Steady-state dynamics and effective temperatures of quantum criticality in an open system
SSteady-state dynamics and effective temperatures of quantum criticality in an opensystem
P. Ribeiro
Russian Quantum Center, Novaya street 100 A, Skolkovo, Moscow area, 143025 Russia andCentro de Física das Interacções Fundamentais, Instituto Superior Técnico,Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
F. Zamani
Max Planck Institute for Physics of Complex Systems, 01187 Dresden, Germany andMax Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany
S. Kirchner ∗ Center for Correlated Matter, Zhejiang University, Hangzhou, Zhejiang 310058, China
We study the thermal and non-thermal steady state scaling functions and the steady-state dy-namics of a model of local quantum criticality. The model we consider, i.e. the pseudogap Kondomodel, allows us to study the concept of effective temperatures near fully interacting as well as weak-coupling fixed points. In the vicinity of each fixed point we establish the existence of an effectivetemperature –different at each fixed point– such that the equilibrium fluctuation-dissipation theo-rem is recovered. Most notably, steady-state scaling functions in terms of the effective temperaturescoincide with the equilibrium scaling functions. This result extends to higher correlation functionsas is explicitly demonstrated for the Kondo singlet strength. The non-linear charge transport is alsostudied and analyzed in terms of the effective temperature.
PACS numbers: 05.70.Jk,05.70.Ln,64.70.Tg,72.10.Fk
The interest in understanding the dynamics of stronglycorrelated systems beyond the linear response regime hasin recent years grown tremendously.The quantum dynamics in adiabatically isolated opti-cal traps has been successfully modeled using powerfulnumerical schemes [1, 2]. In open systems mainly di-agrammatic techniques on the Schwinger-Keldysh con-tour have been employed. For nanostructured systemsseveral techniques exist to describe the ensuing out-of-equilibrium properties. These approaches, however, areeither perturbative in nature [3], centered around hightemperatures and short times [4–8], or approximate thecontinuous baths by discrete Wilson chains [9–11]. Thesituation might be simpler for non-linear dynamics thatarises in the vicinity of a quantum critical point (QCP),where a vanishing energy scale leads to scaling and uni-versality [12–19].For the dynamics near classical continuous phase transi-tions a well-established theoretical framework exists, ty-ing the dynamics to the statics and the conserved quan-tities [20]. In addition, the concept of effective temper-ature (T eff ) was established as an useful notion for therelaxational dynamics of classical critical systems [21–24], although it appears somewhat less useful for fullyinteracting critical points [23]. T eff is commonly definedby extending the equilibrium fluctuation-dissipation the-orem to the non-linear regime. The existence of effectivetemperatures in quantum systems was recently investi-gated [18, 25–28]. For a recent review see [25]. In compar-ison to classical criticality, at a QCP, dynamics already enters at the equilibrium level. For a QCP that can bedescribed by a Ginzburg-Landau-Wilson functional in el-evated dimensions, it was found that the voltage-driventransition is in the universality class of the associatedthermal classical model with voltage acting as T eff [12].Unconventional QCPs in contrast are not described solelyin terms of an order parameter functional [29, 30].In this letter we address the following general ques-tions within a model system of unconventional quantumcriticality: Is the existence of T eff tied to dynamical (or ω/T -)scaling? Does T eff have meaning for higher correla-tion functions? How unique is T eff at a given fixed pointonce boundary conditions have been specified? Can crit-ical scaling functions be expressed through T eff and if so,how do these scaling functions relate to the equilibriumscaling functions? The model system is the pseudogapKondo (pKM) model that describes a quantum spin anti-ferromagnetically coupled to a conduction electron bathpossessing a pseudogap near its Fermi energy, character-ized by a powerlaw exponent. Depending on the couplingstrength, the quantum spin is either screened or remainsfree in the zero temperature ( T ) limit. The two phasesare separated by a critical point dispaying critical Kondodestruction, see Fig.1. The pKM has been invoked todescribe non-magnetic impurities in the cuprate super-conductors [31] and point-defects in graphene [32]. Itunderlies the pseudogap free moment phases occurringin certain disordered metals [33] and can also be realizedin double quantum-dot systems [34]. The quantum crit-ical properties of the pKM in equilibrium have been ad- a r X i v : . [ c ond - m a t . s t r- e l ] D ec Figure 1. (a) Sketch of the model: a spin interacts with twofermionic leads which are characterized by their respectivedensity of states ρ − c,L/R ( ω ) and chemical potential µ L/R . (b)Phase diagram of the multichannel pKM with gap exponent r < r max : A QCP (C) separates the multichannel Kondo fixedpoint (MCK) from the (weak-coupling) local moment fixedpoint (LM). dressed in [35–45]. Our main findings are that the steadystate dynamic spin susceptibility, the conductance, andthe Kondo-singlet strength, a 4-point correlator, repro-duce their equilibrium behavior in the scaling regimes ofthe fixed points of the model when expressed in terms ofa fixed-point specific T eff . The model.
