Master equation based steady-state cluster perturbation theory
Martin Nuss, Gerhard Dorn, Antonius Dorda, Wolfgang von der Linden, Enrico Arrigoni
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Master equation based steady-state cluster perturbation theory
Martin Nuss, ∗ Gerhard Dorn, Antonius Dorda, Wolfgang von der Linden, and Enrico Arrigoni
Institute of Theoretical and Computational Physics,Graz University of Technology, 8010 Graz, Austria (Dated: August 23, 2018)A simple and efficient approximation scheme to study electronic transport characteristics ofstrongly correlated nano devices, molecular junctions or heterostructures out of equilibrium is pro-vided by steady-state cluster perturbation theory. In this work, we improve the starting point ofthis perturbative, nonequilibrium Green’s function based method. Specifically, we employ an im-proved unperturbed (so-called reference) state ˆ ρ S , constructed as the steady-state of a quantummaster equation within the Born-Markov approximation. This resulting hybrid method inheritsbeneficial aspects of both, the quantum master equation as well as the nonequilibrium Green’sfunction technique. We benchmark the new scheme on two experimentally relevant systems in thesingle-electron transistor regime: An electron-electron interaction based quantum diode and a triplequantum dot ring junction, which both feature negative differential conductance. The results of thenew method improve significantly with respect to the plain quantum master equation treatment atmodest additional computational cost. PACS numbers: 71.15.-m, 71.27+a, 73.63.-b, 73.63.Kv
I. INTRODUCTION
Electronic transport in the realm of molecular scalejunctions and devices has become a subject of intensestudy in recent years.
Nowadays the controlled as-sembly of structures via electro migration, the con-tacting in mechanical break-junction setups, elec-tronic gating and measurement via scanning tun-nelling microscopy have become established tools,ultimately opening routes from elementary understand-ing to device engineering. Prompted by these formidableadvances in experimental techniques, the characteriza-tion of transport through e.g. molecules bound byanchor groups to metal electrodes, heterostruc-tures or nano structures on two- dimensional sub-strates has become feasible. These constitute thefoundation for future applications in electronic devicesbased on single electron tunnelling, quantum interfer-ence effects, spin control or even quantum many-body effects like Kondo behaviour. Typically the electronic transport through such de-vices is significantly influenced by electronic correlationeffects, which may become large due to the reduced ef-fective dimensionality and/or confined geometries. Thisis reflected, for instance, in major discrepancies betweenexperimental and theoretical current-voltage characteris-tics obtained with (uncorrelated) nonequilibrium Green’sfunction calculations based on ab-initio densityfunctional theory states.
The inclusion of many-body effects in the theoretical description of fermionicsystems out of equilibrium is challenging and anactive area of current research.
Suitable approxima-tions need to be devised in order to solve a finite stronglycorrelated quantum many-body problem out of equilib-rium coupled to an infinite environment. Typically, thenonequilibrium setup consists of a correlated central re-gion (system) attached to two leads (environment). A well-established method for treating such open quan-tum systems is by means of quantum master equations(Qme).
Herein, the environment-degrees of freedomare integrated out and usually incorporated in a pertur-bative manner. The Qme approach allows a detailed in-vestigation of transport phenomena and recent self-consistent extensions attempt to cure some of its long-standing limitations. In the framework of nonequilibrium Green’s functions(NEGF) various schemes exist to approximately calcu-late the electronic self-energy of the correlated region,see e.g. Ref. 68,82–87. In cluster approaches, such ascluster perturbation theory (CPT) and its improvement,the variational cluster approach (VCA), the whole sys-tem is partitioned into parts which can be treated ex-actly and determine the self-energy. Originally devisedfor strongly correlated systems in equilibrium, bothapproaches have recently been extended to nonequilib-rium situations in the time dependent case as wellas in the steady-state. In previous work we appliedthe steady-state CPT (stsCPT) to obtain transport char-acteristics of heterostructures, quantum dots andmolecular junctions and obtained good results evenin the challenging Kondo regime. A key issue in the CPT approach is to identify an ap-propriate many-body state for the disconnected corre-lated cluster in the central region, as a starting point ofperturbation theory, the so-called reference state. Up tonow, a common choice in stsCPT is to use an equilibriumstate at some temperature T S (often T S = 0) and chemi-cal potential µ S in-between the values of the leads. Suchan ad-hoc choice is clearly unsatisfactory. Furthermore,it fails to describe certain quantum interference effectsin transport phenomena as for example so-called currentblocking effects. The purpose of the present work is to improve onstsCPT by constructing a consistent and conceptuallymore appropriate reference state, given by the steady-state reduced many-body density matrix ˆ ρ S obtainedfrom a Qme in the Born-Markov approximation. Withinthis quantum master equation based stsCPT (meCPT),the ambiguity in defining µ S and T S for the central re-gion is resolved. The equilibrium case, in which µ S and T S coincide with those of the environment, is automati-cally included. In contrast to standard Qme approaches,lead induced level-broadening effects are accounted forand the noninteracting limit is reproduced exactly, as inthe original stsCPT. In addition, meCPT is able to cap-ture the previously mentioned current blocking effects,as shown below.Other NEGF/Qme hybrid methods exist in theliterature. For instance, in a recent work we have proposed a so-called auxiliary master equation(AME) approach, whereby a Lindblad equation is intro-duced which models the leads by a small number of bathsites plus Markovian environments. The AME is suitedto address steady-state properties of single impurityproblems as encountered in the framework of nonequilib-rium dynamical mean field theory.
In con-trast, the meCPT presented in this work is more appro-priate to treat non-local self-energy effects which cannotbe captured by single-site DMFT.This paper is organized as follows: After defining themodel Hamiltonian in Sec. II, the meCPT is introducedin detail in Sec. III. We present results obtained withthe improved method for two experimentally realizabledevices: i) In Sec. IV A, an electron-electron interactionbased quantum diode, ii) and in Sec. IV B, a triple quan-tum dot ring junction which both feature negative differ-ential conductance (NDC).For ring systems, extensive Qme results and an ex-planation of the NDC in terms of quantum interferencemediated blocking are available in Ref. 45,46.
