Full counting statistics in the self-dual interacting resonant level model
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Full counting statistics in the self-dual interacting resonant level model
Sam T. Carr,
1, 2
Dmitry A. Bagrets,
3, 4 and Peter Schmitteckert
2, 4 Institut f¨ur Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany DFG Center for Functional Nanostructures, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, 50937 K¨oln, Germany Institute of Nanotechnology, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany (Dated: November 9, 2018)We present a general technique to obtain the zero temperature full counting statistics of chargetransfer in interacting impurity models out of equilibrium from time-dependent simulations on alattice. We demonstrate the technique with application to the self-dual interacting resonant levelmodel, where very good agreement between numerical simulations using the density matrix renor-malization group and those obtained analytically from the thermodynamic Bethe ansatz is found.We show from the exact form of counting statistics that the quasiparticles involved in transportcarry charge 2 e in the low bias regime, and e/ PACS numbers: 73.63.-b, 72.70.+m, 05.40.Ca, 05.60.Gg
As nano-devices decrease in size and electron corre-lations become more important, the study of out-of-equilibrium transport properties on the nanoscale be-comes of fundamental importance. Of great worth arepowerful numerical approaches to interacting many-bodysystems, such as the density matrix renormalizationgroup (DMRG) [1]. While recent work using the DMRGhas shown remarkable development in calculation of I - V [2] and noise [3] characteristics of impurity models, thegrail of such an enterprise is to calculate all moments ofcharge transfer when the system is driven out of equilib-rium, known as the full counting statistics (FCS) [4, 5].The electron FCS in mesoscopic transport concentrateson a charge distribution p n , which is the probability that n electrons are transmitted from the left to the right leadduring the measurement time t m . Knowledge of all prob-abilities p n gives full knowledge of the statistics of chargetransfer. It is usually more convenient to work with thecumulant generating function (CGF) of the distributionwhich depends on the counting field χ and is defined as Z t m ( χ ) = X n e inχ p n = h e iχQ/e i . (1)The last equality here is well defined in the classical limit,when the charge Q is a random variable associated withsome stochastic process [6]. In the quantum case, one hasto supplement this definition with a prescription for howthe measurements are time-ordered; it was shown in [4]that in terms of a simple model of a spin- galvanometeras a measurement device that the correct prescription is Z t m ( χ ) = h ˜ T t e i χ e R tm ˆ I ( t ) dt T t e i χ e R tm ˆ I ( t ) dt i , (2)where T t means time-ordered, ˜ T t means anti-time or-dered, and the average h . . . i is taken over some ini-tial (in general non-equilibrium) state. We deal with F ( χ ) = − ln Z ( χ ), as this generates the irreducible cu-mulants of charge transfer C n = − (cid:16) ∂i∂χ (cid:17) n F ( χ ) (cid:12)(cid:12)(cid:12) χ =0 . In the long-time limit, each of these is proportional to themeasurement time t m , for example C = It m where I isthe current flowing in the system, and C = St m where S is the zero-frequency shot noise.While the FCS for non-interacting electrons is fairlycomprehensively understood [4, 7], far less is known inthe presence of interactions [8, 9], particularly when theinteractions drive the system into a strongly correlatedregime. In this paper we describe a general technique toobtain the CGF for quantum impurity models at finitebias by means of time-dependent numerical simulation.As a concrete example, we take the interacting reso-nant level model (IRLM), described by the Hamiltonian H = − t X n = L,R L/ X i =0 (cid:16) c † n,i c n,i +1 + H.c. (cid:17) + ( ǫ − U ) d † d (3)+ X n (cid:16) t ′ n c † n, d + H.c (cid:17) + U X n (cid:18) d † d − (cid:19) c † n, c n, . Here, c † n,i creates a Fermion on the i ’th