# Two-channel Kondo phases and frustration-induced transitions in triple quantum dots

###### Abstract

We study theoretically a ring of three quantum dots mutually coupled by antiferromagnetic exchange interactions, and tunnel-coupled to two metallic leads: the simplest model in which the consequences of local frustration arising from internal degrees of freedom may be studied within a 2-channel environment. Two-channel Kondo (2CK) physics is found to predominate at low-energies in the mirror-symmetric models considered, with a residual spin- overscreened by coupling to both leads. It is however shown that two distinct 2CK phases, with different ground state parities, arise on tuning the interdot exchange couplings. In consequence a frustration-induced quantum phase transition occurs, the 2CK phases being separated by a quantum critical point for which an effective low-energy model is derived. Precisely at the transition, parity mixing of the quasi-degenerate local trimer states acts to destabilise the 2CK fixed points; and the critical fixed point is shown to consist of a free pseudospin together with effective 1-channel spin quenching, itself reflecting underlying channel-anisotropy in the inherently 2-channel system. Numerical renormalization group techniques and physical arguments are used to obtain a detailed understanding of the problem, including study of both thermodynamic and dynamical properties of the system.

###### pacs:

71.27.+a, 72.15.Qm, 73.63.Kv## I Introduction

Quantum dot devices have recently been the focus of intense investigation, due to the impressive experimental control available in manipulating the microscopic interactions responsible for Kondo physicsHewson (1993); Kouwenhoven et al. (1997); Pustilnik and Glazman (2004); Goldhaber-Gordon et al. (1998); Cronenwett et al. (1998). In particular the classic spin- Kondo effectHewson (1993) — in which a localized spin is fully screened by coupling to itinerant conduction electrons in a single attached metallic lead — has now been widely studied experimentally (see eg [Goldhaber-Gordon et al., 1998; Cronenwett et al., 1998; Jeong et al., 2001; van der Wiel et al., 2000; Nygard et al., 2000]).

But arguably the most diverse and subtle Kondo physics results from the interplay between internal spin and orbital degrees of freedom in coupled quantum dots. A wide range of strongly correlated electron behavior is accessible in such systems, with variants of the standard Kondo effectFerrero et al. (2007); Borda et al. (2003); Kuzmenko et al. (2006); Galpin et al. (2005); Mitchell et al. (2009); Vernek et al. (2009); Numata et al. (2009), quantum phase transitionsGalpin et al. (2005); Mitchell et al. (2009); Dias da Silva et al. (2008); Pustilnik et al. (2004); Žitko and Bonča (2007) and non-Fermi liquid phasesGeorges and Sengupta (1995); Oreg and Goldhaber-Gordon (2003); Zarand et al. (2006); Anders et al. (2004) having been studied theoretically. In particular, both doubleKikoin and Avishai (2001); Vojta et al. (2002); Borda et al. (2003); Galpin et al. (2005); Mitchell et al. (2006); Anders et al. (2008); López et al. (2005); Zaránd et al. (2006); Dias da Silva et al. (2008) and tripleLazarovits et al. (2005); Oguri et al. (2005); Žitko et al. (2006); Kuzmenko et al. (2006); Lobos and Aligia (2006); Wang (2007); Delgado and Hawrylak (2008); Žitko and Bonča (2008); Ingersent et al. (2005); Vernek et al. (2009); Numata et al. (2009) quantum dots have been shown to demonstrate low-temperature behavior quite different from their single-dot counterpartsFerrero et al. (2007). Advances in nanofabrication techniquesGoldhaber-Gordon et al. (1998); Cronenwett et al. (1998); Jeong et al. (2001); van der Wiel et al. (2000); Nygard et al. (2000); Blick et al. (1996); Roch et al. (2008); Gaudreau et al. (2006); Schröer et al. (2007); Vidan et al. (2005); Rogge and Haug (2008); Grove-Rasmussen et al. (2008); Potok et al. (2007), and atomic-scale manipulation using scanning tunneling microscopyJamneala et al. (2001); Uchihashi et al. (2008), now allow for the construction of coupled quantum dot structures, in which the geometry and capacitance of the dots can be controlled, and their couplings fine-tunedKouwenhoven et al. (1997); Gaudreau et al. (2006). Experimental access to a rich range of Kondo and related physics is thus within reach.