II. MODEL
We consider a model of spin- fermions, having in mindthe electronic degrees of freedom of a contacted nanostructure, heterostructure or a molecular junction. TheHamiltonian consists of three parts:ˆ H = ˆ H S + ˆ H E + ˆ H SE . (1a)i) The “system” ˆ H S represents the interacting centralregion i.e. the nano device or molecule consisting ofsingle-particle as well as interaction many-body terms.It is described by electronic annihilation/creation opera-tors f iσ /f † iσ at site i = [1 , . . . , N S ] where N S is typicallysmall and spin σ = {↑ , ↓} . We will specify the par-ticular form of ˆ H S in the respective results section. ii)The “environment” Hamiltonian ˆ H E describes the two noninteracting electronic leadsˆ H E = X λ =1 X kσ ǫ λkσ c † λkσ c λkσ , (1b)where c λkσ /c † λkσ denote the fermion operators of the in-finite size lead λ with energies ǫ λkσ and electronic den-sity of states (DOS) ρ λσ ( ω ) = N λ P k δ ( ω − ǫ λkσ ) where N λ → ∞ are the number of levels in the leads. Thedisconnected leads are held at constant temperatures T λ and chemical potentials µ λ so that the particles obeythe Fermi-Dirac distribution p FD λ ( ω, T λ , µ λ ). iii) Fi-nally the system and the environment are coupled by thesingle-particle hoppingˆ H SE = X λ =1 X ikσ (cid:16) t ′ λikσ f † iσ c λkσ (cid:17) + h.c. . (1c) III. MASTER EQUATION BASED CLUSTERPERTURBATION THEORY
Our goal is to obtain the steady-state transport charac-teristics of the Hamiltonian ˆ H , Eq. (1a) in a nonequilib-rium situation induced by environment parameters likea bias voltage V B or temperature gradient ∆ T . Theimportant step consists in evaluating the steady-statesingle-particle Green’s function in Keldysh space e G inthe well established Keldysh-Schwinger nonequilibriumGreen’s function formalism. In general ˆ H is bothinteracting and of infinite spatial extent. Therefore ex-plicit evaluation of e G is prohibitive in all but the mostsimple cases which motivates the introduction of approx-imate schemes.One such scheme is CPT, in which one performsan expansion in a ’small’ single-particle perturbation,for example the system-environment coupling ˆ H SE ofEq. (1c). The unperturbed Hamiltonian ˆ H S + ˆ H E canbe solved exactly. While in the noninteracting case CPTbecomes exact, results obtained in the presence of in-teraction are approximate and depend on the referencestate for the unperturbed system. A common practicewithin stsCPT is to use a pure state given by theequilibrium ground state | Ψ i S of the disconnected in-teracting system Hamiltonian ˆ H S . In a nonequilibriumsituation, this is still ambiguous, as it depends on an ar-bitrary choice of the chemical potential µ S and/or tem-perature T S for the interacting finite system.The goal of this work is to provide an unambiguousand conceptually more rigorous criterion for the choiceof the reference state for the interacting central region.Ideally, the reference state is selected such that it resem-bles best the situation of the coupled system, i.e. for thefull Hamiltonian, Eq. (1a) in the steady-state. An appro-priate choice in equilibrium is to use the grand-canonicaldensity operator ˆ ρ Sgc as reference state. In this case, T S and µ S are uniquely determined by the equilibriumsituation. Equivalently, ˆ ρ Sgc is given by the steady-statesolution of a Qme in the Born-Markov approximation(see Sec. III B), when coupling the system to one thermalenvironment. From this viewpoint a natural extension tothe nonequilibrium situation is to make use of a Qme aswell in order to obtain a consistent reference state, giventhen by the steady-state reduced density operator of thesystem ˆ ρ S . In this work, a second order Born-MarkovQme is employed, which yields the correct zeroth orderreduced density operator ˆ ρ S (adjusted to ˆ H SE ). Subsequently, ˆ H SE is included within the CPT approx-imation, in order to obtain improved results for theGreen’s function and in turn for the transport observ-ables.In summary, the meCPT method consists of the fol-lowing three main steps, analogous to a standard CPTtreatment:1. Decompose the whole system into a small inter-acting central region (system) and noninteractingleads of infinite size (environment), see ˆ H S and ˆ H E in Eq. (1a).2. The new step introduced in this work is to solve aQme for the system in order to obtain the reduceddensity operator ˆ ρ S , which serves as a referencestate to calculate the cluster (retarded) Green’sfunction g Rijσ ( τ ) = − iθ ( τ )tr (cid:26) ˆ ρ S h f iσ ( τ ) , f † jσ i + (cid:27) . (2)3. Reintroduce the system-environment coupling ˆ H SE perturbatively, see Sec. III A and Eq. (4), to deter-mine the Green’s function of the coupled system. A. Steady-state cluster perturbation theory
Here we briefly recall the main, well-established CPTconcepts and equations, as this is the starting point forthe formalism presented in this work. For an in depthdiscussion of CPT and its nonequilibrium extensionwe refer the reader to the literature.
The central element of stsCPT is the steady-statesingle-particle Green’s function in Keldysh space e G = (cid:18) G R G K G A (cid:19) , (3)where R denotes the retarded, A the advanced, and K theKeldysh component. In the present formalism, G R/A/K become matrices in the space of cluster sites and dependon one energy variable ω since time translational invari-ance applies in the steady-state.As explained above, in order to compute e G ( ω ) withinstsCPT one partitions ˆ H , Eq. (1a) in real space, into in-dividually exactly solvable parts, in this case, the system ˆ H S and the environment ˆ H E , which leaves the couplingHamiltonian H SE as a perturbation. The single-particleGreen’s function of the disconnected Hamiltonian is de-noted by e g ( ω ), which obviously does not mix the dis-connected regions. For the noninteracting environment,the respective block entries of e g ( ω ) are available analyti-cally. For the interacting part the respective entriesof e g ( ω ) are calculated via the Lehmann representationwith respect to the reference state. This can be com-puted e.g. based on the Band Lanczos method. The full steady-state Green’s function in the CPT ap-proximation is found by reintroducing the inter-clustercoupling perturbatively e G ( ω ) − = e g ( ω ) − − f M ; M R = M A = M , M K = 0 ,(4)where we denote by the matrix M the single-particleWannier representation of ˆ H SE . CPT is equivalent tousing the self-energy e Σ of the disconnected Hamiltonianas an approximation to the full self-energy. Therefore,the quality of the approximation can in principle be sys-tematically improved by adding more and more sites ofthe leads to the central cluster. However, in doing so thecomplexity for the exact solution of the central clustergrows exponentially. Independent of the reference state,this scheme becomes exact in the noninteracting limit.
