Quantum Dot in Interacting Environments
QQuantum dot in interacting environments
Colin Rylands ∗ and Natan Andrei † Department of Physics, Rutgers University Piscataway, New Jersey 08854. (Dated: January 12, 2018)A quantum impurity attached to an interacting quantum wire gives rise to an array of of newphenomena. Using Bethe Ansatz we solve exactly models describing two geometries of a quantum dotcoupled to an interacting quantum wire: a quantum dot that is (i) side-coupled and (ii) embeddedin a Luttinger liquid. We find the eigenstates and determine the spectrum through the BetheAnsatz equations. Using this we derive exact expressions for the ground state dot occupation. Thethermodynamics are then studied using the thermodynamics Bethe Ansatz equations. It is shownthat at low energies the dot becomes fully hybridized and acts as a backscattering impurity ortunnel junction depending on the geometry and furthermore that the two geometries are relatedby changing the sign of the interactions. Although remaining strongly coupled for all values of theinteraction in the wire, there exists competition between the tunneling and backscattering leading toa suppression or enhancement of the dot occupation depending on the sign of the bulk interactions.
I. INTRODUCTION
Coupling a quantum impurity to an interacting one di-mensional lead produces some of the most striking phe-nomena of low dimensional physics. A simple backscat-tering impurity is known to cause the wire to be split ifthe interactions are repulsive while a junction betweentwo leads can lead to perfect conductance in the pres-ence of attractive interactions [1]. More interesting stillare scenarios in which the impurity has internal degrees offreedom. These allow for richer and more exotic phases toappear [2]. Among these, systems of quantum dots cou-pled to interacting leads have attracted much attention[1][3][4][5][6][7][8][9][10][11][12]. The low energy descrip-tion of the leads is typically given by Luttinger liquidtheory which is the effective low energy description of alarge number of interacting systems [2, 13]. Here the in- t td † ψ ± ψ ± d † t ψ ± (a)(b) 1 FIG. 1: We consider two geometries of Luttinger dot system;(a) embedded and (b) side-coupled. The embedded geome-try also includes a Coulomb interaction between the dot andleads. Once unfolded the side-coupled and embedded geome-tries are the same but with the latter containing non localinteractions (2) ∗ Electronic address: [email protected] † Electronic address: [email protected] dividual electrons are dissolved and the excitations arebosonic density modes. In contrast, the relevant degreesof freedom on the dot are electronic. A competition en-sues between the tunneling from lead to dot which iscarried out by electrons and the energy cost of reconsti-tuting an electron from the bosons in the lead.Such systems are readily achievable in many exper-imental settings allowing for confrontation of theorywith experiment. Luttinger liquids provide the effec-tive description of carbon nano tubes [14][15], frac-tional quantum Hall edges [16][17][18], cold atomic gases[19][20][21][22] or He flowing through nano pores [23][24]to name but a few. Additionally they are known to de-scribe tunneling processes in higher dimensional resistiveleads [25][26] and more generally are the archetype of anon-Fermi liquid. Luttinger liquid-quantum dot systemshave successfully been realized in a number of experi-ments [27][28]. These realize the embedded geometry,see Fig. 1(a) of a dot placed between two otherwise dis-connected leads. Measurement of the conductance hasrevealed interesting non-Fermi liquid scaling as well asMajorana physics.Building on the work of [29][30] we use Bethe Ansatzto solve exactly Luttinger liquid-quantum dot systems inboth the embedded (see Fig 1(a)) and side coupled (seeFig 1(b)) geometries. The exact solution shows that thespectra of the two geometries are related by changingthe sign of the bulk interaction, a fact previously