One of the most delicate effects, however, arises in the two-channel Kondo (2CK) model proposed by Nozières and BlandinNozières and Blandin (1980), which describes a spin- symmetrically coupled to two independent metallic leads. The standard, strong coupling Fermi liquid fixed point common in single-channel models is destabilized here. Much studied theoretically (for a review see [Cox and Zawadowski, 1998]), the dot spin in the 2CK model is overscreened at low temperatures, with each channel competing to compensate the local moment. The nontrivial intermediate-coupling fixed point which controls the low-temperature behavior of the system has a number of unusual non-Fermi-liquid (NFL) propertiesAndrei and Destri (1984); Tsvelick (1985); Sacramento and Schlottmann (1989); Cox and Zawadowski (1998), including a residual entropy of and a magnetic susceptibility which diverges logarithmically at low temperature.

The key ingredient of this 2CK physics is the suppression of charge transfer between the two symmetrically coupled leadsNozières and Blandin (1980); Cox and Zawadowski (1998); Affleck and Ludwig (1993) — the necessity in essence of a central spin. Experimental realizations of 2CK physics have been variously sought in eg heavy fermion systems containing UraniumCox (1987); Seaman et al. (1991); Andrei and Bolech (2002), scattering from 2-level systems using ballistic metal point contactsRalph and Buhrman (1992); Ralph et al. (1994), systems with tunneling between nonmagnetic impurities in metalsValdár and Zawadowski (1983a, b), and from impurities in grapheneSengupta and Baskaran (2007); albeit that the interpretation of observed behavior in terms of 2CK physics is not always unambiguous.

Recently however, Potok et al have demonstratedPotok et al. (2007) clear two-channel behavior in a coupled quantum dot device in which one small and one large dot are tunnel-coupled, with the small dot also coupled to a metallic lead. The large dot acts as an interacting second lead, but is fine-tuned to the Coulomb blockade regime so that charge transfer is energetically disfavoured. A small degree of inter-lead charge transfer must nonetheless occur, so ultimately the system is a Fermi liquid, with the 2CK fixed point destabilized below some low-temperature scale (which crossover has been widely studied theoreticallyAffleck and Ludwig (1993); Zaránd et al. (2006); Žitko and Bonča (2007), as has the instability of the 2CK fixed point to channel asymmetryNozières and Blandin (1980); Affleck and Ludwig (1990); Andrei and Jerez (1995); Sacramento and Schlottmann (1989)). But at finite temperatures, 2CK physics is clearly observedPotok et al. (2007).

Studying coupled quantum dot systems in a two-channel environment has attracted considerable theoretical interest recentlyIngersent et al. (1992); Pustilnik et al. (2004); Zarand et al. (2006); Kikoin and Oreg (2007); Kuzmenko et al. (2003); Žitko and Bonča (2007, 2008), in part because NFL states are accessible. The two-impurity two-channel Kondo modelJayaprakash et al. (1981); Jones and Varma (1987); Ingersent et al. (1992) (where two antiferromagnetically coupled dots are each coupled to their own lead) is a prime example, in which the tendency to form a local singlet state on the two dots competes with the formation of two separated single-channel Kondo statesJayaprakash et al. (1981); Jones and Varma (1987); Ingersent et al. (1992). The quantum critical point separating these phases is again the 2CK fixed pointAffleck et al. (1995); Gan (1995). Triple quantum dot (TQD) models with three dots coupled to two leads in a mirror-symmetric fashion have also been studiedMoustakas and Fisher (1997); Kuzmenko et al. (2003); Žitko and Bonča (2007, 2008); Vernek et al. (2009); Numata et al. (2009), recent workŽitko and Bonča (2008) by Žitko and Bonča showing in particular that a range of fixed points familiar from simpler quantum impurity models are accessible in a ring model. Indeed both two-channel and two-impurity Kondo effects are realised, on tuning the interdot couplings as a third dot is coupled to the two-impurity two-channel systemŽitko and Bonča (2008).