B. Born-Markov equation for the reference state
In the following we outline how to obtain the refer-ence state ˆ ρ S by using a Born-Markov-secular (BMsme),or more generally a Born-Markov master equation(BMme). Although this approach is standard, forcompleteness we present here the main aspects and no-tation. We loosely follow the treatment of Ref. 40,78,79.The real time τ evolution of the full many-body den-sity matrix ˆ ρ is given by the von-Neumann equation˙ˆ ρ = − i h ˆ H , ˆ ρ i − . Typically the large size of the Hilbertspace of ˆ H prohibits the full solution in the interact-ing case. One thus considers the weak coupling limit | ˆ H SE | ≪ | ˆ H E | and performs a perturbation theory interms of | ˆ H SE | . In the usual way one obtains an equation for the re-duced many-body density matrix of the system ˆ ρ S ( τ ) =tr E { ˆ ρ } by working in the interaction picture ˆ ρ I ( τ ) = e + i ( ˆ H S + ˆ H E ) τ ˆ ρ (0) e − i ( ˆ H S + ˆ H E ) τ with respect to the cou-pling Hamiltonian, Eq. (1c). One then performs threestandard approximations: i) Within the Born approxi-mation, valid to lowest order in | ˆ H SE | , the density ma-trix is factorized ˆ ρ I ( τ ) ≈ ˆ ρ SI ( τ ) ⊗ ˆ ρ EI . Furthermore, theenvironment ˆ ρ EI is assumed to be so large that it is notaffected by | ˆ H SE | and thus independent of time. ii) TheMarkov approximation implies a memory-less environ-ment, that is, the system density matrix varies muchslower in time than the decay time of the environmentcorrelation functions C αβ ( τ ). Upon transforming back tothe Schr¨odinger picture this yields the BMme, which istime-local, preserves trace and hermiticity, and dependson constant coefficients. iii) To obtain an equation ofLindblad form which also preserves positivity one typi-cally employs the secular approximation, which averagesover fast oscillating terms, yielding the BMsme. The system-environment coupling can be quite gener-ally written in the form ˆ H SE = P α ˆ S α ⊗ ˆ E α , with ˆ S α = ˆ S † α and ˆ E α = ˆ E † α . This hermitian form is convenient for fur-ther treatment.The tensor product form can be achievedeven for fermions by a Jordan-Wigner transformation, see App. B. For our coupling Hamiltonian, Eq. (1c) andparticle number conserving systems, the coupling opera-tors take the formˆ S iσ = 1 √ f iσ + f † iσ ) , ˆ E λiσ = 1 √ c λiσ + c † λiσ ) (5)ˆ S iσ = i √ f iσ − f † iσ ) , ˆ E λiσ = i √ c λiσ − c † λiσ ) .In the energy eigenbasis of the system Hamiltonianˆ H S | a i = ω a | a i , the BMme in the Schr¨odinger repre-sentation reads ˙ˆ ρ S ( τ ) = − i h ˆ H S + ˆ H LS , ˆ ρ S ( τ ) i − + X abcd Ξ ab,cd | a i h b | ˆ ρ S ( τ ) | d i h c | − (cid:20) | d i h c | | a i h b | , ˆ ρ S ( τ ) (cid:21) + ! ,(6)withΞ ab,cd = X αβ ξ αβ ( ω ba , ω dc ) h a | ˆ S β | b i h c | ˆ S α | d i ∗ , (7)where ω ba = ω b − ω a . The Lamb-shift Hamiltonian ˆ H LS and the environment functions ξ αβ ( ω , ω ) are definedin App. A. When employing the secular approximation,the terms in the BMsme simplify and in Eq. (7) one canreplace ξ αβ ( ω ba , ω dc ) → ξ αβ ( ω b − ω a ) δ ω b − ω a ,ω d − ω c . Dueto the secular approximation the BMsme can only lead tointerference between degenerate states. The more generalBMme also couples non-degenerate states at the cost ofloosing the Lindblad structure of the Qme, see Sec. IV Band Ref. 40. Single-particle Green’s function
As discussed above, for meCPT, the Green’s function e g ( ω ) of the isolated system is evaluated from the refer-ence state ˆ ρ S . The retarded component Eq. (2) takes theexplicit form g Rij ( σ ) ( ω ) = X abc ρ Sab × (8) (cid:18) h b | f iσ | c i h c | f † jσ | a i ω + i + − ( ω c − ω b ) + h b | f † jσ | c i h c | f iσ | a i ω + i + − ( ω a − ω c ) (cid:19) , where i, j denote indices of system sites. The advancedcomponent follows from g A = (cid:0) g R (cid:1) † and the Keldyshcomponent g K of the finite, unperturbed system is notrelevant for the CPT equation, Eq. (4). Once e g is ob-tained, the full Green’s function is again approximatelyobtained within CPT by Eq. (4). Notice that for U = 0, e G is independent of the reference state, which is whystsCPT, stsVCA as well as meCPT coincide (and becomeexact) in the noninteracting case. C. Numerical implementation
From a numerical point of view, the two main steps areto first obtain the reference state ˆ ρ S by solving the Qmeand then to evaluate the Green’s functions using Eq. (8)and Eq. (4). For the solution of the BMme, Eq. (6) oneneeds to carry out the following: i) Full diagonalizationof the interacting system Hamiltonian which is done in LAPACK , making use of the block structure in ˆ N and ˆ S z .ii) Evaluation of the coefficients of the BMme in Eq. (6),which involves coupling matrix elements h a | ˆ S α | b i andnumerical integration of the bath correlations functions,see App. A, C, for which an adaptive Gauss-Kronrodscheme is employed. iii) The steady-state ˆ ρ S is finally ob-tained from the unique eigenvector with eigenvalue zeroof Eq. (6), which we determine by a sparse Arnoldi diag-onalization. Again, a block structure is related to ˆ N andˆ S z . The numerical effort for the exact diagonalizationscales with the size of the Hilbert space, and thereforeexponentially with the system size N S . In the secondmajor step, the Green’s function of the disconnected sys-tem is calculated by Eq. (8). Finally, the meCPT Green’sfunction e G ( ω ) is found using Eq. (4). We outline howto evaluate observables within meCPT and the Qme inApp. D. IV. RESULTS
In this section we present results obtained from themeCPT approach. In all calculations, except the ones inSec. IV B, the secular approximation is applied for thereference state ˆ ρ S . The main improvements of meCPTwith respect to bare BMsme are i) the inclusion of lead in-duced broadening effects, ii) the correct U = 0 limit andiii) a correction for effects missed by an improper treat-ment of quasi degenerate states in the BMsme (see be-low). In comparison to the previous “standard” stsCPT,meCPT also captures current blocking effects, which arediscussed in detail in Ref. 39 and Ref. 40 within a Qmetreatment. Γ = Γ/2 Γ (cid:0)
Γ/2 V B + /2 (cid:2)(cid:1) L (cid:3)(cid:4) R V B - /2V G -U/2+ (cid:5)(cid:6) ↑ U ↓ T L L T RR left lead right leadcentral molecule =T =T ↑↓ ↑ ↑ FIG. 1: (Color online)
Quantum dot diode:
Schematic repre-sentation, see Sec. IV A. Single quantum dot with Hubbardinteraction U and gate voltage V G (particle-hole symmetric at V G = 0), coupled via Γ L / R = Γ2 to a left and right lead. Theright lead is fully polarized, i.e. only spin- ↑ DOS is present.An external bias voltage V B shifts the chemical potentials by µ L / R = ± V B . The leads are in the wide band limit and atthe same temperature T . A. Quantum dot diode
We first discuss a quite simple model system: a quan-tum diode based on electron-electron interaction effects.Fig. 1 depicts this junction consisting of a single interact-ing orbital described by a Hubbard interaction and anon-site term to allow for a gate voltage V G : ˆ H S = U (cid:18) ˆ n f ↑ − (cid:19) (cid:18) ˆ n f ↓ − (cid:19) + V G X σ ˆ n fσ ,where ˆ n fσ = f † σ f σ . The environment Eq. (1b), consistsof two spin dependent, conducting leads. We modelboth, the left (L) and the right (R) lead by a flat DOSwith local retarded single-particle Green’s function g RL/R ( ω ) = − D ln (cid:16) ω + i + − Dω + i + + D (cid:17) , with a half-bandwidth D much larger than all other energy scales in the model,mimicking a wide band limit. We keep both leads at thesame temperature T L = T R = T and at chemical poten-tials µ L = − µ R = V B corresponding to a symmetricallyapplied bias voltage V B . The right lead is fully spin po-larized, i.e. tunnelling of one spin species ( ↓ ) into theright lead is prohibited while both spin species can tun-nel to the left lead. The system is coupled to the twoleads via a single-particle hopping amplitude t ′ in ˆ H SE ,Eq. (1c) which results in a lead broadening parameter ofΓ ↑ L = Γ ↓ L = Γ ↑ R = Γ2 = π | t ′ | D , Eq. (C1), and Γ ↓ R ≡ ω = 0) as definedin Eq. (C1). For meCPT we use H SE , see Eq. (1c), asperturbation.Such a system could be realized in: i) A “metal - artifi-cial atom - half-metallic ferromagnet“ nano structurewhere spin- ↑ DOS is present at the Fermi energy whilethe respective spin- ↓ DOS is zero. ii) A graphene nanostructure with ferromagnetic cobalt electrodes. iii)A one dimensional optical lattice of ultra cold fermions in a quantum simulator where the hopping of spin- ↓ par-ticles into the right reservoir is suppressed. For all threesystems spin- ↓ particles cannot reach the right lead, inthe first two due to a vanishing DOS, in the third onedue to a vanishing tunnelling amplitude.We consider parameters such that the junction is op-erated in a single electron transistor (SET) regime, i.e.temperatures above the Kondo temperature. In thisregime we expect an interaction induced - magnetizationmediated blocking due to the fact that the system fillsup with spin- ↓ particles. On the one hand they cannotescape, yielding a vanishing spin- ↓ current, and on theother hand they suppress the spin- ↑ occupation, at fi-nite repulsive interaction U , resulting also in a vanishingspin- ↑ current. Fig. 2 (A) shows the meCPT stability diagram of theinteracting system in the V B − V G plane. When applyinga particle-hole transformation for all particles, leads andsystem, along with t ′ → − t ′ we easily find the symmetryproperties j ( − V B , − V G ) = − j ( V B , V G ) , h n fσ i ( − V B , − V G ) = 1 − h n fσ i ( V B , V G ) .From the continuity equation it is clear that only spin- ↑ steady-state current can flow which limits the maximumcurrent to Γ2 . The energies ω N of the isolated quantumdot can be labelled by the total particle number N andare for V G = U given by ω = 0, ω = U and ω = 2 U .This gate voltage corresponds to the dashed line, markedby (X) in Fig. 2 (A). The corresponding energy differ-ences ∆ = 0 . U between the single-occupied and theempty dot and ∆ = 1 . U between double-occupied andsingle-occupied dot are associated with a further trans-port channel opening as soon as the bias V B reaches twicetheir values. The meCPT result for the current exhibitsthe well known Coulomb diamond close to V B = 0 and V G = 0, where current is hindered because all systemenergies are far outside the transport window ± V B , seeEq. (D2). At V G = 0 a current sets in at V B = ±| U | , i.e.when transport across the system’s single particle levelbecomes allowed. The point, at which the current setsin, shifts with V G linearly to higher bias voltages. Thistransition is broadened ∝ max(Γ = 0 . U, T = 0 . U ).However, not only the transport window and possibleexcitations in the system energies determine the current-voltage characteristics. The particular occupation of thesystem states may lead to more complicated effects, suchas current blocking.Our first main result is that in contrast to stsCPT theblocking is correctly reproduced in meCPT. The currentblocking is visible in Fig. 2 (A) in region (Y), see also thedetailed data in subplot (C1). It is asymmetric in V B and therefore responsible for the rectifying behaviour for | V G | > | U | . This feature is easily understood from theplots of the spin resolved densities in Fig. 2 (C2). In theregion of interest, for positive V B , h n ↓ i = 1 which hindersspin- ↑ particles from the left lead to enter the system, due V B / U-5 0 5 < n σ > meCPT
Quantum dot diode: (A)
Stability diagram, based on the total current j = h j ↑ i + h j ↓ i as a function ofbias voltage V B and gate voltage V G , obtained within meCPT. Note that h j ↓ i ≡
0. Results are depicted for T = 0 . U andΓ = 0 . U . (Y) marks the current blocking region. The green dashed line (X) at V G = U indicates the parameter regime for thepanels (B) and (C). (B) Diagonal part of the reduced density matrix ρ Saa obtained by BMsme. (C1)
Spin- ↑ current j ↑ withinmeCPT compared to BMsme. Solid lines are for the same parameters as line (X) in panel (A). Blue dashed and solid lines forBMsme are indistinguishable. (C2) Spin resolved densities h n ↑ i and h n ↓ i for the same parameters as in panel (C1), see solidlines in the legend. to the repulsive interaction U and suppresses the current.For negative V B , the situation is reversed. A direct com-putation of the current in the framework of BMsme, seeApp. D 2, also predicts the blocking, which is however notthe case if we use stsCPT based on the zero temperatureground state | Ψ i S . The blocking is evident in Fig. 2 (B),where we observe that in the blocking regime, the reduceddensity is ρ S = |↓i h↓| . Independent of the value of U > V B in meCPTand BMsme. Fig. 2 (C1) shows that within BMsme thisregime is entered after a U independent hump in thecurrent while within meCPT the hump is broader andweakly U dependent. The current blocking disappearsat a bias voltage V B ∝ U in both methods. Immedi-ately apparent are the much broader features in meCPT,which leads to a less pronounced effect in contrast tothe total blocking predicted by BMsme. In BMsme thebroadening parameter Γ enters merely as prefactor of thecurrent, and broadening is solely induced by the temper-ature. This temperature induced broadening is correctlytaken into account in both methods. For T >
Γ the lat- ter dominates and the meCPT results are similar to theplain BMsme solution. A comparison of the three meth-ods is given in Tab. I. In this simple model the blockingcan be captured even by a straight forward steady-statemean-field theory in the Keldysh Green’s function withself-consistently determined spin densities or in stsVCA.This is not the case for the more elaborate system studiedin the next section.
TABLE I: Comparison of steady-state cluster perturbationtheory (stsCPT), the Born-Markov-secular master equation(BMsme) and the quantum master equation based stsCPT(meCPT) with respect to their ability to capture temperature( T ) or lead (Γ) induced level broadening, current blocking andwhether the noninteracting limit is fulfilled.method T -broadening Γ-broadening blocking U = 0stsCPT yes yes no exactBMsme yes no yes approx.meCPT yes yes yes exact B. Triple quantum dot
In this section we discuss a more elaborate modelsystem: a triple quantum dot ring junction which fea-tures negative differential conductance (NDC) based onelectron-electron interaction effects mediated by quan-tum interference due to degenerate states as outlined indetail in Ref. 45,46. Fig. 3 (A) depicts the triple quantumdot ring junction, described by the following HubbardHamiltonian ˆ H S = X i =1 U (cid:0) ˆ n fi ↑ − (cid:1)(cid:0) ˆ n fi ↓ − (cid:1) + V G X i =1 X σ ˆ n fiσ + t X h ij i X σ f † iσ f jσ . (9)In addition to the model parameters described inSec. IV A, a nearest-neighbour h ij i hopping t is present.The environment, Eq. (1b) and coupling, Eq. (1c) are nowboth symmetric in spin. Moreover, we use µ L = − µ R = V B , T = T L = T R and Γ L = Γ R = Γ2 = π | t ′ | D .Such a junction can be experimentally realized: i) Vialocal anodic oxidation (LAO) on a GaAs/AlGaAs het-erostructure which enables tunable few electron con-trol. ii) In a graphene nano structure. Experimen-tally the stability diagram has been explored alongsidecharacterisation and transport measurements. Thenegative differential conductance has been observed in adevice aimed as a quantum rectifier.
Theoretically thestudy of the nonequilibrium behaviour of such a devicehas become an active field recently.
We investigate transport properties for values of theparameters such that the junction is in a single electrontransistor (SET) regime, i.e. temperatures above theKondo temperature. In this regime we expect an inter-action induced - quantum interference mediated blockingas discussed in Ref. 45,46. The rotational symmetry en-sures degenerate eigenstates labelled by a quantum num-ber of angular momentum. In situations where these de-generate states participate in the transport they providetwo equivalent pathways through the system and lead toquantum interference. The blocking sets in at a biasvoltage, where the degenerate states start to participatein the transport. It then becomes possible that a super-position is selected which forms one state with a node atthe right lead. In the long time limit this state will befully occupied while the other one will be empty due toCoulomb repulsion, for reasons very similar to the onesdiscussed in the previous section.
