Universal zero-bias conductance for the single-electron transistor M. Yoshida Departamento de Física, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, 13500 Rio Claro, SP, Brazil A. C. Seridonio* and L. N. Oliveira Departamento de Física e Informática, Instituto de Física de São Carlos, Universidade de São Paulo, 369 São Carlos, SP, Brazil �Received 26 May 2009; revised manuscript received 16 November 2009; published 16 December 2009� The thermal dependence of the zero-bias conductance for the single electron transistor is the target of two independent renormalization-group approaches, both based on the spin-degenerate Anderson impurity model. The first approach, an analytical derivation, maps the Kondo-regime conductance onto the universal conduc- tance function for the particle-hole symmetric model. Linear, the mapping is parametrized by the Kondo temperature and the charge in the Kondo cloud. The second approach, a numerical renormalization-group computation of the conductance as a function the temperature and applied gate voltages offers a comprehensive view of zero-bias charge transport through the device. The first approach is exact in the Kondo regime; the second, essentially exact throughout the parametric space of the model. For illustrative purposes, conductance curves resulting from the two approaches are compared. DOI: 10.1103/PhysRevB.80.235317 PACS number�s�: 73.21.La, 72.15.Qm, 73.23.Hk I. INTRODUCTION Nearly five decades ago, Anderson conceived a Hamil- tonian to describe the interaction between a magnetic impu- rity and otherwise free conduction electrons.1 Once a daunt- ing theoretical challenge, the Anderson Hamiltonian yielded to an essentially exact numerical diagonalization,2 followed by an exact analytical diagonalization.3,4 From these and al- ternative approaches, physical properties were extracted, which eased the interpretation of experimental data,5 theoret- ical results provided unifying views of apparently unrelated phenomena,6 quantitative comparisons brought forth novel perceptions,7 and dissections of the Anderson model brought to light the physics of nanoscale devices.8–12 The last ten years were remarkably fruitful. Parallel ad- vances in scanning tunneling spectroscopy and in the fabri- cation of nanostructured semiconductor devices enhanced the interest in transport properties.13–25 In both areas, numer- ous experimental breakthroughs and theoretical analyses were reported, and the Anderson Hamiltonian proved spec- tacularly successful in more than one occasion.26,27 Notwithstanding the substantial volume of exact results, certain aspects of the model remain obscure. Consider uni- versality, a concept important in its own right and by virtue of its diverse applications. Universal relations serve as benchmarks checking the accuracy of numerical data, as re- sources promoting the convergence of theoretical findings, and as instruments bridging the gap between the theorist’s tablet and the laboratory logbook. The conditions under which the Anderson model exhibits universal thermodynami- cal properties were identified.2–4 Although one expects all properties of the model to be universal in the same domain, few firm results for the dynamical and transport properties can be found in libraries.28 The early effort of Costi et al. showed that the transport coefficients for the particle-hole symmetric Anderson model are universal.29 For asymmetric models—even ones that display universal thermodynamical properties—nonetheless, the universal curves fail to fit the numerical data, the disagreement growing with the asymme- try. Puzzled by such contrasts, we have conducted a system- atic study of the transport properties for the Anderson Hamil- tonian. We combined analytical and numerical- renormalization group �NRG� tools and paid special attention to universality. In a preliminary report,30 we have discussed an Anderson model for a quantum dot side coupled to a quantum wire, a device comprising two conduction paths whose transport properties are marked by interference.31–35 Notwithstanding the constructive or destructive effects, we have been able to identify universal behavior throughout the Kondo regime, the parametrical domain favoring the forma- tion of a magnetic moment at the quantum dot, and its pro- gressive screening by the conduction electrons as the tem- perature is lowered past the scale set by the Kondo temperature TK. Specifically, we found the thermal depen- dence of the conductance to map linearly onto a universal function of the temperature T scaled by the Kondo tempera- ture TK. The mapping is itself universal; i.e., it depends on a single physical property, the ground-state phase shift �, into which the contributions from all model parameters are lumped. This paper examines the alternative experimental setup in which a quantum dot or molecule, instead of side coupled to, is embedded in the conduction path.5,18,26,36–40 We show that the thermal dependence of the conductance maps onto the same universal function. Although linear, the mapping now depends explicitly on a model parameter—an external poten- tial applied to the conduction electrons—and hence contrasts with the conclusion in our previous report. This dependence accounts for distinctions between the transport properties in the embedded and side-coupled arrangements. At high tem- peratures, for instance, potentials appropriately applied to the conduction electrons in the side-coupled geometry drive the conductance from nearly zero up to the ballistic limit G2 =2e2 /h. If the quantum dot is embedded in the conduction path, by contrast, the high-temperature conductance is pinned at low values and virtually insensitive to potentials applied to the conduction electrons. Our analysis shows that, in the embed- PHYSICAL REVIEW B 80, 235317 �2009� 1098-0121/2009/80�23�/235317�22� ©2009 The American Physical Society235317-1 http://dx.doi.org/10.1103/PhysRevB.80.235317 ded configuration, the dot charge parametrizes the mapping to the universal conductance curve. That charge being always close to unity in the Kondo regime, the mapping is never very far from the identity so that the conductance is always less than 25% below the universal curve. To discuss the thermal dependence of the conductance in the Kondo regime more specifically, we have to know the Kondo temperature and the ground-state dot charge as func- tions of the model parameters. The Kondo regime is, more- over, only one of the physically relevant domains in the para- metric space of the Anderson Hamiltonian. To offer a more comprehensive view of zero-bias conduction through the single electron transistor �SET�, we have gone beyond the mapping. The second half of this paper, Secs. VI and VII, covers the Kondo range and the domains contiguous to it with a large number of temperature-dependent conductance curves. To discuss the numerical data, we take advantage of the mapping to the universal function, which links the salient features of the conductance plots to the rapid changes in dot occupation that accompany the transitions between regimes. The transition from the Kondo regime to the neighboring domains makes the mapping to the universal function pro- gressively less accurate. To document its decay, we also dis- play the conductance against temperature for various gate voltages across the transition and compare the plots with the conductance curve predicted by the mapping. All illustra- tions considered, the numerical study provides a unified de- scription of conduction in the single-electron transistor. The text is divided in eight sections, more technical as- pects of the analysis having been confined to the three Ap- pendixes. Section II defines the model. Section III derives an expression relating the conductance to the spectral density of the quantum dot level. Section IV is dedicated to universal- ity, and Sec. V to the fixed points of the model Hamiltonian and to an extension of Langreth’s exact expression for the ground-state spectral density. Section VI then shows that, in the Kondo regime, the thermal dependence of the conduc- tance can be mapped onto the symmetric-SET universal con- ductance. The numerical survey is reported next. Section VII de- scribes the numerical procedure, and Sec. VIII displays and discusses the results. Finally, Sec. IX summarizes the con- clusions drawn from the analytical derivation and from the numerically computed conductances. II. SINGLE-ELECTRON TRANSISTOR Figure 1 depicts a SET, the prototypical example of em- bedding. The subject of numerous experimental studies, the SET comprises two independent conduction bands coupled by a localized level. In the laboratory, the quantum dot is unevenly coupled to the electron gases represented by the two rectangles. The consequences of this asymmetry being well understood8–11 conciseness recommends that we ana- lyze only the evenly coupled device. Qualitatively, the physics of Fig. 1 was understood long before the first device was developed.8,9 To recapitulate, it is convenient to start from Fig. 2, which depicts the spectrum of the SET Hamiltonian H in the weak-coupling limit. With the dot levels decoupled from the conduction bands, the eigenstates and eigenvalues of H can be labeled by the dot quantum numbers, among which the dot occupation nd is chiefly important. For fixed nd, the product of the lowest dot state by the conduction-band ground state is shown as a bold dash. The gray levels above it represent the excited states consistent with the same nd label. A small transition amplitude V between the quantum dot and the wires is sufficient to modify this picture. The ampli- tude V strongly couples each gray level to the degenerate or nearly degenerate states in the neighboring columns. Excep- tions are the lowest levels in the column labeled nd=N in Fig. 2, which are energetically distant from their neighbors and thus remain unperturbed to first order in the coupling. At low temperatures, with kBT small in comparison with the energy �E separating the ground state from the closest level in the neighboring columns, the dot occupation is frozen at nd=N, a constraint that raises the Coulomb blockade against conduction through the dot. Suitably adjusted, the gate potential Vd in Fig. 1 lifts the blockade. The potential shifts the dot energies. Adjusted to the condition �E�0, it levels the bold dashes in the nd=N and nd=N+1 columns in Fig. 2 so that an infinitesimal bias suffices to induce electronic flow between the wires through the dot. The zero-bias conductance peaks whenever the gate potential Vd tunes the ground-state expectation value of nd to a half-integer, e.g., ���nd���→N+1 /2 as �E→0 in Fig. 2. Each peak identifies a resonance at the Fermi level. As the gate voltage is swept past �E=0, the ground-state occupation FIG. 1. �Color online� Single electron transistor. A quantum dot �circle� bridges two noninteracting quantum wires �rectangles�. A gate potential Vd controls the dot energy, while the symmetric po- tentials Vw shift the energy of the wire orbitals close to the dot. n d = N − 2 N − 1 N N + 1 N + 2 δE FIG. 2. �Color online� SET energies in the weak-coupling limit. The dot-level occupation nd labels the energies. For each nd, the bold dash represents the conduction-band ground state, while the thinner lines represent excitations. The coupling between the dot and the two quantum wires mixes each level to the neighboring columns. YOSHIDA, SERIDONIO, AND OLIVEIRA PHYSICAL REVIEW B 80, 235317 �2009� 235317-2 changes rapidly from nd=N to nd=N+1, and as required by the Friedel sum rule, so does the ground-state phase shift. At moderately low temperatures, for thermal energies smaller than the average spacing between the bold dashes in the fig- ure, the plot of the conductance against gate