′ [Eq. (12b)]: G q • = η √ d ′ , ( − n • ] − q (56a)= η ( − n +1 √ d ′ • + q • d ′ ) . (56b)The causal field superoperators in Liouville-Fock spaceare thus simply the commutator and anticommutator ofthese Hilbert-Fock space fields operators (cf. Ref. 50),but only after the fermion-parity has been applied toits argument. In Eq. (56a), taking the commutatorand anticommutator of an operator is the superoperator-equivalent of performing the Keldysh rotation. Oneuseful aspect of the form (56a) is that for q = + the twofundamental properties (54)-(55) are immediately clear:the property (54) follows from the vanishing of the traceof a commutator and (55) from the vanishing of any com-mutator with the unit operator. These are dual prop-erties with respect to the scalar product in Liouville-Fockspace, see Sec. II D 2. The form (56a) is also convenientfor a direct comparison with expressions which occur informalisms using the local dot Green’s functions.
2. Causal basis for Liouville-Fock space: Super-Pauliprinciple
As was shown in Ref. 30, the causal field superopera-tors G ± [Eq. (46)] generate a basis for the Liouville spaceof the quantum dot, in close analogy with the construc-tion of the usual fermion Fock-basis in the many-particleHilbert space. The 4-dimensional Hilbert-Fock space ofthe quantum dot is spanned by the orthogonal state vec-tors | i , | ↑i = d †↑ | i , | ↓i = d †↓ | i , | i = d †↑ d †↓ | i . (57)Operators A = P k,l =0 , ↑ , ↓ , A k,l | k ih l | acting on this spacethemselves form a 16 dimensional linear space L with theinner product ( A | B ) = Tr D A † B , which we refer to as the Liouville space of the quantum dot. By | A ) we denotean operator A considered as a supervector in L , and usethe rounded bracket notation to avoid possible confusionwith Hilbert space state vectors | ψ i . A set of mutuallyorthogonal supervectors, i.e., operators A i , i = 1 , ..., A i | A j ) = Tr D A † i A j = δ i,j (58)form an orthonormal basis in the quantum-dot Liouvillespace. Superoperators are linear maps S : L → L andcan be expressed in this basis as: S = X i,j S i,j | A i )( A j | , i, j = 0 , ..., . (59)where ( A |• = Tr D A † • denotes the linear function onoperators • built from the operator A . The super-Hermitian conjugation Eq. (53) is defined with respectto the Liouville-space inner product, i.e., ( A | S † | B ) =( B | ( SA )) ∗ .For a Liouville space of a many-particle system, a Liouville-Fock space , a super Fock-basis can be con-structed starting from some operator defining a vacuumsuperstate. (In Sec. II D 3, we indicate that some caremust taken when using the term “superstate”.) For thecausal field superoperators (46) the vacuum superstateof the quantum dot is given by | Z L ) = , (60a)which is indeed destroyed by the annihilation superop-erators G − = ˜ G : ˜ G ησ | Z L ) = 0 for all η , σ by the su-perhermitian conjugate of the identity (54). From thisvacuum another seven bosonic operators are created byapplication of all possible products of an even numberof fermionic creation superoperators G +1 = ¯ G ησ . Thedoubly occupied superstates are | χ σ ) = ¯ G + σ ¯ G − σ | Z L ) = − e iπn σ , (60b) | T + ) = ¯ G + ↑ ¯ G + ↓ | Z L ) = d †↑ d †↓ , (60c) | T − ) = − ¯ G −↑ ¯ G −↓ | Z L ) = d ↓ d ↑ , (60d) | S σ ) = ¯ G + σ ¯ G − ¯ σ | Z L ) = d † σ d ¯ σ , (60e) where σ = ± = ↑ , ↓ , cf. Eq. (14). The operators χ σ are proportional to the fermion-parity operator fora single spin- σ orbital of the dot, e iπn σ = 1 − n σ .The “most filled” superstate | Z R ) (quadruply occu-pied) equals the total fermion-parity operator of the dot e iπn = Q σ e iπn σ = (1 − n ↑ )(1 − n ↓ ), normalized to theLiouville-space scalar product: | Z R ) = ¯ G + ↑ ¯ G −↑ ¯ G + ↓ ¯ G −↓ | Z L ) = e iπn . (60f)In addition, another eight fermionic superoperators, arecreated by the action of products of odd numbers offermionic creation superoperators ¯ G ησ , either with oneexcitation | α + ησ ) = σ − η ¯ G η, ( ση ) | Z L ) = √ σ − η d η, ( ση ) , (61a)or with three (¯ ση = σ ¯ η = − ση ) | α − ησ ) = σ − η ¯ G η, ( ση ) ¯ G η, (¯ ση ) ¯ G ¯ η, ( σ ¯ η ) | Z L )= e iπn √ σ − η d η, ( ση ) . (61b)Using the anticommutation relations (50), one shows thatthe above 16 supervectors (operators) (60)-(61) form acomplete, orthonormal basis in the sense of Eq. (58) forthe Liouville-Fock space. For the central applications ofthis paper, we do not need the fermionic part of this basisexcept for the next Sec. II D 3. However, since second-quantized expressions for superoperators act on the entireLiouville space, one must be aware of this linear subspace,as we will illustrate in several cases in the following.In the above construction of Liouville-Fock space, thefermion-parity operator ( − n = 2 Z R plays a fundamen-tal role. First, it was included into the definition of thefield superoperators to ensure fermionic anticommutationrelation. This, in particular, results in( ¯ G ) = 0 , (62)which expresses that one cannot doubly occupy a super-state (labeled by 1 = η, σ ). We will refer to Eq. (62) asthe super-Pauli exclusion principle by analogy with thePauli principle for fermionic fields in Hilbert-Fock space,for which ( d † σ ) = 0. A consequence of central impor-tance to the paper is that any product of more than fourcreation (or destruction) superoperators necessarily con-tains at least one duplicate and therefore vanishes:¯ G m . . . ¯ G = 0 for m > . (63)This is to be compared with the vanishing of productsof more than two field creation operators in Hilbert-Fockspace, i.e., d † σ m ..d † σ = 0 for m >
2. Equation (63) can begeneralized to an Anderson model with N spin-orbitalsby replacing the condition with m > N . Second, thesingly and triply occupied superstates are constructedfrom the same set of four Hilbert-Fock field operators d ησ , but they differ by the application of the parity op-erator, | α − ησ ) = ( − n | α + ησ ). It is this factor that ensures2their orthogonality. Finally, the left multiplication bythe fermion-parity operator implements a superoperatoranalog of a particle-hole transformation, mapping basissupervectors with N superparticles onto those with 4 −N superparticles. In analogy to the usual second quantization, one canintroduce a super occupation operator [cf. Eq. (53)]: N ησ = ¯ G ησ ˜ G ¯ ησ = ¯ G ησ ¯ G † ησ . (64)By construction, due to the anticommutation relations itsatisfies [ N ησ , ¯ G ησ ] = ¯ G ησ . (65)It therefore simply counts the number of times that thecreation superoperators ¯ G ησ appears in a basis superket,which is restricted to 0 or 1 by Eq. (62): with N := N ησ N i ¯ G m . . . ¯ G | Z L ) = ( δ i,m + . . . + δ i, ) ¯ G m . . . ¯ G | Z L ) . (66)Finally, we note that operators Eqs. (60b)-(60e) areclosely related to the group generators of the spin-( S ) and charge-rotation ( T ) symmetry of the Andersonmodel (they transform as irreducibly under the symme-try group). By working in the causal Liouville-Fock spacebasis (60)-(61) one thus not only profits from the funda-mental causal properties of interest here, but one alsomaximally exploits these model-specific symmetries, seethe study Ref. 30, where this was of crucial importance.
3. Examples of second quantization in Liouville space
Before we move on, we first illustrate the above intro-duced second quantization in Liouville-Fock space usingcausal superfermions. We discuss the expansion of a su-pervector, using the density operator as an example, andthe expansion of a superoperator, the Liouvillian L . a. Mixed state supervector ρ . We can construct themost general form of the reduced density operator forthe quantum dot by accounting for the restrictions ona physical mixed state: the operator ρ must (i) bepositive, (ii) be self-adjoint, (iii) have unit trace, and(iv) satisfy the fermion-parity superselection rule (univa-lence). The latter requires that any density oper-ator ρ has no off-diagonal matrix elements with respectto the fermion-parity quantum number [cf. Eq. (3)]:[ ρ, ( − n ] − = 0 . (67)The linear space containing such operators satisfying (ii)-(iv) is spanned by the bosonic operators in Eqs. (60).The reduced density operator is a supervector in thisspace, and is thus generated by application of prod-ucts of an even number of creation superfields fromthe vacuum superket with, in general, seven coefficients Ω ± ( t ) , Φ ± ( t ) , Υ ± ( t ) , and Ξ( t ): ρ ( t ) = n + X σ Φ σ ( t ) ¯ G + σ ¯ G − σ + X σ Ω σ ( t ) ¯ G + σ ¯ G − ¯ σ + X η η Υ η ( t ) ¯ G η ↑ ¯ G η ↓ + Ξ( t ) ¯ G + ↑ ¯ G −↑ ¯ G + ↓ ¯ G −↓ o | Z L )= | Z L ) + X σ Φ σ ( t ) | χ σ ) + Ξ( t ) | Z R )+ X η Υ η ( t ) | T η ) + X σ Ω σ ( t ) | S σ ) . (68)With appropriate restrictions imposed by the positivitycondition (i), these coefficients thus parametrize an ar-bitrary dot state, e.g., the complicated time-dependentdensity operator ρ ( t ) of the U = 0 Anderson model[Eq. (45b)]. The coefficients are the non-equilibrium av-erages h•i ( t ) = Tr D [ • ρ ( t )] of the complete set of localobservable operators (60). The coefficientΦ σ ( t ) = ( χ σ | ρ ( t )) = − h e iπn σ i ( t ) (69)gives the average occupation: h n σ i ( t ) = 1 / σ ( t ) byusing e iπn σ = (1 − n σ ). The coefficientΞ( t ) = ( Z R | ρ ( t )) = h e iπn i ( t ) , (70)the average of the fermion-parity operator, Ξ( t ) = h Q σ e iπn σ i ( t ) = 2 h n ↑ n ↓ i ( t ) − h n i ( t ) + 1 /
2, takes intoaccount the correlations of the occupancies: h n ↑ n ↓ i ( t ) = h n ↑ i ( t ) h n ↓ i ( t ) is equivalent to Ξ( t ) = 2 Q σ Φ σ ( t ). Fur-thermore, the average of an “anomalous” and a spin-flipcombination of Hilbert-Fock space field operatorsΥ ¯ η ( t ) = ( T ¯ η | ρ ( t )) = η h d η ↑ d η ↓ i ( t ) , (71)Ω ¯ σ ( t ) = ( S ¯ σ | ρ ( t )) = h d † σ d ¯ σ i ( t ) , (72)describe the transverse spin ( ↑ - ↓ ) coherence and theelectron-pair (0-2) coherence of the state at time t . Atthe initial time t such coherences can be prepared: thetransverse spin coherence by contact with a ferromagnetwith a polarization transverse to the magnetic field B and the electron-pair coherence by contact with a super-conductor. At finite times t such coherences will persist,but in the stationary limit t → ∞ they must vanish sincethe Anderson model has spin-rotation symmetry (withrespect to the magnetic field axis) and charge-rotationsymmetry. Likewise, two-particle correlations can beinitially present on the quantum dot if it has been incontact with an interacting system. These will decay, inthe sense that lim t →∞ Ξ( t ) = 2 Q σ lim t →∞ Φ σ ( t ), if thequantum dot is noninteracting ( U = 0).Of the eight bosonic operators (60), only Z L has anonzero trace, and the physical requirement Tr ρ = 1completely fixes its coefficient in Eq. (68). By itself, theoperator ρ ∞ := 12 | Z L ) = 14 (73)3represents the physical stationary state of the quantumdot coupled to reservoirs at infinite temperature (i.e., T much larger than any other energy scale, i.e., U , ǫ − µ r , B ). In any finite-temperature mixed state (68), thereare in general two- and four-superfermion excitations. Inour formalism, such super excitations correspond to a“cooling” relative to the infinite-temperature supervac-uum (73). Although this point of view is opposite to thatin the Hilbert-Fock space (where excitations rather de-scribe a “heating” of the zero-temperature vacuum | i ),the causal superfermion approach is thus entirely physicaland brings definite insights and advantages in the studyof open quantum systems. However, care must be takento import physical concepts from second quantization inHilbert-Fock space. For instance, it should be noted thatof the basis supervectors only | Z L ) can represent a phys-ical state on its own : the other 15 basis supervectors,such as ¯ G | Z L ), are traceless by construction [Eq. (54)]and cannot fulfill the probability normalization condi-tion Tr D ρ = 1 by themselves. Moreover, the fermionicbasis vectors (61), such as ¯ G | Z L ), do not have the rightfermion-parity. It is only in superpositions with | Z L ) ofthe form (68) that the bosonic basis supervectors (60)take part in real mixed states described by a density op-erator, whereas the fermionic basis vectors (61) only playa role in virtual intermediate mixed states, see discussionof Eq. (76) in the following. This should be kept in mindwhen speaking formally about “superstates”, “superpar-ticles” or “superexcitations”, a terminology which we doconsider to be useful. b. Liouvillian superoperator L . As a next illustra-tion, we discuss the second quantized form of the Liou-villian superoperator of the isolated Anderson model interms of the field superoperators: L = X η,σ η (cid:16) [ ǫ + U/
2] + σB/ (cid:17) ¯ G ησ ˜ G ¯ ησ + (74a)+ U X η,σ (cid:16) ¯ G ησ ˜ G ¯ ησ ˜ G η ¯ σ ˜ G ¯ η ¯ σ + ¯ G ησ ¯ G ¯ ησ ¯ G η ¯ σ ˜ G ¯ η ¯ σ (cid:17) , (74b)which is verified by substitution of Eq. (46) to give L = [ H, • ] − with H given by Eq. (13). Similar to theusual second quantization technique, this expression di-rectly reveals a number of general properties. For in-stance, particle number conservation is expressed by the fact that in the field superoperators only conjugate pairsof η , ¯ η appear. Furthermore, since only products ofan even number of field superoperators appear, the su-peroperator L preserves the fermion-parity superselec-tion rule of the density operator Eq. (67): The off-diagonal supermatrix elements between a bosonic [ | B ),Eq. (60)] and a fermionic [ | F ), Eq. (61)] basis operatorvanish, ( B | L | F ) = ( F | L | B ) = 0, simply because theseare created from | Z L ) by the action of an even and anodd number of field superoperators, respectively. As aresult, if initially ρ ( t ) satisfies Eq. (67) then so does ρ ( t ) = e − iL ( t − t ) ρ ( t ) at later times t > t for a closedsystem. A property more specific to the use of causal superfermions in the second quantized form Eq. (74), isthat the conservation of probability is immediately ob-vious term-by-term : each term ends with a destructionsuperoperator ¯ G on the left, ensuring by Eq. (54) thatthe trace is preserved Tr D e − iL ( t − t ) ρ ( t ) = Tr D ρ ( t ).More specific to the Anderson model is that Eq. (74)automatically achieves an interesting decomposition ofthe interaction term. In particular, the quartic term(74b) ∝ U commutes with the simple quadratic term(74a) which also contains U . Importantly, the quarticterm acts only in the fermionic sector of the Liouville-Fock space [spanned by the superkets (61)]: for any twobosonic operators B and B ′ we have ( B | L | quartic | B ′ ) = 0.This follows from( B | X η,σ ¯ G ˜ G ¯1 ˜ G ˜ G ¯2 | B ′ ) = 0 , (75)where 1 = η, σ and 2 = η, ¯ σ and the same for thesecond term in Eq. (74b) (which is the Hermitian su-peradjoint of this). Equation (75) immediately followsfrom the structure of the quartic term using reasoningvery similar to that used in the usual second quanti-zation in Hilbert-Fock space. The result (75), togetherwith the fermion-parity superselection rule (67), now im-plies that in the time-evolution expansion Eq. (45b) thequartic interaction term Eq. (74b) plays no role in thefree quantum-dot propagator e − iL ( t k +1 − t k ) when it oc-curs after an even number ( k ) of tunneling Liouvillians L V . For example, inserting in the fourth-order expres-sion of Eq. (45b) a complete set of superstates (expansionof the unit superoperator) for the quantum dot between L V ( t ) and L V ( t ) and substituting Eq. (48) we obtainthe structure · · · G q · · · G q e − iL ( t − t ) G q · · · G q · · · ρ ( t ) = X B,B ′ · · · G q · · · G q | B )( B | e − iL ( t − t ) | B ′ )( B ′ | G q · · · G q · · · ρ ( t ) , (76)where only the quantum-dot part of the expressions areshown. The sums over | B ), | B ′ ) are restricted to thebosonic superkets Eq. (60): since ρ ( t ) is a bosonic oper- ator, application of an even number of superfields bringsus back to the bosonic sector. Now the quartic U -termdrops out in the matrix elements ( B | e − iL ( t − t ) | B ′ ) due4to Eq. (75). The propagation of the bosonic virtual in-termediate states in Eq. (45b) is thus defined entirelyby the quadratic part of the dot Liouvillian, Eq. (74a),and is thus effectively noninteracting , with a renormal-ized single-particle energy level: ǫ → ǫ + U/
2. This gen-eral rule leads to very useful simplifications in perturba-tive and nonperturbative studies of the interactingAnderson model that will not be explored further here.Another point revealed by the second quantization of L , Eq. (74), is that the the essential two-particle operatorto which the interaction couples in the Hamiltonian H ,Eq. (13), is the fermion-parity operator ( − n = 2 Z R : L | quartic = (cid:2) U ( − n , • (cid:3) − = U X ν,η,σ | α νησ )( α ¯ νησ | . (77)The term (77) captures the essential many-particle effectof the interaction U since in the quadratic term Eq. (74a)the effect of U can be compensated by tuning the levelposition to the particle-hole symmetry point ǫ = − U/ U ( − n contained in H . This also shows that it acts only in the Liouville-Fockspace spanned by fermionic operators Eq. (61), againleading to Eq. (75), since the fermion-parity operator byconstruction (anti)commutes with all bosonic (fermionic)operators by Eq. (60) [Eq. (61)]. The second rewriting interms the fermionic superbras and superkets [Eq. (61)]explicitly confirms this.Finally, we emphasize that a particularly, useful aspectof the above reasoning, based directly on the Liouville-Fock representation Eq. (74), is that it also allows oneto infer general physical properties of a superoperatordescribing an open fermionic system, even when it doesnot have the commutator form which L has.