knownthrough bosonization [8], and are described in terms ofcharge and chiral degrees of freedom. At low energies weshow that the dot becomes fully hybridized and acts as abackscattering impurity for the side-coupled model andas a tunnel junction for the embedded system. This cre-ates a competition between the charge and chiral degreesof freedom when the back scattering or tunnel junction isirrelevant, leading to non Fermi liquid exponents in theground state dot occupation. We then go on to study thefinite temperature properties of the system deriving theThermodynamic Bethe Ansatz equations and using thisto obtain the finite temperature dot occupation.This paper is organised as follows: in section II we a r X i v : . [ c ond - m a t . s t r- e l ] J a n introduce the Hamiltonians and construct their exacteigenstates. We derive the exact spectrum of both sys-tems through their Bethe Ansatz equations by means ofthe off diagonal Bethe Ansatz method (ODBA) [31]. InSection III we find the ground state of the system and de-rive the exact dot occupation. From this we extract therenormalization group picture of the system and find theleading relevant and irrelevant operators in section IV.The thermodynamics of the system are studied in sec-tion V where we find the free energy contribution fromthe dot and use it to obtain the dot occupation at finitetemperature. In the final section we conclude. II. MODELS AND EIGENSTATES
The systems we consider consist of a quantum dot at-tached to an interacting lead, a Luttinger liquid, theattachment being either in the embedded or the side-coupled geometry. The Hamiltonian of a Luttinger liquidis given by, H LL = − i (cid:90) dx ( ψ † + ∂ x ψ + − ψ †− ∂ x ψ − )+4 g (cid:90) dx ψ † + ( x ) ψ †− ( x ) ψ − ( x ) ψ + ( x ) (1)where ψ †± are right and left moving fermions which in-teract with a point like interaction of strength g [2]. Forthe side-coupled geometry we have x ∈ [ − L/ , L/
2] whilefor the embedded geometry we take two Luttinger liquidsrestricted to x ∈ [ − L/ ,
0] and x ∈ [0 , L/ H embLL = − i (cid:90) dx ( ψ † + ∂ x ψ + − ψ †− ∂ x ψ − )+4 g (cid:88) σ = ± (cid:90) dx ψ † σ ( x ) ψ † σ ( − x ) ψ σ ( − x ) ψ σ ( x )(2)The embedded system now consists of one branch ofleft-movers and one branch of right movers restrictedto x ∈ [ − L/ , L/
2] but unlike the side-coupled systemwhere the left and right fermions interact locally witheach other, in the embedded system after unfolding theinteraction is between particles of the same chirality andis non local. Further, the spectrum being linear a cut-off needs to be imposed to render the energies finite.We shall impose it on the particle momenta: k ≥ −D .All physical quantities are taken to be small comparedwith the cutoff and at the end of the calculation we send D → ∞ , to obtain universal results.The quantum dot is modelled by a resonant level withenergy (cid:15) described by, H dot = (cid:15) d † d, (3) (a) d † A [10]+ A [10] − A [01]+ A [01] − − ( b ) + −D −D−K −K FIG. 2: (a) The single particle wavefunction given by (6) is de-picted. Particles are either incoming on the left or right withamplitudes A [10]+ , A [01] − or outgoing on the left or right withamplitudes A [10] − , A [01]+ . (b) The linear derivative requires thatwe cutoff the bottom of the Dirac sea so that k > −D whichwe will take to infinity in the end. When the rapidity notationis used the dot energy acts as a local chemical potential andin the ground state levels are filled up to −K = −D e − B/ ,with B = B ( (cid:15) ). coupled to Luttinger liquid via a tunnelling term, H t = t ψ † + (0) + ψ †− (0)) d + h.c (4)which mediates both forward and backscattering in themodel, the latter changing left movers to right moversand vice versa. Furthermore in the embedded system weadd a Coulomb interaction between the ends of the leadsand the dot which is the same strength as the Luttingerinteraction, H c = gd † d (cid:88) σ = ± ψ † σ (0) ψ σ (0) . (5)Both energy scales in the dot Hamiltonian are small com-pared the the cut-off, (cid:15) , Γ (cid:28) D , where Γ = t is the levelwidth.We shall determine the spectrum and the full set ofexact eigenstates of both Hamiltonians, H sc = H LL + H t + H dot and H emb = H emb LL + H t + H dot + H c , using theBethe Ansatz approach, and then proceed to the groundstate (T=0) and thermodynamic ( T >