site of the leftor right lead, while d † is the creation operator on theresonant level. The parameter t = 1 is the hopping pa-rameter of the leads, ǫ is the energy of the resonantlevel, t ′ n is the hybridization between the resonant leveland the leads (here, we will assume a symmetric coupling t ′ R = t ′ L ), and U is the interaction between the resonantlevel and the leads. In the present work, we concentrateon the point U = 2 t where the model shows a certainself-duality [2, 10]. The separation of the above Hamilto-nian into two leads and the quantum impurity howeveris much more general than this model and the methoddescribed below may be readily applied to other setups.The counting field is added via the substitution t ′ L ( R ) → t ′ e ± iχ/ and if we define the resulting Hamilto-nian after this substitution as H χ , it can be shown thatthe CGF defined in Eq. 2 may be rewritten as Z t m ( χ ) = h Ψ(0) | e i H − χ t m e − i H χ t m | Ψ(0) i . (4) π I m ( F ’) / χ / V S D measuring time t m M=60, χ =0.01M=60, χ =0.20M=96, χ =0.20V SD =0.2; analyt. V SD =0.8; analyt.V SD =1.2; analyt.V SD =1.5; analyt.V SD =1.7; analyt. FIG. 1: [Color online] Time evolution of the imaginary partof − ∂ t m F ( χ ) / ( χV SD ) for various values of bias voltage V SD ,system size M , and small values of counting field χ . As ex-plained in the text, this converges in the long time limit tothe conductance of the model. The analytic result for conduc-tance is also plotted. It is seen that after an initial transient,excellent agreement is seen between the numerical simulationin the presence of the counting field and the analytic result. This is now very convenient for a numerical time evo-lution approach, as | Ψ( t m ) i = e − i H χ t m | Ψ(0) i is sim-ply the time evolution of the starting state Ψ(0) i –see also Ref. 7. Following Ref. 11, we prepare an ini-tial state | Ψ(0) i at time τ = 0 by applying a poten-tial ± V SD / χ = 0) until some time t , when the largest transientshave died away, and we are roughly in a steady state.We then switch on χ and perform two time evolutions | Ψ ± χ ( τ ) i = exp( − i ( H ± χ − E )( τ − t )) | Ψ( t ) i , where E is the eigen-energy of the initial state. After each timestep, we look at the overlap between these two states h Ψ − χ ( τ ) | Ψ + χ ( τ ) i , thus producing from Eq. 4 Z ( χ ) as afunction of the measuring time t m = τ − t . To obtainthe long time t m → ∞ steady state F ( χ ) = t m ˜ F ( χ ) fromthe finite time numerical data, we fit to ∂F ( χ ) ∂t m = ˜ F ( χ )+ A cos( V SD t m + α )+ Be − βt m cos( ωt m + γ ) , (5)where ∂F ( χ ) ∂t m is obtained by the derivative of cubic splinesfitted to F ( χ ), taking care with the complex logarithmso as to get a continuous function. Here, the first cosineterm is the Josephson current that appears from a finitesize gap [2], while the final term is a decaying transient.We now look at the results of applying this method tothe self-dual IRLM, defined by Hamiltonian (3). For thetime-evolution, we use the full time-dependent extension[12] of the DMRG where we optimize the target spacefor all time steps simultaneously avoiding the run-awayerrors in the wave function in adaptive time evolutionschemes [13]. The time stepping is performed by a ma- π R e ( F ’) / χ measuring time t m M=60, χ =0.01M=60, χ =0.20M=96, χ =0.20V SD =0.2; M=60 fscV SD =0.2; M=96 fsc V SD =0.8; M=60 fscV SD =0.8; M=96 fscV SD =1.2; M=60 fscV SD =1.5; M=60 fscV SD =1.7; M=60 fsc FIG. 2: [Color online] Time evolution of the real part of ∂ t m F ( χ ) /χ for various values of V SD , M and small valuesof χ . This gives the zero frequency shot-noise of the model,which is known to have a noticeable finite-size correction (fsc)[3]. Excellent agreement is found between the time evolvedsimulation and the fsc analytic result, which is also plotted. trix exponential using the Krylov space representation[12]. We first perform short time dynamics up to t m = 2and successively restart the DMRG procedure to longermeasuring times up to t m = 25 using time steps of 0 . · . We would like toemphasize that the general framework is not tied to theDMRG and could be applied within other numerical ap-proaches, or even to bosonic problem such as the photondynamics in wave guiding structures [14].We begin the discussion by looking at the overlapfor small values of χ , where one can write F ( χ ) ≈− iC χ + C χ /