4. Interaction picture of causal superfermions
Using the second quantization, we can also easilywork out the explicit form of interaction Liouvillian L V [Eq. (48)] in the interaction picture, L V ( t ) = P q e i ( L + L R )( t − t ) G q J q ¯1 e − i ( L + L R )( t − t ) , which is requiredin the next section. For a noninteracting quantum dot( U = 0) we have the following simplifying property: for1 = η, σ h L, G q i − = ηǫ σ G q , for U = 0 (78) ǫ σ = ǫ + Bσ/ , (79)which follows from the quadratic part (74a) of L using thesuperfermion commutation relations (50). Note that theright-hand side is independent of q , i.e., the creation andannihilation superoperators have the same frequency ηǫ σ ,in contrast to Hilbert-Fock space fields. The interaction- picture field superoperators G q ( t ) := e iL ( t − t ) G q e − iL ( t − t ) (80a)= e iηǫ σ ( t − t ) G q for U = 0 (80b)in the noninteracting case ( U = 0) are then simply pro-portional to those in the Schr¨odinger picture, G q , sincewe can commute e − iL ( t − t ) through G q using Eq. (78),resulting only in a phase factor. This is the crucial sim-plification, which allows the exact solution of the non-interacting problem to be obtained quite simply oncewe have taken the wideband limit, as discussed in thenext section. The field superoperators of the noninter-acting reservoirs, have the same simple time dependenceas those of the dot for U = 0. Since by our definitionsin Eq. (46) and (47) J ¯ q and G q with opposite q indexplay the same role, one can write analogous to Eq. (74),accounting for a factor due to the rescaling (24), L R = [ H R , • ] − = 2 π Γ η ( ω + µ r ) ˜ J ¯ J ¯1 , (81)(with the usual implicit integration over the index ω and summation over indices η, σ, r of the multi-index1 = η, σ, ω, r ) from which it follows that[ L R , J q ] − = η ( ω + µ r ) J q . (82)As a result, for ¯1 = − η, σ, ω, rJ q ¯1 ( t ) := e iL R ( t − t ) J q ¯1 e − iL R ( t − t ) (83a)= e − iη ( ω + µ r )( t − t ) J q ¯1 . (83b)Therefore, implicitly summing (integrating) over discrete(continuous) indices, L V ( t ) = e − iη ( ω + µ r )( t − t ) J q ¯1 G q ( t ) (84a)= e − iη ( ω + µ r − ηǫ σ )( t − t ) J q ¯1 G q , for U = 0 . (84b) III. TIME-EVOLUTION AND CAUSALSUPERFERMIONS
We now first set up the time-dependent perturbationtheory for the general, interacting case ( U = 0), explic-itly incorporating the wideband limit from the start onthe level of superoperators . This leads to a renormalizedversion of the perturbation series for which the crucialresult Eq. (80b) can be directly exploited to solve thenoninteracting problem ( U = 0) exactly by a next-to-leading-order perturbative calculation. A. Real-time perturbation expansion for thereduced propagator
We are now in a position to exploit the causal super-operator second quantization technique to the expansion5
FIG. 2. Real-time perturbation expansion in the widebandlimit: Diagrams contributing to the full time-evolution prop-agator Π( t, t ) in the first three loop orders of perturbationtheory as given by Eq. (85), Π = e − iL ( t − t ) , Π and Π .Within each column, the two-loop diagrams have the samecontraction configuration but differ by the contraction func-tions involved (¯ γ or ˜ γ ). Diagrams and expressions for the cor-responding terms for the self-energy Σ( t, t ) = Σ + Σ + . . . are obtained by (i) retaining only the diagrams in the secondand third column and (ii) discarding from these diagramsthe free time-evolution parts before the first and after thelast vertex. The full evolution Π( t, t ) is then generated byΣ( t , t ) through the Dyson equation (38). As indicated, theretarded contractions are “Markovian”, i.e., in the widebandlimit they act instantaneously and do not allow for internaltime-integrations ( δ -function constraint), cf. Eq. (94). Weindicate the number of two-loop diagrams that give zero dueto this constraint, but do not draw them. These are the onlycontractions that survive in the T → ∞ limit and are consid-ered further in Fig. 3a. for the propagator Π = P ∞ m =0 Π m defined by Eq. (45a).For each term in the expansion of order m in L V , de-noted by Π m , one can collect all reservoir superoperatorsby commuting them to the left through the G q ’s:Π m = ( − i ) m e − iL ( t − t ) Tr R (cid:2) L V ( t m ) ...L V ( t ) ρ tot ( t ) (cid:3) =( − i ) m h J q m ¯ m ( t m ) ...J q ¯1 ( t ) i R e − iL ( t − t ) G q m m ( t m ) ...G q ( t ) , (85)where we implicitly perform a time-ordered integrationsuch that t ≥ t m ≥ ... ≥ t ≥ t , as well as a summationover all dummy indices m, ..,
1. Here, h S i R denotes thereservoir average of a super operator S : h S i R = Tr R ( Sρ R ) . (86)To eliminate the reservoirs, we need the multi-particle correlation functions of the reservoirs.Their time-dependence amounts to a simplephase factor by Eq. (83b), h J q m ¯ m ( t m ) ...J q ¯1 ( t ) i R = Q mi =1 e − iη i ( ω i + µ ri )( t i − t ) h J q m ¯ m ...J q ¯1 i R , and the remaining equal time correlation functions follow from the Wicktheorem for the J q superoperators: for even m h J q m m ...J q i R = X contr ( − P Y h i,j i h J q i i J q j j i R . (87)Here, ( − P denotes the usual fermionic sign of the per-mutation P that disentangles the contractions over whichwe sum, h i, j i denoting the product over contracted pairs.For odd m the average vanished by the fermion-paritysuperselection rule. The Wick theorem (87) was shownin Ref. 30 to follow from the standard derivation ofGaudin when using the superoperator expression forthe equilibrium fluctuation-dissipation theorem for thereservoirs: ¯ J | ρ R ) = tanh( η ω / T ) ˜ J | ρ R ) . (88)The field superoperators (47) are called “causal” sincethey make the constraints imposed by causality ex-plicit: there are only two possible types of contractionfunctions h J q J q i R in the expansion Eq. (87) that arenonzero. These are the retarded function˜ γ , ( η ω ) := h ¯ J ˜ J i R = Γ π δ , ¯1 , (89)and the Keldysh function¯ γ , ( η ω ) := h ¯ J ¯ J i R = Γ π tanh( η ω / T ) δ , ¯1 , (90)while all other possible pair contractions are equal tozero. These properties of the contractions give a cor-responding causal structure to the perturbation theorywhich is revealed only when using the causal field super-operators (46), as we will see in the following. We havethus explicitly integrated out the reservoir degrees of free-dom, and obtained the real-time perturbation theory for the reduced-time evolution superoperator:Π m = ( − i ) m X contr ( − P Y h i,j i γ q i i,j ( t i − t j ) × (91) e − iL ( t − t ) G q m m e − iL ( t m − t m − ) G q m − m − . . . G q e − iL ( t − t ) . An individual term consists of a sequence of free dot evo-lutions, generated by L [Eq. (74)], interrupted by thepair-wise action of quantum dot field superoperators G q [Eq. (80a)], which is weighted by the time-dependentreservoir correlation function (Fourier transform) γ q i i ′ ,j ′ ( t i − t j ) := Z dω i e − iη i ( ω i + µ i )( t i − t j ) γ q i i,j ( η i ω i ) . (92)On the right hand side, we also make use of both the q -index as well as the “bar-tilde” notation, as in Eq. (49): γ q j i,j := h J + i J q j j i =˜ γ i,j δ q j , − + ¯ γ i,j δ q j , + . (93)We note that the initial time t cancels out in the reser-voir dynamical phase factor since γ q j i,j ∝ δ ∝ δ ( ω i − ω j ) δ ¯ η i ,η j [by Eqs. (89)-(90)]. The primed multi-indices i ′ , j ′ on the left hand side of Eq. (92) indicate that we leaveout the reservoir frequencies ω i and ω j from the multi-indices i , j , respectively. At this stage these frequencieshave been integrated out of the theory, and from hereonwe omit the primes, i.e., the multi-indices (1, ¯1, etc. inEq. (91)) do not contain ω i anymore, unless stated oth-erwise.In Fig. 2, we represent individual terms in Eq. (91)diagrammatically and the total evolution is the sumof such terms over all possible Wick pairings of an evennumber of discrete indices m, . . . ,
1, integrated over or-dered times, i.e., t ≥ t m . . . ≥ t ≥ t . We will referto an m -th order diagram contributing to Π m with m/ γ ) as a m/ causal structure : thedestruction superoperator G − = ˜ G can never appear onthe left of the field superoperator (either G ± = ¯ G or ˜ G )with which it is contracted. One implication of this struc-ture is that ¯ G always stands on the far left, at the latesttime t m . This ensures by Eq. (54) that term-by-term Tr Π( t, t ) = Tr and therefore probability is conserved, Tr ρ ( t ) = Tr ρ ( t ) = 1, since Tr Π = Tr and Tr Π m = 0for m ≥
1. We now turn to further implications of thiscausal structure in the wideband limit.
B. Wide-band limit
The perturbation theory Eq. (91) applies generallywithout further assumptions to the interacting Andersonmodel ( U = 0). However, even when considering the non-interacting limit ( U = 0) in combination with the wide-band limit (WBL) for the stationary state ( t → ∞ ) it isnot directly obvious how to explicitly evaluate Eq. (91).To obtain the exact solution in that case one needs toidentify which contributions vanish in each loop order ofthe time-evolution superoperator, and then sum up theremaining ones from all orders. We now show howin the wideband limit the time-dependent perturbationseries (91) for the interacting case ( U = 0) can be trans-formed with the help of our causal superoperators. Inthis new formulation, the solution of the noninteractinglimit ( U = 0) also becomes obvious, even allowing for thedirect calculation of the full time-evolution Π( t, t ).
1. Retarded reservoir correlations, - elimination ofannihilation superfields
The key simplification in the wideband limit, in whichthe rates Γ rσ are constant, is that the retarded con-traction function becomes energy independent, corre-sponding to a δ -function in time [cf. Eq. (41)] (see also Refs. 77, 79, and 115):˜ γ , ( t , t ) = Γ π δ , ¯2 Z dω e − iη ( ω + µ r )( t − t ) = Γ δ ( t − t ) δ , ¯1 = Γ δ ( t − t ) δ , ¯1 , (94)where Γ = Γ rσ does not depend on the frequency ω or time t . By working with causal field superoperatorswe thus automatically collect a Markovian part of thedynamics: the “Markovian contraction” ˜ γ appears onlywhen a destruction superoperator ˜ G is contracted witha ¯ G (necessarily so by the causal structure). This allowsone to easily eliminate the ˜ G from the perturbation series(85), thereby isolating the remaining, nontrivial part ofthe time-evolution. To do this, we note that “processes”described by ˜ γ occur instantly in time. Therefore, allΠ m -diagrams vanish in which one or more vertices appearbetween any two vertices connected by a ˜ γ -contraction:there is no phase space left for the integration of the timevariable of such vertices due to the δ -function constraint(94). This means that in the surviving diagrams the ˜ γ contractions form a ladder series, see Fig. 3a, which canbe summed up. The skeleton diagram for this resumma-tion is shown in Fig. 3b and consists of a single term:with Eq. (94)˜Σ( t − t ) = − i X r ¯ G Γ δ ( t − t ) e − iL ( t − t ) ˜ G ¯1 (95a)= 2 ˜Σ δ ( t − t ) = ˜Σ ¯ δ ( t − t ) , (95b)with the time-independent superoperator˜Σ = − i X
12 Γ ¯ G ˜ G ¯1 = − i X σ
12 Γ σ X η ¯ G ησ ˜ G ¯ ησ . (96)Note that in the sum over 1 = η, σ, r , Γ = Γ rσ does notdepend on η and we again introduced the function ¯ δ ofEq. (41). The sum of the spin-resolved tunnel rates overthe reservoirs is denoted byΓ σ = X r Γ rσ . (97)The superoperator (96) is just the (constant) Laplacetransform of ˜Σ( t − t ) and is skew-adjoint, ˜Σ † = − ˜Σ,since ˜ G ¯1 = ¯ G † . By resumming diagrams as illustratedin Fig. 3b, we can now simply leave out all terms withretarded contractions ˜ γ from the series and we can incor-porate their effect into a simple renormalization of thebare dot Liouvillian by the skeleton term (95): L → ¯ L = L + ˜Σ . (98)In this way, we have eliminated the annihilation fieldsuperoperators ˜ G of the quantum dot which enter onlythrough the retarded reservoir correlation function ˜ γ [cf.Eq. (89)]. This elimination was first pointed out in themore general framework of the real-time renormalization-group as formulated in Ref. 51 (where it is referred to a7 FIG. 3. Wide-band limit: (a) Free dot evolutions (the blackhorizontal lines) interrupted by a sequence of k retarded con-tractions ˜ γ (the blue curved lines) are resummed to definea renormalized Liouvillian ¯ L := L + ˜Σ [Eq. (98)]. The re-tarded “Markovian” contractions give rise to instantaneous ˜Σblocks [Eq. (95)], cf. Fig. 2. The sum defines a renormalized unperturbed evolution ¯Π (the red horizontal line) which is dissipative , see Eq. (116), and provides a starting point for anew perturbation theory. (b) Next, diagrams with a fixed con-figuration of Keldysh contractions ¯ γ (the black curved lines)are summed over all possible insertions of retarded contrac-tions, here illustrated for two Keldysh loops. What remains isa renormalized perturbation theory in which only ¯ L and cre-ation superoperators ¯ G appear explicitly with Keldysh con-tractions ¯ γ , see Fig. 4. discrete RG step), which is not limited to the widebandlimit and which explicitly constructs the corrections dueto the frequency dependence (e.g., vertex renormaliza-tion). However, the above simpler derivation may beof broad practical interest since in most studies the wide-band limit is assumed from the start anyway. Also, theuse of δ -restrictions on time integrations reveals a mathe-matical analogy to the theory of disordered metals wherespatial δ -correlations of the disorder suppress crossingimpurity contractions.
2. Renormalized perturbation theory for finite temperature
Having eliminated the destruction superoperators ˜ G and the retarded reservoir correlation functions ˜ γ by thereplacement Eq. (98), we obtain a new time-ordered ex-pansion for the propagator, Π( t, t ) = P ∞ m =0 ¯Π m ( t, t ), for which the m th-order term is analogous to Eq. (85)¯Π m =( − i ) m h ¯ J ¯ m ( t m ) ... ¯ J ¯1 ( t ) i R e − i ¯ L ( t − t ) ¯ G ′ m ( t m ) ... ¯ G ′ ( t )=( − i ) m X contr ( − P Y h i,j i ¯ γ i,j ( t i − t j ) (99a) × e − i ¯ L ( t − t ) ¯ G ′ m ( t m ) ... ¯ G ′ ( t )=( − i ) m X contr ( − P Y h i,j i ¯ γ i,j ( t i − t j ) (99b) × e − i ¯ L ( t − t m ) ¯ G m e − i ¯ L ( t m − t m − ) . . . ¯ G e − i ¯ L ( t − t ) . with the same conventions as in Eq. (91), but with a renormalized interaction picture for the causal creationsuperoperators¯ G ′ j ( t ) = e i ¯ L ( t − t ) ¯ G j e − i ¯ L ( t − t ) (100)whose difference from Eq. (80a) is indicated by the prime.The renormalized perturbation theory for Π( t, t ) isexpressed entirely in terms of the wideband limit form ofKeldysh contraction function (90) (see App. B),¯ γ , ( t − t )= δ , ¯1 Γ π Z dω e − iη ( ω + µ r )( t − t ) tanh( η ω / T )= δ , ¯1 − i Γ T sinh ( πT ( t − t )) e − iη µ r ( t − t ) , (101)the creation field superoperators ¯ G , and the renormal-ized Liouvillian ¯ L generating the renormalized free evo-lution ¯Π ( t, t ) = e − i ¯ L ( t − t ) . The diagrammatic expan-sion, shown in Fig. 4, is much simpler than the originalone in Fig. 2. Since all appearing creation superoperators¯ G anticommute, the key difficulty in the superoperatorstructure of Eq. (99b) lies in the failure of the ¯ G to com-mute with the renormalized Liouvillian, more precisely,[ ¯ G, L ] − ¯ G due to the quartic interaction term (74b) in L for U = 0.The expansion (99b) captures the time-dependent,finite-temperature effects. For T → ∞ the renormal-ized perturbation theory is exact already in zeroth order:in this limit, all higher order m ≥ T → ∞ [even without taking the wideband limit,cf. Eq. (90)]. Thus, ¯ L generates the exact, dissipative,Markovian effective Liouvillian [cf. Eq. (40)] in the infi-nite temperature limit:lim T →∞ L ( t, t ′ ) = ¯ L ¯ δ ( t − t ′ ) , (102)lim T →∞ Π( t, t ) = e − i ¯ L ( t − t ) , (103)with ¯ L = L + ˜Σ. Since this renormalized time-evolutionserves as a reference for the renormalized perturbationtheory (99b), it will be considered in more detail in thenext section.8 FIG. 4.
Renormalized real-time perturbation expansion inthe wideband limit: contributing to the full time-evolutionpropagator Π( t, t ) in the first few loop orders of renormal-ized expansion (99b), ¯Π = e − i ¯ L ( t − t ) , ¯Π and ¯Π . Diagramsand expressions for the renormalized self-energy ¯Σ( t, t ) =¯Σ + ¯Σ + . . . are obtained by the same steps indicated inFig. 2. The full evolution Π( t, t ) is now obtained by ¯Σ( t, t )from the alternative Dyson equation (104) in which the unper-turbed evolution is generated by the renormalized Liouvillian¯ L = L + ¯Σ, see Fig. 3a. In contrast to the series in Fig. 2, therenormalized series for both Π( t, t ) and ¯Σ( t, t ), and there-fore also for Σ( t, t ) = ˜Σ( t, t ) + ¯Σ( t, t ), terminates at looporder two for the noninteracting Anderson model ( U = 0) dueto the super-Pauli principle (62)-(63), see Eq. (122)-(123). Although the causal superfermion approach is crucialin setting up the renormalized series (99b), one can cal-culate ˜Σ, and therefore ¯ L , using any equivalent densityoperator technique (Nakajima-Zwanzig, etc.)simply by taking the leading order in the tunnel couplingin the wideband limit and then letting T → ∞ . C. Infinite temperature limit and fermion-parity
Before we continue our analysis of the renormalizedperturbation theory (99b) for the noninteracting limit( U = 0) in Sec. III D, we point out an interesting im-mediate consequence of the above general structure of(99b) which applies to the interacting Anderson model( U = 0). In fact, it applies to a broad class of quantum-dot models, i.e., for other model Hamiltonians insteadof H , Eq. (13), coupled to the reservoirs by a bilinear,particle-conserving H T .We have taken the T → ∞ limit to define the start-ing point for both the construction of Liouville-Fockspace [namely, the vacuum superket | Z L ) = ] andfor the renormalization of the perturbation theory (99b)[ ˜Σ¯ δ ( t − t ′ ) = lim T →∞ Σ( t, t ′ )]. The result for ˜Σ holdsnonperturbatively in all parameters in the limit T → ∞ even though it results from the leading term in the per-turbation theory in Γ rσ . It is all the more surprising thatit has observable implications in a finite -temperature ex-periment, for arbitrary values of Γ rσ , the interaction U , applied voltages and magnetic field (only restrictedby the wideband limit). This result was first noted inthe perturbative study Ref. 31 and subsequently relatedto the T → ∞ limit and the fermion parity, generaliz- ing it nonperturbatively in Γ and arbitrary Anderson-type models. We now analyze this fermion-parity pro-tected decay mode within the time-dependent perturba-tion theory in order to directly compare with the anal-ysis in Sec. IV, which avoids the Laplace space analysisof Ref. 30 altogether. Moreover, we now also include thespin-dependent tunneling which Ref. 31 also considered.For this discussion and that following in Sec. III C 2, itis useful to elaborate more on the self-energy, althoughmost parts of this work emphasize the possibility of cal-culating the time-evolution propagator Π( t, t ) directlyfrom Eq. (99b) by using field superoperators. The self-energy also facilitates comparison with results from real-time RG and other density operator approaches.