0) properties. TheBethe Ansatz method we employ here is distinct fromthat which has been typically used for quantum impuritymodels [33][34]. As the problem contains both forwardand back scattering we must formulate it in an in-outbasis with the configuration space being partitioned inregions labelled by both the order of the particles andby their closeness to the origin. The large degeneracypresent in the bulk system due to the linear derivative isthen used to find a consistent set of wave functions [29].We illustrate this by explicitly constructing the one andtwo particle eigenstates from which we can determine the N -particle states.After the unfolding procedure the two systems differonly in the two particle interaction meaning the singleparticle eigenstates are the same in both models. Thetunnelling to and from the dot takes place at the originhence we may expand the wavefunction in plane waveson either side of it, the most general form for the singleparticle state of energy E = k being, (cid:88) σ = ± (cid:90) e σikx (cid:104) θ ( − x ) A [10] σ + θ ( − x ) A [01] σ (cid:105) ψ † σ ( x ) | (cid:105) + Bd † | (cid:105) , (6)where θ ( ± x ) are Heaviside functions. The amplitudes A [10]+ and A [01] − are those of an incoming particle and arerelated to the outgoing amplitudes A [10] − and A [01]+ (seeFig. 2(a)) by the bare single particle S-matrix - S , whichtakes an incoming particle to an outgoing one. Tradingin the particle momentum k for the rapidity variable z ,defined as k − (cid:15) = D e z/ , we have, (cid:32) A [01]+ A [10] − (cid:33) = S ( z ) (cid:32) A [10]+ A [01] − (cid:33) (7) S ( z ) = (cid:32) e z/ e z/ + ie c − ie c e z/ + ie c − ie c e z/ + ie c e z/ e z/ + ie c (cid:33) (8)with e c = Γ / D . In addition the dot amplitude B is B = (cid:88) σ = ± e ( c − z ) / (cid:16) A [10] σ + A [01] σ (cid:17) . (9)From here periodic boundary conditions can be imposed ψ †± ( − L/
2) = ψ †± ( L/
2) resulting in e − i D e z/ L − i(cid:15) L (cid:32) A [10]+ A [01] − (cid:33) = S ( z ) (cid:32) A [10]+ A [01] − (cid:33) (10)which can then be solved for the allowed values of therapidity z .We now proceed to the two particle case wherein thebulk interaction g enters differently in both models. Weshall first consider the side-coupled model and discussthe embedded model subsequently. Since the two parti-cle interaction is point-like as is the tunnelling to the dotwe may divide configuration space into regions such thatthe interactions only occur at the boundary between tworegions. Therefore away from these boundaries we writethe wavefunction as a sum over plane waves. For two par-ticles we require 8 regions which are specified not only bythe ordering of the particle positions x , x and the im-purity but also according to which position is closer tothe origin. For example if x is to the left of the impu-rity, x to its right with x closer to the impurity thenthe amplitude in this region is denoted A [102 B ] σ σ , σ j = ± being the chirality of the particle at x j . The region inwhich x is closer is denoted A [102 A ] σ σ . The consequencefor the wavefunction is that we include Heaviside func-tions θ ( x Q ) which have support only in the region Q , e.g θ ( x [102 B ] ) = θ ( x ) θ ( − x ) θ ( − x − x ) and θ ( x [102 A ] ) = θ ( x ) θ ( − x ) θ ( x + x ). The most general two particlestate with energy E = k + k = (cid:80) j =1 D e z j / + 2 (cid:15) is (a) (b) FIG. 3: (Color online) The amplitudes in the two particlewavefunction are arranged into 8 vectors given by (12) andaccording to whether the particles are incoming or outgoingas well as their ordering with respect to the impurity. (a)The amplitudes in (cid:126)A consist of both particles incoming butparticle 2 (black) closer to the impurity than particle 1 (red).