TQDs arranged in a ring geometry also provideVidan et al. (2005); Mitchell et al. (2009); Žitko and Bonča (2008) the simplest and most direct means of studying the effect of local frustration on Kondo physics. In mirror-symmetric systems, all dot states can be classified by their parity under leftright interchangeKuzmenko et al. (2006); Mitchell et al. (2009). This symmetry permits a level crossing of states in the isolated trimer on varying the interdot couplings, with a pair of degenerate doublets comprising the ground state when all dots are equivalent. It was recently shownMitchell et al. (2009) that this level-crossing is preserved in the full many-body system when a single lead is attached. The situation is however more subtle on coupling the trimer to two leads, which is the focus of the present paper. We study a two-channel TQD ring model, shown schematically in Fig. 1 (and discussed below), as a function of the interdot exchange couplings; using the density matrix extensionPeters et al. (2006); Weichselbaum and von Delft (2007) of Wilson’s numerical renormalization group (NRG) techniqueWilson (1975); Krishnamurthy et al. (1980); Bulla et al. (2008).

We show that 2CK physics predominates in this model, but that two distinct 2CK phases must in fact arise since local trimer states of different parity can couple to the leads. In consequence, one expects a quantum phase transition to occur between the two 2CK phases. This is indeed shown to arise, with the 2CK phases separated by a nontrivial quantum critical point, the nature of which is uncovered explicitly and analysed in detail.

The paper is organized as follows. In Sec. II we discuss the two-channel TQD Hamiltonian, and develop low-energy effective models to describe the behavior of the system when deep in each 2CK phase. Symmetry arguments indicate the presence of a quantum phase transition near to the point of inherent magnetic frustration in the TQD, and an effective low-energy model valid in the vicinity of the transition is derived. Sec. III presents NRG results for the full system, considering both thermodynamics and dynamical properties of the 2CK phases. The transition itself is investigated in Sec. IV, and the nature of the critical point elucidated, employing heuristic physical arguments in addition to direct calculation. In Sec. V the effective low-energy model describing the transition is itself studied directly using NRG. The paper concludes with a brief summary.

Before proceeding, we point out that the stability of 2CK physics in the model studied here is of course delicate, just as it is for the standard two-channel Kondo modelNozières and Blandin (1980); Cox and Zawadowski (1998); Affleck and Ludwig (1993). A small degree of charge transfer, which would arise in a real TQD device from interdot tunnel-couplings (as opposed to exchange couplings), will ultimately lead to a crossoverAffleck and Ludwig (1993); Zaránd et al. (2006); Žitko and Bonča (2007) from a 2CK to a stable Fermi liquid fixed point, below some characteristic low-temperature scale. The same situation is of course well known to occur for channel anisotropyNozières and Blandin (1980); Affleck and Ludwig (1990); Andrei and Jerez (1995); Sacramento and Schlottmann (1989), as would arise in the TQD model upon destruction of overall leftright symmetry via eg different exchange couplings (see Fig. 1).

## Ii 2CK trimer model

We consider a system of three (single-level) quantum dots, arranged in a triangular geometry, as illustrated in Fig. 1. Dot ‘2’ is coupled to dots ‘1’ and ‘3’, which are coupled to each other and to their own metallic lead. We model the central dot 2 strictly as a spin- to prevent inter-lead charge transfer, but dots 1 and 3 are Anderson-like sites, permitting variable occupation. Tunneling is allowed between these terminal dots and their connected leads, but the dots are coupled to each other by an antiferromagnetic (AF) exchange interaction to form a Heisenberg ring. We focus explicitly on a system tuned to left/right mirror symmetry (see Fig. 1), with Hamiltonian . Here refers to the two equivalent non-interacting leads (), which are tunnel-coupled to dots 1 and 3 via . describes the trimer itself, with exchange couplings and ,

(1) |

where is a spin- operator for dot . For dots , is the number operator, is the level energy and is the intradot Coulomb repulsion. The full Hamiltonian is thus invariant under simultaneous and permutation.