1. Renormalized self-energy
The self-energy superoperator is defined either by thekinetic equation (39) for ρ ( t ) or the equivalent Dysonequation for Π( t, t ), Eq. (38). Diagrammatically it is de-fined by collecting those parts of diagrams of Π( t, t ) thatare irreducibly contracted (i.e., diagrams pieces obtainedby only cutting through free time-evolutions, withoutcutting contractions) and summing these. The pertur-bation theory for the self-energy superoperator is thensimply obtained from the perturbation theory for Π( t, t )by (i) restricting the sum to irreducible contractions and(ii) omitting the initial ( t → t ) and final ( t m → t ) freetime-evolutions. The perturbation theory can then be re-summed in terms of these self-energy diagram blocks. Ifthis is done for the original perturbation theory Eq. (85),taking e − iL ( t − t ) as the free time-evolution, we obtainthe Dyson Eq. (38) with self-energy Σ( t, t ) (equal to theNakajima-Zwanzig kernel). However, the renormalizedperturbation theory Eq. (99b) takes e − i ¯ L ( t − t ) as a ref-erence. This series can be resummed as well in termsof different self-energy diagram blocks, now denoted by¯Σ( t, t ′ ). This gives an equivalent Dyson equation for thesame superoperator Π( t, t ),Π( t, t ) = e − i ¯ L ( t − t ) − i Z dt dt t ≥ t ≥ t ≥ t e − i ¯ L ( t − t ) ¯Σ( t , t )Π( t , t ) , (104)The renormalized self-energy superoperator ¯Σ( t, t ) is obtained from Eq. (99b) by keeping irreducible Keldyshcontractions. This corresponds to a decomposition of theeffective dot Liouvillian, L ( t, t ′ ) = ¯ L ¯ δ ( t − t ′ ) + ¯Σ( t, t ′ ) , (105)alternative to Eq. (40). Since the kinetic equation (39)only depends on the sum of the reference Liouvillian L ¯ δ ( t − t ′ ) [ ¯ L ¯ δ ( t − t ′ )] and the self-energy Σ( t, t ′ ) [ ¯Σ( t, t ′ )]appearing in the Dyson equation Eq. (38) [Eq. (104)], thisresults in the same time-evolution Π( t, t ) in the wide-band limit.9
2. Fermion-parity protected decay mode
We now discuss how the infinite temperature self-energy ˜Σ affects the finite-temperature time-evolution ofthe density operator ρ ( t ). The key observation is that bythe causal structure [cf. Sec. III A] of Eq. (99b), also therenormalized self-energy ¯Σ always has a creation superop-erator ¯ G standing on the far right , i.e., at the time t ofthe initial tunnel “process”. An immediate consequenceof our Liouville-Fock space construction using causal su-perfermions is that the maximally filled superket, i.e.,the fermion-parity operator | Z R ) = ( − n , is an exactright zero eigenvector of the nontrivial self-energy since¯ G m | Z R ) = 0 [Eq. (55)],¯Σ( t, t ) | Z R ) = 0 for any t, t . (106)This a consequence of the super-Pauli principle in Liou-ville space, Eq. (62). The time-evolution of the excitationmode | Z R ) is then completely determined by ˜Σ which weobtained exactly in the wideband limit. We emphasizethat it is determined completely by the leading order termin Γ in the limit T → ∞ . The action of ˜Σ on this modefollows directly from the superfermion anticommutationrelation and ¯ G | Z R ) = 0 [Eq. (55)],˜Σ( t, t ′ ) | Z R ) = − i ¯ δ ( t − t ′ ) X Γ ¯ G ˜ G ¯1 | Z R ) (107)= − i ¯ δ ( t − t ′ ) X Γ (1 − ˜ G ¯1 ¯ G ) | Z R )= − i ¯ δ ( t − t ′ )Γ | Z R ) for all t ≥ t ′ ≥ t . The fermion-parity eigenvalue is simply the sum of alltunnel rates over both reservoirs and spinsΓ = X σ Γ σr = X σ Γ σ (108)times − i . The renormalized time-dependent perturba-tion theory directly shows that for the fermion-paritymode | Z R ) the T → ∞ evolution remains exact at allfinite temperatures :Π( t, t ) | Z R ) = e − Γ( t − t ) | Z R ) for all t ≥ t . (109)All higher order corrections given by Eq. (99b), re-sponsible for dependence on U , ǫ , B , µ r , and T , van-ish: ¯Π m ( t, t ) | Z R ) = 0 for m ≥
1. This follows since | Z R ) is a super eigenvector of ¯ L and ¯ G m | Z R ) = 0by the super-Pauli principle. The former follows from L | Z R ) ∝ [ H, ( − n ] − = 0 by the superselection rule (3)and Eq. (107). The exact result (109) can be also formu-lated for the original self-energy: Σ has | Z R ) as an exacteigenmode with eigenvalue − i Γ,Σ( t, t ′ ) | Z R ) = − i Γ | Z R ) for all t ≥ t ′ ≥ t . (110)For multi-orbital models this generalizes to Γ = P rσl Γ rσl and | Z R ) = Q σl e iπn σl /N for N spin-orbitals,where l is the orbital quantum number. It should be noted that Eqs. (109)-(110) hold nonper-turbatively both in the tunneling Γ as well as in Coulombinteraction U and down to zero temperature T = 0: onlythe wideband limit is used here. Due to the fundamentalfermion-parity superselection principle the eigenvalue isthus prevented from picking up any dependence on ener-gies other than Γ = P rσ Γ rσ [Eq. (108)] and the decayremains strictly exponential. Since only the sum of spin-dependent rates enters, the spin-polarization of the tun-neling also has no influence. Because we explicitly usedthis fermion-parity superselection principle in the con-struction of the causal field superoperators [cf. Eq. (55)and following discussion], the property (109) becomes di-rectly clear on the superoperator level once the widebandlimit has been taken [Eq. (99b)].We note that ¯ G has only one nontrivial zero righteigenvector, | Z R ), for all values of the multi-index 1: inanalogy to usual the second quantization, only the max-imally filled state is a common zero eigenvector of all creation superoperators. Therefore the above argumentapplies only to the special fermion-parity superket | Z R ).The exact result (109) implies for the time-evolutionof ρ ( t ) = Π( t, t ) ρ ( t ), starting from an initial state ρ ( t )with the general form Eq. (68) at t = t , that ρ ( t ) = h Ξ( t ) e − Γ( t − t ) + . . . i | Z R ) + . . . , (111)see the introductory discussion of Eq. (1) and Fig. 1. Thedecay of the initial two-particle correlation Ξ( t ) on thequantum dot thus happens on a time scale t − t . Γ − which is independent of all further energy scales men-tioned above. As pointed out in Ref. 31, the appearanceof the sum of all rates (108) in the decay rate of the thetwo-particle correlation seems to have a simple origin inthe noninteracting limit ( U = 0): using a Markovianapproximation, h n ↑ n ↓ i = h n ↑ ih n ↓ i ∝ Q σ e − Γ σ ( t − t ) = e − Γ( t − t ) . However, as emphasized there, it is all themore surprising that the decay maintains this exponen-tial form and the value of the decay rate in the interactinglimit, even when attaching a ferromagnet or supercon-ductor or when the dot is initially in a correlated state,where this factorization breaks down. Also, note that theMarkovian approximation remains exactly valid for thisdecay mode.The real-time renormalization group approach, wasfound to be consistent with the eigenvalue equation (107),even without any truncations of the exact hierarchy ofRG equations or any approximations other than the wide-band limit (as it should). In the continuous RG-flow ofthe effective Liouvillian towards low energies, the coeffi-cient of | Z R )( Z R | is given by the eigenvalue − i Γ and thiscoefficient does not flow.Finally, we recall that the superket | Z R ) in Eq. (109)on its own does not represent a physical state, only inlinear combination with the vacuum superket | Z L ) of theform Eq. (68) it does. Measurement setups that can tar-get specifically the fermion-parity protected decay modecontained in this mixed state were analyzed in Refs. 310and 121. In the next section, we will calculate the fulltime dependence of such physical states for T = ∞ , andfinite interaction U and in Sec. IV for finite T but U = 0.
3. Infinite temperature limit and Markovian relaxation
As mentioned in Sec. III B 2, the decay of all modesis Markovian and exponential in the infinite-temperaturelimit, which surprisingly continues to hold for the specialfermion-parity mode | Z R ) at any finite T as we have justseen. To calculate the T → ∞ time evolution, ¯Π( t, t ) = e − i ¯ L ( t − t ) [Eq. (103)], which serves as a reference for therenormalized perturbation theory (99b), we need (someof) the super eigenvectors of ¯ L = L + ˜Σ. These followeasily from the second quantized forms (96) and (74) ofthe superoperators, as we now show.First, the self-energy ˜Σ = − i P ησ Γ σ N ησ [Eq. (96)]simply counts the mode occupation through the super-operator N ησ [Eq. (64)], and multiplies it with the halfof the spin-resolved decay rate (97), Γ σ = P r Γ rσ . Thebosonic basis superkets Eqs. (60) are thus super eigenvec-tors of ˜Σ, and the latter can be expressed in projectorsonto the bosonic part of the basis (60)˜Σ = − i X σ Γ σ | χ σ )( χ σ | (112a) − i Γ | Z R )( Z R | (112b) − i Γ h X σ | S σ )( S σ | + X η | T η )( T η | i + F, (112c)where again Γ = P σ Γ σ [Eq. (108)] and F denotes theirrelevant fermionic part [cf. Sec. II D 2]. Since ˜Σ † = − ˜Σ[cf. Eq. (95)], the eigenvalues are necessarily imaginary orzero [bosonic eigenvector | Z L )], and the left super eigen-vectors are the super adjoints of the right ones with thesame eigenvalues.Similarly, we can rewrite the Liouvillian: L = X η,σ ( η [ ǫ + U/ σB/ N ησ + L | quartic (113a)= X σ σB | S σ )( S σ | + X η η (2 ǫ + U ) | T η )( T η | + F, (113b)where F again denotes the irrelevant fermionic part. In writing Eq. (113b) we used that the quartic part of L [Eq. (74b)] acts only on the fermionic part of theLiouville-Fock space [cf. Eq. (77)], whereas the quadraticpart has a component in both the bosonic and thefermionic part.For the particular case of the Anderson model, thebosonic blocks of L and ˜Σ commute, as Eqs. (113b) and(112) explicitly show. Therefore, when working in thebasis naturally provided by the causal superfermions, weobtain the diagonal form for (bosonic part of) ¯ L = L + ˜Σ when simply adding Eqs. (112) and (113b), which wequote here for future reference:¯ L = − i X σ Γ σ | χ σ )( χ σ | (114a) − i Γ | Z R )( Z R | (114b)+ X σ (cid:0) σB − i Γ (cid:1) | S σ )( S σ | (114c)+ X η (cid:0) η (2 ǫ + U ) − i Γ (cid:1) | T η )( T η | + F, (114d)Notably, this implies that for the time-evolution in thelimit T → ∞ – fully determined by the bosonic partof ¯ L = L + ˜Σ – the nontrivial quartic part (77) has noinfluence: the interaction parameter U only enters viathe quadratic part of the interaction Liouvillian. From the diagonal form (114) we immediately obtainthe T → ∞ time evolution of the reduced density oper-ator expressed in terms of the coefficients of the initialdot density operator ρ ( t ) [cf. Eq. (68)]:lim T →∞ ρ ( t ) = e − i ¯ L ( t − t ) ρ ( t ) (115)= 12 | Z L ) + X σ e − Γ σ ( t − t ) Φ σ ( t ) | χ σ )+ e − Γ( t − t ) Ξ( t ) | Z R )+ X η e − h iη (2 ǫ + U )+ 12 Γ i ( t − t ) Υ η ( t ) | T η )+ X σ e − h iσB + 12 Γ i ( t − t ) Ω σ ( t ) | S σ ) . (116)The result (116) explicitly illustrates that renormalizedfree evolution dissipative, involving exponential decaywith rates Γ σ , Γ / ↑ - ↓ excitations (frequency B ) or coherent electron-pair 0-2excitations (frequency 2 ǫ + U ) in the initial state ρ ( t ).At infinite temperature, we obtain from Eq. (116) inthe stationary limit the maximum von Neumann entropystate (73), lim T →∞ ρ ( ∞ ) = ρ ∞ = = | Z L ), the vac-uum superket (60a) in our Liouville-Fock space construc-tion. Indeed, | Z L ) is the right zero eigenvector of ¯ L since | Z L )( Z L | is missing in Eq. (114), a point that will beimportant later on [cf. discussion of Eq. (156)]. Gen-erally, the left zero super eigenvector ( Z L | guaranteeingprobability conservation, ( Z L | ¯ L = 0, only implies the existence of at least one right zero supervector, the sta-tionary state | ρ st ) (assuming it is unique). However, ingeneral this stationary state is not equal to the (properlynormalized) super adjoint of ( Z L | . The above discussionnow shows that in general the deviation of | ρ st ) from | Z L ) is generated by the finite temperature correctionsdescribed by ¯Π m for m ≥ D. Noninteracting limit
Nearly all considerations of the perturbation seriesEq. (85) and (99b) so far apply to the interacting An-derson model ( U = 0) and, where mentioned, its gener-alizations. We now identify the simplifications that thenoninteracting limit U = 0 brings.In Sec. II D 4, we found that the interaction pic-ture field superoperators G q ( t ) = e iL ( t − t ) G q e − iL ( t − t ) [Eq. (80b)] simplify for U = 0 since in this case L isquadratic in these fields. However, even in this sim-ple limit, the original perturbation theory (85) still con-tains an infinite series of terms. Although this seriescan be resummed, this can be avoided if one insteadstarts from the physically motivated renormalized per-turbation series (99) as we now show. For U = 0 therenormalized interaction picture of the creation super-fields ¯ G ′ ( t ) = e i ¯ L ( t − t ) ¯ G e − i ¯ L ( t − t ) [Eq. (100)] with re-spect to the quadratic renormalized Liouvillian¯ L = X η,σ (cid:0) ηǫ σ − i Γ σ (cid:1) ¯ G ησ ˜ G ¯ ησ , (117)simplifies, since now h ¯ L, ¯ G i − = (cid:0) ηǫ σ − i Γ σ (cid:1) ¯ G , (118)which follows from the anticommutation relations (50).Analogous to Eq. (80b) we obtain:¯ G ′ ( t ) = e (cid:0) iηǫ σ + 12 Γ σ (cid:1) ( t − t ) ¯ G , (119)and therefore these field superoperators also anticom-mute: [ ¯ G ′ ( t ) , ¯ G ′ ( t )] + = 0 . (120)Inserting Eq. (119) into (99b) we obtain the key result¯Π m = ( − i ) m X contr ( − P × Y h i,j i ¯ γ i,j ( t i − t j ) e iη i ǫ σi ( t i − t j ) (121a) × m Y k =1 e
12 Γ σk ( t k − t ) (121b) × e − i ¯ L ( t − t ) ¯ G m ... ¯ G . (121c)Note that the dissipative factors (121b) depend explicitlyon t , in contrast to the coherent phase factors (121a),in which t cancels out [cf. Eq. (92)]. Their exponentialincrease leads to no problems as it is always dominated bythe exponentially decaying term e − i ¯ L ( t − t ) in Eq. (121c).The result (121) shows that all terms of order m > identically zero due to their superoperator structure .Like Eq. (106), this is another manifestation of the super-Pauli principle (62)-(63). The exact result for the case U = 0 is thus obtained in just the first two nonvanish-ing loop orders of the renormalized perturbation theory,represented by precisely those diagrams shown in Fig. 4:Π( t, t ) = ¯Π ( t, t ) + ¯Π ( t, t ) + ¯Π ( t, t ) , (122)¯Σ( t, t ) = ¯Σ ( t, t ) + ¯Σ ( t, t ) , (123)and the exact total self-energy follows from Σ( t, t ) =˜Σ( t, t ) + ¯Σ( t, t ). The super-Pauli principle also showsthat the renormalized Dyson equation can be solved ex-actly, relating Eq. (122) to Eq. (123): ¯Π ( t, t ) = ¯Π ∗ ¯Σ ∗ ¯Π ( t, t ) , (124)¯Π ( t, t ) = ¯Π ∗ (cid:2) ¯Σ + ¯Σ ∗ ¯Π ∗ ¯Σ (cid:3) ∗ ¯Π ( t, t ) , (125)where by A ∗ B ( t, t ) = R tt dt A ( t, t ) B ( t , t ) we denotethe time convolution. This illustrates the computationaladvantage of working with field superoperators directlyin Liouville space, in particular when incorporating thecausal structure into these fields. Equations (122)-(123)show that it becomes practical to avoid the calculationof the self-energy in the limit U = 0, since the structureof the time propagator Π is no less complicated. In fact,as we will see explicitly in the next section [Eq. (138),Fig. 5 and Eq. (164b)], the inclusion of the second re-ducible term in ¯Π on the right-hand side in Eq. (125)makes it a simpler object than ¯Σ in the U = 0 limit.We emphasize that the simple structure of the noninter-acting problem appears only in the renormalized series,i.e., after explicitly exploiting the wideband limit. Simi-lar, but less drastic, simplifications on the superoperatorlevel can be exploited for interacting problems as well. We emphasize that the truncation in Eqs. (122)-(123)does not rely on the spin- and charge-rotation symme-try of the Anderson model: only the number of spin or-bitals of the model (=2) and the absence of quartic termsand higher in L matter ( U = 0). The crucial observationfor this was that in Eq. (121) we were able to commute ¯ L through all the ¯ G fields to the left side, thereby only gen-erating c -number factors, without changing the structureof the appearing field superoperators. For the interact-ing Anderson model, however, the expansion (91) mustbe used: here ¯ L cannot be commuted through the fields ¯ G without generating additional, higher order terms. Theneven the renormalized perturbation theory has nonzeroterms beyond the two-loop order. (Note however, thatthe self-energy ˜Σ remains quadratic even for finite U dueto the wideband limit.) Yet even in this case the causalsuperfermions bring simplifications, cf. Eq. (76). The result Eqs. (122)-(123) thus explicitly shows thatthe higher order terms in the renormalized perturba-tion theory are generated by the nontrivial part of theCoulomb interaction, Eq. (74b), which is the part thatcouples U to the fermion-parity operator, Eq. (77). Thereal-time renormalization group in the one- plus two-loopapproximation encodes these higher order terms into arenormalization of ¯ L and ¯ G . An important property ofthis approach is that while it provides a good approxima-tion for large U as well, it exactly includes the solution2for the noninteracting limit U = 0. The above simpleanalysis explains why at least the one- and two-loop or-ders must be included to exactly recover the noninteract-ing limit (i.e., for the full self-energy and the full densityoperator, not just selected quantities ).Finally, we note that Eq. (123) shows that thequantum-dot self-energy Σ and therefore the effective Li-ouvillian (105) is quartic in the field superoperators,even in the noninteracting limit ( U = 0). This is to becontrasted with, e.g., Green’s function and path-integralapproaches where only quadratic expressions appear fornoninteracting problems. To understand better how thisdifference arises we turn to the explicit evaluation of allcontributions in Eq. (122) for U = 0, which will reveal afurther simplification. IV. EXACT NONINTERACTINGTIME-EVOLUTION