(b) The amplitudes in (cid:126)A consist of particle two outgoing.These vectors are related by S ( z ). therefore | E (cid:105) = (cid:88) Q (cid:88) σ ,σ = ± (cid:90) θ ( x Q ) A Qσ σ (cid:89) j e iσ j k j x j ψ † σ j ( x j ) | (cid:105) + (cid:88) σ = ± (cid:90) (cid:104) θ ( − x ) B [10] σ + θ ( x ) B [01] σ (cid:105) ψ † σ ( x ) d † | (cid:105) . (11)The amplitudes A Qσ σ are related to each other by S-matrices which are fixed by the Hamiltonian and in turnfix B [10] ± and B [01] ± . To define these S-matrices we formcolumn vectors of the amplitudes, (cid:126)A = A [120 B ]++ A [102 B ]+ − A [201 B ] − + A [021 B ] −− (cid:126)A = A [210 A ]++ A [102 A ]+ − A [201 A ] − + A [012 A ] −− (cid:126)A = A [201 A ]++ A [012 A ]+ − A [210 A ] − + A [102 A ] −− (cid:126)A = A [201 B ]++ A [021 B ]+ − A [120 B ] − + A [102 B ] −− (cid:126)A = A [021 B ]++ A [201 B ]+ − A [102 B ] − + A [120 B ] −− (cid:126)A = A [012 A ]++ A [201 A ]+ − A [102 A ] − + A [210 A ] −− (cid:126)A = A [102 A ]++ A [210 A ]+ − A [012 A ] − + A [201 A ] −− (cid:126)A = A [102 B ]++ A [120 B ]+ − A [021 B ] − + A [201 B ] −− (12)which have the following interpretation: (cid:126)A ( (cid:126)A ) are theamplitudes where both particles are incident on the im-purity but particle 2 (1) is closer, (cid:126)A ( (cid:126)A ) are the ampli-tudes in which both particles are outgoing with particle2 (1) closer to the impurity, (cid:126)A ( (cid:126)A ) describes particle2 (1) having scattered off the impurity and is still closerto the impurity than 1 (2) while (cid:126)A ( (cid:126)A ) also describesparticle 2 (1) having scattered but with 1 (2) is closer. (cid:126)A and (cid:126)A are explicitly depicted in Fig. 3. After applyingthe Hamiltonian to (11) we find that it is an eigenstateprovided, (cid:126)A = S ( z ) (cid:126)A , (cid:126)A = S ( z ) (cid:126)A , (13) (cid:126)A = S ( z ) (cid:126)A , (cid:126)A = S ( z ) (cid:126)A , (14) (cid:126)A = S (cid:126)A , (cid:126)A = S (cid:126)A , (15) (cid:126)A = W ( z − z ) (cid:126)A , (cid:126)A = W ( z − z ) (cid:126)A . (16)The S-matrices S and S which take a particle pastthe impurity, i.e. from incoming to outgoing are S ( z ) = S ( z ) ⊗ , S ( z ) = ⊗ S ( z ) , (17)with S ( z ) the same as in the single particle state (8),the S-matrix S scatters an incoming particle past anoutgoing particle and is S = e iφ e iφ
00 0 0 1 . (18)where φ = − g ) encodes the bulk interaction and W ( z − z ) which scatters an incoming (outgoing) par-ticle past another incoming (outgoing) particle is givenby W ( z ) = sinh ( z )sinh ( z − iφ ) − sinh iφ sinh ( z − iφ ) − sinh iφ sinh ( z − iφ ) sinh ( z )sinh ( z − iφ )
00 0 0 1 . (19)In addition the dot amplitudes are given by B [10] ± = 12 e ( c − z ) / (cid:88) σ = ± (cid:104) A [210 A ] σ ± + A [201 A ] σ ± (cid:105) − e ( c − z ) / (cid:88) σ = ± (cid:104) A [120 B ] ± σ + A [102 B ] ± σ (cid:105) , (20) B [01] ± = 12 e ( c − z ) / (cid:88) σ = ± (cid:104) A [102 A ] σ ± + A [012 A ] σ ± (cid:105) − e ( c − z ) / (cid:88) σ = ± (cid:104) A [201 B ] ± σ + A [021 B ] ± σ (cid:105) . (21)Inserting these expressions for the amplitudes into (11)we get the two particle eigenstate of the side-coupledmodel. Since all amplitudes are generated from (cid:126)A bysuccessive application of the various S-matrices, as de-picted in Fig. 4, there are two ways to obtain each (cid:126)A j both of which must be equivalent for the construc-tion to be consistent. This consistency imposes that theS-matrices satisfy a generalised Yang Baxter equationwhich takes the form of the reflection equation S S S W = W S S S (22)which can be checked to hold by substitution. ~A ~A ~A ~A ~A ~A ~A ~A W S S S S S S W FIG. 4: (Color Online)The amplitudes are related by applyingthe operators as in (13) and depicted here. For consistencywe require the amplitudes obtained by proceeding clockwiseor counter-clockwise are the same resulting in (22).