2. Hence the imaginary part at χ ≪ V SD and counting fields χ , where wesee relatively weak transient effects, and a clear conver-gence to the known analytic value, previously discussedin Ref. 2. We will return to a discussion of the analyticresult later. Correspondingly, the real part gives shotnoise, which is plotted in Fig. 2, and can be compared tothe results in Ref. 3. Despite the rather larger transientbehavior, the asymptotic behavior is quite clear, and isin perfect agreement with the previously discussed ana-lytic result for noise (which should be corrected for finitesize effects [3]). We note that within the present method,the extraction of the (zero-frequency) noise is a lot easierthan obtaining the correlation function directly, as noiseis obtained directly from the large time limit of ˜ F ( χ )without having to perform a Fourier transformation in-cluding the associated finite size extrapolation [15].Having examined a few time scans, we now turn to theFCS – but before examining the numerical plots let usgive the analytical result for the CGF of this model. Ithas been long known [16] that the IRLM is integrable –that is all eigenvalues and eigenstates of the interactingmany-body problem may be constructed via the Betheansatz. However it was only much later realized [17]that at the special interaction value given by the self-dual point, the voltage bias operator is also diagonal inthis Bethe basis, allowing the exact solution to be foundout of equilibrium via the methods of Ref. 18 by mappingto a boundary sine-Gordon (BSG) model. We note thatthis is not a generic feature of integrable models, a moregeneral extension of the Bethe Ansatz to non-equilibriumsystems is currently being discussed in the literature [19].The current and shot noise in the self-dual IRLMhave previously been found and compared to numericalwork in Refs. 2, 3. One can extend the previous analy-sis [20, 21] to obtain all cumulants of charge transfer inthe zero temperature limit. We find that the results arebest expressed as a recurrence relation [22] C n +1 = 13 (cid:18) C n − V SD ∂C n ∂V SD (cid:19) . (6)Taking the known expression for the current C [2] andresumming the cumulants, we find for small V SD < V c :˜ F ( χ ) = − iV SD χ π − V SD X m> a ( m )2 m (cid:18) V SD T ′ B (cid:19) m (cid:0) e − miχ − (cid:1) , (7a)and in the opposite circumstance V SD > V c :˜ F ( χ ) = V SD X m> a ( m ) m (cid:18) V SD T ′ B (cid:19) − m (cid:16) e i mχ − (cid:17) . (7b)Here V c = √ / T ′ B is the convergence radius of each ofthe series, and T ′ B ∼ ( t ′ ) / is the natural energy scaleof the problem. The non-universal parameter of propor-tionality relates the regularization of the field theory (onwhich the Bethe ansatz solution is based) to that of theoriginal lattice model, and can be taken from previousworks [2]. This leaves zero free fitting parameters for allcomparisons between analytic and numerical results inthe current study. The series coefficients are given by a K ( m ) = ( − m +1 Γ (1 + Km )4 √ πm !Γ (cid:0) + ( K − m (cid:1) . (8)We note here that the above results could be conjecturedfrom only the self-duality of the model [10]. We also pointout that the formal mapping between the IRLM and theBSG model is only proven at the self-dual point of theformer model, which forces K = 4 , / I - V characteristics of Josephson junctions [23]. F ~ [ T B ] χ [ π ] M=72, V SD =0.2, Re(F~)M=72, V SD =0.2, Im(F~)M=72, V SD =0.4, Re(F~)M=72, V SD =0.4, Im(F~)M=60, V SD =0.4, Re(F~)M=60, V SD =0.4, Im(F~)M=72, V SD =0.5, Re(F~)M=72, V SD =0.5, Im(F~)M=96, V SD =0.5, Re(F~)M=96, V SD =0.5, Im(F~)Im F~( χ ), V SD =0.2Im F~( χ ), V SD =0.4Im F~( χ ), V SD =0.5 FIG. 3: [Color online] Cumulant generating function ˜ F ( χ ) asdetermined by the DMRG, for various bias voltages V SD < V c .The data above is for t ′ = 0 . V c = 0 .