### ii.1 The isolated trimer

We are interested in the TQD deep in the Coulomb blockade valley, where each dot is in practice singly occupied. To this end we consider explicitly , the or states being much higher () in energy. (and the full ) is then particle-hole symmetric, although this is incidental: we require only that the singly-occupied manifold of TQD states lies lowest. This manifold comprises two doublets and a spin quartet, which is always higher than the doublets for AF exchange couplings.

For any , , the lowest doublets of the isolated TQD are

(2) |

with and for spins , and the raising/lowering operator for the spin on dot 2. defines the ‘vacuum’ state of the local (dot) Hilbert space, in which dots 1 and 3 are unoccupied, and dot 2 carries a spin-.

The energy separation of the two doublets is , with the ground state of the isolated TQD for and lowest for . When is dominant, dots 1 and 3 naturally lock up into a singlet (see , Eq. 2), leaving a free spin on dot 2; with and . For by contrast, dots 1 and 3 are now in a triplet configuration ( in Eq. 2), with and .

The states are degenerate precisely at , reflecting the inherent magnetic frustration at that point. A level crossing of the doublets at is permitted because each has different symmetry under permutation. We define a parity operator which exchanges orbital labels and (as discussed further in the Appendix). From Eq. 1 it is clear that commutes with the isolated TQD Hamiltonian, . Thus all states of can be classified according to parity, the eigenvalues of being only (since ). In the spin-only (‘singly-occupied’) regime, the parity operator may be expressed concisely as Dirac (1930) , with the total spin of dots 1 and 3. Thus describes the parity of the doublet states of . The full lead-coupled Hamiltonian is not of course invariant to interchange alone, but rather to simultaneous exchange of the dot labels and the left/right leads (embodied in , see Appendix); which we refer to as ‘overall ’ symmetry.

### ii.2 Effective low-energy models

On tunnel-coupling to the leads, effective models describing the system on low-energy/temperature scales in the -electron valley of interest may be obtained by standard Schrieffer-Wolff transformationsHewson (1993); Schrieffer and Wolff (1966), perturbatively eliminating virtual excitations into the - and -electron sectors of to second order in (and neglecting retardation effects Hewson (1993) as usual). The calculations are lengthy, so rather than giving full details we sketch below a somewhat simplified, but physically more transparent, account of the key results (Eqs. 7,13 below).

First we consider the effective low-energy model appropriate to the temperature range , in which all dots become singly-occupied. Here the appropriate unity operator for the TQD Hilbert space is , with the spin of dot . To second order in the dot-lead tunneling , a spin-model of form arises; where describes the leads as above and:

(3) |

Here the effective exchange coupling is , with the lead density of states per orbital at the Fermi level; and is the hybridization, with total lead density of states , and the number of orbitals/k-states in the lead (such that , and hence , is finite in the continuum limit ). In Eq. 3, is the spin density of lead at dot given by

(4a) | ||||

(4b) |

with the Pauli matrices and the creation operator for the ‘0’-orbital of the Wilson chain.

As above, the lowest states of are the doublets given in Eq. 2. Provided they are not near-degenerate, only the lower doublet need be retained: for and for . To first order in , an effective low-energy model is then obtained simply by projecting into the reduced Hilbert space of the lowest doublet, using

(5) |

for the appropriate or doublet ground state. The resultant Hamiltonian follows as

(6) |

using the symmetry .

Eq. 6 is of two-channel Kondo form,

(7) |

where is the spin- operator representing the appropriate doublet or , with components and . At this level of calculation the effective exchange coupling is given by ; so from Eq. 2 an AF effective Kondo coupling then arises for the ground state appropriate to , while for the doublet (lowest for ), results. In the latter case, there is in fact a weak residual AF coupling: a full Schrieffer-Wolff calculation gives precisely the two-channel Kondo model Eq. 7 as one would expect, but with given by

(8a) | ||||

(8b) |

yielding to leading order in , and

a much smaller but non-vanishing
for the singlet-locked doublet , reflecting the residual AF coupling between the spin on dot 2 and the leads.