The steady-state charge distribution and current-voltage characteristics of the interacting triple quantumdot are presented in Fig. 3 (B, C) in a wide bias voltagewindow. The current, depicted in panel (C), in generalincreases in a stepwise manner and is fully antisymmet-ric with respect to the bias voltage direction. A block-ing effect occurs at V B ≈ . | t | as can be observed inthe BMsme and meCPT data. The previous versionof stsCPT based on the pure zero temperature ground state | Ψ i S misses this region of NDC. In contrast tothe simpler model presented in the previous section, aself-consistent mean-field solution does not capture theblocking effects correctly in this more elaborate system.The BMsme solution shows many more steps in the cur-rent than the stsCPT one, which is due to transitions inthe reference state ˆ ρ S of the central region. The meCPTresults in general follow these finer steps, correcting theirwidth to incorporate also lead induced broadening effectsin addition to the pure temperature broadening. As canbe seen in panel (B1), meCPT predicts a large chargeincrease at the site connected to the high bias lead. Notethat the charge density at site 2, which is connected tothe right lead is simply: h n i ( V B ) = h n i ( − V B ). Thecharge density at site 3 is symmetric with respect to thebias voltage origin.Next we study the impact of a gate voltage on theblocking. Results obtained by meCPT are depicted asstability diagram in Fig. 4. Upon increasing | V G | , the on-set of the blocking shifts linearly to higher V B (Y). Wefind a Coulomb diamond for 2 V G ' V B − | t | (D). Uponincreasing the bias voltage out of the Coulomb diamond,see e.g. line (X), a current sets in but is promptly hin-dered by the blocking so that the current diminishes aftera hump of width ∝ max( T, Γ). Interestingly this devicecould be operated as a transistor in two fundamentallydifferent modes. In mode (T1), at a source-drain voltageof ≈ | t | the current is on for a gate voltage of V G = 0and off for V G ≈ . | t | due to the Coulomb blockade. Inmode (T2), at a source-drain voltage of ≈ . | t | the cur-rent is off for a gate voltage of V G = 0 due to quantuminterference mediated blocking and on for V G = 0 . | t | .Next we discuss the current characteristics in the vicin-ity of the blocking in more detail, as well as the impactof the interaction strength U . The first row of Fig. 5shows the total current through the device for differentvalues of U . The blocking region shifts to lower bias volt-ages with increasing U . As discussed earlier, structuresin the BMsme results are only broadened by tempera-ture effects in the steady-state density (compare e.g. thewidth of the structures in the local density in the secondrow of Fig. 5), while meCPT additionally takes into ac-count the finite life time of the quasi particles due to thecoupling to the leads, given by 1 / Γ. This can be seenby solving Eq. (4) for the local Green’s function at de-vice sites. Especially for higher lead broadening Γ thisgives rise to significant differences in the meCPT resultscompared to the BMsme data. From the bottom rowof Fig. 5 we see that, before the blocking regime is en-tered, the steady-state changes from a pure N = 2 stateto a mixed N = 2 / N = 3 state at the hump in thecurrent. Obviously, blocking arises because the systemreaches a pure N = 3 state for U = 2 | t | and U = 3 | t | at V B ≈ . | t | . For U = | t | the current is only partiallyblocked, because the contribution of the N = 2 state isnot fully suppressed. For all U -values, however, we findNDC. As far as the meCPT current is concerned, thecomplete blocking at higher interaction strengths, pre- FIG. 3: (Color online)
Triple quantum dot (A)
Schematic representation, see Sec. IV B. System Hamiltonian as defined inEq. (9). Site 1 couples to the left lead and site 2 to the right one, both with Γ L / R = Γ2 . The leads are held at the sametemperature T L / R = T and the chemical potentials µ L / R = ± V B are shifted by the bias voltage. (B) Local charge density h n i i as a function of bias voltage V B . The results are obtained by meCPT, BMsme and stsCPT, see color code of panel (C). (C) Total current j = P σ h j L σ i into the system at site 1 as a function of bias voltage V B . Results, shown in panels (B,C), are for U = 2 | t | , T ≈ . | t | , Γ = 0 . | t | and V G = 0, corresponding to line (X) in Fig. 4. (Y) (Y) (X)(T1) (T2)(D) V B /|t| V G / | t | FIG. 4: (Color online)
Triple quantum dot: stability diagram.
Total current entering the system as a function of bias voltage V B and gate voltage V G , obtained within meCPT. The block-ing region is indicated by (Y), the Coulomb diamond by (D).The two arrows (T1) and (T2) mark two device operationmodes as discussed in the text. All results are for U = 2 | t | , T = 0 . | t | and Γ = 0 . | t | . Dashed line (X) for V G = 0 marksthe parameter region depicted in Fig. 3 (C). dicted by BMsme, is reduced to a partial blocking due tothe lead induced broadening effects in meCPT. Although ρ Sab changes significantly twice in the blocking region (for U = 2 and U = 3), the charge density h n i i just increasesonce from h n i ≈ .