It is important to note that while no interaction be-tween two incoming (outgoing) particles is present inthe Hamiltonian, W is introduced in order to ob-tain the correct eigenstates and satisfy the generalisedYang-Baxter consistency conditions. To do so we ex-ploit the freedom to introduce discontinuities of the form θ ( ± ( x − x )) into the the part of the wave functionthat describes two right movers or two left movers (or θ ( ± ( x + x )) into the the part of the wave function thatdescribes one left mover and one right mover). The ki-netic term in the Hamiltonian referring to these particlesis of the form ± i ( ∂ x + i∂ x ) (or ± i ( ∂ x − i∂ x )) and van-ishes when acting on these discontinuities. This freedomarises from the linear spectrum that brings about a in-finite degeneracy of the energy levels, the level k + k being degenerate with ( k + q )+( k − q ) for any q . The in-troduction of the discontinuities corresponds then to thecorrect choice of basis states in this degenerate subspacefrom which the perturbation can be turned on, as we areinstructed to do carrying out perturbation theory from adegenerate level. For more detail see [33].We can then go on to impose periodic boundary con-ditions giving e − ik L (cid:126)A = S S W (cid:126)A (23) e − ik L W (cid:126)A = S S (cid:126)A (24)which can be solved to determine z , .The eigenstates for higher particle number are con-structed similarly and the N particle state with energy E = (cid:80) Nj =1 k j = (cid:80) Nj =1 D e z j / + N (cid:15) is, | E (cid:105) = (cid:88) Q (cid:88) (cid:126)σ (cid:90) θ ( x Q ) A Q(cid:126)σ N (cid:89) j e iσ j k j x j ψ † σ j ( x j ) | (cid:105) + (cid:48) (cid:88) P (cid:48) (cid:88) (cid:126)σ (cid:90) θ ( x P ) B P(cid:126)σ (cid:48) (cid:89) j e iσ j k j x j ψ † σ j ( x j ) d † | (cid:105) (25)Here θ ( x Q ) are Heaviside functions which partition con-figuration space into 2 N N ! regions. As before Q are la-belled by the ordering of the N particles as well as ac-cording to which particle is closest to the origin while (cid:126)σ = ( σ , . . . , σ N ) with σ j = ± . In the second line theprimed sums indicate that one particle is removed - be-ing on the dot - and the sums are over the remaining N − N dimensional space S j = S j ( z j ) ⊗ k (cid:54) = j , (26) S ij = e iφ e iφ
00 0 0 1 ij ⊗ k (cid:54) = i,j , (27) W ij = sinh ( z j − z i )sinh ( z j − z i − iφ ) − sinh iφ sinh ( z j − z i − iφ ) − sinh iφ sinh ( z j − z i − iφ ) sinh ( z j − z i )sinh ( z j − z i − iφ )
00 0 0 1 ij ⊗ k (cid:54) = i,j . (28)where the subscripts denote which particle spaces theoperators act upon. In order for this wavefunction to beconsistent it must satisfy the following Yang-Baxter andreflection equations, S k S jk S j W jk = W jk S j S jk S k (29) W jk W jl W kl = W kl W jl W jk (30) W jk S jl S kl = S kl S jl W jk . (31)The first of these being the generalisation to N particlesof (22) while the remaining two come from the consis-tency of the wavefunction away from the dot. These areindeed satisfied by (26),(27) and (35) which is a sufficientcondition for the consistency of the wave function [35].The expressions for B P(cid:126)σ in terms A Q(cid:126)σ can also be foundand are straightforward generalisations of (20) and (21).Therefore we have successfully constructed the N particleeigenstates of the side-coupled model.The spectrum can then be determined by imposingperiodic boundary conditions ψ †± ( − L/