Hence, sufficiently deep in either regime or , the low-energy behavior of the system is that of a 2CK model. The lowest spin- state of the TQD is thus overscreened by conduction electrons, embodied in the infrared 2CK fixed point describing the non-Fermi liquid ground stateNozières and Blandin (1980); Cox and Zawadowski (1998); Affleck and Ludwig (1993), in which the partially quenched spin is characterised by a residual entropy of (); overscreening setting in below the characteristic two channel Kondo scale , determined from perturbative scaling asNozières and Blandin (1980)

(9) |

Since commutes with all components of in Eq. 7, , whence parity is conserved in the effective low-energy model. Since that parity is determined by or , there are two distinct 2CK phases, which one thus expects to be separated by a quantum phase transition (QPT).

In the vicinity of the transition, ie close to , neither of the two 2CK models in Eq. 7 is of course sufficient to describe the low-energy physics: the states and are now near-degenerate, so both must be retained in the low-energy trimer manifold. Hence, defining and proceeding in direct parallel to the discussion above, an effective low-energy model in the vicinity of the transition is obtained from . From Eq. 3 for , using such that and hence , one obtains

(10) |

The final term in Eq. 10 (arising from ) is simply the energy difference between the doublets, and the first term is (see Eq. 6) the 2CK coupling of each doublet to the leads. It is helpful to recast Eq. 10 in terms of spin- operators for real spin () and pseudospin for the local Hilbert space, defined by

(11a) | ||||

(11b) |

and

(12a) | ||||

(12b) |

From Eq. 11, the TQD doublets are each eigenstates of and ; in particular, the eigenvalues of correspond simply to (half) the parity of the appropriate doublet. By contrast, the doublets are interconverted by (Eq. 12b), and , acting to switch parity.

After simple if laborious algebra, Eq. 10 reduces to

(13) |

in terms of spin/pseudospin operators, with

(14) |

and using Eq. 2. [A full Schrieffer-Wolff calculation again gives precisely the effective Hamiltonian Eq. 13, with

(15) |

recovering to leading order in , and given by Eq. 8.]

Eq. 13 is the essential low-energy model applicable to the vicinity of the QPT; we study it directly via NRG in Sec.V. The pseudospin operators can naturally be classified according to parity, the components of having different parity under : using , Eqs. 11b,12b give , while for the raising/lowering components. By contrast, all components of spin (Eqs. 11a,12a) commute with . Hence, since the global (overall ) parity must be conserved (), interactions involving the component of pseudospin can only couple to even combinations of the lead spin densities (symmetric under interchange of the lead labels), as in the first term of Eq. 13 (or 10); while by the same token interactions involving must be associated with the odd (antisymmetric) combination , as in the second term of Eq. 13. In the vicinity of the QPT the latter is of course the key interaction in , since in switching the parity of the TQD states it in essence drives the transition between the two 2CK phases.

Finally, note that the last term of Eq. 13, equivalent to a magnetic field acting on the pseudospin, energetically favors the doublet () when their energy separation , and () for . Hence, when is sufficiently large that only one of the doublets need be retained in the low-energy TQD manifold, the terms are obviously suppressed; and Eq. 13 then reduces as it must to one or other of the 2CK models Eq. 7. In fact, as shown in Sec. IV ff, for any , the term in Eq. 13 ensures that one or other of the 2CK fixed points ultimately remains the stable low-temperature FP.

## Iii Properties of the 2CK Phases

The physical picture thus indicates that 2CK physics dominates the low-energy behavior of the model; with a QPT occurring as a function of between two 2CK phases of distinct parity.

We now analyse the properties of each 2CK phase of the full model; using Wilson’s NRG techniqueWilson (1975); Krishnamurthy et al. (1980), employing a complete basis set of the Wilson chainPeters et al. (2006) to calculate the full density matrixPeters et al. (2006); Weichselbaum and von Delft (2007) (for a recent review see [Bulla et al., 2008]). Calculations are typically performed for an NRG discretization parameter , retaining the lowest states per iteration. As above we choose for convenience , and consider a symmetric constant density of states for each lead, with density of states per conduction orbital , and bandwidth (such that results shown are essentially independent of ). In all calculations shown explicitly, we use a fixed and , varying the exchange (see Fig. 1) not (a).