75 to h n i ≈ V B = 0 . | t | )the system is in a pure state with N = 2, which corre-sponds to the zero temperature ground state | Ψ i S inthe N = 2 sector. Here the transmission function T ( ω ),Eq. (D3), of meCPT agrees with the one of stsCPT. A small current is obtained due to the N = 2 → ω ≈ . | t | . Increasing the bias voltage has no in-fluence on the reference state in stsCPT, which thereforeremains in the N = 2 particle sector. Consequently, thetransmission function in stsCPT does not change. Onlythe transport window increases linearly with increasing V B . For V B = 1 . | t | it includes the peak at ≈ . | t | andresults in a significant increase in the current obtainedin stsCPT (see stsCPT result in Fig. 3). This is in starkcontrast to the BMsme current, depicted in Fig. 5, whichexhibits perfect blocking for V B = 1 . | t | . The reasonfor the current-blocking is that only two states, both inthe N = 3 sector and doubly degenerate, have significantweight in ρ Sab . The meCPT solution is based on the mod-ified density matrix and therefore the current is dimin-ished, since the next possible excitation is at ω ≈ . | t | ( N = 2 → W ( ω ) ≈ ( − . | t | , . | t | ), Eq. (D2). Due to the lead in-duced broadening of T ( ω ) and the temperature inducedbroadening of the transport window, the current is how-ever only partially blocked. For V B = 2 . | t | this excita-tion falls into the transport window and the current isno longer blocked. In this case, the state ρ Sab is a mix-ture of N = 2 , ,
4. The dominant excitation responsiblefor this current is again the ground state excitation at ω ≈ . | t | from N = 2 →
3. This is why in this regimethe stsCPT current, based on the pure two particle stateis again similar to the meCPT current.Our results on the Qme level have been checked withthose presented by Begemann et al. in Ref. 39 and Darau et al. in Ref. 40 for a six orbital ring which shows similarblocking effects. Different types of blocking effects invarious parameter regimes have been discussed in detailin a Qme framework also for the three orbital ring byDonarini et al. in Ref. 45,46. bias voltage V B /|t| meCPTBMsme U=1|t| U=2|t| U=3|t|
T=0.02 |t|, Γ =0.1|t|T=0.02 |t|, Γ =0.5|t|T |t|, Γ =0.1|t|=0.10 0.5 1 1.5 2 2.500.511.52 0 0.5 1 1.5 2 2.500.511.520 0.5 1 1.5 2 2.500.511.52 0 0.5 1 1.5 2 2.500.20.40.60.81 V B /|t| V B /|t| N=2N=3N=4N=5 V B /|t| w N < n > FIG. 5: (Color online)
Triple quantum dot:
Dependence of the current blocking on the interaction strength U . (Top row) Total current j as a function of bias voltage V B . (Middle row) Charge density h n i at site 1. The color code of the top rowis valid. (Bottom row) Summed diagonal elements of the density matrix w N = P a ∈ N ρ Saa per particle number N . The blackmarkers in the mid panel ( U = 2 | t | ) indicate for which V B detailed results are given in Fig. 6. Solid lines in all panels are for T = 0 . | t | , Γ = 0 . | t | and V G = 0. Results for T = 0 . | t | are depicted in the central panels by dotted lines and those forΓ = 0 . | t | in the right panels by dashed lines. V =0.4|t| B V =1.4|t| B V =2.4|t| B FIG. 6: (Color online)
Triple quantum dot:
Dynamic transmission function T ( ω ), Eq. (D3), as obtained by meCPT and stsCPT.Same parameters as in Fig. 5 (bottom mid) at the three indicated bias voltages: V B = 0 . | t | (left), V B = 1 . | t | (middle) and V B = 2 . | t | (right). The temperature broadened transport window W ( ω ), Eq. (D2), is depicted as a dashed black line. Quasi-degenerate states
Next we study the reliability of the secular approxima-tion in the case of quasi degeneracy of the isolated en-ergies of the system and benchmark its applicability to create a reference state for meCPT. To this end we ap-ply a second gate voltage that couples only to the thirdorbital, see Fig. 3 (left), and leads to an additional term V G, ˆ n f in the system Hamiltonian. This lifts the degen-eracy of states present at V G, = 0 and therefore requires0a treatment within the BMme, see Ref. 40.In the following we discuss the same parameter regimeas above. In Fig. 7 we present results obtained usingmeCPT (solid lines) and Qme results (dashed lines) forthe BMsme (A) and for the BMme (B). The meCPT re-sults of each panel are obtained using the respective Qme.In the BMsme data a very small | V G, | has a drastic ef-fect on the current-voltage characteristics. The block-ing present at V G, = 0 is immediately lifted by verysmall | V G, | and the current jumps to a plateau. Forlarger | V G, | the current stays on this plateau until fur-ther transport channels open up. This ”jump“ at small | V G, | arises due to the improper treatment of quasi de-generacies in BMsme. MeCPT results based on BMsmeshow a smooth change of the current-voltage character-istics. BMme on the other hand correctly accounts forthe coupling of the quasi-degenerate states and also ex-hibits a smooth dependence on V G, . For meCPT basedon BMme we find qualitative similar results to meCPTbased on BMsme, which emphasizes the robustness ofthe meCPT results in general. From this it is appar-ent that meCPT is capable of repairing the decouplingof quasi-degenerate states in the BMsme to some degree.However, to study blocking effects at quasi degeneratepoints it is of advantage to make use of the BMme inmeCPT.As discussed below in Sec. IV C, the BMme is not ofLindblad form and does not necessarily result in a pos-itive definite reduced many-body density matrix ρ Sab ingeneral. Using a not proper density matrix in Eq. (8)may result in non-causal Green’s functions when thesteady-state ρ Sab is obtained from the BMme. This canbe avoided by using a modified reference state ρ Sab → ρ Sab
Θ(∆ − | ω a − ω b | ), with Θ( x ) the Heaviside step func-tion and ∆ a small quantity, being e.g. ≈ − , in Eq. (8),which renders the Green’s functions causal. This is some-what an ad-hoc approximation and should be seen simplyas a way to explore the effects of continuously breakingdegeneracy in the problem. C. Current conservation
Finally we comment on conservation laws in meCPT.Within BMsme and BMme the current conservation(continuity equation) is always maximally violated in asense that the current within the system is zero. Thisis due to the zeroth order ˆ ρ S as discussed in App. D 2.In BMsme the inflow from the left lead into the systemhowever always equals the outflow from the system to theright lead. Without the secular approximation the quan-tum master equation (BMme) is not of Lindblad formand the final many-body density matrix is not guaran-teed to be positive definite. This in turn can leadto slightly negative currents in regions where they arerequired to be positive by the direction of the bias volt-age . Furthermore, the inflow can be slightly differentfrom the outflow. j / Γ V G,3 =0.00|t|V
G,3 =0.01|t|V
G,3 =0.10|t| ( B) Born-MarkovmeCPTQme(A) Born-Markov-secular j / Γ V B /|t|V B /|t| FIG. 7: (Color online)
Triple quantum dot:
Effects of liftingdegeneracies in the system energies by a third gate voltage.Total current j as a function of bias voltage V B , for threedifferent gate voltages V G, applied to site 3. Results based onthe Born-Markov-secular approximation are compared withthose of the Born-Markov approximation. Solid/dashed linesindicate the meCPT/BM(s)me result. All results are for U =3 | t | , T = 0 . | t | and Γ = 0 . | t | . In the noninteracting case, meCPT fully repairs the vi-olation of the continuity equation present in the referencestate. For increasing interaction strength, the violationof the continuity equation typically grows also in meCPT.In particular, the overall symmetry of the current staysintact (in our case, inflow equals outflow), while the cur-rent on bonds between interacting sites does not exactlymatch the current between noninteracting sites. Thistypically small violation of the continuity equation canbe attributed to the violation of Ward identities inthe non-conserving approximation scheme of CPT.
V. SUMMARY AND CONCLUSIONS
We improved steady-state cluster perturbation theorywith an appropriate, consistent reference state. This ref-erence state is obtained by the reduced many-body den-sity matrix in the steady-state obtained from a quantummaster equation. The resulting hybrid method inher-its beneficial aspects of steady-state cluster perturbationtheory as well as from the quantum master equation.1We benchmarked the new method on two experimen-tally realizable systems: a quantum diode and a triplequantum dot ring, which both feature negative differen-tial conductance and interaction induced current block-ing effects. meCPT is able to improve the bare quantummaster equation results by a correct inclusion of lead in-duced level-broadening effects, and the correct noninter-acting limit. In contrast to previous realizations of thesteady-state cluster perturbation theory, meCPT is ableto correctly predict interaction induced current blockingeffects. It is well known that the secular approximation(BMsme) is not applicable to quasi degenerate problems,which is corroborated by our results for the steady-statecurrent. However, meCPT based on the BMsme density,is able to repair most of the shortcomings of BMsme. Theresults are very close to those obtained by meCPT basedon the density of BMme, where the quasi-degeneratestates are treated consistently.The computational effort of meCPT beyond that ofthe bare quantum master equation scales with the num-ber of significant entries in the reference state densitymatrix but is typically small. In the presented formu-lation the new method is flexible and fast and thereforewell suited to study nano structures, molecular junctionsor heterostructures also starting from an ab-inito calcu-lation.
Acknowledgments
The authors acknowledge fruitful discussion with A.Rosch. This work was partly supported by the AustrianScience Fund (FWF) Grants No. P24081 and P26508as well as SFB-ViCoM projects F04103 and F04104 andNaWi Graz. MN, GD and AD thank the Forschungszen-trum J¨ulich, in particular the autumn school on corre-lated electrons, for hospitality and support.