We consider a pKM with a density ofstates that vanishes in a power-law fashion with ex-ponent ≤ r ≤ at their respective Fermi level, ρ − c,l ( ω ) ∼ | ω | r Θ( D − | ω | ) , with half-bandwidth D . Here, l = L, R labels the two leads, see Fig.1(a). In themultichannel version of the model the spin degree offreedom ( S ) is generalized from SU (2) to SU ( N ) andthe fermionic excitations ( c ) of the leads transform un-der the fundamental representation of SU ( N ) × SU ( M ) with N spin and M charge channels. At T = 0 and r < r max < , a critical point (C) separates a multichan-nel Kondo (MCK)-screened phase from a local moment(LM) phase at a critical value J c of the exchange cou-pling J > , see Fig.1(b). The characterization of thephases and the leading power law exponents of observ-ables of the pKM have been obtained by perturbativeRG, large- N methods, and NRG [35–38, 40, 43]. Withinthe large- N approach, at T = 0 , scaling arguments areable to predict the critical exponents of dynamical ob-servables [39, 46]. Non-equilibrium steady-states (NESS)are obtained by applying a time-independent bias voltage V = ( µ L − µ R ) / | e | , where µ l is the chemical potentialof lead l , see Fig. 1(a). As T characterizes the fermionicreservoirs, it remains well-defined even for V (cid:54) = 0 .A similar setup has been considered in a perturbativeRG-like study adapted to the NESS condition [47]. Thismodel has also been invoked in a variational study of thedynamics following a local quench where it was foundthat quenches in the Kondo phase thermalize while thisin not the case for quenches across the QCP into the LMregime [48]. The system is described by the Hamiltonian H = (cid:88) p,ασl ε pl c † pασl c pασl + 1 N (cid:88) ll (cid:48) (cid:88) α J ll (cid:48) S . s α ; ll (cid:48) , (1) Figure 2. (a) χ (cid:48) (0) − vs J for different T . (b) φ s vs J fordifferent T . The T = 0 curve is approached from below inthe MCK and from above in the LM phases. (c) Scaling T /T eff vs V /T at fixed points LM, C, and MCK: T eff ∼ V for V (cid:29) T . (d) FDR − χ ( ω ) vs ω/T eff near fixed point C,shown for V /D = 10 − , − , − , − , − . The grey lineis FDR − ( ω ) = tanh( βω/ . where σ = 1 , . . . , N and α = 1 , . . . , M are, respec-tively, the SU ( N ) -spin and SU ( M ) -channel indices, l labels the leads and p is a momentum index. The co-tunneling term [49] in Eq. (1) contains the local opera-tors s iα ; ll (cid:48) = n c (cid:80) pp (cid:48) σσ (cid:48) c † pασl t iσσ (cid:48) c † p (cid:48) ασ (cid:48) l (cid:48) with t the fun-damental representation of SU ( N ) and n c is the num-ber of fermionic single-particle states. In a totally anti-symmetric representation, one can decompose the spinoperator as S σσ (cid:48) = f † σ f σ (cid:48) − qδ σσ (cid:48) , where q is subject tothe constraint ˆ Q = (cid:80) σ f † σ f σ = qN and the f † σ , f σ (cid:48) obeyfermionic commutation relations.We employ a dynamical large-N limit [39, 50], suitablygeneralized to the Keldysh contour [16, 18] while keeping q = QN and κ = M/N constant. This results in Σ >,, A possible order parameter for the transi-tion from the overscreened Kondo to local-moment phaseis given by lim T → T χ ( ω = 0 , T ) , where χ ( ω, T ) is theFourier transform of the local (impurity) spin-spin cor-relation function χ ( t − t (cid:48) ) , see Fig. 2(a). We work onthe Keldysh contour where the lesser and greater com-ponents are defined in the usual way as χ > ( t − t (cid:48) ) = Figure 3. Scaling of observables with T eff at different fixedpoints for the values of V as in Fig. 2(d): (a) χ (cid:48) (0) − vs T eff ;(b) ∂ ln ω ln χ (cid:48)(cid:48) ( ω ) vs ω/T eff ; (c) φ s vs T eff . For each fixedpoint, the