voltage shows a succession of peaks. Conductances measured at moderately low temperatures do display a sequence of resonances. At very low temperatures, however, the conductance pattern changes to a sequence of intervals alternating between insu- lating valleys and conducting plateaus. The conducting plateaus are due to the Kondo effect. For gate voltages corresponding to odd ground-state dot occupa- tions, the magnetic moment of the resulting dot spin interacts antiferromagnetically with the conduction electrons. As the device is cooled past the Kondo temperature, the screening of the moment creates the Kondo resonance, a spiked en- hancement of the density of states anchored at the Fermi level. The pinned resonance defeats the Coulomb blockade and allows ballistic conduction through the quantum dot. III. ANDERSON MODEL A variant of the Anderson Hamiltonian encapsulates the physics of the device in Fig. 1. A spin degenerate level cd represents the dot level, and two structureless half-filled con- duction bands, labeled L �left� and R �right�, represent the two quantum wires. The L�R� wire comprises N state ckL�ckR� with energies defined by the linear dispersion relation �k = �k−kF�vF�0�k�2kF� so that the bandwidth is 2D=2vFkF. The per-particle per-spin density of conduction states is � =1 /2D, and we will let ��D /N denote the energy splittings in the conduction bands. The model Hamiltonian is then the sum of three terms, H=Hw+Hd+Hwd, where the first term describes the wires, Hw = � k� �kck� † ck� + W N � kq� ck� † cq�, �1� with an intrawire scattering potential W, fixed by the poten- tial Vw in Fig. 1, and �=L ,R. The Hamiltonian Hd describes the dot, Hd = �dnd + Und↑nd↓, �2� where U represents the Coulomb repulsion between electrons in the dot orbital, and the dot energy �d is controlled by the gate potential Vd in Fig. 1. Finally, the Hamiltonian Hwd couples the wires to the dot, Hwd = V 2N � k� �ck� † cd + H.c.� . �3� A. Parity To exploit the inversion symmetry of Fig. 1, we define the normalized even �ak� and odd �bk� operators as ak = 1 2 �ckL + ckR� , �4a� bk = 1 2 �ckL − ckR� . �4b� The projection of the model Hamiltonian on the basis of ak’s and bk’s splits it in two decoupled pieces, H=HA+HB, where HA = � k �kak †ak + Wf0 †f0 + V�f0 †cd + H.c.� + Hd, �5� with the traditional NRG shorthand f0 � � k ak/ N , �6� and HB = � k �kbk †bk + W N � kq bk †bq. �7� B. Conductance The odd Hamiltonian HB is decoupled from the quantum dot. It is, moreover, quadratic and hence easily diagonaliz- able. Appendix C determines its spectrum, analyzes the re- sponse of the conduction and dot electrons to the application of an infinitesimal bias, and turns the result into the follow- ing linear-response expression for the conductance: G�T� = G2 W −D D �d��,T��− �f��� �� �d� , �8� where f��� is the Fermi function; G2�2e2 /h, the conduc- tance quantum, W = 1 + 2�2W2 , �9� is the width = �V2 of the cd level, here renormalized by the scattering potential W, and �d��,T� = 1 f����mn e−�Em Z ��n�cd †�m��2���mn − �� �10� is the spectral density for the dot level. Here �m� and �n� are eigenstates of HA with eigenvalues Em and En, respectively, �mn�Em−En, and Z is the partition function for the Hamil- tonian HA. As one would expect, given that the odd Hamiltonian HB commutes with cd, only the eigenvalues and eigenvectors of HA are needed to compute the right-hand sides of Eqs. �8� and �10�. The following discussion will hence focus on the even Hamiltonian, Eq. �5�, which is equivalent to the con- ventional spin-degenerate Anderson Hamiltonian.1 C. Characteristic energies Four characteristic energies govern the physical properties of the Anderson Hamiltonian. Two of them are the charge- excitation energies catching the eye in Fig. 3: the energy −�d needed to remove an electron from the dot level and the energy �d+U needed to add an electron to the level. The UNIVERSAL ZERO-BIAS CONDUCTANCE FOR THE… PHYSICAL REVIEW B 80, 235317 �2009� 235317-3 particle-hole transformation ck→ck †, cd→−cd † swaps the two energies so that the transformed dot Hamiltonian is given by the right-hand side of Eq. �2� with �d→−��d+U�. If 2�d+U=0, the dot Hamiltonian remains invariant under the particle-hole transformation. If, in addition, W=0, Eq. �5� reduces to the symmetric Hamiltonian HA S = � k �kak †ak + V�f0 †cd + H.c.� − U 2 �nd↑ − nd↓�2. �11� With V�0, two other energies arise: the level width W Eq. �9�� and the Kondo energy kBTK, given by TK � �J exp�− 1/�J� , �12� where J is the antiferromagnetic interaction between the con- duction electrons and the dot magnetic moment,41 �J = 2 W ��d� U �d + U . �13� In the Kondo regime, thermal and excitation energies are much smaller than min���d� ,�d+U�. In Fig. 3, only the low- est levels in the central columns are energetically accessible. The energy W, associated with transitions from the central to the external columns in the figure �i.e., with cd 1→cd 2 and cd 1→cd 0 transitions� becomes inoperant. Instead, at very low excitation and thermal energies, smaller than the Kondo en- ergy kBTK, the dot spin binds antiferromagnetically to the conduction spins. In Fig. 3, the lowest states in the left and right central columns hybridize to constitute a Kondo singlet. IV. UNIVERSALITY The concepts recapitulated in Sec. III C emerged over three decades ago, with the first accurate computation of the magnetic susceptibility of the Anderson model,2 long before the first essentially exact computation of the conductance. A particularly important result in the more recent survey of transport properties of Costi et al.29 is the thermal depen- dence of the conductance for the symmetric Hamiltonian HA S , the universal curve GS�T /TK�, depicted by the solid line in Fig. 4. For kBT�D and any pair � ,U� satisfying �U in Eq. �11�, proper adjustment of the Kondo temperature TK gives a conductance curve G�T /TK� that reproduces GS�T /TK�. In Fig. 4, for instance, the solid line was computed from the eigenvalues and eigenvectors of HA S with =0.1D and U=3D. The definition G�TK��0.5G2 yielded the Kondo temperature TK=2.4 10−6D. When the calculation was re- peated for U=0.6D and the same , the Kondo temperature grew four orders of magnitude to TK=2.2 10−2D. Still, for kBT�0.1D, the plot of G�T /TK� resulted indistinguishable from the solid curve. While TK is model-parameter depen- dent, G�T /TK� is not. Particle-hole asymmetry drives G away from GS. For U +2�d�0 or W�0, the universal curve GS�T /TK� no longer matches G�T /TK�. An example is the dashed curve in Fig. 4, calculated with =0.1D, U=3D, �d=−0.3D, and W=0. The definition G�TK�=0.5G2, which in this case yields TK=4 10−3D, forces the solid and the dashed lines to agree at T=TK; the conductance for the asymmetric model nonethe- less undershoots �overshoots� the universal curve for T �TK�T�TK�. To reconcile this discrepancy with the concept of universality, the following sections rely on renormalization-group concepts. ΓK FL LM n d = 0 1 1 2 εd + U εd ΓW ΓW FIG. 3. �Color online� Spectrum of the spin-degenerate Ander- son model, displayed as in Fig. 2. In the weak-coupling limit, the eigenstates are labeled by the occupation nd and spin component of the dot configuration displayed at the bottom. For V�0, each level in the left and right columns hybridizes with nearly degenerate lev- els in the central columns and acquires the width W in Eq. �9�. At low energies, the levels in the two central columns combine into a singlet and acquire a width K�kBTK. The vertical arrows near the right border mark the domains of the LM and FL fixed points. 10−2 10−1 1 10 103 0.0 0.5 1.0 G G2 T/TK FL LM ASYMMETRIC SYMMETRIC FIG. 4. �Color online� Thermal dependences of the conductance for two sets of model parameters, computed by the procedure in Sec. VII. The solid line depicts the universal conductance curve �Ref. 29� for the symmetric Hamiltonian �11�. Here, it was com- puted with =0.1D and U=3D. The temperatures were scaled by the Kondo temperature TK=2.4 10−6D /kB, fixed by the require- ment G�TK�=0.5G2. The dashed curve is the conductance for Hamiltonian �5� with =0.1D, U=3D, �d=−0.3D, and W=0, which yielded TK=4.0 10−3D. To keep the data within the tem- perature range kBT�0.1D, the dashed plot stops at T=25TK. The horizontal arrows pointing to the vertical axes indicate the corre- sponding fixed-point conductances, given by Eqs. �22a� and �22b�. YOSHIDA, SERIDONIO, AND OLIVEIRA PHYSICAL REVIEW B 80, 235317 �2009� 235317-4 V. FIXED POINTS Renormalization-group theory probes the spectrum of Hamiltonians in search of characteristic energies and scaling invariances. The wire Hamiltonian �1�, for instance, exhibits a single trivial characteristic energy: the conduction band- width 2D. For energies ��D, therefore, its spectrum is in- variant under the scaling transformation Hw→�Hw, for ar- bitrary scaling parameter ��1. Accordingly, for ��D, the wire Hamiltonian is a stable fixed point of the renormalization-group transformation in Ref. 2. Latent in the Anderson Hamiltonian �5�, by contrast, are the four nontrivial characteristic energies discussed in Sec. III C. Part of the spectrum of HA lies close to fixed points; the remainder is in transition ranges. In the vicinity of a fixed point, the spectrum remains approximately invariant under scaling; in the transition intervals, the eigenvalues are com- parable to one or more characteristic energies and hence change rapidly under scale transformations. In particular, the portion of the spectrum pertinent to the Kondo regime com- prises two lines of fixed points and a crossover region. For given thermal or excitation energy E, the inequality max�E , W��min���d� ,�d+U ,D� defines the Kondo regime. As Fig. 3 shows, the dot occupation is then nearly unitary. In the energy range kBTK�E�min���d� ,�d+U ,D�, the Hamil- tonian HA is near the local moment �LM� fixed point. At very low energies, E�kBTK, i.e., below the energy scale defined by the narrow set of levels at the center of Fig. 3, the spec- trum becomes asymptotically invariant under scaling as the Hamiltonian approaches the frozen level �FL� fixed-point. In the intermediate region E�kBTK, the Hamiltonian crosses over from the LM to the FL. A. Fixed-point Hamiltonians As the two central columns in Fig. 3 indicate, the LM is an unstable fixed-point consistent of a conduction band and a free spin-1 /2 variable. In the FL, a singlet replaces the spin, and the Hamiltonian is equivalent to a conduction band—a stable fixed point. In their most general form, the fixed-point conduction bands mimic the wire Hamiltonian, i.e., H LM * = � k �kak †ak + WLMf0 †f0, �14� and H FL * = � k �kak †ak + WFLf0 †f0, �15� with scattering potentials WFL and WLM dependent on V, W, U, and �d. Equations �14� and �15� identify two lines of fixed points, parametrized by WLM and WFL, respectively. The Schrieffer-Wolff transformation offers an approxima- tion for the LM potential, �WLM = �W + 2 W ��d� 2�d + U �d + U . �16� For most applications, this expression is insufficiently accu- rate, and an NRG computation is necessary to determine WLM and WFL. The exception is Hamiltonian �11�, for which WLM =0, as required by particle-hole symmetry. B. Fixed-point phase shifts Appendix A diagonalizes the quadratic Hamiltonians �14� and �15�. For the LM, the diagonal form reads H LM * = � k ��g� †g�, �17� with phase-shifted energies �� = �� − �LM � . �18� Here � is the energy splitting defined in Sec. III. At the LM, all conduction states are uniformly phase shifted, with tan �LM = − �WLM . �19� For HA=HA S , in particular, �LM =0, and the low-energy eigen- values �k coincide with the �k. The FL eigenvalues are likewise uniformly phase shifted, H FL * = � k �̃kg̃k †g̃k, �20� where �̃k=�k− �� / ��. From the Friedel sum rule, it follows that42 � = �LM + 2 . �21� For HA=HA S , in particular, �= /2. C. Conductance at the fixed points The LM is the fixed point to which the Anderson Hamil- tonian would come if =0, i.e., if the dot were decoupled from the electron gases. For 0� �min���d� ,�d+U ,D�, al- though the renormalization-group flow never reaches the LM, it brings HA close to the fixed point. The substantial portion of the spectrum of HA marked by the thin double- headed arrow in Fig. 3 is approximately described by the many-body eigenvalues of H LM * , and in the pertinent energy range, the physical properties of HA and H LM * are approxi- mately the same. Likewise, at low temperatures, the proper- ties of HA approach those of H FL * . The renormalization-group flow of the Hamiltonian con- trols the thermal dependence of its physical properties. The electrical conductance is no exception. As the temperature is reduced from T�TK to T�TK, as illustrated by the curves in Fig. 4, the conductance crosses over from a lower plateau to a higher one. The extension of Langreth’s expression42 de- rived in Appendix B determines the plateau conductances, GLM = G2 sin2��LM − �W� = G2 cos2�� − �W� , �22a� GFL = G2 sin2�� − �W� , �22b� where �W is the ground-state phase shift for V=0. According to the analysis in Appendix A, UNIVERSAL ZERO-BIAS CONDUCTANCE FOR THE… PHYSICAL REVIEW B 80, 235317 �2009� 235317-5 tan �W = − �W . �23� The solid curve in Fig. 4 was computed for HA=HA S so that �w=0, while the ground-state �i.e., FL� phase shift is � = /2. The high-temperature �low-temperature� tendency of the plot, G�T�TK�→0 G�T�TK�→G2�, agrees with Eq. �22a� Eq. �22b��. The dashed curve was also computed with �w=0, but the ground-state phase shift is smaller than /2 because the Hamiltonian now lacks particle-hole symmetry. From the low-energy eigenvalues in the NRG run that gen- erated the conductance curve we find �=0.43 . The conduc- tances predicted by Eqs. �22� are indicated by the two hori- zontal arrows in Fig. 4. Since the width =0.1D and the dot energy ��d�=0.3D fail to satisfy the condition � ��d�, the renormalization- group flow of the asymmetric Hamiltonian in Fig. 4 bypasses the LM. The Kondo thermal energy for the asymmetric Hamiltonian is kBTK=4 10−3D. This relatively high energy leaves no room in the renormalization-group path for the thermal bracket D�kBT�kBTK, which defines the LM. In- stead of crossing from the LM to the FL, the Hamiltonian therefore crosses over from the band-edge regime kBT�D to the FL, Kondo screening takes place before the dot moment is fully formed, and even at the highest temperatures in Fig. 4, the conductance is well above the horizontal arrow point- ing to GLM. At low temperatures, by contrast, the renormalization-group flow drives HA toward H FL * , and the dashed curve in Fig. 4 rises to GFL. VI. CROSSOVER In the Kondo regime, the Schrieffer-Wolff transformation41 brings the Anderson Hamiltonian HA to the Kondo form HJ = � k �kak †ak + WLMf0 †f0 + J� �� f0� † ���f0� · S , �24� with J defined in Eq. �13�. To eliminate the scattering potential on the right-hand side, it is convenient to project HJ upon the basis of the eigenoperators gk of the LM, which yields43 HJ = � k ��g� †g� + JW� �� �0� † ����0� · S , �25� where JW=J cos2 �LM, and �0 = 1 N � � g�. �26� In the symmetric case �LM vanishes, and the operator �0 reduces to f0. In the Kondo regime, the second term on the right-hand side of Eq. �25� drives the Hamiltonian from the LM to the FL. Along the resulting trajectory, the eigenvalues of HJ scale with TK.3,4,44,45 Let TK and T̄K�TK be the Kondo tem- peratures corresponding to two sets of model parameters in the Kondo regime: M�� ,W ,U ,�d� and M̄ �� ̄ ,W̄ , Ū , �̄d�, to which correspond the antiferromagnetic couplings J and J̄, respectively. If �m� is an eigenvector of HJ with eigenvalue Em, then a corresponding eigenvector �m̄� of HJ̄, the scaling image of �m�, can always be found, with the same quantum numbers and eigenvalue Ēm such that Em /TK= Ēm / T̄K. The matrix elements of any linear combination of the op- erators gk are moreover universal. Given two eigenstates �m� and �n� of HJ and their scaling images �m̄� and �n̄�, then the matrix elements of �0, for example, are equal: �m��0�n� = �m̄��0�n̄�. Likewise, the matrix elements of the normalized operator �1 = 3 N � � �� D g� �27� are universal: �m��1�n�= �m̄��1�n̄�. A. Thermal dependence of the conductance By contrast, the matrix elements �m�cd�n� on the right- hand side of Eq. �10� are nonuniversal. Even at the lowest energies, as Eq. �B14� shows, they depend explicitly on the model parameters. To discuss universal properties, therefore, we must relate them to universal matrix elements, such as �m��0�n�, �m��1�n�, or �m�g��n�. As a first step toward that goal, we evaluate the commutator HA,aq †� = �qaq † + V N cd † + W N � p ap †, �28� and sum the result over q to find that HA, f0 †� = 1 3 f1 † + Vcd † + Wf0 †. �29� Here we have defined another shorthand f1 = 3 N � q �q D aq. �30� Equation �29� relates the matrix elements of cd † between two �low-energy� eigenstates �m� and �n� of HA to those of the operators f0 and f1, V�m�cd †�n� = �Em − En − W��m�f0 †�n� − 3D�m�f1 †�n� . �31� In the Kondo regime, with max�Em ,En��D, the first two terms within the parentheses on the right-hand side can be dropped. In the symmetric case, since f0�f1� coincides with �0��1�, Eq. �31� shows that the product V�m�cd�n� is universal, in line with the firmly established notion that �d�� /kBTK ,T /TK� and GS�T /TK� are universal functions.29,28 To discuss asym- metric Hamiltonians, we have to relate the operators f0 and f1 to �0 and �1. This is done in Appendix A 2, which shows that, in the Kondo regime, a linear transformation with model-parameter dependent coefficients relates the matrix el- ements of both f0 and f1 to those of �0 and �1. When Eq. �A21� is substituted for f0 and f1 on the right-side of Eq. �31�, it results that YOSHIDA, SERIDONIO, AND OLIVEIRA PHYSICAL REVIEW B 80, 235317 �2009� 235317-6 � W�m�cd †�n� = �0�m��0 †�n� + �1�m��1 †�n� . �32� Here, the constants �0 and �1 are combinations of the �un- known� linear coefficients on the right-hand side of Eq. �A21�, the parameter W on the right-hand side of Eq. �31�, and the ratio � W /V, by which we multiplied Eq. �31� to shorten the following algebra. Substitution in Eq. �10� yields an expression relating the spectral density �d to universal functions, � W�d��,T� = �0 2�0��,T� + �1 2�1��,T� + �0�1��01���,T� , �33� where � j��,T� = � mn e−�Em Zf��� ��n�� j�m��2��Em − En − �� �j = 0,1� , �34� and ��01���,T� = � mn e−�Em Zf��� ��m��0 †�n��n��1�m� + c.c.� ��Em − En − �� . �35� Next, we substitute Eq. �33� on the right-hand side of Eq. �8� to split the conduction into three pieces, G�T� = �0 2G0�T� + �1 2G1�T� + �0�1G�01��T� , �36� where Gj�T� = G2 � −D D � j��,T��− �f��� �� �d� �j = 0,1� , �37� and G�01��T� = G2 � −D D ��01���,T��− �f��� �� �d� . �38� B. Universal contributions to the conductance Given the universality of the energies Em and of the ma- trix elements �m�� j�n��j=1,2� on the right-hand sides of Eqs. �34� and �35�, we see that the spectral densities � j�� ,T��j=0,1� and ��01��� ,T� are universal. Inspection of the right-hand sides of Eqs. �37� and �38� shows that the functions Gj�j=0,1� and G�01� are likewise universal. To compute them, we are free to consider any convenient Kondo-regime Hamiltonian. Particle-hole symmetry makes HA S especially convenient. To show that the cross terms make no contribution to the conductance, i.e., that G�01��T�=0, we only have to notice that, while leaving HA S unchanged, the particle-hole transfor- mation cd→−cd †, gk→gk † �i.e., ak→ak †� reverses the sign of the product of matrix elements �m��i †�n��n�� j�m�+c.c. on the right-hand side of Eq. �35�. We see that ��01��� ,T� is an odd function of � so that the integral on the right-hand side of Eq. �38� vanishes. To evaluate G0 and G1, we start out from the closed form resulting from the diagrammatic expansion �in the coupling V� of the conduction-electron retarded Green’s function for the symmetric Hamiltonian, Gkk� S ��� = Gk �0�����kk� + V2 N Gk �0����Gd S���Gk� �0�, �39� where Gd S is the retarded dot-level Green’s function for the symmetric Hamiltonian, and Gk �0���� = 1 � − �k + i� �40� is the free conduction-electron retarded Green’s function. From Gkk� S , it is a simple matter to obtain the spectral densities on the right-hand side of Eq. �33�, �0��,T� = − 1 N J� kk� Gkk� S ��� , �41� and �1��,T� = − 3 ND2J� kk� �k�k�Gkk� S ��� . �42� To compute the conductances at temperatures T satisfying kBT�D, we only need the spectral densities for ��D. It is an excellent approximation, therefore, to expand the right- hand side of Eq. �40� to linear order in � /D, Gk �0���� = 2� D − i ��� − �k� �� � D� . �43� The sums over momenta on the right-hand side of Eqs. �41� and �42� are then easily computed. Among the resulting terms, only the even powers of � contribute to the integral on the right-hand side of Eq. �37�. To compute the conductance to O �kBT /D�2� we hence neglect the terms of O�� /D�. Equation �41� then gives �0��,T� = � − � �d S��,T� . �44� Substitution on the right-hand side of Eq. �37� then shows that the universal function G0 is related to the universal con- duction for the symmetric Hamiltonian, G0�T� = G2 − GS�T� . �45� Equation �45� becomes exact, asymptotically, at low tem- peratures. In the Kondo regime, the deviations, of O �kBT /D�2�, are insignificant. As an illustration, the open circles in Fig. 5 show NRG data for the conductance G0�T�, Eq. �37�, in excellent agreement with the solid line represent- ing the right-hand side of Eq. �45�. To the same accuracy, we can neglect the O�� /D� terms resulting from the summation on the right-hand side of Eq. �42�, which yields �1��,T� = 6� �d S��,T� . �46� Equation �37� then shows that G1 is also related to the con- ductance for the symmetric Hamiltonian, UNIVERSAL ZERO-BIAS CONDUCTANCE FOR THE… PHYSICAL REVIEW B 80, 235317 �2009� 235317-7 G1�T� = 6 2GS�T� , �47� C. Mapping to the universal conductance We next combine Eqs. �45� and �47� with the result G�01�=0 to reduce Eq. �36� to the equality G�T� = �0 2 „G2 − GS�T�… + �1 2 6 2GS�T� . �48� To determine the coefficients �0 and �1, we need only compare the right-hand side with the fixed-point expressions for the conductance. At the LM, GS=0, and Eq. �22a� shows that �0 2=cos2��−�W�. At the FL, GS=G2, and Eq. �22b� shows that �6 / 2��1 2=sin2��−�W�. These two results substi- tuted on its right-hand side, Eq. �48� reads G� T TK � − G2 2 = − �GS� T TK � − G2 2 �cos 2�� − �W� . �49� Central in this paper, Eq. �49� maps the conductance G�T /TK� to the universal function GS�T /TK� linearly. The mapping is controlled by the argument 2��−�W� of the trigo- nometric function on the right-hand side. According to the Friedel sum rule,42 2��−�W� / is the screening charge in- duced by the coupling to the dot, which must equal the dot occupation nd to insure electrically neutrality. In particular, when the gate potential Vg is such that dot occupation is unitary, the phase-shift difference is �−�W= /2, the cosine on the right-hand side of Eq. �49� is equal to −1, the second terms on the right- and left-hand sides cancel out of Eq. �49�, and the remaining terms are equal, G�T /TK�=GS�T /TK�. A particular case is the symmetric Hamiltonian �11�, for which �= /2, �w=0, and G�T /TK�=GS�T /TK�.29 For other gate potentials in the Kondo regime, the ground- state dot occupation is approximately unitary, as Fig. 3 ex- plained. The phase shift difference is never too far from /2. If, by contrast, it were �−�w= /4, the conductance in Eq. �49� would be flat: G�T�=G2 /2. For the intermediate differ- ences /4��−�w� /2 observed in the Kondo regime, the conductance lies between the universal