Appendix A: Born-Markov and Pauli masterequation
Here we provide the detailed expressions for the co-efficients in the BMme and BMsme of Eq. (6) and dis-cuss the equations governing the time evolution into thesteady-state.The Lamb-shift Hamiltonian is defined as ˆ H LS = P ab Λ ab | a i h b | , withΛ ab = 12 i X αβ X c λ αβ ( ω bc , ω ac ) h c | ˆ S β | b i h c | ˆ S α | a i ∗ .(A1)Note that [ ˆ H LS , ˆ H S ] − = 0. In the secular approxima-tion (BMsme) one can replace λ αβ ( ω bc , ω ac ) → λ αβ ( ω b − ω c ) δ ω b ,ω a . The expressions for the BMme and BMsmeEq. (6) are valid if h ˆ H E , ˆ ρ E i − = 0 and tr n ˆ E α ˆ ρ E o = 0. The environment functions ξ αβ and λ αβ in Eq. (A1) andEq. (7) are determined by the time dependent environ-ment correlation functions C αβ ( τ ) = tr n ˆ E α ( τ ) ˆ E β ˆ ρ E o , (A2)where the Heisenberg time evolution in the environmentoperators is ˆ E α ( τ ) = e + i ˆ H E τ ˆ E α e − i ˆ H E τ .For the BMme, ξ αβ and λ αβ are given by a sum ofcomplex Laplace transforms ξ αβ ( ω , ω ) = Z ∞ dτ C αβ ( τ ) e + iω τ + Z −∞ dτ C αβ ( τ ) e + iω τ ,(A3) λ αβ ( ω , ω ) = Z ∞ dτ C αβ ( τ ) e + iω τ − Z −∞ dτ C αβ ( τ ) e + iω τ ,(A4)whereas for the BMsme ( ω = ω ) the expressions sim-plify to the full even and odd Fourier transforms ξ αβ ( ω ) = ∞ Z −∞ dτ C αβ ( τ ) e + iωτ , (A5) λ αβ ( ω ) = ∞ Z −∞ dτ sign( τ ) C αβ ( τ ) e + iωτ = iπ ∞ Z −∞ P dω ′ ξ αβ ( ω ′ ) ω − ω ′ .(A6)The coupled equations for the real time evolution of thecomponents of the reduced system many-body densitymatrix ρ Sab = h a | ˆ ρ S | b i according to the BMsme read˙ ρ Sab ( τ ) = i ( ω b − ω a ) ρ Sab ( τ ) (A7)+ i X c (cid:18) ρ Sac ( τ )Λ cb − Λ ac ρ Scb ( τ ) (cid:19) + X cd Ξ ac,bd ρ Scd ( τ ) −
12 Ξ cd,ca ρ Sdb ( τ ) −
12 Ξ cb,cd ρ Sad ( τ ) ! .The equations simplify further for system Hamiltoni-ans ˆ H S with non-degenerate eigenenergies ω a . Then thediagonal components φ a = ρ Saa decouple from the off-diagonals and one recovers the Pauli master equation forclassical probabilities˙ φ a ( τ ) = X c (cid:18) Ξ ac φ c ( τ ) − Ξ ca φ a ( τ ) (cid:19) , (A8)with simplified coefficientsΞ ab := Ξ ab,ab = X αβ ξ αβ ( ω b − ω a ) h a | ˆ S β | b i h a | ˆ S α | b i ∗ .2In this case the dynamics of the decoupled off-diagonalterms ( a = b ) is given by˙ ρ Sab ( τ ) = i ( ω b + Λ b − ω a − Λ a ) − X c (cid:18) Ξ ca + Ξ cb (cid:19)! ρ Sab ( τ ) ,where the simplified Lamb shift terms areΛ a := Λ aa = 12 i X αβ X c λ αβ ( ω a − ω c ) h c | ˆ S β | a i h c | ˆ S α | a i ∗ . Appendix B: Hermitian tensor product form of thecoupling Hamiltonian
For the BMsme (see Sec. III B) it is necessary to bringthe fermionic system-environment coupling Hamiltonian,Eq. (1c) to a hermitian tensor product form, which re-quires [ ˆ S α , ˆ E α ] − = 0. For the fermionic operators inEq. (1c) we however have [ f † iσ , c λkσ ] − = 2 f † iσ c λkσ . A so-lution is provided in Ref. 79 by performing a Jordan-Wigner transformation on the system and environ-ment operators f iσ = Y σ (cid:0) ξ z ⊗ . . . ⊗ ξ zi − ξ − i i +1 ⊗ . . . ⊗ L S (cid:1) S,σ ⊗ Y λ (11 ⊗ . . . ⊗ L E ) E,λσ , c λjσ = Y σ (cid:0) ξ z ⊗ . . . ⊗ ξ zL S (cid:1) S,σ ⊗ Y λ (cid:0) η z ⊗ . . . ⊗ η zj − η − j j +1 ⊗ . . . ⊗ L E (cid:1) E,λσ ,where ξ i and η j denote local spin- degrees of freedomat the system and environment sites respectively and theoverall ordering of operators is important. L S /L E denotethe size of the system / environment. Reordering Eq. (1c)we find ˆ H SEλ = P ijσ t ′ λijσ f † iσ c λjσ − t ′∗ λijσ f iσ c † λjσ , where theminus sign arises due to the fermionic anti-commutator.Plugging in the Jordan-Wigner transformed operatorsleads toˆ H SEλ = X ijσ (cid:18) t ′ λijσ (cid:20) ξ + i ⊗ [ − ξ zi +1 ⊗ . . . ⊗ ξ zL S ⊗ η z ⊗ . . . ⊗ η zj − ] ⊗ η − j (cid:21) σλ + t ′∗ λijσ (cid:20) ξ − i ⊗ [ − ξ zi +1 ⊗ . . . ⊗ ξ zL S ⊗ η z ⊗ . . . ⊗ η zj − ] ⊗ η + j (cid:21) σλ (cid:19) = X i (cid:16) ¯ f † i ⊗ ¯ c i + ¯ f i ⊗ ¯ c † i (cid:17) , where in the last line we have defined new operators¯ f iσ = (cid:2) ξ − i ⊗ [ − ξ zi +1 ⊗ . . . ⊗ ξ zL S ] (cid:3) σ ,¯ f † iσ = (cid:2) [ − ξ zi +1 ⊗ . . . ⊗ ξ zL S ] ⊗ ξ + i (cid:3) σ ,¯ c λiσ = X j t ′ λijσ (cid:2) [ η z ⊗ . . . ⊗ η zj − ] ⊗ η − j (cid:3) λσ ,¯ c † λiσ = X j t ′∗ λijσ (cid:2) η + j ⊗ [ η z ⊗ . . . ⊗ η zj − ] (cid:3) λσ .Note that the phase operator ˆ P i ( jλ ) σ = (cid:2) − ξ zi +1 ⊗ . . . ⊗ ξ zL S ⊗ [ η z ⊗ . . . ⊗ η zj − ] λ (cid:3) σ =( − P λ ′ LS P m = i +1 ˆ n m + ˆ N jλ ′ counts the particles betweensystem site i and environment site j for spin σ de-pending on the ordering of the environments λ . Itis straight forward to show that the bar operatorsfulfil fermionic anti-commutation rules. Furthermore[ ¯ f iσ , ¯ c λiσ ] − = 0, which allows us to write the couplingHamiltonian in a tensor product form. Note that ingeneral [ ¯ f iσ , ¯ c λ ′ jσ ] − = 0 for i = j which is however notrelevant for the coupling Hamiltonian where only thesame i couple.The new operators in hermitian form are given inEq. (5) by replacing c → ¯ c and f → ¯ f . Next we show, byexamining the BMsme, that in most cases the additionalphase operator in ¯ c drops out of the calculations and weare even allowed to use the original f and c operatorsinstead of the barred ones. The operators ¯ c only enterthe equations in the environment correlation functions C αβ ( τ ) as defined in Eq. (A2). Plugging in the barredoperators we obtain for normal systems which preserveparticle number C αβ ( τ ) ∝ tr n e + i ˆ H E τ f † λjσ e − i ˆ H E τ ˆ P i ( jλ ) σ c λjσ ˆ ρ E o ,with ˆ P ij = 11, where we required that [ ˆ H E , ˆ P ij ] − = 0.The dropping out of the phase operators implies thatfor normal systems where the disconnected environmentsconserve particle number we can omit the Jordan-Wignertransformation and do all calculations as is with the orig-inal environment creation/annihilation operators in her-mitian form. Appendix C: Bath correlation functions