In this section, we specialize to the noninteracting limit U = 0, unless stated otherwise. A. Time-evolution
1. Time-evolution propagator
We now calculate the one- and two-loop corrections tothe renormalized free propagator (103), which is definedby the time-local renormalized Liouvillian (114)¯Π ( t, t ) = lim T →∞ Π( t, t ) = e − i ¯ L ( t − t ) (126)= e − Γ( t − t ) | Z R )( Z R | + X σ e − Γ σ ( t − t ) | χ σ )( χ σ | + X σ e − (cid:16) iσB + 12 Γ (cid:17) ( t − t ) | S σ )( S σ | + X η e − (cid:16) iη ǫ + 12 Γ (cid:17) ( t − t ) | T η )( T η | , (the fermionic part is omitted) and then discuss the time-dependent density operator ρ ( t ). a. One-loop propagator. The one-loop ( m = 2) con-tribution to Eq. (99b) can be written as:¯Π ( t, t ) (127)= − e − i ¯ L ( t − t ) X Z dt dt t ≥ t ≥ t ≥ t ¯ G ′ ( t ) ¯ G ′ ( t )¯ γ ( t − t )= − e − i ¯ L ( t − t ) X r,σ,η ¯ G ¯ G ¯2 (128) × Z dt dt t ≥ t ≥ t ≥ t ¯ γ , ¯2 ( t − t ) e iηǫ σ ( t − t ) e
12 Γ σ ( t + t − t ) , making use of the factor δ , ¯1 in the contraction ¯ γ , andwriting 2 = η, σ, r . The one-loop contributions only gen-erate transitions which increase the superfermion numberby two. This is expected since in Eq. (128) only creation superoperators appear. In the bosonic sector (60) thesetransitions proceed from the supervacuum state | Z L ) tothe doubly excited superkets | χ σ ) and from there to themost occupied superket | Z R ). The corresponding ma-trix elements of the superoperator ¯ G ¯ G ¯2 are again eas-ily determined using the algebra of superfermions: first,we note that ¯ G ησ ¯ G ¯ ησ = η ¯ G + σ ¯ G − σ since the fields an-ticommute. Next, we see that for transitions between 0and 2 superparticles, by definition ¯ G + σ ¯ G − σ | Z L ) = | χ σ ),and between 2 and 4 superparticles, ( Z R | ¯ G + σ ¯ G − σ =[ ˜ G + σ ˜ G − σ | Z R )] † = [ ¯ G +¯ σ ¯ G − ¯ σ | Z L )] † = [ | χ ¯ σ )] † = ( χ ¯ σ | , us-ing Eq. (53). As a result,¯ G ησ ¯ G ¯ ησ = η (cid:16) | χ σ )( Z L | + | Z R )( χ ¯ σ | (cid:17) + F, (129)where F is again an irrelevant fermionic part. Insertingthis in Eq. (128), we obtain after summing over η :¯Π ( t, t ) = − e − i ¯ L ( t − t ) X σ h | χ σ )( Z L | + | Z R )( χ ¯ σ | i (130) × X r rσ Z dt dt t ≥ t ≥ t ≥ t T sin ( ǫ rσ ( t − t ))sinh( πT ( t − t )) e
12 Γ σ ( t + t − t ) , where we denote the quantum-dot energies ǫ σ = ǫ + σB/ µ r by ǫ rσ = ǫ σ − µ r . (131)Next, the diagonal form (114) of the renormalized freepropagator is used¯ L | χ σ ) = − i Γ σ | χ σ ) , ¯ L | Z R ) = − i Γ | Z R ) , (132)and we change variables, Θ = t + t − t and τ = t − t in the integration, R tt R t t dt dt = R ∆0 dτ R − ττ d Θ,denoting ∆ = t − t , and then perform the Θ-integral,¯Π ( t, t ) = X σ (cid:8) | χ σ )( Z L | + e − Γ ¯ σ ∆ | Z R )( χ ¯ σ | (cid:9) × X r rσ Γ σ (cid:0) F + rσ (∆) + F − rσ (∆) (cid:1) . (133)The explicit expressions for the time-dependent coeffi-cients (see App. D), F + rσ (∆) := − Z ∆0 dτ T sin( ǫ rσ τ )sinh ( πT τ ) e −
12 Γ σ τ = 1 π Im Ψ (cid:18) − iǫ rσ − Γ σ πT (cid:19) (134a)+ 1 π Im e ( − πT + iǫ rσ −
12 Γ σ )∆ Φ (cid:16) e − πT ∆ ; 1; 12 + iǫ rσ −
12 Γ σ πT (cid:17) ,F − rσ (∆) := e − Γ σ ∆ Z ∆0 dτ T sin( ǫ rσ τ )sinh( πT τ ) e
12 Γ σ τ = − e − Γ σ ∆ · F + rσ (∆) (cid:12)(cid:12) Γ σ →− Γ σ , (134b)3can be expressed through the digamma-function Ψ( z ),with Im Ψ( z ) = − Im P ∞ n =0 / ( n + z ) and the Lerchtranscendent function Φ( z ; s ; ν ) = P + ∞ n =0 z n ( n + ν ) s . Theasymptotic values are F + rσ ( ∞ ) = 1 π Im Ψ (cid:18) − iǫ rσ − Γ σ πT (cid:19) , (135a) F − rσ ( ∞ ) = 0 . (135b)The decay to these stationary values, given by the sec-ond term on the right-hand side of Eq. (134a), has theasymptotic form: F + rσ (∆) − F + rσ ( ∞ ) = sin (cid:18) ǫ rσ ∆ + arctan (cid:16) ǫ rσ ς σ (cid:17)(cid:19) (136) × T p ( πT ) + ( ǫ rσ ) e − πT ∆ ∆ ≫ Γ − σ ≫ T − T p (Γ σ / + ( ǫ rσ ) e −
12 Γ σ ∆ sinh( πT ∆) ∆ & T − ≫ Γ − σ , where ς σ = max { Γ σ , πT } . Thus for low temperature T ≪ Γ rσ , the oscillatory decay of both F + rσ (∆) and F − rσ (∆) [following from Eq. (134b)] is at least as fast as e − (Γ σ / . b. Two-loop propagator. The two-loop ( m = 4) con-tribution to Eq. (99a) reads as¯Π ( t, t ) = e − i ¯ L ( t − t ) Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t ¯ G ′ ( t ) ¯ G ′ ( t ) ¯ G ′ ( t ) ¯ G ′ ( t ) × X h i,j,k,l i ( − P ¯ γ ij ( t i − t j )¯ γ kl ( t k − t j ) , (137)where h i, j, k, l i denotes the sum over the following possi-ble contractions: i, j, k, l = 4 , , , , , , , , , U = 0),besides the truncation (122) is that this contribution canbe factorized as (see App. E and Fig. 5):¯Π ( t, t ) = ¯Π ( t, t ) e i ¯ L ( t − t ) ¯Π ( t, t ) . (138)This general insight is again readily obtained by mak-ing use of the causal superfermions: since without in-teraction, the time-dependent interaction-picture fieldsanticommute [Eq. (120)], relabeling of dummy indicesand integration variables in Eq. (137) directly leads tothe factorization (138) on the superoperator level. Thisis shown diagrammatically in Fig. 5 and written out inApp. E. Our approach thus clarifies how in the many-body Liouville-space simplification arises for U = 0, which isimportant since interacting theories are necessarily for-mulated in this large space. Clearly, when making useof the collinearity of the spin-dependence of the tunnel-ing and due to the magnetic field, the Liouville spaces of FIG. 5. Factorization (138) of the renormalized two-loop propagator ¯Π ( t, t ): diagrammatic proof, see App. Efor explicit expressions. We use the modified conventionthat diagrams represent the same expressions Eq. (121) asin Fig. 4 but now without the overall renormalized unper-turbed propagator e − i ¯ L ( t − t ) in Eq. (121c). First equality:Two loop terms in Fig. 4 with the time-ordered integrations t ≥ t ≥ t ≥ t ≥ t ≥ t . Second equality: We firstduplicated the terms while compensating with a factor 1 / t i thus stands for ¯ G i ( t i ) with multi-index i = η i , σ i . In each term the integrations are such that ver-tices maintain their order, e.g., in the second term we inte-grate over t ≥ t ≥ t ≥ t ≥ t ≥ t . Since the vertexsuperoperators anticommute [Eq. (120)], we can factorize the integrand superoperators and bring together the black andthe red parts. For the second and fifth diagram (crossed con-traction), this introduces a quantum-dot fermion sign ( − − P inEq. (137)]. As a result all integrands are given by the samesuperoperator. Third equality: by summing all the integralswith all possible time-orderings, the resulting integral also fac-torizes. The resulting expression in the factors is precisely theone-loop propagator (i.e., the one-loop terms in Fig. 4, againusing the mentioned convention). Multiplying the equationby e − i ¯ L ( t − t ) gives Eq. (138). superfermions with different spin can be considered in-dependently and one arrives also at Eq. (138). However,this consideration, formulated in App. F, does not makeclear how in the formalism applicable to interacting casesthis factorization comes about. Moreover, it is unneces-sary when one makes use of the causal superfermions.Independent of the result Eq. (138), the structure of thesuperoperator (137) can also be determined easily us-ing the Liouville-space second quantization. InsertingEq. (119), the superoperator part of the expression (137)is ∝ e − i ¯ L ( t − t ) ¯ G ¯ G ¯ G ¯ G . The product of four creationsuperoperators can only be nonzero if all multi-indices1 , , , always propor-tional to the transition superoperator taking the vacuumsuperket | Z L ) into the most filled fermion-parity superket4 | Z R ) [Eq. (60f)]:¯Π ( t, t ) = ϑ ( t − t ) | Z R )( Z L | . (139)All that remains is to calculate the coefficient ϑ as afunction of ∆ = t − t by substituting Eq. (133) intoEq. (138): ϑ (∆) =2 X σ X r,r ′ Γ rσ Γ σ (cid:16) F + rσ (∆) + F − rσ (∆) (cid:17) × Γ r ′ ¯ σ Γ ¯ σ (cid:16) F + r ′ ¯ σ (∆) + F − r ′ ¯ σ (∆) (cid:17) (140a)=4 X r Γ r ↑ Γ ↑ (cid:16) F + r ↑ (∆) + F − r ↑ (∆) (cid:17) × X r ′ Γ r ′ ↓ Γ ↓ (cid:16) F + r ′ ↓ (∆) + F − r ′ ↓ (∆) (cid:17) . (140b) That the fourfold, time-ordered integral Eq. (137) re-duces to the simple product of the two spin-resolvedfunctions depending only on the difference ∆ = t − t is expected from the considerations in App. F. However,without the renormalized formulation of the perturbationtheory in the causal superfermion framework the originof such simplifications in real-time calculations remainunclear.Finally, we note that the superoperator form (139) aswell as the truncation of the perturbation series (122)both remain valid for the case of the noncollinear mag-netizations of the (ferromagnetic) reservoirs and / or oftunnel junctions. They are based on the very generalcausal structure of the perturbation series, which is in-dependent of the spin-rotation and other symmetries ofthe problem. Summarizing, the full time-evolution prop-agator reads asΠ( t, t ) = [ ¯Π ( t, t ) + ¯Π ( t, t ) + ¯Π ( t, t )] (141)= lim T →∞ Π( t, t ) + X r,σ rσ Γ σ h F + rσ ( t − t ) + F − rσ ( t − t ) in | χ σ )( Z L | + e − Γ ¯ σ ( t − t ) | Z R )( χ ¯ σ | o + ϑ ( t − t ) | Z R )( Z L | , where the first term is given by Eq. (126), F ± rσ byEq. (134) and ϑ by Eq. (140).
2. Time-dependent density operator
Evaluating ρ ( t ) = Π( t, t ) ρ ( t ), where the initialstate ρ ( t ) has the general form (68) with t = t , we ob-tain the exact time-dependent density operator for U = 0in the wideband limit: ρ ( t ) = lim T →∞ ρ ( t ) + X r,σ Γ rσ Γ σ h F + rσ ( t − t ) + F − rσ ( t − t ) i | χ σ ) (142a)+ X r,σ e − Γ ¯ σ ( t − t ) rσ Γ σ h F + rσ ( t − t ) + F − rσ ( t − t ) i Φ ¯ σ ( t ) + ϑ ( t − t ) ! | Z R ) , (142b)Here, the first term on the right hand side of Eq. (142a)is the exact T → ∞ result (116) discussed in Sec. III C 3. a. Electron-pair and spin coherence. We first notethe absence of corrections to the T → ∞ decay as givenby Eq. (116) of the electron-pair coherence [Eq. (71)] andtransverse spin coherence coefficients [Eq. (72)] of theinitial state ρ ( t ). For U = 0 these therefore decay ex-ponentially to zero with rates which are independent oftemperature and equal: Υ η ( t ) = e − (Γ / t − t ) Υ η ( t ) andΩ σ ( t ) = e − (Γ / t − t ) Ω σ ( t ), respectively. This is muchlike the fermion parity discussed below, but in contrastto the latter, here the decay is altered when U = 0. The stationary values are zero by charge and spin-rotationsymmetry, see Sec. IV B. The finite temperature givesrise to corrections to both the stationary values and tothe decay, which we now discuss. b. Spin-orbital occupancies.
The second term in(142a) modifies the decay of the level occupancies:Φ σ ( t ) = h n σ i ( t ) − = e − Γ σ ( t − t ) Φ σ ( t ) (143)+ X r Γ rσ Γ σ h F + rσ ( t − t ) + F − rσ ( t − t ) i . rσ = ˜Γ r and zero magnetic field, B = 0, after calculat-ing the ω -integral expression left unevaluated in Ref. 77.Additionally switching off the reservoir dependence of thetunnel coefficients, i.e., ˜Γ r = ˜Γ, in Eq. (143) and assum-ing that initially the dot is unoccupied, h n σ i ( t ) = 0,we find agreement with the result of Ref. 134 obtainedwithin these assumptions. The stationary value, ob-tained using Eq. (135a),Φ σ ( ∞ ) := h n σ i ( ∞ ) − = 1 π X r Γ rσ Γ σ Im Ψ (cid:18)
12 + Γ σ − iǫ rσ πT (cid:19) , (144)under corresponding simplifications also agrees with thatobtained in Refs. 77 and 134. Both the stationary valueand the decay towards it depend on the spin σ : forspin-independent tunneling Γ rσ := ˜Γ r , this is a con-sequence of the Zeeman splitting B on the quantumdot. For B = 0, however, this is due to nonequilib-rium spin accumulation on the quantum dot, h S z i ( t ) = P σ h σn σ i / P σ σ Φ σ /
2, caused by the spin-dependenttunneling. Only when both B = 0 and Γ rσ := ˜Γ r dowe have full spin-rotation symmetry. In this case, theoccupancies are equal h n ↑ i = h n ↓ i , and there is no cor-rection to the longitudinal spin h S z i ( t ) as given by the T → ∞ value (116): h S z i ( t ) decays exponentially to zero,in agreement with the spin-rotation symmetry ( z -axis).The decay rate, Γ σ = P r ˜Γ r , is identical to the rateΓ / P r ˜Γ r of the spin- and electron-pair coherencesΥ η ( t ) and Ω σ ( t ), respectively, all of which are tempera-ture independent. c. Fermion-parity and two-particle correlations. The full time-evolution of the fermion-parity operatorexpansion coefficient [Eq. (70)] thus reads as follows:Ξ( t ) = e − Γ( t − t ) Ξ( t ) + X σ e − Γ σ ( t − t ) × X r rσ Γ ¯ σ h F + r ¯ σ ( t − t ) + F − r ¯ σ ( t − t ) i Φ σ ( t )+ ϑ ( t − t ) . (145)This coefficient takes account of the correlations of theoccupancies through the average of the fermion-parityoperator: Ξ( t ) = h e iπn i = 2 h Q σ ( n σ − / i = 2 h n ↑ n ↓ i−h n i +1 /
2. The result (145) is valid for an arbitrary initialstate of the quantum dot with two-particle correlations: h n ↑ n ↓ i ( t ) = h n ↑ i ( t ) h n ↓ i ( t ), equivalent toΞ( t ) = 2 Y σ Φ σ ( t ) . (146)The first contribution to Eq. (145) is the exponential de-cay determined by the T → ∞ limit. As explained inSec. III C 2, this part of the time-dependent-decay never has any finite temperature corrections: it is, in fact, in-dependent of all parameters except the sum of all rates Γ = P rσ Γ rσ , even when the interaction is switched on( U = 0). The second term describes the transient ef-fect of the initial occupancies on the correlations throughΦ σ ( t ) = h n σ i ( t ) − /
2. This term decays to zero in anoscillatory fashion, for small temperatures with an expo-nential envelope ∝ e − (Γ ¯ σ / σ )( t − t ) . Thus, besides therates encountered so far, there are two additional decayrates: Γ ¯ σ + Γ σ for σ = ↑ , ↓ . (147)When all tunnel rates are spin and reservoir independentand equal to Γ rσ = ˜Γ, this rate reduces to Γ σ + Γ ¯ σ =3˜Γ, in contrast to the other rates encountered so far,Γ = 4˜Γ, Γ σ = 2˜Γ or Γ σ = ˜Γ. For this simple case thisadditional energy scale was noted in Ref. 67 and relatedto the rates of a virtually excited particle-hole pair and anincident particle. Our spin-resolved result (147), writtenas (Γ ¯ σ +Γ σ ) / σ /
2, agrees with this. Here, we find thatthis scale is related to decay of two-particle correlationsarising from the initial spin-orbital occupations.Finally, the coefficient ϑ ( t − t ) defines the station-ary value Ξ( ∞ ). It is defined exclusively by the station-ary two-loop propagator (139) and its nonzero stationaryvalue (140) factorizes into the stationary values Φ σ ( ∞ )[see Eq. (144)]:Ξ( ∞ ) = ϑ ( ∞ ) = 2 Y σ Φ σ ( ∞ ) . (148)This relation simply expresses that the stationarynonequilibrium averages of the total fermion-parity op-erator factorize into the averages of the fermion parity ofthe two spinorbitals:lim t →∞ h e iπn i ( t ) = lim t →∞ Y σ h e iπn σ i ( t ) . (149)Rewritten using e iπn σ = (1 − n σ ), this is equivalentto the factorization of the correlator of the occupancies,lim t →∞ h n ↑ n ↓ i ( t ) = lim t →∞ h n ↑ i ( t ) · h n ↓ i ( t ), which is ex-pected for the noninteracting limit ( U = 0): each spin- σ “channel” can be averaged independently, see App. F.This relation always holds in the stationary limit: theinitial correlations between the orbital occupancies, ex-pressed by the inequality Eq. (146), are “forgotten” in thelong time limit. For a two-particle correlated initial stateon the quantum dot, satisfying Eq. (146), the relationEq. (149) is violated on time scales t − t . Γ − . Thistimescale is set by the fermion-parity protected eigen-value, and is therefore independent of all other parame-ters in the problem, even for nonzero interaction U , asdiscussed in Sec. III C 2. For an initially uncorrelatedquantum dot the relation Ξ( t ) = 2 Q σ Φ σ ( t ) or equiva-lently, h n ↑ i ( t ) h n ↓ i ( t ) = h n ↑ n ↓ i ( t ) at t = t , continues tohold for all times t ≥ t .Although the effects of the initial dot state for U = 0have been studied previously all these worksare based on spinless electrons or, equivalently, on a sin-gle spin-orbital model. For the a single-spin orbital the6electron-parity operator practically coincides with thelevel occupancy operator n σ : ( − n σ ∝ n σ − /
2. Thusby the simplicity of this model the effect of the initialtwo-particle correlators h n ↑ n ↓ i cannot appear. However,for electrons with spin it makes a crucial difference if thedot was initially prepared in the nonfactorizable form,Eq. (146), or not, as our result show. We are not awareof any work presenting an exact analytical result for thenonfactorizing time evolution of this correlator in thislimit. B. Stationary limit
We now illustrate how simplifications arise in Laplacespace in the noninteracting limit and directly calculatethe exact U = 0 stationary density operator. This in-dependent calculation is also of more general interestsince it involves the direct, explicit calculation of thestationary self-energy Σ( i
0) and the effective Liouvillian L ( i
0) = L + Σ( i
0) for U = 0 using the renormalized per-turbation theory but now formulated in Laplace space.This independent result was used (but not derived) inRef. 30 to check that the real-time renormalization groupexplicitly recovers the noninteracting limit for the sta-tionary current and the corresponding self-energy parts[see Eq. (176)]. Here, we provide the derivation of thisimportant benchmark not only for the current, but alsofor the full density operator and the self-energy.