Fig. 2 shows the evolution of the lowest energy levels of the system as a function of NRG iteration number , exemplifying RG flow between different FPs of the model. Panel (a) is for a system deep in the regime (specifically ), while panel (b) shows the energy levels for (). For comparison, panel (c) is for a pure (single spin-) 2CK model of form Eq. 7, ie , with a Kondo coupling chosen to be the same as the effective coupling of the ground state TQD doublet in (a) (as obtained from Eq. 8(b)). In both cases (a) and (b), the levels are seen to converge quite rapidly to their values, which are clearly those of the 2CK FP in Fig. 2(c). These levels are of course characteristic of the 2CK FP, and — after a trivial rescaling by a factor which depends on the NRG discretization parameter — are described by the fractions , , , , … as determined from a conformal field theory analysis of the FPAffleck and Ludwig (1990); Ye (2006).

The iteration number by which the levels converge to the set of 2CK FP levels is however clearly different in (a) and (b), reflecting the different Kondo scales characteristic of the two cases. Case (b) () flows close to the local moment (LM) FP between to (with levels naturally characteristic of the LM FP in that range), and approaches the stable 2CK FP by . By contrast, convergence to the 2CK FP in (a) and (c) both occur at (with a much shorter range of close to the LM FP). Since the iteration number is related to an effective temperature through Wilson (1975); Krishnamurthy et al. (1980) , the 2CK Kondo scales of the three examples in Fig. 2 are thus exponentially small (as expected from Eq. 9); but with for (b) being some 8 orders of magnitude smaller than that of (a), reflecting the distinct nature of the coupling in the two cases, as discussed in Sec. II.2.

### iii.1 Thermodynamics

NRG results for thermodynamics in each phase are now considered. We focus on the ‘impurity’ (TQD) contributionKrishnamurthy et al. (1980); Bulla et al. (2008) to the entropy, , and the uniform spin susceptibility, (here refers to the spin of the entire system and with denoting a thermal average in the absence of the TQD); the temperature () dependences of which provide clear signatures of the underlying FPs reached under renormalization on progressive reduction of the temperature/energy scaleKrishnamurthy et al. (1980); Bulla et al. (2008).

#### iii.1.1

For the regime the effective 2CK model Eq. 7 should describe the system for , where the lowest TQD doublet, in this case the odd parity state , couples symmetrically to the leads. 2CK physics is thus expected below .

For the full model, Fig. 3 shows both [panel (a)] and [panel (b)] for fixed and , with , and , all deep in the regime. In each case, at high temperatures the behavior is governed by the free orbital Krishnamurthy et al. (1980) (FO) FP, with all possible states of the TQD thermally accessible, and hence . For the dots become in essence singly occupied, and hence an entropy of is expected not (b). Below all but the lowest trimer doublet is projected out. Thus approaches , signifying the LM FP where the lowest doublet is essentially a free spin- disconnected from the leads (Eq. 7 with ).

The LM FP is however unstable, and the system always flows to the stable 2CK FP below , recovering (Fig. 3(a)) the residual entropy known to be characteristic of the 2CK FPAndrei and Destri (1984); Tsvelick (1985); Sacramento and Schlottmann (1989). In practice we may define a Kondo temperature through the entropy, via (suitably between and ); or alternatively through the spin susceptibility via (as in [Krishnamurthy et al., 1980]). Deep in the 2CK phases the two definitions are of course equivalent (), probing as they do the common characteristic scale associated with flow to the 2CK FP. The inset to Fig. 3(a) shows the entropy of the three systems rescaled in terms of ; showing scaling collapse to a common functional form, ie the universality characteristic of the crossover from the LM FP to the stable 2CK FPSacramento and Schlottmann (1989).

The underlying FPs of the model are likewise evident from , Fig. 3(b). The highest behavior corresponds to two uncorrelated sites (dots 1 and 3 of the TQD) and a free spin (dot 2). Hence , readily understood as the mean of the quasidegenerate states. On decreasing the LM FP is again rapidly reached, the lowest TQD doublet following the free spin- Curie law . Below the spin susceptibility is quenchedAndrei and Destri (1984); Affleck and Ludwig (1990); Sacramento and Schlottmann (1989) in the sense that as . In the inset to Fig. 3(b) the data are rescaled in terms of , again showing universality in the approach to the 2CK FPSacramento and Schlottmann (1989).