In the wide band limit, analytical expressions for thebath correlation functions are available in Ref. 39. Forarbitrary environment DOS, explicit evaluation of the en-vironment correlation functions becomes convenient forhermitian couplings, Eq. (5) as outlined in App. B. Es-sentially the environment functions can all be obtainedvia integrals of the environment DOS ρ ( ω ). Care has tobe taken when going to very low temperatures and solv-ing the integrals with finite precision arithmetic to avoidunderflow errors.3The time dependent environment correlation functions C αβ ( τ ), Eq. (A2) become C ( τ ) = C ( τ ) = 14 π X λσ ∞ Z −∞ dν Γ λσ ( ν ) × (cid:18) e − iντ + 2 ip FD ( ν, T λ , µ λ ) sin ( ντ ) (cid:19) , C ( τ ) = − C ( τ ) = i π X λσ ∞ Z −∞ dν Γ λσ ( ν ) × (cid:18) − e − iντ + 2 p FD ( ν, T λ , µ λ ) cos ( ντ ) (cid:19) ,where C αβ ( τ ) = C † βα ( − τ ) and the coefficientΓ λσ ( ν ) =2 π | t ′ λσ | X k δ ( ν − ω λkσ ) , (C1)is proportional to the lead DOS.For the BMsme, the respective full even Fourier trans-forms ξ αβ ( ω ), Eq. (A5) we find ξ ( ω ) = ξ ( ω ) =12 X λσ Γ λσ ( − ω ) p FD ( − ω, β λ , µ λ ) + Γ λσ ( ω ) p FD ( ω, T λ , µ λ ) , ξ ( ω ) = − ξ ( ω ) = i X λσ Γ λσ ( − ω ) p FD ( − ω, β λ , µ λ ) − Γ λσ ( ω ) p FD ( ω, T λ , µ λ ) ,where p FD ( ω, T, µ ) = 1 − p FD ( ω, T, µ ).The odd Fourier transforms λ αβ ( ω ), Eq. (A6) are givenby λ ( ω ) = λ ( ω ) = i π X λσ ∞ Z −∞ P dν Γ λσ ( ν ) (cid:18) p FD ( ν, β λ , µ λ ) ν + ω − p FD ( ν, β λ , µ λ ) ν − ω (cid:19) , λ ( ω ) = − λ ( ω ) = − π X λσ ∞ Z −∞ P dν Γ λσ ( ν ) (cid:18) p FD ( ν, β λ , µ λ ) ν + ω + p FD ( ν, β λ , µ λ ) ν − ω (cid:19) . Appendix D: Evaluation of steady-state observables1. Steady-state cluster perturbation theory
Within meCPT single-particle observables are avail-able by integration of e G ( ω ), Eq. (4). Its easy to showthat the single-particle density matrix κ ijσ = δ ij − i ∞ R −∞ dω π G Kijσ ( ω ) can be expressed in terms of the re- tarded CPT Green’s function κ ijσ = δ ij − i ∞ Z −∞ dω π G Rinσ ( ω ) P njσ ( ω ) − P inσ ( ω )( G Rjnσ ( ω )) ∗ + G Rinσ ( ω ) (cid:0) [ P σ ( ω ) , M σ ] − (cid:1) nm ( G Rjmσ ( ω )) ∗ ! ,where M σ is the inter-cluster perturbation defined inEq. (4). Here we use the Einstein summation conven-tion, the last line holds within CPT and P ijσ ( ω ) = δ ij (1 − p FD ( ω, T i , µ iσ )).From the real part of the single-particle density-matrixwe read off the site occupation h n i i = P σ κ iiσ the spinresolved occupations h n iσ i = κ iiσ and the magnetization h m i i = ( κ ii ↑ − κ ii ↓ ).The current h j h ij i i between nearest-neighbour sites h ij i is related to the imaginary part of κ ijσ and reads in sym-metrized form h j h ij i i = e ~ ( h ijσ κ ijσ − h jiσ κ jiσ ) ,which is of Meir-Wingreen form and h ijσ is the single-particle Hamiltonian.Equivalently, the transmission current between two en-vironments λ = 1 , h j / i = e ~ ∞ Z −∞ dω π W ( ω )tr {T ( ω ) } , (D1)with the transport window W ( ω ) = p FD ( ω, T , µ ) − p FD ( ω, T , µ ) , (D2)and where the transmission function T ( ω ) = G R ( ω )Γ ( ω ) (cid:0) G R ( ω ) (cid:1) † Γ ( ω ) , (D3)is given in terms of G R ( ω ) = (cid:16) ( g R ( ω )) − − ( e Σ + e Σ ) (cid:17) − with the lead broaden-ing functions of lead λ projected onto the system sites i, j is e Σ λij = M iλ g Rλλ M λj and Γ λ = − ℑ m (cid:16)e Σ λ (cid:17) ,compare also Eq. (C1).
2. Born-Markov master equation
Within the Qme, basic single-particle observables areavailable in terms of the reduced system many-body den-sity matrix ˆ ρ S . The single-particle density matrix κ reads κ ijσ = tr (cid:18) f † iσ f jσ ˆ ρ S (cid:19) = X ab h b | f † iσ f jσ | a i ρ Sab , (D4)4where a and b denote eigenstates of the system Hamilto-nian. Note that within the BMme/BMsme κ ijσ is purelyreal and therefore does predict zero current.However, an expression for the current to reservoir λ can be found by making use of the operator of total sys-tem charge ˆ Q and total system particle number ˆ N , where q denotes the charge of one carrier X λ j λ ( τ ) = ddτ h ˆ Q ( τ ) i = q tr (cid:16) ˆ N ˙ˆ ρ S ( τ ) (cid:17) .Taking ˙ˆ ρ S ( τ ) from the Qme we obtain j λ = q X abc (cid:18) n c − n b − n a (cid:19) Ξ λca,cb (cid:19) ρ Sab , and for non-degenerate systems, in the Pauli limit we findfrom the BMsme j λ non-deg = q X ab ( n a − n b ) Ξ λab φ b . ∗ [email protected] G. Cuniberti, G. Fagas, and K. Richter,
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