1. Frequency space perturbation expansion for theself-energy
To keep the paper self-contained, we first briefly outlinethe renormalized perturbation theory in Laplace spacewhich is also valid for U = 0. Although this theorywas formulated in Ref. 30, it was not used to explic-itly calculate the U = 0 limit, but the RG approach wasused instead to analytically verify the current and therelevant self-energy parts in this limit. We proceed inclose analogy to the above time-dependent formulationand start with the Laplace transform of the (nonrenor-malized) perturbation expansion Eq. (45b) for the time-evolution propagator. Alternatively, we transform thegeneral solution Eq. (37) and then expand the resolvent1 / ( z − L tot ) in powers of L V using Tr R L R = 0:Π( z ) := Z ∞ t dte izt Π( t, t ) = Tr R iz − L tot ρ R (150)= ∞ X k =0 iz − L Tr R (cid:18) L V z − L R − L (cid:19) k ρ R . Inserting Eq. (48) for L V , we have for the m th-order termof the expansion of the Laplace-space resolvent, Π( z ) = P ∞ m =0 Π m ( z ) [cf. Eq. (85)]:Π m ( z ) = i Tr R (cid:0) J q m ¯ m ...J q ¯1 ρ R (cid:1) × (151)1 z − L G q m m z − L − X m . . . z − L − X G q z − L .
Here, we have commuted all J q to the far right, using[ L R , J q ] − = η ( ω + µ r ) J q and collected all reservoir ener-gies x k = η k ( ω k + µ k ) of the G q k k ( J q k ¯ k ) originally standingto the left to the resolvent i in X i = P mi x k . Evaluatingthe reservoir average using the Wick theorem (87), weobtain terms that can be represented by the same dia-grams as in Fig. 2, where the free propagators connectingvertices stand for 1 / ( z − L − X i ) and the contraction lineconnecting a pair of vertices ¯ G - ¯ G and ¯ G - ˜ G stands forthe contraction function (89) and (90), where the line isnow assigned a frequency x k . The irreducible parts ofthese diagrams [cf. Sec. III C 1] are collected into the su-peroperator self-energy Σ( z ), which is just the Laplacetransform of Σ( t − t ′ ) in Eq. (38):Σ( z ) = Z ∞ e izτ Σ( τ, dτ. (152)After grouping the diagrams into Σ( z ) blocks, we canresum the resulting geometric series and obtain theLaplace-space solution of the Dyson equation (38) / thekinetic equation (39): ρ ( z ) = iz − L ( z ) ρ ( t ) , (153)where the effective Liouvillian in the Laplace representa-tion [the Laplace transform of Eq. (105)] is decomposedas in Eq. (105): L ( z ) = L + Σ( z ) . (154)The required expansion for Σ( z ) thus has the form:Σ( z ) = ( − P (cid:16)Y γ (cid:17) irr × (155)¯ G m z − X m − L ......G q z − X − L G q . The causal structure of this expansion [cf. discussionof Eq. (91)] enforces that the leftmost vertex in eachcontraction is always a creation superoperator ¯ G m as aconsequence of Eq. (54). We use the same conventionsas in Eqs. (91) and (85), suppressing all sums and inte-grations over reservoir frequencies. Note that unlike inthe time representation, these integrals cannot be pulledinto the contraction functions, i.e., the γ qi,j are given byEqs. (89) and (90). This is because in the Laplace trans-form the frequencies are convoluted with the dot evolu-tion in the propagators ( z − L + X ) − . The main ad-vantage of Eq. (155) is that we can now directly work inthe stationary limit by taking the limit z → i L ( i
0) = L + Σ( i γ contraction can be neglected, and that one canintegrate out the ˜ γ contractions and the ˜ G vertices, in-corporating them into the Laplace transform ˜Σ of theself-energy ˜Σ( t, t ′ ) [Eq. (95)], which is simply the time-independent factor given by Eq. (96). What remains is toevaluate the perturbative expansion for ¯Σ( z ) = Σ( z ) − ˜Σ,with the simplified diagram rules schematized by¯Σ( z ) =( − P Y i ¯ γ i ! irr × (156)¯ G m z − X m − ¯ L ... z − X − ¯ L ¯ G , where we sum over irreducible ¯ γ contractions and ¯ L = L + ˜Σ as before [Eq. (98)]. We can now find the sta-tionary state as the right zero eigenvector of L ( i
0) = L + Σ( i
0) = ¯ L + ¯Σ( i set z = 0 since ¯ L is a dissipative Liouvillian which automat-ically regularizes all propagators.
We note how thesetechnical properties neatly tie in with the physical mean-ing. That the denominators in Eq. (156) contain no zeroswas a key point in setting up the real-time renormaliza-tion group flow in Laplace space for the calculation ofeffective Liouvillians, referred to as the “zero-eigenvectorproblem”.
2. Noninteracting limit and super-Pauli principle
We now work out the simplifications that occur for theself-energy in the limit U = 0, in full analogy with thetime-representation. The part ˜Σ is obtained by setting U = 0 in Eq. (96) or (112). In the remaining calculationof ¯Σ, [Eq. (156)], using Eq. (118) we commute the ¯ G through the resolvents: this gives the analog of Eq. (121):¯Σ( z ) = ( − P Y i ¯ γ i ! irr ¯ G m ... ¯ G × (157)1 z − X m − E m − ¯ L ... z − X − E − ¯ L , where ¯ L is given by Eq. (114) with U = 0 and E i = P ik =2 ǫ k is the sum over renormalized, single-particle quantum dot energies ǫ i = η i ǫ σ i − i Γ σ i . (158)These have acquired an imaginary part by the inclusionof broadening of the T → ∞ limit through the self-energyterm ˜Σ in the renormalized Liouvillian ¯ L . Thus, also inthe Laplace representation, the super-Pauli principle (62)directly reveals that in the U = 0 limit nonzero terms ofthe renormalized self-energy of order m > z ) = ¯Σ ( z ) + ¯Σ ( z ), cf. Eq. (123). a. One loop self-energy: stationary occupancies andcurrent. We first calculate the one-loop diagram for ¯Σ by substituting Eq. (129) into¯Σ ( i
0) = X η,σ,r ¯ G ¯ G ¯2 Z d ¯ ω ¯ γ (¯ ω ) i − ¯ ω − ¯ µ − ǫ ¯2 − ¯ L = X η,σ,r η Γ rσ π (cid:18) | χ σ )( Z L | Z d ¯ ω tanh(¯ ω ) i − ¯ ω − ¯ µ − ǫ ¯2 − ¯ L + | Z R )( χ ¯ σ | Z d ¯ ω tanh(¯ ω ) i − ¯ ω − ¯ µ − ǫ ¯2 − ¯ L (cid:19) + F. (159)Here and in the following, we abbreviate ¯ ω i ≡ η i ω i = x and ¯ µ i := η i µ r i and we write this in the form (omittingthe fermionic part)¯Σ ( i
0) = − i X σ { ψ σ | χ σ )( Z L | + φ σ | Z R )( χ σ |} . (160)Using ( Z L | ¯ L = 0 and summing over η explicitly, thisgives after replacing x → − x in the second integral thestationary value of the first two coefficients (see App. C): ψ σ = − i X r Γ rσ π Z dx (cid:20) tanh( x/ T ) i Γ σ − x + ǫ rσ − tanh( x/ T ) i Γ σ − x − ǫ rσ (cid:21) = − X r rσ π Im Ψ (cid:18)
12 + Γ σ − iǫ rσ πT (cid:19) , (161)where, [cf. Eq. (131)] ǫ rσ = ǫ + σB/ − µ r . (162)We will see that the coefficients (161) of the one-loop self-energy ¯Σ ( i
0) completely determine the stationarycurrent. As a result, the current only shows the usualbroadening Γ σ / P r Γ rσ / σ .Analogously, we calculate the remaining coefficients φ σ ( χ σ | ¯ L = − i Γ( χ σ | , cf. Eq. (114a): φ σ = − i X r Γ r ¯ σ π Z dx (cid:20) tanh( x/ T ) i ( Γ ¯ σ + Γ σ ) − x + ǫ r ¯ σ − tanh( x/ T ) i ( Γ ¯ σ + Γ σ ) − x − ǫ r ¯ σ (cid:21) = − X r r ¯ σ π Im Ψ (cid:18)
12 + Γ ¯ σ + Γ σ − iǫ r ¯ σ πT (cid:19) . (163)This part of the self-energy involves a quite differentbroadening, Γ σ + Γ ¯ σ instead of Γ σ , an energy scalethat we noted earlier in Eq. (147). However, we will seethat the exact U = 0 stationary current is not sensitiveto this quantity. We note that for U = 0, it can be shownthat this broadening, although modified by the interac-tion, does enter into the stationary current. b. Two loop self-energy: stationary average fermion-parity. The two-loop contribution to ¯Σ is calculatedin close analogy to the two-loop contributions to ¯Π inEq. (137):8¯Σ ( i
0) = X σ,r,r ′ Γ rσ Γ r ′ ¯ σ (2 π ) ¯ G l ¯ G ¯ l ¯ G l ′ ¯ G ¯ l ′ Z d ¯ ω l Z d ¯ ω l ′ tanh(¯ ω l / T ) tanh(¯ ω l ′ / T )( η l ǫ rσ + i ( Γ σ + Γ ¯ σ ) − ¯ ω l ) × (164a) η l ′ ǫ r ′ ¯ σ + i Γ ¯ σ − ¯ ω l ′ ) 1( η l ǫ rσ + η l ′ ǫ r ′ ¯ σ + i Γ2 − ¯ ω l − ¯ ω l ′ ) + 1( η l ǫ rσ + i Γ σ − ¯ ω l ) 1( η l ǫ rσ + η l ′ ǫ r ′ ¯ σ + i Γ2 − ¯ ω l − ¯ ω l ′ ) ! = X σ,r,r ′ Γ rσ Γ r ′ ¯ σ (2 π ) ¯ G l ¯ G ¯ l ¯ G l ′ ¯ G ¯ l ′ Z d ¯ ω l tanh(¯ ω l / T )( η l ǫ rσ + i ( Γ σ + Γ ¯ σ ) − ¯ ω l ) 1( η l ǫ rσ + i Γ σ − ¯ ω l ) Z d ¯ ω l ′ tanh(¯ ω l ′ / T )( η l ′ ǫ r ′ ¯ σ + i Γ ¯ σ − ¯ ω l ′ ) (164b)Here, we made use of the fact that for ¯ γ ij the multi-indices are related as i = ¯ j , and we relabeled 4 → l = ω l , η l , r, σ and 2 → l ′ = ω l ′ , η l ′ , r ′ , ¯ σ for compactness.We also explicitly made use of relation σ l = σ = ¯ σ l ′ ,which expresses the fact that in the product ¯ G l ¯ G ¯ l ¯ G l ′ ¯ G ¯ l ′ all operators must be different [otherwise the productis zero due to the super-Pauli principle (62)-(63)]. Equa-tion (164a) has two types of contributions, represented bythe irreducible parts of the last two diagrams in Fig. 4.To obtain the form (164a) we have anticommuted thefield superoperators to have the same order in each typeof contribution. This cancels the Wick sign, similar tothe rewriting of Eq. (137) into Eq. (138), cf. Fig. 5 andApp. E. Notably, when combining these two types of con-tributions in Eq. (164b), the double integral over frequen-cies ¯ ω l and ¯ ω l ′ becomes factorizable but only for zero quantum-dot frequency z = 0. Therefore the completeexpression (164b) is a sum over factorizable integrals. Asin Eqs. (137) and (139), the superoperator structure of(164b) is very simple:¯Σ ( i
0) = ζ | Z R )( Z L | . (165)Using ¯ G l ¯ G ¯ l ¯ G l ′ ¯ G ¯ l ′ = η l η l ′ | Z R )( Z L | and Eq. (C1) [App. C]we obtain: ζ = X σ,r,r ′ rσ Γ r ′ ¯ σ π × (166) (cid:20) Im Ψ (cid:18)
12 + Γ σ + Γ ¯ σ − iǫ rσ πT (cid:19) Im Ψ (cid:18)
12 + Γ ¯ σ − iǫ r ′ ¯ σ πT (cid:19) − Im Ψ (cid:18)
12 + Γ σ − iǫ rσ πT (cid:19) Im Ψ (cid:18)
12 + Γ ¯ σ − iǫ r ′ ¯ σ πT (cid:19)(cid:21) . Summarizing, the exact zero-frequency effective dot Li-ouvillian in the wide-band limit for the noninteractingAnderson model ( U = 0) is iL ( i
0) := i (cid:2) L + ˜Σ + ¯Σ ( i
0) + ¯Σ ( i (cid:3) =Γ | Z R )( Z R | + ζ | Z R )( Z L | + (167) X σ h φ σ | Z R )( χ σ | + ψ σ | χ σ )( Z L | i + X σ Γ σ | χ σ )( χ σ | + X σ h iσB + Γ i | S σ )( S σ | + X η h iη (2 ǫ + U ) + Γ i | T η )( T η | , with ψ σ given by Eq. (161), φ σ given by Eq. (163), ζ givenby Eq. (166) and not writing the irrelevant fermionicpart.
3. Stationary density operator
The stationary state, the unique right zero eigenvec-tor of L ( i L ( i generally, the effective Liouvillian writtenin the basis Eqs. (60) must have the form (ignoring thefermionic part again): iL ( i
0) = Γ | Z R )( Z R | + ζ | Z R )( Z L | + (168) X σ h φ σ | Z R )( χ σ | + ψ σ | χ σ )( Z L | i + X σ,σ ξ σ,σ ′ | χ σ )( χ σ ′ | + X σ E σ | S σ )( S σ | + X η M η | T η )( T η | . This form follows from the causal structure, the wide-band limit (fixing the first coefficient to the constant,fermion-parity eigenvalue − i Γ, cf. Sec. III C 2) andthe symmetries of the Anderson model. (Notably, it does not require U = 0.) As in Ref. 30, we have conservationof the charge and of the spin along the direction of themagnetic field B , even though we now also include spin-dependent tunneling (we assume that the magnetic field B and the polarization vectors are collinear, preservingthe spin-rotation symmetry about this axes). The sta-tionary state can be expressed in terms of the effective Li-ouvillian coefficients and by comparing Eq. (167) withthe general form Eq. (168), we obtain in terms of thematrix ξ σ,σ ′ = Γ σ δ σ,σ ′ , and the vectors [Eq. (161) andEq. (163)]: ρ ( ∞ ) = | Z L ) − X σ,σ ′ ξ − σ,σ ′ ψ σ ′ | χ σ )+ ( X σ,σ ′ φ σ ξ − σ,σ ′ ψ σ ′ − ζ ) | Z R ) (169a)= | Z L ) + X σ Φ σ ( ∞ ) | χ σ ) + Ξ( ∞ ) | Z R ) , (169b)9where the expansion coefficients areΦ σ ( ∞ ) = 1 π X r Γ rσ Γ σ Im Ψ (cid:18)
12 + Γ σ − iǫ rσ πT (cid:19) , (170)Ξ( ∞ ) = 2Φ ↑ ( ∞ ) · Φ ↓ ( ∞ ) . (171)We note that the additional broadening scale Γ σ + Γ ¯ σ that we noted already in Eqs. (147) and (163) is alsopresent in ζ [Eq. (166)] but drops out in the calculationof Ξ( ∞ ) when one sums over σ . Equation (170) repro-duces the stationary spin-orbital occupancies Eq. (144)through h n σ i = + Φ σ . Moreover, Eq. (171) confirmsthe factorization (148) of the stationary value Ξ( ∞ ) intothe coefficients Φ σ ( ∞ ). On the superoperator level, therenormalized two-loop self-energy ¯Σ does not factorizefor any frequency z , not even at z = 0, see Eqs. (165) and(166). One can verify that to achieve such a factorizationa reducible term needs to be added to ¯Σ ( t, t ). This isprecisely what happens in Eq. (125) and produces essen-tially the Laplace transform of superoperator ¯Π ( t, t ),which indeed factorizes at any time t by Eq. (138). Thisshows that for ¯Σ ( z ) itself no such factorization is tobe expected at any frequency. There are thus advan-tages of working directly with full propagators in time-space in comparison with working with self-energies inLaplace-space, when considering the noninteracting limitof the Anderson model and its generalizations, Further-more, Eqs. (169b) and (171) show the physical impor-tance of the stationary two-loop self-energy superoper-ator ¯Σ ( i quartic term (164b) appearing inthe effective theory despite the absence of two-particleinteractions ( U = 0). To obtain the correct two-particlecorrelations for U = 0 in the stationary limit it is crucialto calculate both the one- and two-loop self-energies, i.e.,to work with the effective Liouvillian which is quartic inthe fields.Finally, we note that in the expansion (169b) of the stationary state, the terms containing superkets | S σ ) and | T η ), describing spin- and electron-pair coherence [cf.Sec. II D 3], do not appear since they are forbidden bycharge and spin rotation symmetry. If such coherencesare prepared in the initial state, they must decay to zeroin time, in agreement with the central result Eq. (142). C. Time-dependent current
In this last section of the paper we illustrate that in thecalculation of observable averages very similar simplifica-tions can be made using the causal field superoperators.We focus on the example of the time-dependent chargecurrent in the noninteracting limit U = 0.