The characteristic low- logarithmic divergenceAndrei and Destri (1984); Affleck and Ludwig (1990); Sacramento and Schlottmann (1989) of itself is evident in Fig. 4 for the same parameters as Fig. 3. The slopes of the divergence vary widely between the three cases, but all collapse to the universal formSacramento and Schlottmann (1989) (with a constant), as seen in the inset of the figure.

#### iii.1.2

From the arguments of Sec. II.2, 2CK physics is again expected at low-energies in the regime, where the even-parity doublet is now the lowest TQD state. This is confirmed in Fig. 5, where (in direct parallel to Fig. 3 for ) thermodynamics are shown for fixed and , but now with , and : the scaling forms for and vs , shown in the insets of Fig. 5, are precisely those arising for and shown in Fig. 3.

### iii.2 Dynamics

We turn now to dynamics, focussing largely on the local single-particle spectrum of dot 1 (or, equivalently by symmetry, dot 3): , with the local retarded Green function. We obtain it through the Dyson equation,

(16) |

where is the non-interacting propagator (ie for ); and is the proper electron self-energy, with (such that ). The non-interacting is given trivially byHewson (1993) (with ), where is the usual one-electron hybridization function for coupling of dot 1 to the left lead: , with () for all inside the band, and .

An expression for the electron self-energy is also readily obtained using equation of motion methodsHewson (1993); Zubarev (1960). It is given by

(17) |

where is the Fourier transform of the retarded correlator , and where the local Green function itself is given by (independent of spin in the absence of a magnetic field). The self-energy can be calculated directly within the density matrix formulation of the NRGPeters et al. (2006); Weichselbaum and von Delft (2007); Bulla et al. (1998, 2008), via Eq. 17; with then obtained from Eq. 16. calculated in this way is highly accurateBulla et al. (1998, 2008), and automatically guarantees correct normalization of the spectrumPeters et al. (2006); Weichselbaum and von Delft (2007).

To motivate study of the spectrum (), we note that it controls the zero-bias conductance through dot 1. The TQD does not of course mediate current from the to leads, since the internal couplings between constituent dots are pure exchange (Fig. 1 and Eq. 1). However the and leads in Fig. 1 can obviously each be ‘split’ in two (symmetrically, to preserve overall symmetry), enabling a current to be driven through dot 1 or dot 3; with the same zero-bias conductance in either case, by symmetry. Following Meir and WingreenMeir and Wingreen (1992), the resultant conductance follows as

(18) |

where is the Fermi function (with the Fermi level) and the conductance quantum. The ()-dependence of thus controls the conductance, and for in particular . From Eq. 16, the local propagator for may be expressed as in terms of the renormalized single-particle level and renormalized hybridization , given by

(19a) | ||||

(19b) |

in terms of the value of the self-energy at ; and hence from Eq. 18:

(20) |

NRG results for single-particle dynamics are considered below, but first a question arises. We have argued above that, sufficiently deep in either regime or , the low-energy physics of the full three-site TQD model must reduce to that of a single spin- 2CK model (of form Eq. 7). The question is: to which dynamical property of a pure 2CK model should the spectral density be compared? To answer this, note first that Eq. 18 may be written equivalently as

(21) |

in terms of the t-matrix, , for the lead; with defined in the usual way in terms of scattering of electron states in the lead, via

(22) |

where is the propagator for the lead states. Using equation of motion methods Hewson (1993); Zubarev (1960) it is straightforward to show that (likewise ); so (recall ), hence the equivalence of Eqs. 18,21.

To compare for the full TQD model to results for a single-spin 2CK model Affleck and Ludwig (1993); Tóth et al. (2007); Johannesson et al. (2005); Anders (2005); Pustilnik et al. (2004); Tóth and Zaránd (2008), we thus require for the latter. Using the definition of the 0-orbital of a Wilson chain (Eq. 4b), the Hamiltonian for a single-spin 2CK model is