equilibrium scaling form (grey curves) is comparedwith the same quantity under non-equilibrium conditions and T substituted by T eff . − i N (cid:80) a (cid:104) S a ( t ) S a ( t (cid:48) ) (cid:105) with t ∈ γ ← and t (cid:48) ∈ γ → and χ < ( t − t (cid:48) ) = − i N (cid:80) a (cid:104) S a ( t (cid:48) ) S a ( t ) (cid:105) , with t ∈ γ → and t (cid:48) ∈ γ ← so that χ R ( t ) = Θ ( t ) [ χ > ( t ) − χ < ( t )] and χ A = χ R + χ < − χ > . Here, γ → ( ← ) is the forward (back-ward) branch of the Keldysh contour, respectively.We also consider the “singlet-strength” φ s , definedthrough the Kondo term contribution to the totalenergy of the system as N (cid:80) ll (cid:48) (cid:80) c J ll (cid:48) (cid:104) S . s c,ll (cid:48) (cid:105) = − Jκ (cid:16) N − N (cid:17) φ s [52]. φ s is a dimensionless quantity,which possesses a well-defined large- N limit and quan-tifies the degree of singlet formation. In terms ofthe fermionic fields, it can be written as the local-in-time limit of a 4-point correlator [51]. Its equilib-rium properties will be discussed below. The steadystate charge current passing through each channel is J P = − ∂ t (cid:68) ˆ N L ( t ) (cid:69) /M , where ˆ N L = (cid:80) pασ c † pασL c pασL is the number of particles in the left lead. The out-of-equilibrium conditions considered here respect particle-hole symmetry which implies a vanishing energy current.Throughout the paper we set κ = 0 . , r = 0 . , and q = 1 / . This results in r max = 0 . . Our choice ofvalues for κ and r ensures a finite static spin suscepti-bility χ (cid:48) ( ω = 0) within the MCK phase as T → . Wedenote the real (imaginary) part of χ R ( ω ) by χ (cid:48) ( χ (cid:48)(cid:48) ). Thermal steady-state . The equilibrium ( V = 0 ) behav-ior of χ (cid:48) ( ω = 0 , T ) in the relaxational regime ( ω (cid:28) T )near the MCK, C, and LM fixed points is shown inFig.2-(a). For J < J c (cid:39) . D , i.e. in the LM phase,one observes Curie-like behavior at lowest temperatures χ (cid:48) ( ω = 0 , T ) ∝ T − . In the MCK phase ( J > J c andwith our choices of κ and r ), the T = 0 susceptibility re-mains finite. The grey lines in Figs.3-(b) show the scalingplots of the logarithmic derivative of χ (cid:48)(cid:48) ( ω ) for different values of the temperature, i.e. ∂ ln ω ln χ (cid:48)(cid:48) ( ω ) for the dif-ferent fixed points. Note that ∂ ln ω ln χ (cid:48)(cid:48) ( ω ) (cid:39) α χ withinthe scaling region where χ (cid:48)(cid:48) ( ω ) ∝ | ω | α χ . The values of α χ in the quantum coherent regime ( ω/T (cid:29) ) agree withthose obtained analytically from a T = 0 scaling ansatz[46] for the MCK ( α χ (cid:39) . ) and C ( α χ = − . )fixed points. These results are compatible with a dy-namical scaling form χ (cid:48)(cid:48) ( ω, T ) = T α χ Φ ( ω/T ) , in termsof an universal scaling function Φ ( x ) possessing asymp-totic values Φ ( x ) ∝ x for x (cid:28) and Φ ( x ) ∝ x α χ for x (cid:29) . Thus, the scaling properties are in line with dy-namical ω/T -scaling for the C and MCK fixed points. Forthe LM fixed point we find α χ = − and a scaling form χ (cid:48)(cid:48) ( ω ) = T α χ (cid:101) Φ (cid:0) ω/T κ (cid:1) , indicative of a weak-couplingfixed point and absence of hyperscaling. These resultswill be further addressed elsewhere [46].The singlet-strength φ s vs. J at different T and at V = 0 is shown in Fig.2-(b). The numerical data at T (cid:54) = 0 sug-gest that φ s ( J, T = 0) is a continuous function of J . Atthe C fixed point we find that φ s ( J, T ) as a function of J crosses for different values of T (for sufficiently low T ). Non-thermal steady-states . We consider a non-equilibrium setup where the two leads, initially decou-pled from