1. Current self-energy
We first present considerations which apply generally,i.e., to the interacting Anderson model ( U = 0), and, in fact, to multiorbital generalizations. Generally, an ob-servable A that is not local to the dot requires the calcu-lation of an additional self-energy Σ A with its own real-time diagrammatic expansion. However, for a quan-tity which is conserved in the tunneling, such as thecurrent I r into reservoir r , this is not necessary: itcan be obtained by simply keeping track of the part ofthe self-energy that is related to reservoir r . That is,we decompose Σ = P r Σ r by splitting up the interac-tion L V = P r L V r into r -contributions L V r = [ V r , • ] − [cf. Eq. (21)] at the latest time t m in each term of theperturbation series [cf. Eq. (45)]. Then, by rewriting I r = − i [ H tot , n r ] = − i [ V r , n r ] = − iL V r n r with fixed r one finds the relation h I r ( t ) i = − i Tr D L n + Tr R L V r ρ tot ( t ) (172a)= − i Tr D L n + Z tt dt ′ Σ r ( t, t ′ ) ρ ( t ′ ) , (172b)where L n + = [ n, • ] + is the anti commutator with the dotparticle number operator n . Besides the computationalsimplification, extended here to the time-dependent case,this result can be used to show very easily nonpertur-batively that the stationary current at zero bias voltageis always zero (as it should be), something which is notalways obvious. This is an obvious physical requirement.However, within the real-time approach (or its equiva-lent, the Nakajima-Zwanzig approach), designed to dealwith strongly interacting models, it is not obvious howto verify explicitly that in general this is actually thecase, in particular, when going to higher orders in theperturbation theory in Γ or when making nonperturba-tive approximations in this framework, as, for instance,in Ref. 30. When properly done, concrete calculationsof this type always seem to comply with zero currentat zero bias, but why this is so in general has not beenclarified before. We found that Eq. (172b) provides thiskey step in explicitly verifying this physical requirement.We can now again take advantage of the causal struc-ture by decomposing Σ r = ˜Σ r + ¯Σ r into the T → ∞ part ¯Σ r and the finite temperature corrections ¯Σ r . Theonly difference with the analysis in Sec. III C is that onesimply does not sum over the reservoir index r of thecontraction with the latest field superoperator ¯ G m in the m/ Tr D L n + = ( Z L | L n + , appearingin Eq. (172b), in the dual Liouville-Fock basis (60): Tr D L n + • = X σ Tr D ( n σ • ) = X σ ( χ σ | + 2( Z L | . (173)When this is inserted into Eq. (172b), it follows from thecausal structure of the perturbation theory, ( Z L | Σ r ∝ Tr D Σ r = 0 ( not from probability conservation) thatthe current is a sum of projections onto two doubly excited | χ σ ) [cf Eq. (60b)]: h I r ( t ) i = − i X σ ( χ σ | h ˜Σ r ( t, t ′ ) + ¯Σ r ( t, t ′ ) i ρ ( t ′ )= ˜ I r ( t ) + ¯ I r ( t ) . (174)
2. Wideband limit and artifacts
The expression for ¯ I r ( t ) is, in general, rather compli-cated since it requires the nontrivial part of the self-energy ¯Σ r . However, in the wideband limit ˜Σ r = − i P σ Γ rσ ¯ G ˜ G ¯1 ¯ δ ( t − t ′ ) is just Eq. (95) without thesum over r . The superoperator expansion of ˜Σ r is giventhen by Eq. (112), where one has to replace Γ σ → Γ rσ ,Γ → P σ Γ rσ . For ˜ I r this gives (compare with the Green’sfunction result of Ref. 115 and Eq. (30) in Ref. 134):˜ I r ( t ) = − i X σ ( χ σ | t Z t dt ′ ˜Σ r ( t, t ′ ) ρ ( t ′ ) (175a)= − X σ Γ rσ Φ σ ( t ) . (175b)This only depends on the coefficients Φ σ ( t ) = h n σ i ( t ) − / T → ∞ values. In the T → ∞ limit the current is givenby the contribution (175b) alone, when substituting forΦ σ ( t ) the value lim T →∞ Φ σ ( t ) = e − Γ σ ( t − t ) Φ σ ( t ) [cf.Eq. (116)]. In the stationary limit, this gives a vanishingcurrent, which is as it should be since the bias voltage isdominated by thermal fluctuations. Note that this holdsgenerally for U = 0 and nonperturbatively in Γ, as inSec. III C 3.For finite temperature, the current requires the calcu-lation of ¯ I r ( t ) but also ˜ I r ( t ) changes since Φ σ ( t ) takesanother value for finite T [cf. Eq. (175b)], also requir-ing a calculation. Before we turn to this, we note thatthe wideband limit result for (175b) has the disconcert-ing property that at the initial time t it can give riseto a nonzero total current for a general initial conditionΦ σ ( t ). This is again clear in the T → ∞ limit men-tioned above: lim T →∞ I r ( t ) = lim T →∞ ˜ I r ( t ) = Φ σ ( t )yields a nonzero value. In the next section, we explicitlyshow that for the U = 0 limit this also occurs at finitetemperature.This nonzero current is inconsistent with our initialassumption that the dot and reservoirs are decoupled at t = t and is unphysical. This is an artifact of the wide-band limit and raises the question on which time-scale(175b) is correct. For the noninteracting case U = 0this has been discussed, e.g., in Refs. 76, 77, and 134 andit was shown by explicit calculation for a finite band-width D that the current starts from zero at t = t as itshould, but then on the time-scale set by the inverse band width, 1 /D , the result rapidly approaches the widebandlimit result.The use of causal field superoperators allows us togeneralize this qualitative understanding to the inter-acting case ( U = 0) since the effect of the widebandlimit can be traced explicitly on the level of superop-erators. The effect is twofold: first, as explained inSec. III B, the δ -function constraint on time integra-tions arising from the large bandwidth energy forbidsdiagrams in which a retarded contraction crosses withany other contraction line. Since experimentally thebandwidth D is usually much larger than any charac-teristic energy of the dot, temperature or transport biasthis should be a good approximation. Second, the time-dependent retarded contraction (94), retained in ˜Σ( t, t ′ ),is basically the Fourier transform of the function Γ ,ω ,˜ γ , ( t, t ′ ) ∝ R dω Γ ,ω e − iη ( ω + µ r )( t − t ′ ) . For large butfinite bandwidth D this therefore has a characteristic fi-nite time support of order 1 /D . If we now correct for thisin Eq. (175a), then the integral over t ′ starts from zeroat t , as it should, and only on the time scale 1 /D doesit acquires the wideband limit value given by Eq. (175b).Note however, that the times on which this difference isnoticeable is below femtoseconds for the typical energies D ∼ . t .
3. Noninteracting current
We now again return to the noninteracting limit U = 0.In this case, the first contribution ˜ I r ( t ) to the current(174) is given by Eq. (175b) with the value of Φ σ ( t ) givenby the central result (143). We see that ˜ I r ( t ) depends onthe initial state through Φ σ ( t ) and decays to a nonzerostationary value. The second contribution ¯ I r ( t ) can nowbe simplified for U = 0 using the super-Pauli principle(62)-(63): in the renormalized perturbation expansion forthe finite T corrections only two terms survive, ¯Σ r ( t, t ′ ) =¯Σ r ( t, t ′ ) + ¯Σ r ( t, t ′ ), in full analogy to Eq. (123). In addi-tion, by the same principle ( χ σ | ¯Σ r ( t, t ′ ) = 0 in Eq. (174)[since ¯Σ r ∝ ¯ G ¯ G ¯2 ¯ G ¯ G ¯1 ∝ | Z R )( Z L | ], and therefore onlythe two-loop self-energy ¯Σ r ( t, t ′ ) contributes to the cur-rent in this limit (here there is no summation over the r -component of the multi-index 2):¯Σ r ( t, t ′ ) = X ¯ G e − i ¯ L ( t − t ′ ) ¯ G ¯2 ¯ γ , ¯2 ( t, t ′ ) (176)= X η,σ − Γ rσ T e (cid:16) iηǫ rσ + 12 Γ σ (cid:17) ( t − t ′ ) sinh ( πT ( t − t ′ )) e − i ¯ L ( t − t ′ ) ¯ G ¯ G ¯2 . Using Eq. (129) together with ( χ σ | ¯ L = ( χ σ | ˜Σ = − i Γ σ ( χ σ | [cf. Eq. (114a)] and summing over η one ob-1tains¯Σ r ( t, t ′ ) = (177) − i Γ rσ T e −
12 Γ σ ( t − t ′ ) sin ( ǫ rσ ( t − t ′ ))sinh ( πT ( t − t ′ )) | χ σ )( Z L | + . . . The terms not written out give no contribution wheninserted in Eq. (174) for the current (either proportionalto | Z R )( χ ¯ σ | or to fermionic projectors), and the termshown gives a contribution independent of the initial dotstate (since ( Z L | ρ ( t ′ ) = 1 / I r = − X σ Γ rσ T Z tt dt ′ e −
12 Γ σ ( t − t ′ ) sin ( ǫ rσ ( t − t ′ ))sinh ( πT ( t − t ′ ))= X σ Γ rσ F + rσ (∆) , (178)where again ∆ = t − t . The total average current throughreservoir r , written for the case of two reservoirs r = ± , µ r = rV b / h I r i ( t ) = X σ Γ rσ Γ ¯ rσ Γ σ (cid:0) F + rσ (∆) − F +¯ rσ (∆) (cid:1) (179) − X r,σ Γ rσ Γ σ F − rσ (∆) − e − Γ σ ∆ t X σ Γ rσ Φ σ ( t ) . For the noninteracting case the time-dependent currenthas already been explicitly calculated in the limit of onespin-orbital, e.g., in Ref. 134 (using the Keldysh Green’sfunction approach), under the assumption that the dotwas initially fully unoccupied, and for an arbitrary oc-cupation in Ref. 75 (using the real-time renormalizationgroup for the interacting resonant-level model in the limitof T → r and do not vanish as V b = 0. They originatefrom the current caused the change of the dot charge( I dis = dn ( t ) /dt ), the displacement current. The dis-placement current decays, as it should, to zero in thestationary limit ∆ → + ∞ , which follows from the asymp-totic relation Eq. (135a). We note the deviation of theresults of Ref. 77 from the above body of works. Close to the initial time, D − ≪ | t − t | ≪ Γ − rσ (cf.discussion above) h I r i ( t ) ≈ ˜ I r ( t ) = X σ Γ rσ [ − n σ ( t )] , (180)i.e., the total current is dominated by the last term inEq. (179), the part of the displacement current comingfrom Eq. (175b). This is again in agreement with thezero-temperature results of Ref. 75 for the spinless, in-teracting resonant level model in the limit of vanishingnonlocal interaction. This also agrees with the result for the initial current in Ref. 134, however, in contrastto that work, we take into account arbitrary initial dotlevel occupancies. The physical picture behind the result(180) again nicely relates to the fundamental importanceof the T = ∞ limit built into our causal superfermiontechnique. Extending the discussions in Ref. 134 [cf.Eq. (36) there] it is as follows. The initial current (180)stems form the part (175), which describes the current inthe T → + ∞ limit [see discussion following Eq. (175)].Due to the wideband limit, the processes described by˜ γ [cf. Eq. (94)] are very fast, taking place on the timesof order D − , and in the wideband limit giving a finiteinstantaneous current response at t = t . In contrast,the temperature-induced processes described by ¯ γ [cf.Eq. (101)] are much slower and do not contribute on suchshort time scales. The current thus “does not know yet”about the actual temperature of the reservoirs on suchtime-scales and therefore behaves such as if T would beinfinite. This is what the physical decomposition (174)of the charge current expresses, which follows naturallyon a general level from our formalism. In the concreteresult (180) the factors h n σ i ( t ) i − / /
2, thestationary value in the limit T → + ∞ , determines theresponse: the empty dot h n σ i ( t ) = 0 will charge up, I r ( t ) = P σ Γ rσ /
2, whereas the filled dot h n σ i ( t ) i = 1will discharge, I r = − P σ Γ rσ / | t − t | ≫ Γ − rσ , is determined by the first termof Eq. (179), which is antisymmetric in the reservoirs(and thus vanishes at zero bias) [cf. Eq. (135a)]: h I r i ( ∞ ) = X r ′ ,σ Γ rσ Γ ¯ rσ π Γ σ r ′ Im Ψ (cid:18)
12 + Γ σ − iǫ r ′ σ πT (cid:19) . (181)Expressed in the fermi-function f ( x ) = e x/T +1 = − tanh( x/ T ) and using Eq. (C2), this can be rewrittenas the more familiar form of a sum of current contribu-tions from the independent spin-orbitals, each broadenedby Γ σ = P r Γ rσ : h I r i ( ∞ ) = X σ Γ rσ Γ ¯ rσ π Γ σ Z ∞−∞ Γ σ / x − ǫ σ ) + (Γ σ / × (cid:16) f ( x + µ r ) − f ( x + µ ¯ r ) (cid:17) dx. (182)This result coincides with either of Ref. 75, 77, and 134in the corresponding limits mentioned above. Finally,we note that the stationary current (179), calculatedhere by explicitly taking the long-time limit, is recoveredfrom our direct calculation of the stationary quantitiesin Sec. IV B: with the help of Eqs. (172b) and (169a) onecan show that the stationary current depends on just twostationary self-energy coefficients: I r ( ∞ ) = X σ Γ ¯ rσ ψ rσ − Γ rσ ψ ¯ rσ σ . (183)2Inserting the U = 0 result for ψ rσ by leaving out the r -sumin Eq. (161) reproduces Eq. (181). This confirms the re-sult for the current in the U = 0 limit obtained in Ref. 30by a real-time RG calculation of these coefficients. Thisis another way of seeing that the additional broadeningscales Γ σ + Γ ¯ σ [Eq. (147)] do not affect the stationarycurrent for U = 0 since the coefficients φ σ [Eq. (163)] and ζ [Eq. (166)] do not appear in Eq. (183).The main objective of the above was to illustrate in atractable example how the causal superfermion techniqueworks for the calculation of an observable, in this case thecurrent. Although we were able to include all possible ini-tial coherences and correlations in the initial density op-erator locally on the quantum dot, the time-dependentcurrent reduces to the sum over its spin-resolved com-ponents. The current is not sensitive to the fermion-parity decay of the quantum-dot mixed state, which canbe detected in ways discussed in the introduction [cf.Eq. (1)]. In the noninteracting and wideband limit theeffect of the spin-polarization of the ferromagnetic leadsis to merely introduce different decay time-scales for dif-ferent spin states (Γ − rσ ). The situation becomes more in-teresting when Coulomb interaction is included since thisgenerates of the effective exchange magnetic field, a nondissipative effect. The time evolution of this fieldafter switching on the tunnel processes between the dotand the ferromagnetic leads is of considerable interest.The method presented in the present paper may serveas a starting point for conveniently addressing how sucheffects develop, in particular even for small Coulomb in-teraction nonperturbatively but strong tunnel coupling(Γ ≫ U ). It is advantageous that this can be done inthe same formalism which can treat the complementarylimit (Γ ≪ U ). V. DISCUSSION AND OUTLOOK
As outlined in the introduction, the time evolution ofstrongly interacting quantum dots is of great experimen-tal interest, but analytical theoretical methods struggleto deal with it. Taking an Anderson model descriptionas a starting point, we focused on improving the real-time approach which has already been successfully ap-plied to explain various experiments. The goal of thispaper was twofold: we wanted (i) to set up from scratchthe real-time approach to time-dependent decay in inter-acting transport problems, systematically exploiting thecausal superfermion technique (Sec. III) and (ii) to high-light its practical advantages by a complete solution ofthe noninteracting Anderson model describing a quan-tum dot with spin-dependent tunneling rates Γ rσ andfor an arbitrarily correlated initial mixed state (Sec. IV).We now summarize these two aspects separately, start-ing with the concrete results (ii), and then turning tothe general framework (i). In the process we generalizethe concrete results to multiorbital models, in both theinteracting and the noninteracting case. We also com- ment on the limitations imposed by the few assumptionsthat we made and provide an outlook on possible furtherapplications which have motivated this work all along. A. Quantum-dot spin-valve: U = 0 and interaction corrections In Sec. IV A we calculated the exact time-evolutionpropagator of the complete two-fermion density operatorin the noninteracting limit ( U = 0). The exact result,nonperturbative in the tunneling rates Γ rσ , is obtainedfrom a simple second-order renormalized perturbationtheory, expanding in the Keldysh reservoir correlationfunction ¯ γ ( ω ) ∝ Γ rσ tanh( ω/ T ) instead of just Γ rσ .Our result (142) includes all possible coefficients of thedensity operator: spin-orbital occupancies ( h n ↑ i , h n ↓ i ),transverse spin coherences (e.g. h d †↑ d ↓ i ) and electron-paircoherences (e.g., h d ↓ d ↑ i ), but also the two-particle corre-lations quantified by the nonequilibrium average of thefermion-parity operator h ( − n i ∼ h n ↓ n ↑ i ( t )+ ... . Thelast three arise only due to the initial preparation of thequantum-dot state.Besides recovering known results for the one-particlequantities, we noted that in general the transient two-particle correlator does not factorize h n ↓ n ↑ i ( t ) = h n ↓ i ( t ) ·h n ↑ i ( t ) until stationarity is reached, h n ↓ n ↑ i ( ∞ ) = h n ↓ i ( ∞ ) · h n ↑ i ( ∞ ). This happens when the quantumdot state is initially prepared in a two-particle correlatedstate. In the stationary state these correlations, how-ever, die out. Another, more striking aspect of thedecay of these initial correlations on the quantum dot, h ( − n i ( t ) ∼ e − Γ( t − t ) h ( − n i ( t ) + ... , is that the strictexponential form and the decay rate Γ = P rσ Γ rσ is inde-pendent of any other parameter in the problem. Withinour superfermion formulation of the real-time approachit is immediately clear that no corrections to this sim-ple “universal” behavior can appear, due to neither fi-nite temperature T (see Sec. III C 2), nor bias voltage V , nor magnetic field B , nor interaction U . Notably, Γdepends only on the sum of the spin-dependent rates,i.e., even the spin-polarization of the tunneling dropsout, an aspect not addressed in Ref. 30. This general-izes an earlier conclusion based on perturbation theory: this absence of corrections holds nonperturbatively in thetunnel coupling Γ rσ for the interacting Anderson modelbut also for the decay in multiorbital generalizations, re-cently studied in Ref. 121. The key point is that by thefundamental fermion-parity superselection rule, any lo-cal quantum-dot Hamiltonian must commute with theoperator ( − n . Therefore, the decay of the initial cor-relations h ( − n i ( t ), appearing in the expansion of thedensity operator, can only come from the tunnel couplingto the reservoirs and has the above mentioned form.In addition, we found that the time evolution of h ( − n i ( t ) ∼ h n ↓ n ↑ i ( t )+ . . . contains additional oscilla-tory decaying terms coming from the initial occupations h n σ i ( t ) with rate Γ ↑ +(Γ ↓ /
2) and Γ ↓ +(Γ ↑ / rσ = ˜Γ as an additional broad-ening scale 3˜Γ in the stationary density operator andin related self-energies. Thus, even in this simple limitthe time-dependent decay of the density operator of thenoninteracting ( U = 0) Anderson model shows four char-acteristic decay rates: ˜Γ, 2˜Γ, 3˜Γ, 4˜Γ.Finally, in Sec. IV C we illustrated the application ofsuperfermions to the calculation of observable quanti-ties for the time-dependent charge current. We showedthat the small-time artifacts of the wideband limit in thetransport current can be discussed on the superoperatorlevel. Also, the T → ∞ limit, built into the field super-operators, naturally appears in the expressions for thedisplacement current. We furthermore confirmed the RGresults for the stationary noninteracting limit in Ref. 30,in particular, we related the observation made there –that only one-loop self-energy corrections matter for thecurrent – to the super-Pauli principle introduced here. B. Superfermions in the real-time approach
The results summarized above served to illustrate threegeneral aspects of superfermions – announced in the titleof the paper – as applied to the real-time transport theorythat we discussed in Sec. III. Therefore these can be alsogeneralized to multiorbital Anderson quantum dots. (i) Causal structure of superfermions.