curve GS�T /TK� and the horizontal G2 /2. Although monotonically decreasing, the function G�T /TK� is therefore flatter than GS. Since �−�w is never too far from /2 in the Kondo re- gime, at the qualitative level we could still treat G�T /TK� as if it were proportional to GS�T /TK�, but the mapping �49� yields much more accurate conductances and affords quanti- tative comparison with numerical or experimental data. To underline this conclusion, the following sections present an NRG survey of electrical conduction through the device in Fig. 1. VII. NUMERICAL PROCEDURE Equation �12� offers an approximation for TK, and Eqs. �16�, �19�, and �21� roughly determine the ground-state phase shift �. Such estimates are far from the accuracy needed to fit numerical or experimental data. In the laboratory, TK and � −�W are adjustable parameters. The former, in particular, is determined by the definition G�TK��G2 /2.5,32,33 In the computer office, the two unknown parameters on the right-hand side of Eq. �49� can be extracted from the conductance itself, or from other properties of the model Hamiltonian. The phase shift � can be obtained from the ground-state eigenvalues of HA, or from the ground-state dot occupation. The Kondo temperature TK has been traditionally derived from fits of the temperature dependent magnetic sus- ceptibility ��T� with the universal curve for kBT��T /TK�.2,4 Here, we break with the tradition and adopt the convention G�TK��G2 /2 so that both sides of Eq. �49� vanish at T=TK. Once TK and � have been determined, by either a Bethe ansatz calculation or an NRG computation,3,4,28 one can rely on the mapping �49� to evaluate the temperature-dependent conductance G�T� in the Kondo regime. Alternatively, one can apply the NRG procedure described in Secs. VII A and VII B to compute equally accurate conductance curves G�T� over the entire parametrical space of the model. Here, we rely on the latter approach to provide a more comprehensive view of the model. Sections VIII A and VIII B will present NRG computations of the conductance as a function of the gate voltage and temperature. We cover the Kondo regime and the neighboring regions of the parametric space of the model to describe charge transport in the single- electron transistor and to examine the behavior of the map- ping �49� beyond the limits of the Kondo regime. A. Numerical-renormalization group method Excellent descriptions of the NRG method being available,2,28,46 one page will be sufficient to recapitulate the four constituents of the procedure. 1. Logarithmic discretization Two dimensionless parameters ��1 and 0�z�1 define the logarithmic discretization of the conduction band.28,47 10−2 10−1 1 10 103 0.0 0.5 1.0 G G2 T/TK G0/G2 (G2 − GS)/G2 U = 3D Γ = 0.1D FIG. 5. �Color online� NRG results for the thermal dependence of the auxiliary conductance G0�T�, associated with the spectral density for the operator �0. The open circles show Eq. �37� for j =0, computed for the symmetric Hamiltonian with the displayed model parameters. The solid line is the right-hand side of Eq. �45�, i.e., the universal curve in Fig. 4 subtracted from the quantum con- ductance G2. YOSHIDA, SERIDONIO, AND OLIVEIRA PHYSICAL REVIEW B 80, 235317 �2009� 235317-8 The infinite energy sequence Em=D�1−z−m�m=0,1 , . . . � de- fines the intervals Im= Em+1 ,Em�. For each interval, a single operator am+=�Im akd�k /nm, with normalization factor nm, is defined. In the negative half of the conduction band, the sequence −Em�m=0,1 , . . . � defines the mirror image am− of each operator am+. The am� form a basis upon which the conduction band Hamiltonian is projected.2 2. Lanczos transformation Next, a Lanczos transformation48 makes tridiagonal the projected conduction Hamiltonian so that the model Hamil- tonian reads HA = �� n=0 � tnfn †fn+1 + Vf0 †cd + H.c.� + Wf0 †f0 + Hd. �50� Here, f0 is the operator defined in Eq. �6�, and fn’s �n =0,1 , . . . � form an orthonormal basis that replaces am�’s �m=0,1 , . . . �. With z=1, we recover the Lanczos transforma- tion in Ref. 2. Otherwise, the codiagonal coefficients tn have to be determined numerically.47 With error O��−n�, it is found49 that tn=D �1−�−1� / log ���1−z−n/2, a result that brings us to the third step in the NRG procedure, the defini- tion of a truncated Hamiltonian. 3. Infrared truncation Given a temperature T and a small dimensionless param- eter �, let N be the smallest integer such that tN��kBT and consider the infinite sum on the right-hand side of Eq. �50�. Compared to kBT, the codiagonal element tN is then negli- gible. To compute G�T�, it is hence safe to drop the term with n=N in the infinite series, an approximation that decouples the subsequent terms from the dot Hamiltonian. Once decou- pled, the terms with n�N will no longer contribute to the conductance; to neglect tN is thus equivalent to introducing an infrared cutoff tN−1. Since kBT and tN−1 are of the same order of magnitude, to emphasize that kBT sets the energy scale we define the reduced bandwidth DN�D �1 −�−1� / log ���−�N−1�/2 and the dimensionless, scaled, trun- cated Hamiltonian HA N, DNHA N � �� n=0 N−1 tnfn †fn+1 + Vf0 †cd + H.c.� + Wf0 †f0 + Hd. �51� In the scaled sum on the right-hand side, tN−1 /DN, the small- est codiagonal coefficient, is of O�1�. 4. Iterative diagonalization and ultraviolet truncation The last step in the NRG procedure is the iterative diago- nalization of the model Hamiltonian. With N=0, the right- hand side of Eq. �51� is easily diagonalized; four eigenvalues Em 0 and four eigenvectors �m�0 �m=1, . . . ,4� result. At this stage, it is equally simple to calculate the matrix elements 0�m�cd�n�0 between the eigenvectors of HA N=0, which will be needed to compute the conductance. Application of the operators f0↑ † , f0↓ † , f0↑ † f0↓ † , and the iden- tity 1 on the eigenvectors of HA N=0 generates 16 states that constitute a basis upon which the Hamiltonian HA N=1 can be projected. Appropriately chosen linear combinations of those operators yield basis states �p�1 �p=1, . . . ,16� that diagonal- ize the charge and spin operators; projected on them, HA N=1 reduces to block-diagonal matrices, which are then diagonal- ized numerically. The matrix elements 0�m�cd�n�0 �m ,n =1, . . . ,4� are projected onto the basis �p�1 and subsequently rotated to the basis of the eigenstates �m�1 �m=1, . . . ,16� of HA N=1. Application of the operators f1↑ † , f1↓ † , f1↑ † f1↓ † , and 1 on the �m�1 creates 64 basis vectors upon which HA N=2 can be projected, and the procedure is iterated. To check the exponential growth of matrix dimensions, a dimensionless parameter � is chosen, an ultraviolet cutoff that controls the cost and the accuracy of the iterative diago- nalization. At the end of iteration N, the eigenvectors with scaled energies Em /DN above � are discarded before the con- struction of the basis states �p�N+1, upon which the Hamil- tonian HA N+1 will be projected. This expedient limits the num- ber of basis states and hence the computational effort in each iteration. The cost of a full NRG run grows linearly with the number of iterations. B. Computation of the conductance In each iteration, the diagonalization in Sec. VII A 4 yields the eigenvalues Em �m=0,1 , . . . ,M, where M is an integer determined by the ultraviolet cutoff� and correspond- ing eigenvectors �m� of the scaled Hamiltonian �51�. Once the matrix elements �m�cd�n� �m ,n=0,1 , . . . ,M� are com- puted, one could in principle combine Eqs. �8� and �10� to compute the conductance at any temperature. In practice, in each iteration the infrared and ultraviolet truncations define a narrow window, in which the computed conductance can be accurately computed. This section explains how one can combine the sequence of thermal intervals thus resulting from the iterative diagonalization to compose a conductance curve. 1. Relation between the conductance and the eigenvalues and eigenvectors of the scaled Hamiltonian Equation �8� relates the conductance to the dot-level spec- tral density �d�� ,T�. Since the temperature T defines the en- ergy window in each iteration, the thermal energy kBT is guaranteed to be between the infrared and ultraviolet cutoffs. Although � may be above �below� the ultraviolet �infrared� cutoff, it is easy to see that only the energies inside the win- dow contribute significantly to the conductance. To this end, substitute Eq. �10� on the right-hand side of Eq. �8�. The resulting integral leads to an expression for the conductance that depends only on the eigenvalues Em of HA N within the energy window and on the corresponding eigenvectors �m�, G�T� = G2 � w Z � mn ��m�cd�n��2 e�Em + e�En . �52� As suggested by this equality, which can be reduced to a few lines of computer code, the computational effort behind a conductance curve G�T� is comparable to the cost of a mag- netic susceptibility plot.2 UNIVERSAL ZERO-BIAS CONDUCTANCE FOR THE… PHYSICAL REVIEW B 80, 235317 �2009� 235317-9 2. Thermal ranges The eigenvalues of the scaled truncated Hamiltonian range from unity to � and hence correspond to energies rang- ing from DN to �DN. Having neglected tN�DN+1, we can only compute conductances for kBT��DN+1, where ��10. At the other extreme, the ultraviolet truncation restricts us to temperatures such that kBT��DN. Thus, provided that �DN�� / ��DN+1�� �, i.e., that ���, the Nth iteration yields reliable conductances in the temperature window �DN+1 �kBT� ��DN+1. If a run is stopped at iteration Nmax, the juxtaposition of the resulting windows yields G�T� for all temperatures above �DNmax+1 /kB. In practice, the conduc- tance is only computed for the kBT�0.1D because irrelevant operators artificially introduced by the logarithmic discreti- zation make the interval 0.1D�kBT�D unreliable. 3. z trick Conductance curves computed with large � show oscilla- tions, which can be traced to a sequence of poles on the J� = � i / log � lines of the complex-energy plane.47 To elimi- nate these artifacts of the discretization, we average the con- ductance curve G�T� computed for given z over a sequence of equally spaced z’s in the interval 0�z�1.50 The expo- nential dependence of the computational effort on 1 / log � makes this averaging procedure far more efficient than com- parably accurate computations with small �. 4. Numerics The conductances in Sec. VIII were computed with � =6 and averaged over two z’s: 0.5 and 1. The amplitude of the residual oscillations encountered after averaging over z, somewhat smaller than 0.001e2 /h, provides an estimate of the error introduced by the logarithmic discretization. The other two parameters controlling the precision of the results were fixed at �=10.5 and �=50, respectively. Spin degenera- cies not counted the number of states below the cutoff in each iteration peaked at 4000 in iteration 6. To estimate the error due to the infrared and ultraviolet truncations, for each N�Nmax we compared the conductance at the lowest tem- perature in the �N−1�th window, �DN�kBT� ��DN, with the conductance at the highest temperature in the Nth win- dow. The mismatch between the two results never exceeding 0.001e2 /h, we conclude that deviations due to the three ap- proximations in the procedure, the logarithmic discretization and the infrared and ultraviolet truncations, are comparable. At any temperature, the estimated absolute deviation in the computed conductances is smaller than 0.05% of the quan- tum conductance. The relatively large discretization parameter expedites the calculation. On a standard desktop computer, a complete run, including �i� the iterative diagonalization of HA and compu- tation of the matrix elements on the right-hand side of Eq. �52� for each z and �ii� the evaluation of the conductance curve in the interval 10−10D�kBT�0.1D, takes less than 30 s. C. Renormalization-group flow Along with the iterative diagonalization procedure, Eq. �51� defines a renormalization-group transformation,2 T HA N� � HA N+2 = �HA N + � n=N N+1 tn DN+2 �fn †fn+1 + H.c.