64. Thecorresponding analytic results for the imaginary part is alsoshown in the plot. The analytic real part is too small to seeon the current scale – the signal seen by the DMRG is almostentirely due to finite size effects.
We now analyze the results, first considering the lowbias regime. We plot the numerical data in Fig. 3 alongwith the analytic result (7a). While the numerical dataand analytic result show reasonable agreement in theimaginary parts, the real part suffers much more seriouslyfrom finite size effects, as was already noticed in Ref. 3.In fact, the true values of the real part of F ( χ ) in thethermodynamic long time limit are very small, so almostall of the numerical signal is due to the finite size andfinite measuring time. We note that while such large ef-fects might be a nuisance for numerical simulations, theyare also extremely relevant for experiments where trans-port measurements through nano-devices may exhibit asimilar behavior. By building a theory of such finite sizeeffects and subtracting this from the data, some prelimi-nary agreement with the analytic result may be seen, inparticular the π periodicity that will be discussed shortly.Details of this will be presented elsewhere.Looking at the analytic expansion, Eq. 7a, we seethat the backscattering current may be thought of asa sum of Poissonian processes of effective quasi-particlesof charges 2 me where m is an integer [20, 22]. As a re-sult of quantum interference, the effective ‘probabilities’in this equivalent Poissonian process are not all positive,however from the periodicity of F ( χ ), it is clear that thefundamental quasi-particles being scattered carry charge2 e . This behavior was already hinted at by the Fanofactor [3], knowledge of the full CGF confirms this.We now turn to the large voltage regime: in Fig. 4 weplot numerical results along with the appropriate analyticcurves from Eq. 7b. Again, we find very good agreementbetween the two results. As in the low bias voltage case,we can interpret the expansion (7b) as effective Poisso- F ~ [ T B ] χ [ π ] Re F~( χ ), V SD =3.11 V c Im F~( χ ), V SD =3.11 V c t’=0.3, V SD =2.0t’=0.3, V SD =2.0t’=0.25, V SD =1.5684t’=0.25, V SD =1.5684 t’=0.3, V SD =1.363t’=0.3, V SD =1.363M=120,t’=0.3, V SD =1.363M=120,t’=0.3, V SD =1.363Re F~( χ ), V SD =2.51 V c Im F~( χ ), V SD =2.51 V c FIG. 4: [Color online] Comparison between numerical andanalytic results for the cumulant generating function at biasvoltages V SD > V c . Data for different values of t ′ such that V SD /T ′ B remains constant collapse onto the same curve, andshow excellent agreement with the analytic result. nian processes, but this time with fractionally chargedquasi-particles me/ π periodic, it is notclear from the numerical technique of explicitly addingthe counting field that we can obtain a result that isnot 2 π periodic. Furthermore, the numerical data is verymessy for χ > π making this parameter regime at presentunobtainable to us. As we discuss shortly, the 4 π period-icity formally arises due to ˜ F ( χ ) becoming double valued– the branch may be chosen such that the function is 2 π periodic with discontinuities, or smoothly continued to be4 π periodic. In an experimental situation, which branchis measured would depend on details of the setup [25].Having found effective quasi-particles with charge e/ e in the low biaslimit, we now discuss how to get from one to the other.In fact, this is easiest to see by studying the countingcurrent I ( χ ) = i∂ ˜ F /∂χ , where we can actually resumthe series and obtain the answer in closed form I ( χ ) = V SD π F (cid:20) { , , } , { , } , − (cid:16) ˜ V e − iχ (cid:17) (cid:21) , (9)where ˜ V = V SD /V c is the normalized voltage, and F is a hypergeometric function. 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