Using vari-ous examples, we illustrated that physical meaning canbe assigned to formal objects appearing in a Liouville-space theory of a strongly interacting, open fermionic sys-tem. Although many concepts carry over from the usualHilbert-Fock space, many others require careful reconsid-eration, e.g., the role of the super-kets in the expansionof a mixed state [Eq. (68)] or the superfermion number[Eq. (64)]. The crucial feature distinguishing quantumfields in Liouville-Fock space from those in Hilbert-Fockspace is what we refer to as the “causal structure”. Onthe one hand, this entails [Eq. (55)]¯ G ησ | Z R ) = 0 , (184)where | Z R ) ∼ ( − n is the fermion-parity operator ap-pearing in the corresponding superselection rule of quan-tum mechanics. Roughly speaking, this imposes the con-straint that “fermions on different Keldysh contours an-ticommute”. On the other hand, the identity [Eq. (54)]( Z L | ¯ G ησ = 0 , (185)where the ( Z L | = Tr represents the trace operation, isinvolved in the probability conservation of the densityoperator. As we showed in Sec. III B, the causal struc-ture implies much more than probability conservation ofthe dynamics. Whereas the former has received muchattention in Green’s function formalism, in density oper-ator approaches much less attention seems to have beengiven to this more fundamental structure. We emphasized the central importance of the unit op-erator | Z L ) ∼ as the Liouville-Fock space vacuum andits physical meaning as the T → ∞ maximally mixedstate. We used the T → ∞ limit as a point of refer-ence, not only in the construction of the Liouville-Fockspace but also in the calculation of the time-evolutionpropagator and its self-energy. This may be comparedwith the limit of infinite bias V b = ∞ , which also admitsan exact analysis. It has recently been studied by Oguriand Sakano and earlier by Gurvitz, while the rel-evance of renormalization corrections at finite bias werepointed out in Ref. 154. In comparison with this, we em-phasize that our formulation using the T → ∞ limit hasthe important technical advantage that it is provides aunique starting point irrespective of the number of reser-voirs. Moreover, it applies irrespective of the asymmetryof the tunnel couplings: the latter spoils the relation be-tween the V b → ∞ and T → ∞ limits for two electrodes,discussed in Ref. 155.Another interesting consequence of incorporating the T → ∞ limit is that the unperturbed evolution (i.e., thereference problem for the renormalized time-dependentperturbation theory) is dissipative and therefore dampedas a function of time. This may prove to be interestingfor numerical schemes that aim to calculate memory ker-nels. This damping depends only on the tunnelcouplings, in contrast to the broadening obtained by arecently proposed dressing scheme , cf. also , whichdepends on the quantum-dot energies and is based on apartial resummation of real-time diagrams that serves adifferent purpose.Finally, the T → ∞ limit also aids the physical under-standing of observables, such as the displacement part ofthe current (180). (ii) Fermion-parity protected decay mode. As shownin Sec. III C the striking independence of the key resultEq. (1), h ( − n i ( t ) ∼ e − Γ( t − t ) h ( − n i ( t )+ . . . , of all re-maining parameters including the interaction U relatesto a formal property of the general theory. Since thecausal superfermion approach uses the T → ∞ limit asa reference point, it reveals that finite-temperature cor-rections to the time evolution only involve creation su-peroperators ¯ G . Clearly then, the time evolution of thesuperket | Z R ) ∼ ( − n in Liouville space cannot haveany such correction: as expressed by Eq. (184), it is the“most filled” superket and simply cannot accommodatemore superfermions. Moreover, it is readily seen that any interacting N -spin-orbital Anderson model (orbitals l = 1 , . . . , N/
2) with quadratic tunnel coupling exhibitsexactly this purely exponential decay mode with rateΓ = P rσ,l Γ rσ,l . Finally, it is interesting to note thathalf of the decay modes that we studied in the noninter-acting case ( U = 0) are, in fact, fixed completely by the T = ∞ calculation. (iii) Super-Pauli principle. The super-Pauli principle(62) states that formal superkets cannot be “doubly oc-4cupied” : ( ¯ G ησ ) = 0 . (186)This simple consequence of the causal Liouville-Fockspace construction provides useful insights in two direc-tions. First, when applying the real-time approach tononinteracting problems, the super-Pauli principle is thekey simplification that keeps the calculations completelytractable on the superoperator level. The renormalizedperturbation theory is simple to set up, and a finite-order N calculation gives the exact result for N spinorbitals, in-cluding all local N -particle nonequilibrium correlations.The higher-order corrections vanish exactly, not by theirscalar magnitude but by their superoperator structure:generalizing Eq. (63) to the case for N spinorbitals wehave ¯ G m ... ¯ G = 0 for m > N, (187)as a direct consequence of the super-Pauli principle.The other major implication of taking the noninteract-ing limit, Eq. (138), can also be generalized to this caseby extending the simple considerations in Fig. 5 to evenorders m = 4 , . . . , N : the m/ m ( t, t ) = (188)1( m/ ( t, t ) e i ¯ L ( t − t ) ¯Π ( t, t ) . . . e i ¯ L ( t − t ) ¯Π ( t, t ) ( m/ . The superoperator algebraic structure thus carries im-portant physical information, which is naturally revealedby the causal superfermions. We emphasize that thesesimple general features of the noninteracting limit remainhidden in the real-time approach unless one starts fromthe renormalized the perturbation theory (99), incorpo-rating the wideband limit. We furthermore showed thatcertain observables, such as the charge current, turn outto be insensitive to corrections beyond the one-loop or-der. This raises a question of practical importance: givena physical M -particle quantity, to which loop order doesone need to calculate the self-energy in order to get theexact noninteracting result?This leads to the second important insight which isrelevant to applications of the real-time renormalizationgroup approach, which aims to provide a good so-lution in both the strong and weak interaction limits.Our complementary frequency-space calculation of thestationary limit in Sec. IV B confirmed that the real-timerenormalization group in the one- plus two-loop approx-imation correctly reproduces the exact noninteractinglimit, in particular the self-energy part relevant to thecurrent, relating this to the super-Pauli principle. Weadditionally calculated the stationary state in this limit,obtaining the exact effective Liouvillian by a two-loop order calculation. For generalized Anderson models with N spin-orbitals, we inferred above that at least a N -loopcalculation in the renormalized perturbation theory will reproduce the exact noninteracting limit. This impliesthat real-time renormalization group schemes must in-clude at least a consistent N -loop RG flow for the Liouvil-lian together with the corresponding vertex corrections inorder to capture the exact noninteracting limit. In viewof the complications encountered in Ref. 30, already atthe N = 2-loop order for the Anderson model, a ques-tion becomes practically relevant under which conditionsmay higher loop orders be avoided, (e.g., for a given ob-servable or specific density operator component)? Herewe should point out that it can be shown that if oneis interested in the evolution of single-particle quantities(expressible through two (super)fields) only one-loop di-agrams are required for the time-dependent decay in thenoninteracting limit. This carries over to multiparticlequantities only under the condition that initial correla-tions on the dot are absent, i.e, these factorize at theinitial time (see main result Eq. (145), cf. Eqs. (146) and(140)). When initial correlations are present, however,higher loop evolution does matter. Note that even whenthe initial density operator contains nonfactorizable cor-relators, our key result Eqs. (138) and (188) shows thatthe time-evolution superoperator can still be factorized.We also found that the noninteracting limit becomesmost transparent when considering the renormalized two-loop propagator ¯Π ( t, t ) in time space (rather than itsLaplace transform), because it factorizes in the limit U = 0. This seems to have no equally simple counter-part in Laplace space for the two-loop renormalized self-energy ¯Σ ( z ). The generalized time-space relations (188)allow for a convenient verification on the superoperatorlevel that a real-time RG scheme correctly reproducesthe noninteracting limit in all nonvanishing loop orders m/ , , . . . , N . C. Limitations and further extensions
Our considerations were quite general. We now endwith comments on the limitations imposed by our as-sumptions and provide an outlook on how these may beovercome.
Noninteracting limit ( U = 0 ) Although we focusedin Sec. IV on the noninteracting limit for illustrativepurposes, the principles demonstrated here, can be ap-plied to interacting problems, as we showed, e.g., inSec. II D 3 b. This is what has motivated our exhaus-tive study all along. A more advanced example is ourRG study Ref. 30, but other approaches may also be de-veloped. For instance, one may consider expanding thetime-evolution propagator Π or its self-energy ¯Σ in the nonlinear part of the Coulomb interaction, i.e., not sim-ply in the parameter U , but in the term U ( − n / U and Γ rσ (the quadratic term(74a) also depends on U , and this part is treated non-5perturbatively). The coefficients for the m th-order termin this nonlinear interaction can be calculated using ourcausal superfermion approach through an expansion inthe Keldysh correlation function ¯ γ , which is truncated asin the noninteracting limit, but now at the (2 + m )-looporder. Also, here the result simplifies due to the super-operator structure dictated by the super-Pauli principle.
Wide-band limit
The wideband limit is another mainsimplifying assumption that we made in Sec. III B. How-ever, beyond this limit the number of Keldysh contrac-tions (¯ γ ) is still limited to two by the super-Pauli princi-ple (62)-(63) in the non-interacting limit ( U = 0). Thisexpresses the general fact that retarded contractions (˜ γ )always connect a creation and an annihilation superoper-ator, and thus its contribution does not change the totalsuper particle number. In contrast, Keldysh contractions¯ γ always connect two creation superoperators, increasingthe total superfermion number by two. Since the super-Pauli principle limits the total number of superfermionsto four, at most, two Keldysh contractions are allowed.This illustrates how common physical reasoning basedon the usual second quantization can be transferred tononequilibrium problems using our causal superfermions,aiding the solution of physical problems. Initial system-reservoir factorization
The assumptionof factorizing system-reservoir correlations at the initialtime is not that restrictive either. Much of the technicaland physical conclusions presented can be generalized toapply also to the case of nonfactorizing system-reservoirinitial conditions and will be discussed in a forthcomingwork. Time-dependent parameters
So far, we have focused onthe time evolution of the quantum dot to the new sta-tionary state after the tunnel couplings Γ rσ experiencea sudden change (quench) at t = t from Γ rσ = 0 to aset of finite values which further remain unchanged for t > t . Although Refs. 75, 77, and 134 also studied thisproblem, the main motivation here was to illustrate theadvantages of the causal superfermion approach in thismost simple setting. However, our formalism can be eas-ily extended to deal with a time dependence of all theparameters involved, i.e., ǫ = ǫ ( t ), B = B ( t ), V b = V b ( t ),and Γ rσ = Γ rσ ( t ) as we briefly outline. First, we notethat once we consider the wideband limit, Eq. (94) re-mains valid if the parameters vary much slower than theinverse band width, which always seems to be experi-mentally given. This allows us to integrate out the re-tarded reservoir contractions also in this case and obtainan infinite-temperature kernel ˜Σ, cf. Eq. (95), but nowwith a time-dependent Γ rσ ( t ) and a corresponding time-dependent renormalized Liouvillian ¯ L ( t ) = L ( t )+ ˜Σ( t ), cf.Eq. (98). The interaction-representation vertices (100)now include a time-ordering superoperator T and reducesin the noninteracting case to a result similar to Eq. (119):¯ G j ( t ) = T e − i R tt dτ ¯ L ( τ ) ¯ G j (cid:16) T e − i R tt dτ ¯ L ( τ ) (cid:17) − (189)= e R tt dτ [ iηǫ σ ( τ )+ 12 Γ σ ( τ )] ¯ G for U = 0 (190) The main ideas of our approach thus remain the same andapply also to multiorbital extensions without any cru-cial complications arising. In particular, the super-Pauliprinciple (62) is also valid for the above field superoper-ators and causes the perturbation series to terminate ata finite order as in Eq. (122), which is one of the centralinsights of this paper. ACKNOWLEDGMENTS
We acknowledge useful discussions with M. Hell, M.Pletyukhov, H. Schoeller, and J. Splettstoesser.
Appendix A: Field superoperators
In this Appendix, we provide further comments onthe construction of field superoperators undertaken inSec. II D 1. As discussed in Sec. II A, we emphasizethe importance of starting this construction from setsof fermionic operators d and b (or a ) for the quantumdot and the reservoirs respectively, which mutually com-mute . The crucial advantage of using such field operatorsis that Eq. (54) holds for the partial traces of the corre-sponding dot or reservoir superoperators G q [Eq. (46)]and J q [Eq. (47)]. This allows one to obtain the reduceddynamics of the dot (by integrating out the reservoir de-grees of freedom), while preserving the causal propertiesEq. (54), which we have shown to bring many computa-tional and physical insights.If one uses in Eq. (46) and (47) instead of d and b mutually anticommuting sets of fermion operators d ′ and b ′ = p Γ rσ / πa ′ , constructed in Sec. II A, thenone obtains the same anticommutation relations (50),(51) for the resulting field superoperators. However, inthis case no definite commutation relations analogous toEq. (52) are obtained (neither commutation, nor anti-commutation relations), which is a major disadvantage.In principle, one can introduce other sets of field super-operators which are free of this problem even thoughone starts again from the anticommuting fields d ′ and b ′ = p Γ rσ / πa ′ . Instead of using Eq. (46) and (47),one defines G q • = 1 √ n d ′ • + q ( − n + n R • d ′ ( − n + n R o , (A1) J q • = 1 √ n b ′ • − q ( − n + n R • b ′ ( − n + n R o . (A2)Here, one uses, in contrast to Eq. (46)-(47), the global fermion parity operator ( − n + n R . The field superoper-ators G and J can be checked to satisfy the same anti-commutation relations Eq. (50)-(51) as G , J and thesame super adjoint relation. In contrast to Eq. (52),they satisfy instead mutual anti commutation relations6Eq. (52):[˜ J , ˜ G ] + = [¯ J , ¯ G ] + = [¯ J , ˜ G ] + = [˜ J , ¯ G ] + = 0 . (A3)However, the disadvantage of this construction is thatinstead of the causal property Eq. (54) we now have Tr ¯ G = 0 , Tr ˜ J = 0 , (A4)where Tr = Tr D Tr R is a global trace, while the crucial local -trace identities Eq. (54) are not valid anymore: Tr D ¯ G , Tr R ˜ J . (A5)This seems to drastically complicate the calculation ofthe partial reservoir trace required in Sec. III. Appendix B: Time representation for the Keldyshcontraction
In the wide-band limit, the explicit form of the time-dependent Keldysh correlation function ¯ γ , ( t − t )[Eq. (90)] can be obtained using the partial fraction ex-pansion for the meromorphic functiontanh( z ) = + ∞ X n = −∞ ( z + iπ ( n + 1 / − . (B1)Closing the contour of integration over ω in the lowerhalf of the complex plane and making use of the residualtheorem, we obtain Eq. (101) of the main text as follows:¯ γ , ( t ) = Γ2 π Z dωe − iη ( ω + µ ) t tanh( ηω/ T ) δ , ¯1 = − i T e − iηµt Γ + ∞ X n =0 e − πT (2 n +1) t δ , ¯1 = − i Γ T sinh ( πT t ) e − iηµt δ , ¯1 . (B2)Here, as in Eq. (94) the multi-indices 2 , ¯1 in the δ -functiondo not contain the reservoir frequencies. Appendix C: Integrals of Keldysh contraction -digamma function
In Eq. (161), (166) and (181), we used the followingresult, obtained from Im Ψ( z ) = − Im P + ∞ n =0 / ( n + z ),the expansion (B1) of tanh( z ), and application of theresidual theorem (closing the integration contour in the lower half of the complex plane): Z + ∞−∞ dx Γ tanh( x/ T )Γ + ( x − ǫ ) = − Im Z + ∞−∞ dx tanh( x/ T ) i Γ + ǫ − x (C1)= − Im + ∞ X n = −∞ Z + ∞−∞ dx x/ T + iπ ( n + 1 /
2) 1 i Γ + ǫ − x = Im + ∞ X n =0 n + 1 / Γ2 πT − iǫ πT == − Im Ψ (cid:18)
12 + Γ − iǫ πT (cid:19) . (C2) Appendix D: Evaluation of the functions F + rσ ( t ) and F − rσ ( t ) Here, we present the calculation of the function F + rσ (∆ t ) given in Eq. (134a). We use the expansion1sinh( x ) = 2 e − x − e − x = 2 ∞ X n =0 e − (2 n +1) x , (D1)which holds for any positive x since e − x < F + rσ (∆ t ) := − Z ∆ t dτ T sin( ǫ rσ τ )sinh ( πT τ ) e − Γ τ (D2)= − T Im ∞ X n =0 Z ∆ t dτ e ( iǫ rσ − Γ) τ − πT τ (2 n +1) =2 T Im ∞ X n =0 − e ( iǫ rσ − Γ)∆ t − πT ∆ t (2 n +1) iǫ rσ − Γ − πT (2 n + 1)= 1 π Im Ψ (cid:18)
12 + Γ − iǫ rσ πT (cid:19) (D3)+ Im (cid:26) e ( iǫ rσ − Γ − πT )∆ t π Φ (cid:18) e − πT ∆ t ; 1; 12 + Γ − iǫ rσ πT (cid:19)(cid:27) , where Ψ( z ) = − γ − P ∞ k =0 [1 / ( z + k ) − / (1 + k )] is thedigamma-function, and Φ( z ; s ; ν ) = P ∞ n =0 z n / ( n + ν ) s isthe Lerch transcendent (see, e.g., Ref. 131). Analogously,we obtain for the function F − rσ ( t ) [Eq. (134b)]: F − rσ (∆ t ) := e − t Z ∆ t dτ T sin( ǫ rσ τ )sinh( πT τ ) e Γ τ (D4a)= − e − t T ∞ X n =0 Im ∆ t Z dτ e ( iǫ rσ +Γ) τ − πT τ (2 n +1) (D4b)= − e − t (cid:0) F + rσ (∆ t ) | Γ →− Γ (cid:1) = Im (cid:26) e ( iǫ rσ − Γ − πT )∆ t π Φ (cid:18) e − πT ∆ t ; 1; 12 − Γ + iǫ rσ πT (cid:19) + e − t π Ψ (cid:18)