the impurity (for t < t ), are held at chemicalpotentials µ L = − µ R = | e | V / ( | e | = 1 in the follow-ing). At t = t the coupling between the leads andthe impurity is turned on. A steady-state is reachedby sending t → −∞ so that any transient behaviorwill already have faded away at (finite) time t . TheNESS fluctuation-dissipation ratio (FDR) for a dynam-ical observable A ( t, t (cid:48) ) = A ( t − t (cid:48) ) is defined throughFDR A ( ω ) = [ A > ( ω ) + A < ( ω )] / [ A > ( ω ) − A < ( ω )] , where A >/< are the Fourier transforms of the greater/lessercomponents of A . At equilibrium, the fluctuation-dissipation theorem implies FDR A ( ω ) = tanh ( βω/ ζ uniquely (with ζ = ± for fermionic (+) and bosonic(-) operators). For a generic out-of-equilibrium sys-tem, the functional form of the FDR differs from theequilibrium one. A frequency-dependent “effective tem-perature”, /β A eff ( ω ) , for the observable A can be de-fined by requiring that tanh (cid:2) β A eff ( ω ) ω/ (cid:3) ζ = FDR A ( ω ) [27, 53]. Following Refs. [18, 21, 26] we define T eff viaFDR χ through its asymptotic low-frequency behavior T − eff = lim ω → β χ eff ( ω ) . In equilibrium T eff = T . On theother hand, a linear-in- V decoherence rate in the non-equilibrium relaxational regime near an interacting QCPis signaled by ω/V -scaling [16]. In this case and at T = 0 one expects T eff = cV , where c characterizes the under-lying fixed point. We thus analyze T /T eff vs V /T . Fig.2-(c) shows the resulting T /T eff as a function of V /T forthe different fixed points computed for different values of V and T . In the non-linear regime, the scaling collapsefor T /T eff implies T eff = cV , where /c is the amplitude ofthe scaling curve in the non-linear regime. A comparisonof FDR − χ with the equilibrium result for fixed point (C) Figure 4. Conductance G normalised to the MCK fixed pointconductance πG = 0 . . (a) G ( T ) vs T computed for thelowest non-zero value of V at different values of J (see colorcoding). (b) G vs V for fixed T . (c) G = J P /V vs T eff atdifferent fixed points. The equilibrium form is given by thegrey curves. is shown in Fig.2-(d). Even for the LM fixed point, wherehyperscaling is violated, T eff ∼ V holds for V (cid:29) T , seeFig. 2-(c), top panel. It is however important to realizethat the properties we see in terms of T eff are a propertyof the flow towards the LM fixed point. Far from equi-librium and outside any scaling regime, χ is a function of ω , T , and V but near a fixed point χ ( ω, T, V ) developsa scaling form in terms of a combination of ω , T , and V . This then raises the question how T eff enters the scal-ing function and leads us to a remarkable result, see Fig.3-(b)-(c): The non-thermal steady-state scaling functionof χ = χ ( ω, T, V ) when scaled in terms of T eff recoversthe equilibrium scaling function of that particular fixedpoint with T eff replacing T . This not only turns out tobe true for χ at each of the fixed points of the model butalso holds for φ s , a higher-order correlation function. Wefirst consider the static susceptibility. Fig. 3-(a) showsthe equilibrium scaling forms of χ (cid:48) (0) − as a function of T eff for different values of T and V for the LM, C andMCK fixed points. The color coding reflects the valuesof T of the system. The equilibrium form (grey lines) isrecovered even for T eff /T (cid:29) .A similar result can be obtained at finite ω : Fig. 3-(b) shows the log-derivative ∂ ln ω ln χ (cid:48)(cid:48) ( ω ) as a functionof ω/T eff for different values of T and V for the LM, Cand MCK fixed points. These should be compared withthe equilibrium results, the underlying grey lines: Theequilibrium scaling