� . �53� The factor � multiplying the first term on the right-hand side magnifies the scale on which the eigenvalues of HA N are ex- amined. On the new scale, the second term is a fine structure. In the absence of characteristic energies, as N grows the magnification compensates the refinement, and the lowest- energy eigenvalues of HA N+2 rapidly become indistinguishable from those of HA N. This indicates that the Hamiltonian has reached a fixed point of T. In the Kondo regime, the condition V=0 turns the Ander- son Hamiltonian HA into the LM fixed point of T. With V �0, as the temperature is reduced past the dominant charac- teristic energy Ec=min���d� ,�d+U ,D�, the Hamiltonian HA N first approaches the LM and then moves away toward the FL fixed point—a strong-coupling fixed point equivalent to Eq. �5� with V→�. Between the LM and the FL lies the Kondo temperature TK, around which the conduction electrons screen the dot moment. If one of the dot-charge excitation energies, �0���d� or �2��d+U, is smaller than the dot width w, the model Hamiltonian enters the mixed-valence regime.55 Instead of min�D , ��d� , �ed+U�, the dominant characteristic energy is now Ec=min�D , w�. The dot moment is only partially formed, as the coupling w drives the model Hamiltonian toward the FL before it can come close to the LM. D. Phase shifts The potential WLM�WFL� on the right-hand side of Eq. �14� Eq. �15�� and the associated phase shift �LM��� depend on the model parameters. With =0, for instance, the Hamil- tonian HA flows directly toward the LM; therefore, WLM =W and �LM =�w. For 0� �D, approximate LM phase shifts can be extracted from the eigenvalues of HA N, where N is such that D�DN�kBTK, i.e., such that the model Hamil- tonian dwells in the vicinity of the LM. For all �0, by contrast, the FL phase shift � can be calculated very accu- rately from the low-energy eigenvalues of HA because as N →� the truncated Hamiltonian approaches the FL Hamil- tonian H FL * = � �,� � �� * g�� † g��. �54� Here, � and � subscripts distinguish the positive eigenval- ues from the negative ones, while �=0,1 , . . . counts the posi- tive �negative� eigenvalues upward �downward� from the Fermi level. Once the eigenvalues of HA N are identified with the many- body energies generated from Eq. �15�, the ground-state phase shift � are extracted from the following approximate expression, which describes all but the � �� * closest to zero very accurately.45 For ��5, in particular, within 0.1% de- viation, � �� * = ���+���/ �� = 1,2, . . . � , �55� where �=1−z��=3 /2−z� for odd �even� N. YOSHIDA, SERIDONIO, AND OLIVEIRA PHYSICAL REVIEW B 80, 235317 �2009� 235317-10 VIII. RESULTS To emulate the conditions under which a SET operates, we fix the Coulomb repulsion U and effective dot-level width w and examine the ground-state phase shift and the thermal dependence of the conductance as a function of the dot energy �d for fixed wire potential W. To mimic the pa- rameters describing typical devices, Sec. VIII B will study model Hamiltonians with U�D. The opposite inequality is, however, more illustrative because it displaces the Kondo regime to higher temperatures. We prefer to discuss it first. A. Conductance for U�D To display a broad range of Kondo temperatures, we choose the Coulomb repulsion U=5D and the effective dot- level width w=0.15D and examine the ground-state phase shift � and thermal dependence of the conductance G�T� as functions of the dot energy �d for five positive wire gate potentials W. We need not study negative wire potentials, which would mirror the conductances and phase shifts cal- culated with positive W, because HA, G, and ��� remain in- variant under the transformation cd→−cd †, ak→ak †, �d+U /2 →−��d+U /2�, and W→−W. Consider, first, the phase shift. Figure 6 displays the argu- ment of the trigonometric function on the right-hand side of the mapping �49�, computed for five wire potentials W, in the dot-energy range 0��d /D�−U. To draw continuous curves and to make the ordinate 2��−�w� / equal to the ground- state dot occupancy nd, we have displaced the domain of definition of � from �− /2, /2� to �0, �, Calculated for W=0, the upright triangles trace a well- known curve,4 one that remains invariant under the particle- hole transformation �d+U /2→−��d+U /2�, �→ /2−�. At the symmetric point �d+U /2=0, which corresponds to Eq. �11�, the phase shift is exactly /2. The arrows above the top axis indicate the Kondo domain, within which the phase shift remains close to /2. As ��d+U /2� grows, the model Hamil- tonian first approaches the limits of the Kondo domain and then invades the mixed-valence domain. In response, � moves away from /2, toward zero for �d+U /2→U /2, or toward for �d+U /2→−U /2. The wire potential reduces the ground-state phase shift throughout the depicted range. For �W=1, for instance, the phase shift at the symmetric dot-level energy �d=−2.5D= −U /2 is reduced from �= /2 to �= /10. In the Kondo regime, as the illustration shows, the difference �−�w is nonetheless pinned at /2. The pinning is due to the Friedel sum rule.42 Since the ground-state phase shift would be �w if were zero, 2��−�w� / is the screening charge due to the coupling to the dot. In the Kondo regime, that charge is nearly unitary, and �−�w� /2. Figure 6 also shows that a positive wire potential tends to displace the Kondo regime toward higher dot energies. For W=0, the rapid decay of the phase shift near �d=0 ��d= −U� marks the resonance between the nd=0 and nd=1 �nd =1 and nd=2� dot-level configurations. The Kondo regime lies between them. As W grows, the two resonances move to higher �d’s and so does the Kondo regime. 1. Conductance landscape According to Eq. �49�, the phase-shift difference �−�w controls G�T�. Consequently, the central features of Fig. 6 are manifest in landscape plots of the conductance. Figure 7�a� shows G�T� in the dot-energy range ��d+U /2��U /2 for U=5D, and =0.15D. The plot surveys the entire Kondo regime and part of the mixed-valence regime. The plane �d =−U /2, which represents the symmetric Hamiltonian �11�, splits the landscape in two symmetric halves, mapped onto each other by the particle-hole transformation cd→−cd †, ak →ak †. At the symmetric point �d=−U /2, the temperature- dependent conductance reproduces the universal function GS�T /TK�. Here and elsewhere in the Kondo regime, the con- ductance at fixed �d rises from zero to ballistic as the tem- perature is reduced past TK, i.e., as one climbs from the high- temperature Coulomb-blockade valley to the low- temperature Kondo plateau. The Kondo temperature depends on �d. Plotted in Fig. 7�b� as a function of the dot energy, TK mirrors the invari- ance of the Hamiltonian under particle-hole transformations and reaches the minimum kBTK=8 10−7D at the symmetric point. As ��d+U /2� grows, the Kondo temperature rises until kBTk�0.1D, an equality indicating proximity to the mixed- valence regime, i.e., to the two resonances centered at �d= −5D and �d=0. As ��d+U /2� grows further, we come into mixed-valence domain. The dot moment shrinks, and so does the Kondo cloud. The Kondo bypass of the Coulomb block- Kondo (W = 0) −5 −4 −3 −2 0 -2 -1 0 εd+U/2 D 2 0.4 0.8 1.2 2.0 ρW = 0.00 ρW = 0.10 ρW = 0.25 ρW = 0.50 ρW = 1.00 2(δ−δw) π εd/D 910Fig. 11 12 FIG. 6. �Color online� Ground-state phase shift �, measured from the phase shift �w obtained from Eq. �B6� for the displayed wire potentials W, as a function of the dot-level energy �d. The phase shifts are defined in the domain 0��� so that the Friedel sum rule makes the ordinate equal the dot occupation nd. The �’s were obtained, with the help of Eq. �55�, from the low-energy spec- trum of HA resulting from NRG runs with U=5D and w=0.15D. The arrows above the top horizontal axis define the Kondo domain for W=0. For W�0, the Kondo domain is displaced to the right. Each vertical arrow pointing to the lower horizontal axis identifies the figure displaying the thermal dependence of the conductance for the indicated dot energy. UNIVERSAL ZERO-BIAS CONDUCTANCE FOR THE… PHYSICAL REVIEW B 80, 235317 �2009� 235317-11 ade becomes less and less effective, and the conductance approaches zero. The steep drops near the �d=0 and �d= −5D planes in Fig. 7�a� mark the fractional dot occupations nd�0.5 and nd�1.5, respectively. 2. Wire potential Figure 8�a� displays the conductance as a function of �d and T for U=5D, w=0.15D, and �W=1. Quantitative dif- ferences distinguish the plot from Fig. 7�a�. In particular, the Kondo temperature is now minimized at the higher dot-level energy �d=−1.9D, the minimum is 30-fold higher, kBTK =2.4 10−6D, the resonance between the nd=1 and nd=2 dot configurations is now centered at �d�−4.2D, and of the resonance between the nd=0 and nd=1 dot configurations only an incipient rise is visible at the high-�d end of the plot. Clearly, the wire potential has displaced the Kondo domain toward higher dot-level energies. This displacement ac- knowledged, we recognize in Fig. 8�a� the salient features of Fig. 7�a�. The two landscapes are similar because the dependence relating the phase-shift difference on the right-hand side of Eq. �49� to the wire potential �W is weak. With �−�w � /2, the conductance curve G�T /TK� is approximately mapped onto GS�T /TK� throughout the Kondo domain. The rise from the high-temperature valley to the Kondo plateau is therefore close to universal, dependent on the model param- eters only through the Kondo temperature TK. Figure 8�b� shows the Kondo temperature as a function of the dot energy, a plot that resembles Fig. 7�b�. TK depends on the antiferromagnetic interaction J between the dot moment and the conduction electrons around it, a constant related to the model parameters by the Schrieffer-Wolff expression,41 0.25 0.5 0.75 1 −5 −4 εd/D −2 −1 0 10−7 10−6 10−5 10−4 kBT/D 10−2 10−1 0 0.25 0.5 G G2 1 −4 −3 −2 −1 0 10−6 10−4 10−2 10−6 10−5 10−4 10−2 −5 −4 −3 −2 −1 0 εd/D kBTK D (a) (b) FIG. 7. �Color online� �a� Conductance as a function of the temperature and dot-level energy for U=5D, =0.15D, and W=0. The plot is symmetric with respect to the �d=−U /2=−2.5D plane. The sharp drops near �d=−5D and �d=0 mark the borders of the Kondo regime, which extends roughly from �d=− to �d+U=− . In the Kondo regime, at fixed �d, the more gradual decay of the conductance with temperature portrays the evaporation of the Kondo droplet. �b� Kondo temperatures resulting from the intersec- tion of the landscape �a� with the horizontal plane G�T=TK� �G2 /2. 0.25 0.5 0.75 1 −5 −4 εd/D −2 −1 0 10−7 10−6 10−5 10−4 kBT/D 10−2 10−1 0 0.25 0.5 G G2 1 −4 −3 −2 −1 0 10−6 10−4 10−2 10−6 10−5 10−4 10−2 −4 −3 −2 −1 0 εd/D kBTK D (a) (b) FIG. 8. �Color online� �a� Conductance as a function of the temperature and dot-level energy for U=5D, w=0.15D, and �W =0.50. The wire potential breaks the particle-hole symmetry visible in Fig. 7. The sharp drop centered at �d=−5D in Fig. 7 is now fully visible, while the one centered at �d=0 is out of sight, an indication that the Kondo regime has been displaced to higher dot energies. The bell-shaped resonance near the bottom left corner of the kBT =10−1D plane stakes the mixed-valence regime. �b� Kondo tem- peratures resulting from the intersection of the landscape �a� with the plane G�T=TK�=G2 /2. YOSHIDA, SERIDONIO, AND OLIVEIRA PHYSICAL REVIEW B 80, 235317 �2009� 235317-12 �J = 2 w � 1 �0 + 1 �2 � . �56� Here, �0 ��2� is one of the two dot-charge excitation ener- gies, the energy needed to remove �add� an electron to the singly-occupied dot level. For the symmetric Hamiltonian �11�, in particular, �0=�2=U /2. For nearly symmetric Hamiltonians, �0= ��d� and �2=U+�d. As �W� or ��d+U /2� grow, the resulting particle-hole asymmetry renormalizes the dot energy51,45 so that �0 and �2 are changed to � 0 *= �� d *� and � 2 *=U+� d *, respectively, where � d * is the effective dot energy at the LM.45 Since both landscapes were computed for the same effec- tive width w=0.15D, only �i� the excitation energies � 0 * and � 2 * and �ii� irrelevant operators make the Kondo tempera- tures in Fig. 8�b� different from those in Fig. 7�b�. The renor- malized excitation energies displace the Kondo domain along the �d axis, while the modified irrelevant operators extend the Kondo plateau toward higher temperatures. This concludes our overview of the numerically computed conductance landscapes. Section VIII A 3 will inspect in more detail the data in four slices of Figs. 7�a� and 8�a� and compare them to Eq. �49�. 