2) = ψ †± ( L/ x = ± L/ k j through e − ik j L A σ ...σ N = ( Z j ) σ (cid:48) ...σ (cid:48) N σ ...σ N A σ (cid:48) ...σ (cid:48) N (32) Z j = W j − j ..W j S j ..S jN S j W jN ..W jj +1 (33)where the matrix Z j takes the j th particle past all othersand past the impurity. By using (22), (30) and (31) onecan show that the Z j commute with each other [ Z j , Z k ] =0 ∀ j, k . They are therefore simultaneously diagonalisableand the spectrum of the side-coupled model is determinedby the eigenvalues of the Z j operators. Before obtainingthese we return to constructing the eigenstates of theembedded model.For the embedded impurity model we note that theunfolding procedure carried out previously allows us toconstruct its eigenstates in the same manner as we didfor the side-coupled model. The N particle eigenstate isof the same form as (25) but owing to the different bulkinteraction in (2) the two particle S-matrices are S ij emb = e iφ e iφ ij ⊗ k (cid:54) = i,j , (34) W ij emb = sinh ( z j − z i )sinh ( z j − z i +2 iφ ) sinh iφ sinh ( z j − z i +2 iφ ) sinh iφ sinh ( z j − z i +2 iφ ) sinh ( z j − z i )sinh ( z j − z i +2 iφ )
00 0 0 1 ij ⊗ k (cid:54) = i,j . (35)and the single particle S-matrices S j the same as (26).The inclusion of the Coulomb term (5) is essential forthis and in its absence the model is not integrable.Imposing the boundary condition ψ ± ( − L/
2) = e iφ ψ †± ( L/
2) we have another eigenvalue problem, e − ik j L A σ ...σ N = (cid:0) Z emb j (cid:1) σ (cid:48) ...σ (cid:48) N σ ...σ N A σ (cid:48) ...σ (cid:48) N (36)where Z emb j is defined similarly to Z j in (32) but using W ij emb and S ij emb and is related to Z j by Z emb j = Z j | φ →− φ . (37)Therefore, the spectrum of the embedded model is ob-tained from the side-coupled model by changing the signof the interaction, φ → − φ .We can replace the bare phase shift φ by the universalLuttinger liquid parameter K using [29] [30] K = (cid:40) φπ side-coupled − φπ embedded (38)meaning that in the thermodynamic limit the two modelsare related by taking K → /K which recovers the dual-ity shown by bosonization [8]. In the subsequent sectionsall calculations will be done for the side-coupled modelthe results of which can then be translated to the em-bedded model by taking K → /K . Note that as φ is aphase shift and restricted to [ − π, π ] we see that the side-coupled system may realize values of K ∈ [0 ,
2] whereasthe embedded system has K ∈ [1 / , ∞ ]. III. DERIVATION OF THE BETHE ANSATZEQUATIONS
Our task now is to determine the eigenvalues of Z j .To this end we note that W ij is actually the R-matrixof the XXZ model, and further that Z j takes the formof the transfer matrix of an inhomogeneous open XXZmodel [36]. The problem of diagonalising this operatorhas recently been achieved by means of the ”Off DiagonalBethe Ansatz” [31]. Inserting these results into (32) andsimplifying the resulting equations using e c (cid:28) e − i D e zα/ L = e iNφ/ i(cid:15) L (cid:20) e z α / − ie c e z α / + ie c (cid:21) × N/ (cid:89) k sinh ( ( z α − λ k − iφ ))sinh ( ( z α − λ k + iφ )) (39) N (cid:89) α sinh ( ( λ j − z α + iφ ))sinh ( ( λ j − z α − iφ )) = − (cid:20) cosh ( ( λ j − c + iφ ))cosh ( ( λ j − c − iφ )) (cid:21) × N/ (cid:89) k sinh ( ( λ j − λ k + 2 iφ ))sinh ( ( λ j − λ k − iφ )) . (40)where the parameters λ j describe the chiral degrees offreedom, z α describe the charge degrees of freedom andthe energy of the system is E = (cid:88) α D e z α / + N (cid:15) . (41)The solution of (39)(40) along with (41) give the exactenergies of the system. IV. GROUND STATE DOT OCCUPATION
Having obtained the Bethe equations governing thesystem we can now construct the ground state. To dothis we first must fill the empty Dirac sea with negativeenergy particles from the cutoff, −D up to some level de-termined by minimisation of the energy (and dependingon (cid:15) , see Fig. 2(b)). After this the thermodynamic limit N, L → ∞ is taken holding the density D = N/L fixedand finally we take the universal limit by removing thecutoff