form is recovered by replacing T by T eff , both for ω (cid:28) T eff and ω (cid:29) T eff [54]. Note that T eff is defined from the FDR of χ in the limit ω/T → .Therefore, the fact that the equilibrium scaling formsof χ (cid:48) (0) and χ (cid:48)(cid:48) ( ω ) are reproduced for T eff /T (cid:29) and ω/T eff (cid:29) , respectively, is remarkable. Fig. 3-(c) depicts φ s as a function of T eff for different values of T and V .Again, the equilibrium scaling behavior (gray curves) isreproduced.Unlike χ and φ s , the conductance G depends on bothpseudoparticle propagators G f and G B . One thus maywonder if T eff can have any meaning for G . In Figs. 4- (a,b) we show the conductance per channel G = J P /V vs T and V respectively. In the linear response regime V, T (cid:28) T K ( J ) of the MCK phase, the current is pro-portional to the applied voltage J P = G V . Outsideof the scaling regime, i.e. for V, T (cid:29) T K ( J ) , G dropsrapidly as V or T increase. The linear and non-linearcurrent-voltage characteristics display power-law behav-ior as T, V → [16, 18]. Near C, i.e. for J = J c , therelation between J P and V is still linear, ( J P = G c V ),however the critical conductance G c is much smaller than G . Fig. 4-(c) shows G vs T eff for different values of T and V for the LM, C and MCK fixed points. The greycurves are obtained by varying T at fixed V for the lowestvalue of V considered in our study, i.e. V min = 10 − D .The temperature dependence of the linear response con-ductance is reproduced at all fixed points when the non-linear conductance is taken as a function of T eff . Thisis true even for values of V several orders of magnitudelarger than V min . In conclusion , we have addressed the steady-state dy-namics near unconventional quantum criticality. Wefound that in the scaling regime of all the fixed pointsconsidered, all observables studied ( χ, φ s , G ) scale interms of the same but fixed point specific effective tem-perature T eff . The local spin-spin correlation function χ and the singlet-strength φ s assume their equilibriumscaling functions even far from equilibrium when scaledin terms of T eff , i.e. T eff replacing T . A similar result re-lates the linear and non-linear conductance. 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G. Green, Phys. Rev. Lett , 091601(2012).[56] M. J. Bhaseen, B. Doyon, A. Lucas, and K. Schalm, Na-ture Phys. , 509 (2015). Steady-state dynamics and effective temperaturesof quantum criticality in an open system:Supplementary Material P. Ribeiro, , F. Zamani, , and S. Kirchner Russian Quantum Center, Novaya street 100 A, Skolkovo, Moscow area, 143025 Russia Centro de Física das Interacções Fundamentais, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais,1049-001 Lisboa, Portugal Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Straße 40, 01187 Dresden, Germany Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany Center for Correlated Matter, Zhejiang University, Hanghzou, Zhejiang 310058, China GENERATING FUNCTIONAL ON THE KELDYSH CONTOUR The generating functional on the Keldysh contour can be written as Z [ ξ ] = (cid:90) Dc (cid:90) Df (cid:90) Dλe i ( c † g − c c + f † g − f f ) e iqN (cid:82) γ dzλ z × e + i N (cid:82) γ dz (cid:80) α J ll (cid:48) z ( (cid:80) σ f † σ,z c ,ασl (cid:48) z )( (cid:80) σ (cid:48) c † ασ (cid:48) lz f σ (cid:48) z ) × e c † ξ c + ξ † c c + f † ξ f + ξ † f f (S.1)where ξ c and ξ f act as sources to the fermionic c and f fields and λ is a scalar Lagrange multiplier enforcing theconstraint (cid:80) σ f † σ f σ = qN . (cid:82) γ dz is the integral over the Keldysh contour γ = γ → + γ ← with its forward ( γ → ) andbackward ( γ ← ) branches.Here, the inverse bare propagators are g − f = ( i∂ z − λ z ) , (S.2) g − c = ( i∂ z − ε pl ) . (S.3)In analogy to the equilibrium procedure –albeit performed on the Matsubara contour– one can introduce a Hubbard-Stratonovich decoupling field B αlz conjugated to (cid:80) σ (cid:48) c † ασ (cid:48) lz f σ (cid:48) z , to decouple the quartic fermionic term in Eq.