3. Thermal dependence of the conductance Figure 9 displays the conductance as a function of the temperature for U=5D, w=0.15D, �d+U /2=0, and five wire potentials: �W=0 and 1, already studied in Figs. 7 and 8, and three intermediate values, �W=0.25, 0.5, and 0.75. With W=0, the open circles represent the symmetric Hamil- tonian �11�, and the solid line through them, the universal function GS�T /TK� first computed in Ref. 29. Notwithstand- ing the wire potentials, the Hamiltonians represented by the squares, triangles, and diamonds lie deep inside the Kondo regime. For each of them, the phase-shift difference ��−�w� in Table I is close to /2. It follows that the right-hand side of Eq. �49� is close to GS�T /TK�. The agreement with the numerical data is excellent. As the Hamiltonian moves away from the symmetric plane �d=−U /2, the particle-hole asymmetry becomes more pronounced, and one might expect the phase shift difference ��−�w� to grow. As Fig. 6 showed, however, the growth is checked by the Friedel sum rule so that ��−�w�� /2 in the Kondo regime. Illustrative results appear in Fig. 10, which displays conductance curves for �d=−3.4D. Even for the strongest wire potential in the legend, �W=1, the difference ��−�w� in Table I is only 6% away from /2. As in Fig. 9, therefore, the conductance curves computed from Eq. �49� are nearly identical to GS�T /TK�. The agreement with the numerical data is again excellent. Since we are now closer to the boundary of the Kondo regime, the Kondo temperature is more sensitive to the renormalization of the dot-level energy induced by strong wire potentials. Compared to Fig. 9, Fig. 10 thus exhibits a substantially broader spread of crossover temperatures. Figure 11 displays numerical results for �d+U /2=−1.5D, a still larger departure from the symmetric condition. For �W�0.5, the agreement with Eq. �49� is excellent; for �W =0.75, it is imperfect only at the highest temperatures shown. For �W=1, however, there is substantial disagree- ment, which justifies a digression. Inspection of Fig. 8 shows that for �d=−4D �and �W=1�, the model Hamiltonian lies well within the mixed-valence regime.56 In the Kondo regime, Eq. �49� is reliable for ther- mal energies that are small on the scale of the dominant characteristic energy EC=min��� d *� ,U+� d * ,D�. If � d * had its bare value, �d=−4D, the mapping would be reliable for kBT�Ec=D. The renormalized dot energy having pushed the model Hamiltonian into the mixed-valence regime, the domi- nant characteristic energy has been reduced to Ec =min� w ,D�= w, which restricts the domain of the mapping to kBT�0.15D. The failure at higher temperatures is due to the contribution �Girr of irrelevant operators, which are siz- able near the characteristic energy. At kBT=0.1D=2 /3 w, for example, the diamonds in Fig. 11 are pushed 0.2e2 /h below the solid line; upon cooling, �Girr decays in proportion to kBT and becomes insignificant below kBT=10−2D. If −�d were steadily increased beyond −�d=4D, the model Hamiltonian would traverse the mixed-valence region. Once �� d *+U�� w, the dot occupation would approach nd=2. The dominant characteristic energy Ec= �� d *+U� would define the crossover energy scale, which would hence rise with −�d. Soon, the model Hamiltonian would be driven to the frozen- level fixed point at the first steps of the renormalization- group flow, and the mapping would be reduced to its FL limit, G�T→0�=sin2��−�w��0. Equation �49� is asymptotically exact at low temperatures, i.e., for kBT�Ec. As the above discussion showed, its prac- tical value is eroded outside the Kondo regime. In the mixed- valence regime, in particular, the asymptotic region lies well below the crossover temperature, i.e., in the vicinity of the FL. To plot the rightmost solid line in Fig. 11, we thus had to match the right-hand side of Eq. �49� to the diamond at G 10−8 10−6 10−4 10−2 0.2 0.4 0.6 1.0 kBT/D G G2 ρW = 0.00 ρW = 0.25 ρW = 0.50 ρW = 0.75 ρW = 1.00 U = 5 D Γw = 0.15 D εd = −2.50 D FIG. 9. �Color online� Thermal dependence of the conductance for �d+U /2=0, and the indicated values of the other model param- eters. The circles, open and filled squares, triangles, and diamonds are the NRG data, while the solid lines through them depict Eq. �49�, with the Kondo temperatures and phase shifts listed in Table I. The curve through the open circles, in particular, is the universal conductance GS�T /TK� for the symmetric Hamiltonian �11� �Ref. 29�. Since ��−�w�� /2, each solid line is close to GS�T /TK�. UNIVERSAL ZERO-BIAS CONDUCTANCE FOR THE… PHYSICAL REVIEW B 80, 235317 �2009� 235317-13 =0.7G2 because the identification G�T=TK�=0.5G2, which defined TK for all the other plots in Figs. 9–12, became un- reliable for �W=1. Table I marks with an asterisk the result- ing Kondo temperature. Near the opposite extreme of the Kondo regime, for fixed small −�d, the wire potential drives the model Hamiltonian toward the center of the Kondo regime. Clear evidence of this displacement is found in Fig. 8�a�: displaced to positive dot-level energies, the mixed-valence domain is no longer visible on the right-hand side of the landscape. The tempera- ture dependence of the conductance is displayed in Fig. 12, which shows the �d=−0.4D plane for the five potentials �W=0, 0.25, 0.5, 0.75, and 1. The �W=0 Hamiltonian is TABLE I. Phase shifts and Kondo temperatures for the 20 NRG runs depicted in Figs. 9–12. The ground-state phase shifts � were obtained from Eq. �55�, the wire phase shifts �w from Eq. �B6�, and the Kondo temperatures from the definition G�T=TK��G2 /2. The Kondo temperature marked with an asterisk belongs to the mixed-valence regime and, as explained in the text, is the output of a different computation. Figure Symbol �W �w / � / ��−�w� / kBTK /D 9 � 0.00 0.00 −0.50 0.50 8.1 10−7 9 � 0.25 −0.21 0.29 0.50 1.1 10−6 9 � 0.50 −0.32 0.18 0.50 2 10−6 9 � 0.75 −0.37 0.13 0.51 3.4 10−6 9 � 1.00 −0.40 0.11 0.51 6 10−6 10 � 0.00 0.00 −0.49 0.49 4.4 10−6 10 � 0.25 −0.21 0.30 0.51 1.1 10−5 10 � 0.50 −0.32 0.20 0.52 3.6 10−5 10 � 0.75 −0.37 0.15 0.52 1.1 10−4 10 � 1.00 −0.40 0.13 0.53 3.6 10−4 11 � 0.00 0.00 −0.48 0.48 8.8 10−5 11 � 0.25 −0.21 0.32 0.53 3.3 10−4 11 � 0.50 −0.32 0.23 0.55 1.6 10−3 11 � 0.75 −0.37 0.22 0.59 7.3 10−3 11 � 1.00 −0.40 0.25 0.65 3.4 10−2* 12 � 0.00 0.00 0.42 0.42 8.0 10−3 12 � 0.25 −0.21 0.24 0.45 2.4 10−3 12 � 0.50 −0.32 0.15 0.46 9.4 10−4 12 � 0.75 −0.37 0.10 0.47 4.0 10−4 12 � 1.00 −0.40 0.08 0.48 1.9 10−4 10−8 10−6 10−4 10−2 0.2 0.4 0.6 1.0 kBT/D G G2 ρW = 0.00 ρW = 0.25 ρW = 0.50 ρW = 0.75 ρW = 1.00 U = 5 D Γw = 0.15 D εd = −3.4 D FIG. 10. �Color online� Thermal dependence of the conductance for �d+U /2=−0.9D. The symbols and lines were computed as de- scribed in Fig. 9. As Table I shows, the argument ��−�w� on the right-hand side of Eq. �49� is close to /2. As a consequence, the solid lines are only slightly different from GS�T /TK�. The agree- ment with the numerical data is, again, excellent. 10−8 10−6 10−4 10−2 0.2 0.4 0.6 1.0 kBT/D G G2 ρW = 0.00 ρW = 0.25 ρW = 0.50 ρW = 0.75 ρW = 1.00 U = 5 D Γw = 0.15 D εd = −4.00 D FIG. 11. �Color online� Thermal dependence of the conductance for �d+U /2=−1.5D. The symbols and lines were calculated as de- scribed in Fig. 9. As discussed in the text, the high-temperature separation between the solid line and the diamonds flags a Hamil- tonian outside the Kondo regime. YOSHIDA, SERIDONIO, AND OLIVEIRA PHYSICAL REVIEW B 80, 235317 �2009� 235317-14 now at the boundary of the Kondo regime, and for kBT �10−2D, irrelevant operators introduce significant deviations �Girr from the solid line. As �W grows, however, the model Hamiltonian sinks deeper into the Kondo regime, and the agreement with the solid lines improves. B. Conductance for U�D We now consider the more realistic Coulomb repulsion U=0.05D, which pushes the Kondo regime to the tempera- ture range kBT�min���d� ,�d+U�. As in Sec. VIII, we con- sider wire gate potentials, in the interval 0��W�1. To gen- erate Kondo temperatures comparable to those in Fig. 7�b�, for each wire potential we choose a dot-level width so that Eq. �9� yields the effective width w=2 10−3D and let the dot energy run from �d+U /2=−0.04D to �d+U /2=0.04D. Ampler than the scope of Figs. 7 and 8, this range provides an encompassing view of the Kondo and mixed-valence re- gimes. 1. Conductance landscape Figure 13 shows the conductance as a function of the temperature and dot energy for �W=0. Analogous to Fig. 7�a�, the plot is symmetric about the �d+U /2=0 plane. In contrast with Fig. 7�a�, however, the landscape displays a ridge, parallel to the �d axis, at high temperatures. The ridge marks the crossover from the free-orbital fixed point,2 the temperature range associated with thermal energies above the charge-excitation energies � 0 *���d� and � 2 *��d+U, and the local-moment fixed point. Since the dot-level spectral density �d�� ,T� peaks at � 0 * and � 2 *,46,52 the conductance first rises and then decays as the model Hamiltonian crosses over from the free-orbital fixed point to the Kondo regime. The Kondo regime occupies the broad low-temperature sector of the plot where the excitation energies � 0 * and � 2 * exceed the dot-level width and the thermal energy kBT. As the model Hamiltonian enters the Kondo regime, Eq. �49� takes control of the conductance and reproduces the gradual rise to the Kondo plateau depicted in Fig. 7�a�. At the outskirts of the Kondo regime, the narrow strip satisfying the inequality �� 0 *�� ��� 2 *�� � defines the mixed-valence regime. As �d+U /2 −��d+U /2�� grows at fixed temperature, the conductance drops sharply in this re- gime. Beyond the mixed-valence strip, � 0 *�� 2 *� is negative, �� 0 *���� 2 * � � exceeds , and the dot occupation nd approaches 0 �2�. The dot magnetic moment vanishes, and the Coulomb blockade obstructs conduction even at T=0. In the entire temperature range �� 0 *��kBT��� 2 *��kBT�, the conductance is close to zero. Figure 13�b� shows the conductance as a function of the dot energy and temperature for �W=1. Comparison with Fig. 10−8 10−6 10−4 10−2 0.2 0.4 0.6 1.0 kBT/D G G2 ρW = 0.00 ρW = 0.25 ρW = 0.50 ρW = 0.75 ρW = 1.00 U = 5 D Γw = 0.15 D εd = −0.40 D FIG. 12. �Color online� Thermal dependence of the conductance for �d+U /2=2.1D. The lines and symbols were computed as de- scribed in Fig. 9. The relatively large separation from the symmetric condition �d+U /2=0 places the W=0 Hamiltonian close to the border of the Kondo regime; at high temperatures, relatively large irrelevant operators, whose influence decays in proportion to kBT /D, introduce deviations from Eq. �49�. Since the wire potential displaces the Kondo regime to higher dot-level energies, the dis- tance from the border grows with �W, and so does the agreement between the numerical data and the solid lines representing Eq. �49�. 