12 + − Γ + iǫ rσ πT (cid:19)(cid:27) . F − rσ (∆ t ) canhave a pole at Γ = 2 πT k ( k = 1 , , ... ) for ǫ rσ = 0, sinceboth Ψ (cid:0) + − Γ+ iǫ rσ πT (cid:1) and Φ (cid:0) e − πT ∆ t ; 1; − Γ+ iǫ rσ πT (cid:1) have it. However, the pole of the Ψ function exactlycompensates the pole of the Φ function giving zero inthat case. That this should be the case is already clearfrom Eq. (D4b) by taking ǫ rσ = 0. Appendix E: Two-loop contributions to thetime-evolution
In this Appendix, we write out the proof of the factor-ization (138), presented diagrammatically in Fig. 5. Themanipulations that we apply are analogous to those ofRef. 41 for two-loop calculations. That reference, how-ever, deals with self-energy diagrams for interacting sys-tems in the zero frequency limit. We start from Eq. (137),repeated here for convenience:¯Π ( t, t ) = e − i ¯ L ( t − t ) × Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t ¯ G ′ ( t ) ¯ G ′ ( t ) ¯ G ′ ( t ) ¯ G ′ ( t ) × X h i,j,k,l i ( − P ¯ γ ij ( t i − t j )¯ γ kl ( t k − t j ) . (E1)Here, we sum over the following possible contractions: i, j, k, l = 4 , , , , , , , , , γ ( t − t )¯ γ ( t − t ). In the irreducible contrac-tions, this changes the order of the vertices from¯ G ′ ( t ) ¯ G ′ ( t ) ¯ G ′ ( t ) ¯ G ′ ( t ), but for each of these we canrestore this order by anticommuting the creation super-operators [Eq. (120)]. This puts the superoperators con-nected by a ¯ γ -contraction adjacent to each other, i.e., onedisentangles the contractions: therefore the sign appear-ing from anticommutation of the creation super operatorsprecisely cancels the fermionic Wick sign ( − P . We areleft with¯Π ( t, t ) = e − i ¯ L ( t − t ) × (E2) h Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t + Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t + Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t i × X ¯ G ′ ( t ) ¯ G ′ ( t ) ¯ G ′ ( t ) ¯ G ′ ( t )¯ γ ( t − t )¯ γ ( t − t ) . By duplicating these terms, while compensating by a fac-tor 1/2, and interchanging the dummy variables t , t ↔ t , t in the duplicates, we obtain a sum of integrals whichcan be factorized as Eq. (138) by comparing with the def- inition of ¯Π ( t, t ) [Eq. (127)]:¯Π ( t, t ) = e − i ¯ L ( t − t ) × (E3a) h Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t + Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t + Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t + Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t + Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t + Z dt dt dt dt t ≥ t ≥ t ≥ t ≥ t ≥ t i × X ¯ G ′ ( t ) ¯ G ′ ( t ) ¯ G ′ ( t ) ¯ G ′ ( t )¯ γ ( t − t )¯ γ ( t − t )= e − i ¯ L ( t − t ) × Z dt dt t ≥ t ≥ t ≥ t X ¯ G ′ ( t ) ¯ G ′ ( t ) Z dt dt t ≥ t ≥ t ≥ t X ¯ G ′ ( t ) ¯ G ′ ( t )= ¯Π ( t, t ) e i ¯ L ( t − t ) ¯Π ( t, t ) . (E3b) Appendix F: Spin-channel decomposition
In this Appendix we outline the calculation of the prop-agator by using the noninteracting limit in the first step and then setting up the perturbation theory. (In con-trast to this, the calculations given in Sec. IV first set upthe perturbation theory and make use of the assump-tion U = 0 in the last step , and rather aim to showhow in a framework applicable to interacting systems,this limit is achieved. The approach we now outline,although shorter, does not make that clear.) In particu-lar, we use that for U = 0 the quantum dot Liouvilliandecomposes into single-spin species, L = P σ L σ . Dueto this special property, the total Liouvillian decomposesinto commuting spin-resolved parts, L tot = P σ L tot σ with L tot σ = L σ + L Rσ + L Vσ , since the reservoirs are noninteract-ing, the tunnel coupling (22) is quadratic, and all spin-dependencies (due the junctions and the magnetic field)are considered to be collinear. Here L σ , L Rσ , and L Vσ are obtained from Eqs. (74), (81), and (48), respectively,by leaving out the sum over the spin-index σ . One nowsplits the propagator (37) into commuting factors relat-ing to different spins:Π( t, t ) = Tr R (cid:16) e − iL tot ( t − t ) ρ R (cid:17) • (F1a)= Tr R (cid:16) e − iL tot ↑ ( t − t ) e − iL tot ↓ ( t − t ) ρ R ↑ ρ R ↓ (cid:17) • (F1b)= Π ↑ ( t, t )Π ↓ ( t, t ) , (F1c)where ρ Rσ = Q r e − T ( H rσ − µ r n rσ ) Z rσ andΠ σ ( t, t ) = Tr Rσ (cid:16) e − iL tot σ ( t − t ) ρ Rσ (cid:17) • . (F2)where the trace runs over one spin-degree of freedom.The superoperator Π σ ( t, t ) can be again calculated usingthe renormalized perturbation series, Eq. (122), whichthe super-Pauli principle now truncates at the one-loop8order: Π σ ( t, t ) = ¯Π σ ( t, t ) + ¯Π σ ( t, t ) . (F3)Here, ¯Π σ ( t, t ) = e − i ¯ L σ ( t − t ) with ¯ L σ = L σ + ˜Σ σ . Inturn, ˜Σ σ and ¯Π σ ( t, t ) are defined by Eqs. 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M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod.Phys. , 1665 (2009) The definitions G ± in Eq. (46) and J ± in (47) differ fromthose in Ref. 30 [Eqs.(51) and (52) respectively] by usingthe opposite convention for the sign of q . This is effectedby placing the field operators d and b to the right ofthe operators ( − n and ( − n R in Eqs. (46) and (47),respectively. In contrast to Eqs.(51) and (52) in Ref. 30,we here explicitly write the superoperators p L n • = p n • p n ,( p = ± , p n = 1). However, the definitions of ¯ G and ˜ G Eq. (49) coincide with those of Ref. 30. Here, we favor q =+ for “creation” ( G + = ¯ G ) and q = − for “destruction”superoperators ( G − = ˜ G ) [but opposite in the reservoirs,cf. Eq. (47)], tying in better with the discussion of secondquantization to Liouville space. The opposite conventionused in Ref. 30 agrees better with that used for “quantum”and “classical” fields obtained by the Keldysh rotation inthe path integral approach of Kamenev et. al. . Although one should in general carefully distinguish be-tween Hermitian and super-Hermitian conjugation, we usethe same notation † for both since it is always clear fromthe context whether we deal with a super- or usual oper-ator. Here, we insert ( − n for the argument of Eq. (56a) (thebullet • ) and make use of ( − n ( − n = 1. R. B. Saptsov and M. R. Wegewijs, (in preparation)(2013)
The fermion-parity guarantees orthogonality,( α +1 | α − ) ∝ Tr D d † ( − n d = Tr D ( − n d d † = 0, since Tr D (( − n • ) = P n ↑ n ↓ ( − n ↑ + n ↓ h n ↑ n ↓ | • | n ↑ n ↓ i = 0 forany one-particle operator since the matrix element isindependent of n ↑ or n ↓ (or both). This connects to the other possible construction of theLiouville-Fock space, starting from the fermion-parity op-erator | Z R ) = ( − n as the vacuum state, which is anni-hilated by the creation operator G +1 = ¯ G by the funda-mental relation (55): ¯ G ησ | Z R ) = 0. See Ref. 30, AppendixE, for a systematic discussion. Note that the total superoperator P η,σ N ησ counting theoccupation of the basis superkets Eqs. (60)-(61) does not equal the superoperator defined by the commutator withthe particle number operator, [ n, • ]. In Eq. (76) all free dot propagators e − iL ( t k +1 − t k ) betweenthe vertices G q k +1 k +1 and G q k k for k = 2 are denoted as el-lipsis “...” for compactness. M. Gaudin, Nucl. Phys. , 89 (1960) A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B , 5528 (1994) Also here one must be careful with physically interpret-ing the expressions: the action of a field superoperator G q results in a Liouville-space superposition of terms with adefinite Keldysh contour index. Only the latter can beidentified with a “process” generated by the total Hamil-tonian H tot [Eq. (32)]. See Secs. II 3b and II 3c in Ref. 30 for a correspondingargument in frequency space.
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For the full expression for Eq. (113b) see Eq. (135) inRef. 30.
The T → ∞ self-energy ˜Σ is quadratic in general dueto the wideband limit. Therefore, the Liouville-Fock basisEqs. (60)-(61) is an eigenbasis, also when generalized tomulti-orbital Anderson models. However, when includingmore general interaction terms in L for such models, itmay be that the bosonic blocks of L and ˜Σ do not com-mute and have no common eigenbasis. In this case, thediagonalization of ¯ L may be less simple. Note that fermionic basis superkets are not super eigen-kets of L for U = 0, due to Eq. (77)], see Ref. 30. For finite temperature and U = 0 the fermionic part of¯ L , not given in Eq. (114), is needed to describe virtualintermediate renormalized time evolution, see the discus-sion of Eq. (76). In the U = 0 illustrations in this paper,it is not needed. To obtain Eqs. (124)-(125), (i) insert the finite expansionsEq. (122) and (123) into the renormalized Dyson equation(104), (ii) use that ¯Σ and ¯Σ commute with ¯Π up tounimportant c-number factors and (iii) use that – ignor-ing ¯Π – by Eq. (62): ( ¯Σ ) α ( ¯Σ ) β = 0 when it contains2 α + 4 β ≥ α, β ) = (0 , , (1 , , (2 , , (1 , For U = 0 Eq. (121) is no longer valid. In this case,nonzero terms arise beyond the second order in the gen- eral expansion Eq. (99b). The reason is that if one triesto commute ¯ L = L + ˜Σ to the left – containing a quarticterm [Eq. (74)] – this generates terms containing destruc-tion superoperators ˜ G which are not required to vanishby the super-Pauli principle. We note that by our discus-sion of Eq. (76) (taking q = q = q = q = +) this effectof the quartic terms in L is restricted to only half of thepropagators in virtual intermediate states. As pointed out in Ref. 30, for selected quantities such asthe charge current a one-loop RT-RG approximation al-ready includes the noninteracting limit. A recent pertur-bative study indicates that for thermoelectric transportsuch fortuitous simplifications are absent . Here “quartic” refers to the total power of field operators¯ G ′ ( t ) in the (renormalized) interaction picture [Eq. (100)].For U = 0 this is proportional to the original creation su-peroperator ¯ G [Eq. (119)]. However, for U = 0 explicitcalculation of ¯ G ′ ( t ) shows it additionally contains de-struction superoperators ˜ G . In this case, the super-Pauliprinciple cannot truncate the perturbation series after thesecond order. Alternatively, one can make use of the explicit basis ex-pansion of the creation superoperator ¯ G defined by Eq.(118) and Eq. (120) in Ref. 30. A. Erd´ely, W. Magnus, F. Oberhettinger, and F. Tricomi,
Higher transcendental functions , Vol. 1 (Mc. Graw-HillBook Company Inc., 1953)
Note that the stationary contributions to ρ ( t ) are gener-ated by terms in Π( t, t ) [Eq. (141)] of the form | A ( t ))( Z L | : since by probability conservation any possible ρ ( t ) con-tains | Z L ) in its expansion [see Eq. (68) with t = t ], wehave | A ( t ))( Z L | ρ ( t ) = | A ( t )), independent of ρ ( t ). This can be seen from Eq. (151) in Ref. 30.
T. L. Schmidt, P. Werner, L. M¨uhlbacher, and A. Komnik,Phys. Rev. B , 235110 (2008) For spin-independent tunneling Γ rσ = ˜Γ this 3˜Γ decayrate already appeared as the imaginary part of one ofthe eigenvalues of the noninteracting Anderson model( U = 0), see Eq. (244) of Ref. 30. This eigenvalue wasfound to drop out of the calculation of the station-ary current and the single-level occupancies. Note thatin Ref. 30 Eq. (244) does not follow from RG considera-tions, but rather directly from the renormalized pertur-bation theory, in particular Eq. (163) there. It also doesnot make use of the T → A. Komnik, Phys. Rev. B , 245102 (2009) That, indeed, no zero can appear in the denominator isrelated to the causal structure Eq. (156) and to the sta-tionary properties of the infinite-temperature self-energydiscussed in Sec. III C 3. By the causal property Eq. (54),the creation superoperators ¯ G have a left zero eigenvec-tor ( Z L | which is unique , as mentioned in Sec. III C 2. Atthe same time, | Z L ), the stationary T → ∞ density op-erator, is the unique right zero eigenvector of ¯ L = L + ¯Σ,even when accounting for the both the bosonic and thefermionic diagonal blocks. Therefore, when inserting acomplete set of superkets in the superoperator expressions( z − ¯ L + X k ) − ¯ G the only zero of ¯ L , occurring for | Z L ),is canceled by the vanishing of the projection ( Z L | ¯ G .Finally, one could worry that insertion of | Z L )( Z L | couldgenerate a zero in the leftmost expression ¯ G ( z − ¯ L + X k ) − in Eq. (156), since for the T → ∞ limit the superad-joint | Z L ) is the corresponding right zero eigenvector of the effective Liouvillian ¯ L [cf. discussion in Sec. III C 3].However, this zero is canceled as well by the adjacent cre-ation superoperator on the right: ¯ G | Z L )( Z L | ( z − ¯ L + X k ) − ¯ G = ¯ G ( z + X k ) − | Z L )( Z L | ¯ G = 0. T. Korb, F. Reininghaus, H. Schoeller, and J. K¨onig,Phys. Rev. B , 165316 (2007) That the rate 3˜Γ drops out for U = 0 was shown in Ref. 30,Sec. III.C.3, due to the vanishing of a propagator super-matrix element, Eq. (246) of that reference. For U = 0,this matrix element does not vanish, and as a result therenormalization group flow is affected by this decay rate. The limit considered in Ref. 30 corresponds to the case:Γ rσ = ˜Γ for all r, σ . The coefficient before the first termin Eq. (168) is then Γ = P r,σ Γ rσ = 4˜Γ. This Γ [from(168)] should not be confused with one used in Ref. 30,the latter corresponds to the ˜Γ in the present notations. Tr D Σ r = 0 does not follow from probability conservation,which only requires Tr D P r Σ r = 0. The causal structureis a stronger constraint. Note that although ˜Σ r ( t, t ′ ) is the (reservoir-resolved) T → ∞ self-energy, ˜ I r ( t ) is not simply the T → ∞ cur-rent. In Ref. 134, also the first-order expansion in U as wellas Monte Carlo simulations for an arbitrary U were dis-cussed. The results of Ref. 77 seem to be at variance withother known solutions and with ours. These authorsused a Feynman path integral to derive a convolution-less master equation. We have carefully checked our re-sult against theirs by using the identity
T / sinh( πT τ ) = i R + ∞−∞ e − iωτ tanh (cid:0) ω T (cid:1) dω π to rewrite our explicitly eval-uated current Eq. (179) in the form of an uneval-uated ω integral, as is done in Ref. 77. We findthat our result, Eq. (179) (and therefore that of oth-ers ) can only be recovered by adding by hand aterm to Eq.(35) of Ref. 77 that seems to be missing:Γ rσ e − (Γ σ / R dω π [ (Γ σ /
2) cos(( ω − ǫ )∆)( ω − ǫ ) +(Γ / − ( ω − ǫ ) sin(( ω − ǫ )∆)( ω − ǫ ) +(Γ / ],considering only one spinorbital σ , as has been done inRef. 77. The result for the initial current in Ref. 77, − Γ rσ n σ ( t ), also deviates from the results of other worksand our Eq. (180), by missing the term Γ rσ / σ ). As a consequence, the result ofRef. 77 also does not seem to fit in the intuitive physicalpicture sketched in Ref. 134 after Eq. (36) and here af-ter Eq. (180). It is puzzling why, despite these differences,the result of Ref. 77 for the average dot occupation num-ber does coincide with that of other works and withour Eq. (143). The authors of Ref. 77, discussing their re-sults for the current, Eqs. (38-39a) in Ref. 77, only men-tion that that “some of their results [for the the current]were also obtained using nonequilibrium Green functiontechnique ” without mentioning or investigating the dis-crepancy with Ref. 134. The results of the present paperand of Ref. 75 confirm the results of Ref. 134 withoutusing the t → −∞ limit which can thus not be usedto explain the difference with the results Ref. 77. The useof this limit, commonly assumed in Keldysh Green’s func-tion techniques, was criticized in Ref. 77 and distinguishestheir method from that of Ref. 134. In Ref. 134 the artificial instantaneous current due to thewideband limit was physically related to the infinite bias limit V b → + ∞ . In Ref. 155 it was shown that for sym- metric tunnel coupling this is equivalent to the T → ∞ limit. Here we discuss spin- and reservoir-dependent tun-neling for which no such equivalence seems be known. Still, the T → ∞ limit provides the essential physicalstarting point. Moreover, our analysis simply extends toan arbitrary number of reservoirs. J. K¨onig and J. Martinek, Phys. Rev. Lett. , 166602(2003) J. Martinek, M. Sindel, L. Borda, J. Barna´s, J. K¨onig,G. Sch¨on, and J. von Delft, Phys. Rev. Lett. , 247202(2003) M.-S. Choi, D. S´anchez, and R. L´opez, Phys. Rev. Lett. , 056601 (2004) M. Braun, J. K¨onig, and J. Martinek, Phys. Rev. B ,195345 (2004) J. Martinek, M. Sindel, L. Borda, J. Barna´s, R. Bulla,J. K¨onig, G. Sch¨on, S. Maekawa, and J. von Delft, Phys.Rev. B , 121302 (2005) Formally, for U = 0 the stationary state has aform that allows for a Wick theorem on the quan-tum dot , although with unknown nonequilibrium pa-rameters Φ σ ( ∞ ) = h n σ i ( ∞ ) − /
2. This is espe-cially clear in terms of superfermions: there is a for-mal analogy between the nonequilibrium dot station-ary state Eq. (169b), which can be written as ρ ( ∞ ) = exp (cid:0) P σ Φ σ ( ∞ ) ¯ G + σ ¯ G − σ (cid:1) | Z L ), and the thermal-equilibrium state, written as ρ eq = exp( − ǫ σ n σ /T ) /Z = exp (cid:0) − P σ tanh( ǫ σ / T ) ¯ G + σ ¯ G − σ (cid:1) | Z L ). The nonequi-librium analog of the fluctuation-dissipation relationEq. (88) for the dot then reads as ˜ G ρ ( ∞ ) = − η Φ σ ¯ G ρ ( ∞ ). The Wick theorem for the dot in thiscase can be proven in full analogy with the equilibrium reservoir Wick theorem, although the dot Keldysh con-traction functions are not, in general, expressible viatanh( ǫ/ T ): ¯ γ dot = h ˜ G ˜ G i ∝ − η Φ σ tanh( η ǫ/ T ).Before stationarity is reached ( t < t < ∞ ) this analogydoes not apply unless one starts at t from a quantum-dotstate without two-particle correlations. S. A. Gurvitz and Y. S. Prager, Phys. Rev. B , 15932(1996) S. A. Gurvitz, Phys. Rev. B , 15215 (1997) B. Wunsch, M. Braun, J. K¨onig, and D. Pfannkuche,Phys. Rev. B , 205319 (2005) A. Oguri, J. Phys. Soc. Jpn. , 2969 (2002) M. Marthaler, Y. Utsumi, D. S. Golubev, A. Shnirman,and G. Sch¨on, Phys. Rev. Lett. , 093901 (2011)