(S.1).Thus, Z [ ξ ] = (cid:90) Df (cid:90) Dλ (cid:90) DBe i ( f † g − f f ) e i (cid:82) dzλ z Q e i ( B † α g − B B α ) × e − i N (cid:82) dz (cid:82) dz (cid:48) (cid:80) σαl B αlz B † αlz (cid:48) ˜ g c,l ( z,z (cid:48) ) f † σ,z f σz (cid:48) × e (cid:104) − √ N (cid:82) γ dz (cid:48) (cid:80) α (cid:48) σ (cid:48) ( ξ † c .g c ) α (cid:48) σ (cid:48) l (cid:48) z B † αl (cid:48) z (cid:48) f σ (cid:48) z (cid:48) − √ N (cid:82) γ dz (cid:80) σα f † σ,z B αlz ( g c .ξ c ) ασlz (cid:105) × e iξ † c g c ξ c e f † ξ f + ξ † f f (S.4)with g − B ; ll (cid:48) = − (cid:104) ˜ J − (cid:105) ll (cid:48) , (S.5)where (cid:104) ˜ J (cid:105) ll = J ll (cid:48) , is the bare inverse propagator of the B field.Finally, with the help of the complex-valued dynamic Hubbard-Stratonovich fields W zz (cid:48) ; l one obtains Z [ ξ ] = (cid:90) Dλ (cid:90) DW e N tr ln [ − iG − f ] − M tr ln [ − i ( G − B + V † ξc G f V ξc )] e iN tr [ W † ∗ [˜ g c ] − ∗ W ] + i (cid:82) dzλ z Q e − iξ † f G f V ξc [ G − B + V † ξc G f V ξc ] − V † ξc G f ξ f + iξ † c g c ξ c + iξ † f G f ξ f (S.6)with G − f ( z, z (cid:48) ) = g − f ( z, z (cid:48) ) − W zz (cid:48) ; l (S.7) G − B ( z, z (cid:48) ) = g − b ( z, z (cid:48) ) − ¯ W z (cid:48) z ; l (S.8)tr (cid:104) W † ∗ [˜ g c ] − ∗ W (cid:105) = (cid:88) l (cid:90) dz (cid:90) dz (cid:48) ¯ W zz (cid:48) ; l W zz (cid:48) ; l ˜ g c,l ( z, z (cid:48) ) (S.9)and V † ξ c and V ξ c are source-dependent terms. Eq.(S.6) is used to derive all correlators by taking derivatives withrespect to the source fields. DYNAMICAL LARGE-N SELF-CONSISTENCY EQUATIONS ON THE KELDYSH CONTOUR In this section we set the sources to zero and compute the saddle-point equations with respect to the bosonic fields W and λ . The generating functional in the absence of sources is Z [ ξ = 0] = (cid:90) Dλ (cid:90) DW e iN S [ W,λ ] (S.10)with S [ W, λ ] = q (cid:90) dzλ z + tr (cid:104) W † ∗ [˜ g c ] − ∗ W (cid:105) − i N tr ln (cid:104) − iG − f (cid:105) + iκ M tr ln (cid:2) − iG − B (cid:3) . (S.11)The saddle point equations are obtained by putting the linear variation of S [ W, λ ] with respect to W and λ to zero: δδW zz (cid:48) ,l S [ W, λ ] = ¯ W zz (cid:48) ,l [˜ g c ( z, z (cid:48) )] − + i N (cid:88) σ G f ( z (cid:48) , z ) = 0 , (S.12) δδ ¯ W zz (cid:48) ,l S [ W, λ ] = [˜ g c,l ( z, z (cid:48) )] − W zz (cid:48) ,l − iκ M (cid:88) α G B : ll ( z, z (cid:48) ) = 0 , (S.13) δδλ z S [ W, λ ] = q + i N (cid:88) σ G f ( z − , z ) = 0 . (S.14)These equations become exact in the large-N limit. These equations are equivalent to ˆ G − f = ˆ g − f − Σ f ˆ G − B = ˆ g − B − Σ B q = − i ˆ G f ( z − , z ) with Σ B ( z, z (cid:48) ) = (cid:18) ¯ W z (cid:48) z,L 00 ¯ W z (cid:48) z,R (cid:19) = − i (cid:18) ˜ g c,L ( z (cid:48) , z ) 00 ˜ g c,R ( z (cid:48) , z ) (cid:19) ˆ G f ( z, z (cid:48) ) (S.15) Σ f ( z, z (cid:48) ) = δ σσ (cid:48) (cid:88) l W zz (cid:48) ,l = δ σσ (cid:48) iκ (cid:88) l ˜ g c,l ( z, z (cid:48) ) ˆ G B ; ll ( z, z (cid:48) ) . (S.16)Note that λ z evaluated at the saddle-point is time independent, i.e. λ t = λ . SINGULAR EXCHANGE COUPLING MATRIX J So far, the treatment has been general and no particular form of the Kondo exchange coupling matrix has beenassumed. For the physically most relevant case where the Kondo Hamiltonian is derived from an Anderson-type modelthrough a