0 0.25 0.5 0.75 1G/G2 kBT/D εd/D −0.06 −0.04 −0.02 0 10−9 10−6 10−3 0 0.25 0.5 0.75 1 −0.06 −0.04 −0.02 0 10−10 10−8 10−6 10−4 10−2 ρW = 0 (a) 0 0.25 0.5 0.75 1G/G2 kBT/D εd/D −0.06 −0.04 −0.02 0 10−9 10−6 10−3 0 0.25 0.5 0.75 1 −0.06 −0.04 −0.02 0 10−10 10−8 10−6 10−4 10−2 ρW = 1 (b) FIG. 13. �Color online� Conductance as a function of the tem- perature and dot energy for U=0.05D, =0.002D, and wire gate potentials �W=0 �a� and �W=1 �b�. In each case, a ridge separates the high-temperature range kBT�U from the Kondo regime. In the Kondo regime, plots �a� and �b� reproduce the landscapes displayed in Figs. 7 and 8, respectively. UNIVERSAL ZERO-BIAS CONDUCTANCE FOR THE… PHYSICAL REVIEW B 80, 235317 �2009� 235317-15 13�a� shows that the wire potential moves the Kondo plateau and mixed-valence regimes to higher dot energies, a dis- placement analogous to the shift distinguishing Fig. 8�a� from Fig. 7�a�. At high temperatures, near the free-orbital fixed point, only in the mixed-valence regime is the conduc- tance significantly affected by the wire gate potential. 2. Thermal dependence of the conductance Figure 14 displays the thermal dependence of the conduc- tance for �d+U /2=0 and four wire potentials, �W=0, 0.5, 0.75, and 1. The symbols represent NRG results, while the solid lines represent Eq. �49� with the Kondo temperature TK extracted from the definition G�TK��G2 /2 and the phase shift � from the low-energy spectrum of the model Hamil- tonian. In each case, the symbols agree impeccably with the solid line representing Eq. �49� in the range kBT�2 10−3D. As indicated by the vertical arrow pointing to the lower horizontal axis, the nd=0 and nd=2 dot configurations become thermally accessible at higher temperatures. Charge excitations then control the dynamics of transport and make it insensitive to the wire potential. The circles, squares, tri- angles, and diamonds therefore depart from the solid lines to form a single, broad conductance peak, which corresponds to the high-temperature ridges in Fig. 13. Figure 15 plots the conductance as a function of the tem- perature for �d+U /2=−0.015D and five wire potentials: �W=0, 0.25, 0.50, 0.75, and 1. The charge excitation ener- gies ��d� and �d+U are now different; the vertical arrow in the figure points to the lowest one and indicates proximity to the mixed-valence regime. The symbols are NRG data, while the solid lines depict Eq. �49�. The open circles and filled squares agree very well with the solid line at temperatures below kBT=10−3D and draw a broad nonuniversal maximum near the indicated excitation energy. The circles and open squares in Fig. 15 follow the pattern set by Fig. 14. For �W�0.5, however, the high-temperature peak in Fig. 15 first turns into an inflexion point and then disappears as the wire potential rises. A brief glance at Fig. 13 is sufficient to see that the dot energy �d=−0.04D moves from the Kondo to the mixed-valence regime as the wire potential grows from �W=0 to �W=1. The filled triangles and open diamonds in Fig. 15 are in the mixed-valence regime. In the same way that the triangles and diamonds in Fig. 11 lie significantly below the pertinent solid lines, only above G2 /2 do the triangles and diamonds in Fig. 15 agree with Eq. �49�, i.e., only below the Kondo tem- perature. This reinforces our finding that besides accurate at all temperatures in the Kondo regime, Eq. �49� is reliable at low temperatures even in the mixed-valence regime. C. Discussion In the Kondo regime, Eqs. �22a� and �22b� fix the high- and the low-temperature conductances, respectively. Equa- tion �49� shows that the universal function GS�T /TK� controls the monotonic transition between the two limits. For W=0, in particular, the fixed-point values depend only on the ground-state phase shift � and are symmetric with respect to G2 /2: GLM =G2 cos2 � and GFL=G2 sin2 �. Thus, depending on �, the transition from GLM to GFL can be steeper or flatter. Since � can never depart much from /2 in the Kondo re- gime, the argument of the trigonometric function on the right-hand side of Eq. �49� can never depart substantially from , and as indicated by the two curves in Fig. 4, G�T /TK��GS�T /TK��20%. By contrast with this crude es- timate, the mapping �49� gives excellent agreement with the symbols in the Kondo-regime curves in Fig. 9–12, 14, and 15. 10−8 10−6 10−4 10−2 0.2 0.4 0.6 1.0 kBT D G G2 ρW = 0.00 ρW = 0.50 ρW = 0.75 ρW = 1.00 U = 0.05 D Γw = 0.002 D εd = −0.025 D U/2D FIG. 14. �Color online� Temperature dependence of the conduc- tance for the displayed model parameters. The vertical arrow points to thermal energy needed to add an electron to or remove an elec- tron from the dot level. Each solid line represents Eq. �49� for the Kondo temperature defined by the equality G�T=TK�=G2 /2 and the phase shift � extracted from the low-energy spectrum of the model Hamiltonian. |εd + U | 10−7 10−5 10−3 10−1 0.2 0.4 0.6 1.0 kBT D G G2 ρW = 0.00 ρW = 0.25 ρW = 0.50 ρW = 0.75 ρW = 1.00 U = 0.05 D Γw = 0.002 D εd = −0.04 D FIG. 15. �Color online� Temperature dependence of the conduc- tance for the displayed model parameters. The vertical arrow points to thermal energy needed to add a second electron to the dot level, smaller than the energy necessary to remove the dot electron. As in Fig. 14, each solid line represents Eq. �49�. YOSHIDA, SERIDONIO, AND OLIVEIRA PHYSICAL REVIEW B 80, 235317 �2009� 235317-16 The wire potential W narrows the dot level and displaces the ground-state phase shift. Depending on the sign and mag- nitude of W, the phase shift can take any value in its domain of definition − /2��� /2. In the Kondo regime, the Frie- del sum rule nonetheless prevents the difference �−�W from straying away from /2. All effects considered, the scatter- ing potential W displaces the conductance curve toward the symmetric limit G�T /TK�=GS�T /TK�. These findings are in line with the experimentally estab- lished notion that in the Kondo regime, SET conductances always decay with temperature.5,14,36,38 This behavior con- trasting with that of the conductance in the side-coupled geometry,31,32 we digress briefly to compare the two arrange- ments. As demonstrated in Ref. 30, a linear mapping analo- gous to Eq. �49� can be established between the side-coupled conductance and GS�T /TK�. Since the coefficient relating the two functions is cos�2��, instead of cos 2��−�w��, the ap- proximate equality �−�w� /2 makes the coefficient sensi- tive to changes in �w. Under a sufficiently strong wire poten- tial, the sign of the coefficient can be reversed. Thus, the thermal dependence of the conductance through the side- coupled device is tunable:35 a wire potential can turn a monotonically increasing function into a monotonically de- creasing one. The embedded geometry of Fig. 1 is much less sensitive to W. In Fig. 1, the charge induced under the symmetric elec- trodes by the potential W is times �w. According to the Friedel sum rule,42 the difference �−�W is the charge of the Kondo cloud, the additional charge that piles up at the wire tips surrounding the dot as the temperature is lowered past TK. Neutrality makes the charge of the Kondo cloud equal to the dot occupancy. Since the symmetric condition nd=1 maximizes the low-temperature conductance, one expects G�T=0� to be ballistic for 2��−�W�= , a conclusion in agreement with Eq. �49�. Since the screening charge is al- ways nearly unitary, one expects the low-temperature con- ductance to be close to the conductance quantum, in agree- ment with the plots in Figs. 9–12, 14, and 15. D. Prospect of comparison with experiment Equation �49� is analogous to the expression relating the universal function GS to the conductance in the side-coupled arrangement.30 Both expressions are linear, and the linearity simplifies the comparison with experimental data. To fit the conductances Gi �i=1, . . . ,N� measured at N temperatures Ti, we follow the procedure illustrated by Fig. 3 in Ref. 30. We start with a trial Kondo temperature TK and compute the scaled temperatures i=Ti /TK �i=1, . . . ,N�. Since we know the universal function GS�T /TK�, it is then a simple matter to compute the set GS� i� �i=1, . . . ,N� of universal conduc- tances at the scaled temperatures. According to Eq. �49�, plotted as a function of GS� i�, Gi should lie on a straight line. If the line is crooked, we have to choose a new trial TK and repeat the procedure. In practice, we seek the Kondo temperature optimizing the least-squares linear fit to the plot of Gi vs GS� i�. In addition to yielding the Kondo temperature, the linear regression determines the argument 2��−�W� of the trigono- metric function on the right-hand side of Eq. �49�. Conve- nient as the SET therefore is to measure the ground-state phase shift �, its interferometric scope is narrow, for the phase shifts in the embedded geometry are always close to �W+ /2. By comparison, the side-coupled device consti- tutes an interferometer of more practical value because it is free from this limitation.30,53 IX. SUMMARY The first part of this report derived our central result, Eq. �49�, which maps the conductance in the embedded geometry onto the universal conductance for the symmetric Anderson model �11� linearly. As our discussion of the result showed, while the Kondo temperature sets the temperature scale, the dot charge controls the mapping, even with a gate potential applied to the wires. If the dot occupation nd is unitary, the mapping reduces to the equality G�T /TK�=GS�T /TK�, the re- sult found by Costi et al.29 Elsewhere in the Kondo regime, the dot occupation being still close to unity G�T /TK� is never qualitatively different from GS�T /TK�. Relative deviations as large as 20% may nevertheless separate the two functions at low temperatures. By contrast with the approximation G�T /TK��GS�T /TK�, Eq. �49� describes the conductance exactly in the Kondo re- gime and is hence the appropriate instrument to describe ex- perimental or numerical data. In particular, once fitted to an experimental curve, the mapping determines the Kondo tem- perature TK, as well as the dot charge. The essentially exact numerical results in the second part of the paper presented an overview of zero-bias conduction through a quantum dot embedded in the conduction path of a nanodevice. Equation �49�, a mapping explicitly param- etrized by the Kondo temperature and dot occupancy, guided our discussion of the computed conduction curves in the Kondo regime. We also surveyed the neighboring domains of the parametrical space: the free-orbital and mixed-valence regimes. In the former, instead of scaling with T /TK, the thermal dependence of the conductance scales with kBT /Ec, where Ec is a charge-excitation energy, Ec=min�� 0 * ,� 2 *�. In the mixed-valence regime, the effective dot-level width w setting the temperature scale, the mapping �49� becomes as- ymptotically exact for kBT� w and only witnesses the final rise of the conductance to its low-temperature limit G�T =0�=G2 sin2��−�w�. By contrast, in the Kondo regime the mapping is asymp- totically exact for kBT�min D ,� 0 * ,� 2 * and hence describes accurately the rise of the conductance throughout the Kondo crossover. The distinction between the Kondo and the mixed- valence regimes is a problem often encountered in the laboratory.5,32 The mapping to the universal curve offers a solution of practical value. ACKNOWLEDGMENTS This work was supported by the CNPq and FAPESP. APPENDIX A: PROPERTIES OF THE FIXED-POINT HAMILTONIANS 1. Diagonalization The LM and FL are described by conduction-band Hamil- tonians of the form UNIVERSAL ZERO-BIAS CONDUCTANCE FOR THE… PHYSICAL REVIEW B 80, 235317 �2009� 235317-17 H* = � k �kak †ak + W*f0 †f0. �A1� We want to bring H* to the diagonal form H* = � � ��g� †g�, �A2� where g� = � q ��qaq. �A3� To this end, we compare the expressions for the commutator g� ,H*� obtained from Eqs. �A1� and �A2�, from which it follows that ��q = 1 �� − �q W* N � k ��k. �A4� Summation of both sides over q then leads to the eigen- value condition, 1 = W* N � q 1 �� − �q . �A5� Inspection of this equality shows that, with exception of a split-off energy, which makes O�1 /N� contributions to the low-energy properties, the �� are shifted by less than � from the �k. We therefore refer to the closest conduction energy �� to label each eigenvalue and define its phase shift �� with the expression �� � �� − � ��. �A6� This definition substituted for ��, a Sommerfeld-Watson transformation54 evaluates the sum on the right-hand side of Eq. �A5�, 1 N � q 1 �� − �q = − � cot �� + � W −D D 1 �� − � d� . �A7� The eigenvalue Eq. �A5� therefore determines the phase shift, cot �� = − 1 �W* + 1 W−D D 1 �� − � d� . �A8� At low energies, the contribution of the last term on the right-hand side, of O�� /D�, can be neglected, and the phase shift becomes uniform, tan � = − �W*. �A9� Next, we square both sides of Eq. �A4� and sum the result over q to determine the coefficients ��q. � q ��q 2 = �� k ��k W* N �2 � q 1 ��� − �q�2 . �A10� The sum on the left-hand side is unitary. To evaluate the sum over q on the right-hand side, we differentiate Eq. �A7� with respect to ��. With relative error O�1 /N�, we find that 1 N2� q 1 ��� − �q�2 = � � sin �� �2 . �A11� Thus, Eq. �A10� reduces to W*� k ��k = − 1 � sin ��, �A12� the negative sign insuring that �kk→1 for W*→0. From Eq. �A4� it then follows that ��q = � �q − �� sin �� ��� � D� . �A13� 2. Energy moments of the matrix elements of the eigenoperators g� This appendix shows that, given two eigenstates �m� and �n� of the Hamiltonian HA, and considered the operators f0, f1, �0, and �1 defined by Eqs. �6�, �30�, �26�, and �27�, respectively, the matrix element �m�f0�n� ��m�f1�n�� is a linear combination of the matrix elements �m��0�n� and �m��1�n� with coefficients that are independent of the eigenstates �m� and �n�, as long their eigenvalues Em and En, respectively, are much smaller than the conduction bandwidth. We define the dimensionless energy Emn��Em−En� /D and assume that �Emn� � 1. �A14� Under this assumption, we consider the energy moments Mmn �p� � 1 N � � ��� D �p �m�g��n� �p = 0,1, . . . � , �A15� where g� ���� is one of the eigenstates �eigenvalues� defined by Eq. �A2�. Since the Mmn p are universal, to evaluate them it is suffi- cient to consider the symmetric Hamiltonian �11�, for which the phase shift �LM =0 so that gk, �k, �0, and �1 coincide with ak, �k, f0, and f1, respectively. From Eq. �11�, we then have that g�,HA S� = ��g� + V N cd. �A16� Multiplication of both sides by ��� /D�p−1 followed by sum- mation over � leads to the coupled recursive relations, Mmn �p� = − EmnMmn �p−1� − V p �m�cd�n� �p = 1,3, . . . � , Mmn �p� = − EmnMmn �p−1� �p = 2,4, . . . � . Reduced to a matrix equation, this system is easily solved, Mmn �p� = − V p �m�cd�n��1 + � r=1 p−2 � �Emn�r r � + Mmn �0��Emn�p �p = 1,3, . . . � , �A17� and YOSHIDA, SERIDONIO, AND OLIVEIRA PHYSICAL REVIEW B 80, 235317 �2009� 235317-18 Mmn �p� = − V p �m�cd�n�� r=1 p−1 � �Emn�r r + Mmn �0��Emn�p �p = 2,4, . . . � , �A18� where the primed sums are restricted to odd r’s. In view of Eq. �A14�, the terms proportional to �Emn�p�p �1� on the right-hand sides of these two equalities can be neglected. From Eq. �A17� we then see that Mmn �1� = −V�m�cd�n� and all the other odd moments are proportional to Mmn �1�, and from Eq. �A18�, we see that the only nonzero even moment is Mmn �0�. More specifically, Mmn �p� = � �m��1�n� p �p = 1,3, . . . � 0 �p = 2,4, . . . � . � �A19� This simple relation suggests that we define the following orthonormal basis of conduction states: �� � 2� + 1 N � p P���p�gp �� = 0,1, . . . � , �A20� where P� denotes a Legendre polynomial. In particular, with �=0 ��=1�, we recover Eq. �26� Eq. �27��. Given two eigenstates �m� and �n� of HA, the matrix ele- ment �m����n� is a linear combination of the moments Mmn �r� �r=1,3 , . . . ,� for odd �, or r=2,4 , . . . ,� for even p�. It fol- lows from Eq. �A19�, then that �m����n���m��1�n� �� =3,5 , . . . �, while �m����n�=0 ��=2,4 , . . . �. More generally, the matrix element of any conduction operator is a linear combination of �m��0�n� and �m��1�n�. In particular, �m�f i�n� = � j=0 1 �ij�m�� j�n� �i = 0,1� , �A21� where f0 and f1 are the operators defined by Eqs. �6� and �30�, respectively, and �ij �i , j=0,1� are model-parameter dependent constants. APPENDIX B: FIXED-POINT CONDUCTANCES This appendix derives an expression for the spectral den- sity �d�� ,T� at the fixed points. We start with Eq. �28�, from which we obtain an expression for the matrix element of the conduction operator aq † between two low-energy eigenstates �m� and �n� of HA, �m�aq †�n� = 1 N V Em − En − �q �m�cd †�n� + W N 1 Em − En − �q �m�� p ap †�n� . �B1� Summation of both sides over q leads to an expression for the matrix element in the last term on the right-hand side, �m�� p ap †�n��1 − WSmn� = NV�m�cd †�n�Smn, �B2� where Smn � 1 N � q 1 Em − En − �q , �B3� which brings Eq. �B1� to the form �m�aq †�n� = �m�cd †�n� N�Em − En − �q� V 1 − WSmn . �B4� Consider now this equality at one of the two fixed points, LM or FL. The fixed-point Hamiltonian has then the qua- dratic form �A2�, which defines the complete basis of the operators g�. The matrix element �m�g� †�n� vanishes unless �m�=g� †�n�, which implies Em=En+��. At a fixed point, there- fore, the sum on the right-hand side of Eq. �B3� reduces to that in Eq. �A7�, i.e., Smn = − � cot �*, �B5� where we have ignored the last term on the right-hand side of Eq. �A7� because at a fixed point the ratio �� /D→0. Equa- tion �B5� suggests that we introduce the phase shift �W, de- fined by tan �W � − �W , �B6� to simplify Eq. �B4�, �m�aq †�n� = V�m�cd †�n� N�Em − En − �q� sin �* cos �W sin��* − �W� . �B7� In analogy with Eq. �A3� we can, moreover, write g� = ��0cd + � q ��qaq, �B8� with normalized coefficients, ��0 2 + � q ��q 2 = 1. �B9� Equation �B8� is easily inverted to yield the expressions aq = � � ��qg�, �B10� and cd = � � ��0g�. �B11� Substitution of Eq. �B10� for aq on the left-hand side of Eq. �B7� and of Eq. �B11� for cd on the right-hand side then yields ��q 2 = V2��0 2 N��� − �q�2� sin �* cos �W sin��* − �W� � 2 . �B12� We divide both sides by N, sum them over q, and substitute Eq. �A11� for the resulting sum on the right-hand side to find that � q ��q 2 = NV2��0 2 � � cos �W sin��* − �W��2 . �B13� UNIVERSAL ZERO-BIAS CONDUCTANCE FOR THE… PHYSICAL REVIEW B 80, 235317 �2009� 235317-19 Substitution in the second term on the left-hand side of Eq. �B9� now shows that, with error O�1 /N�, the fixed-point matrix elements �m�cd †�n� are constants, dependent only on the phase shift and scattering potential, ��m�cd †�n��2 = 1 NV2 sin2��* − �W� 2�2 cos2 �W . �B14� The fixed-point spectral density, as one would also expect, is temperature independent, �d��,T� = 1 NV2Z� m,n e−�Em sin2��* − �W� 2�2 cos2 �W ���� − �� , �B15� equivalent to �d��� = sin2��* − �W� cos2 �W . �B16� With W=0, we recover the celebrated expression42 �d��� = sin2 �* . �B17� More generally, however, to obtain the fixed-point spectral densities, we set �*=� at the FL, and �*=�− /2 at the LM, from which it results that �d LM = cos2�� − �W� w , �B18a� �d FL = sin2�� − �W� w . �B18b� Substitution of Eqs. �B18a� and �B18b� for �d on the right- hand side of Eq. �8� leads to Eqs. �22a� and �22b�, respec- tively. APPENDIX C: ZERO-BIAS CONDUCTANCE By contrast with the coupling to the impurity, which is independent of the odd operators bk defined by Eq. �4b�, the Hamiltonian describing a bias voltage couples to bk’s. Pre- liminary to the discussion of the conductance, it is therefore convenient to derive results for bk’s analogous to those in Appendix A. Specifically, given the formal equivalence be- tween Eqs. �7� and �A1�, we can follow the steps in that Appendix to write HB in the diagonal form HB = � �̃�g̃� †g̃�, �C1� with g̃� = � k �̃�,kbk, �C2� and derive a result analogous to Eq �A12�. At low energies, in particular, i.e., for ��̃���D, the eigenvalues �̃� are uni- formly spaced, with the phase shift �w defined by Eq. �B6�, and � k �̃�,k = cos �W. �C3� Multiplication of both sides by g̃� and summation over � then shows that � k �m̃�bk�ñ� = cos �w� � �m̃�g̃��ñ� �C4� for any pair �m̃�, �ñ� of low-energy eigenstates of HB. It is likewise convenient to compute the following com- mutator: H,ak †bk� = V N cd †bk + W N � q �aq †bk − ak †bq� , �C5� from which we see that, given two eigenstates �!m� and �!n� of HA with eigenvalues Em and En, respectively, �!m�ak †bk�!n� = V N �!m�cd †bk�!n� Em − En + W N � q �!m�aq †bk − ak †bq�!n� Em − En . �C6� 1. Current To calculate the conductance, we can, for instance, exam- ine the current flowing into the R wire, Î = dqR dt = − ie " �H,� k ckR † ckR� , �C7� i.e., Î = − ie 2"�H,� k „ak †ak + bk †bk − �ak †bk + H.c.�…� , �C8� which reduces to Î = ie 2" V N cd †� k �ak + bk� + H.c., �C9� because summed over k, the last term on the right-hand side of Eq. �C5� vanishes. 2. Conductance To induce a current, we add to the model Hamiltonian an infinitesimal slowly growing perturbation that lowers the chemical potential of the R wire relative to that of the L wire, H� � ��h��t� = − e �� 2 � k �ckR † ckR − ckL † ckL�e�t/", �C10� with an infinitesimal shift ��. Projected on the basis of ak’s and bk’s, h� reads h��t� = − e 2� k �ak †bk + H.c.�e�t/", �C11� and Eq. �C6� shows that YOSHIDA, SERIDONIO, AND OLIVEIRA PHYSICAL REVIEW B 80, 235317 �2009� 235317-20 �!m�h��t��!n� = − eV 2 N e�t/"� k �!m�cd †bk − bk †cd�!n� Em − En . �C12� Standard linear response theory relates h� to the conduc- tance, G�T� = − i Z" −� 0 � m e−�Em�!m� Î,h��t���!m�dt , �C13� where Z is the partition function at the temperature T. Comparison with Eq. �C11� shows that the operators ak within the parentheses on the right-hand side of that equality make no contribution to the conductance. We therefore de- fine Îb � ie 2" V N cd †� k bk + H.c., �C14� and rewrite Eq. �C13�, G�T� = − i Z" −� 0 � m e−�Em�!m� Îb,h��t���!m�dt . �C15� Following the insertion of a completeness sum �n�n��n� on the right-hand side of Eq. �C15�, straightforward manipu- lations lead to the familiar expression, G�T� = 1 Z� m,n �e−�Em − e−�En� �!m�Îb�!n��!n�h��0��!m� Em − En + i� . On the right-hand side, we now substitute Eq. �C12� Eq. �C14�� for h� �Îb�. This yields G�T� = − i e2 4" V2 NZ � m,n,k,q � �!m�bq †cd�!n��!n�cd †bk�!m� Em − En + i� + �!m�cd †bk�!n��!n�bq †cd�!m� Em − En + i� � e−�Em − e−�En Em − En . �C16� Aided by Eq. �C4�, we can now trade the sum over the con- duction operators bk for a sum over the eigenoperators g̃�, G�T� = − i e2 2h W N�Z � m,n,�,�� � �!m�g̃�� † cd�!n��!n�cd †g̃��!m� Em − En + i� + �!m�cd †g̃��!n��!n�g̃�� † cd�!m� Em − En + i� � e−�Em − e−�En Em − En . �C17a� Since the g̃� diagonalize HB, only the terms with �=�� con- tribute to the sum on the right-hand side. We interchange the indices m and n in the second term within the parentheses on the right-hand side to show that G�T� = e2 h � w �NZ � m,n,� e−�Em��!m�cd †g̃��!n��2��Em − En� . �C18� Since �!m�= �m��m̃�, where �m���m̃�� is an eigenstate of HA �of the quadratic Hamiltonian HB�, the right-hand side splits into two coupled sums, G�T� = e2 h � W N�Z � m,n,� e−�Em��m�Vcd †�n��2 ��Em − En − �̃��� m̃,ñ e−�Em̃�m̃�g̃��ñ��ñ�g̃� †�m̃� . �C19� The second sum is equal to Zb 1− f��̃p��, where f��� is the Fermi function and Zb is the partition function for the Hamiltonian Hb. The identity − 1 f��� �f �� = �„1 − f���… �C20� then turns Eq. �C19� into G�T� = e2 hZa W �N � m,n,� e−�Em f��̃�� �− �f �� � �̃� ��m�cd †�n��2 ��Em − En − �̃�� , �C21� where Za is the partition function for the Hamiltonian HA. 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