Schrieffer-Wolff transformation, the exchange matrix J ll (cid:48) ( l, l (cid:48) = L, R ) takes the from J = (cid:18) J L √ J L J R √ J L J R J R (cid:19) . Thus, the exchange coupling matrix is singular, det( J ) = 0 . In this case, where one of the eigenvalues of J vanishes,we can write ˜ J = | u + (cid:105) ( J L + J R ) (cid:104) u + | with | u − (cid:105) = − (cid:114) J R J L + J R | L (cid:105) + (cid:114) J L J L + J R | R (cid:105)| u + (cid:105) = (cid:114) J L J L + J R | L (cid:105) + (cid:114) J R J L + J R | R (cid:105) . As the exchange matrix is singular, the component u − of the B field has to vanish and thus ˆ G B = | u + (cid:105) ˆ G B + (cid:104) u + | . In this case the self-consistent equations simplify to Σ B + ( z, z (cid:48) ) = − i ˜ g c, + ( z (cid:48) , z ) ˆ G f ( z, z (cid:48) ) , (S.17) Σ f ( z, z (cid:48) ) = iκ ˜ g c, + ( z, z (cid:48) ) ˆ G B, + ( z, z (cid:48) ) , (S.18)with ˆ G − B + = ˆ g − B + − Σ B , ˆ g − B + = − J L + J R , ˜ g c, + = J L ˜ g c,L + J R ˜ g c,R J L + J R . Using Langreth’s rules, we obtain Σ >,c ( t (cid:48) , t ) ˆ G >, The steady-state condition implies that the system is time translationally invariant so that G R,A,K ( t, t (cid:48) ) = G R,A,K ( t − t (cid:48) ) . Therefore, it is advantageous to solve the self-consistent equations in the frequency domain. Theconventions of the Fourier transform used by us are A ( t ) = (cid:90) dω π A ( ω ) e − iωt ,A ( ω ) = (cid:90) dt A ( t ) e iωt . Eq.(S.19-S.21) take the form Σ >,c ( ν − ω ) G >, With the definitions of the previous sections, Dyson’s equation translates to ρ ± f ( ω ) = σ ± f ( ω ) (cid:26)(cid:104) ω − ˜ λ + πσ Hf ( ω ) (cid:105) + (cid:104) πσ − f ( ω ) (cid:105) (cid:27) − ,ρ ± B ( ω ) = σ ± B ( ω ) (cid:110)(cid:2) − ( J L + J R ) + πσ HB ( ν ) (cid:3) + (cid:2) πσ − B ( ω ) (cid:3) (cid:111) − , with ˜ λ = λ t − κ ( J L + J R ) (cid:82) dωρ + c ( ω ) being a renormalized chemical potential, and Eq.(S.22-S.24) translate to σ ± B ( ω ) = ∓ (cid:90) dν (cid:104) ρ ± c ( ν − ω ) ρ + f ( ν ) − ρ ∓ c ( ν − ω ) ρ − f ( ν ) (cid:105) , (S.33) σ ± f ( ω ) = κ (cid:90) dν (cid:2) ρ ± c ( ω − ν ) ρ + B ( ν ) + ρ ∓ c ( ω − ν ) ρ − B ( ν ) (cid:3) , (S.34) q = 12 (cid:20) − (cid:90) dωρ + f ( ω ) (cid:21) . (S.35)In the particle-hole symmetric case ( q = 1 / ) and for a particle-hole symmetric DOS of the leads ( ρ − c ( ω ) = ρ − c ( − ω ) )the quantities ρ ± f,B and σ ± f,B are real. Details of the numerical treatment The explicit form of the pseudogap density of states of the leads is taken to be ρ − c,l ( ω ) = 1 √ (cid:0) r +12 (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ω √ (cid:12)(cid:12)(cid:12)(cid:12) r e − ω , with l = R, L and Λ = 1 specifies the soft high-energy cutoff. The self-consistent equations were solved iterativelyon a logarithmically discretized grid with points ranging from − to . The criterium for convergence ofthe selfconsistency loop was that the relative difference of two consecutive iterations was less than − . The resultswere benchmarked by the conditions that the fluctuation dissipation ratios of the Green’s functions have to accuratelyreproduce the equilibrium fluctuation dissipation relations demanded by the fluctuation-dissipation theorem. Forall the fixed points we studied a range of temperatures T /D = 10 − , − . , − , . . . , − and a range of voltages T /D = 10 − , − . , − , . . . , − . However convergence of the numerical solution of the self-consistent equationswas not always achieved for all combinations of parameters.0 OBSERVABLES Cross 4-point function In order to compute the currents and the Kondo singlet strength we will need to evaluate the connected 4-pointfunction (cid:68) T γ c † p α σ l c p α σ l f † s f s (cid:69) C . Here, C denotes the connected part of a correlation function and T γ is thetime-ordering operator on the Keldysh contour. Using the procedure outlined above, one obtains (cid:68) T γ f s ( t ) f † s ( t ) c p α σ l ( t ) c † p α σ l ( t ) (cid:69) C = i N n c δ s σ δ s σ δ α α F p l ; p l ( t , t , t , t ) and F p l ; p l ( t , t , t , t ) = (cid:112) J l J l (cid:112) ( J L + J R ) ( J L + J R ) × (cid:90) dz (cid:48) (cid:90) dz G f ( t , z ) g c ; p l ( z, t ) G B ( z, z (cid:48) ) g c ; p l ( t , z (cid:48) ) G f ( z (cid:48) , t ) with g c ; p l ( t, t (cid:48) ) = (cid:104) tp l | g c | t (cid:48) p l (cid:105) . For equal times we have (cid:68) c † p α σ l ( t ) c p α σ l ( t ) f † s ( t ) f s ( t ) (cid:69) C = i N n c δ s σ δ s σ δ α α F p l ; p l ( t ) , where the time-ordering for the equal-time limit is defined through F p l ; p l ( t ) =lim t , , , → t F p l ; p l ( t , t , t , t ) (cid:12)(cid:12) t >t >t >t . F p l ; p l ( t ) can be explicitly evaluated using Langreth rulesand making use of the fact that we describe a steady-state. This procedure is straightforward but involved and yields F p l ; p l ( t ) = 4 iπ (cid:104) I (1) l p ,l p + I (2) l p ,l p (cid:105) , with I (1) p ,p = 18 (cid:112) J l J l ( J L + J R ) (cid:90) dω π (cid:8)(cid:2) − i H (cid:2) A − ++ − l (cid:3) ( ω ) + A − ++ − l ( ω ) (cid:3) (cid:2) ρ + B ( ω ) (cid:3) (cid:2) i H (cid:2) A − ++ − l (cid:3) ( ω ) + A − ++ − l ( ω ) (cid:3) + (cid:2) − i H (cid:2) A − ++ − l (cid:3) ( ω ) + A − ++ − l ( ω ) (cid:3) (cid:2) − iρ HB ( ω ) + ρ − B ( ω ) (cid:3) (cid:2) i H (cid:2) A −− ++ l (cid:3) ( ω ) + A −− ++ l ( ω ) (cid:3) + (cid:2) − i H (cid:2) A −− ++ l (cid:3) ( ω ) + A −− ++ l ( ω ) (cid:3) (cid:2) iρ HB ( ω ) + ρ − B ( ω ) (cid:3) (cid:2) i H (cid:2) A − ++ − l (cid:3) ( ω ) + A − ++ − l ( ω ) (cid:3)(cid:9) I (2) p ,p = 12 (cid:112) J l J l πi (cid:90) dω π (cid:8)(cid:2) − i H (cid:2) A − ++ − l (cid:3) ( ω ) + A − ++ − l ( ω ) (cid:3) A −− ++ l ( ω ) − (cid:2) − i H (cid:2) A −− ++ l (cid:3) ( ω ) + A −− ++ l ( ω ) (cid:3) A − ++ − l ( ω ) (cid:9) , where we defined A Σ l ( ω ) = (cid:90) dν π (cid:104) ρ Σ(1) f ( ν ) ρ Σ(2) c,p l ( ν − ω ) − ρ Σ(3) f ( ν ) ρ Σ(4) c,p l ( ν − ω ) (cid:105) , H [ A ] ( ω ) = − π P (cid:90) dν A ( ν ) ω − ν . Currents The currents of particles and energy through the system are obtained from the change in particle number andenergy of e.g. the left lead through a continuity equation for the conserved charge (particle number or energy), J b = − ∂ t (cid:104)Q b ( t ) (cid:105) = − i (cid:104) [ H ( t ) , Q b ( t )] (cid:105)J E → Q E = H L = (cid:88) p,ασ ε pL c † pασL c pασL J P → Q P = N L = (cid:88) p,ασ c † pασL c pασL (cid:2) c † α c β , c † γ c δ (cid:3) = δ β,γ c † α c δ − δ α,δ c † γ c β and the fact that the Hamiltonian can be decomposed as H = H L + H R + H J with H J = 1 N (cid:88) ll (cid:48) J ll (cid:48) (cid:88) σσ (cid:48) (cid:88) α (cid:0) f † σ f σ (cid:48) − qδ σσ (cid:48) (cid:1) c † ,ασ (cid:48) l (cid:48) c ασl , one obtains J P ( t ) /M = 2 (cid:112) J L,t J R,t Re n c (cid:88) pp (cid:48) F Rp (cid:48) ,Lp ( t ) , J E ( t ) /M = 2 (cid:112) J L,t J R,t Re n c (cid:88) pp (cid:48) ε pL F Rp (cid:48) ,Lp ( t ) . Susceptibility On the Keldysh contour the impurity spin susceptibility is defined by χ ( z, z (cid:48) ) = − i N (cid:88) a (cid:104) T γ S a ( z ) S a ( z (cid:48) ) (cid:105) , where T γ is the time-ordering operator on the Keldysh contour.For a steady state, we obtain χ ± ( ω ) = − (cid:90) dν (cid:104) ρ + f ( ν − ω ) ρ ± f ( ν ) − ρ − f ( ω − ν ) ρ ∓ f ( ν ) (cid:105) , where χ ± f ( ω ) = − πi [ χ > ( ω ) ± χ < ( ω )] . Kondo singlet strength