D → ∞ while holding some other scale, which hasbeen generated by the model, fixed. We will see belowthat this scale is the level width Γ. Once the ground state has been found we will use it to derive exact expressionsfor the occupation of the dot, n d = (cid:10) d † d (cid:11) as a functionof (cid:15) .The form of the possible negative energy states en-tering the ground state depends upon the value of K ,whether it is greater or less than 1 and so the groundstate must be constructed separately in each case. Nev-ertheless we will find a single expression for the dot oc-cupation valid in both regimes. A. K > We begin with φ ∈ [0 , π ] which corresponds to K ∈ [1 , z j = z ∗ N +1 − j = λ j + 2 πi + iφ. (42)with each pair having bare energy − φ/ D e λ j , seeFig. 7.Inserting these expressions into (39) and (40) we obtainequations for the real parts of the pairs, λ j . In the ther-modynamic limit we are not interested in the solutionsper se, but in their distribution, ρ ( λ j ) = 1 L ( λ j − λ j − )on the real line. The distribution has contributions fromthe bulk as well as from an O (1 /L ) term from the dot,allowing us to write it as ρ ( λ ) = ρ b1 ( λ ) + L ρ d1 ( λ ). Thedot occupation is then given as, n d = 2 (cid:90) ρ d1 ( λ ) . (43)The factor of 2 appears here as each λ corresponds to apair of rapidities. These distributions, ρ b1 ( λ ) , L ρ d1 ( λ ) aredetermined by the Bethe equations in their continuousform which for the bulk part is,cos φ/ π D e λ/ = ρ b1 ( λ ) + (cid:90) ∞− B a ( λ − µ ) ρ b1 ( µ ) (44) a ( x ) = i π dd x log sinh ( ( x − niφ ))sinh ( ( x + niφ )) (45)where B = B ( (cid:15) ) is the λ value of the highest filled level.When the dot energy vanishes we have that B (0) = ∞ and bulk distribution is found to be ρ b1 ( λ ) = D e λ/ π cos ( φ/
2) (46)with the bulk part of the ground state energy being E = − (cid:90) ∞−∞ φ/ D e λ/ ρ b1 ( λ ) . (47)To confirm this is indeed the ground state one can in-troduce excitations and check the energy is increased,the simplest type of which consists of adding holes tothe distribution. As is typical for Bethe ansatz mod-els, the energy of a hole turns out to be proportional tothe ground state distribution i.e. a hole at λ = λ h hasenergy ε h ( λ h ) = 4 πρ b1 ( λ h ), increasing the energy. Theother excitations consist of breaking up a pair and plac-ing them above the Fermi surface such they have realrapidity. Each particle then has energy ε p ( z ) = 2 D e z/ in addition to the hole introduced in the ρ ( λ ) distribu-tion.When (cid:15) (cid:54) = 0 the additional term in the energy (see(41)) needs to be balanced by the addition of holes tothe ground state with rapidities starting at − B ( (cid:15) ). Theform of the hole energy, ε h ( λ ) gives us that [30] B ( (cid:15) ) = log (cid:18) α D (cid:15) (cid:19) (48)where α is a constant.Considering now the dot part of the Bethe equations,the dot contribution to the density satisfies, f ( λ − c ) = ρ d1 ( λ ) + (cid:90) ∞− B a ( λ − µ ) ρ d1 ( µ ) , (49)with f n ( x ) = 12 π (cid:90) ∞−∞ e iωx sinh ( π − nφ ) ω sinh 2 πω . (50)The solution is obtained by the Wiener-Hopf method (see[34],[38] or [37] and references therein). Upon integrat-ing over the result as in (43) we find that the exact dotoccupation in the ground state is, n d = − i √ π (cid:90) ∞−∞ dω e − iω (2 log ( (cid:15) ) + a ) sinh (2 πω ) × Γ( + i ( K − ω )Γ(1 + iω )Γ(1 − i (2 − K ) ω ) . (51)where Γ( x ) is the Gamma function, a is a non-universalconstant and we have used (38) to write n d in terms of theLuttinger K . As there is no dependence on the cutoff wecan safely take the universal limit D → ∞ while hold-ing the level width Γ fixed. The width serves as boththe coupling constant and as the strong coupling scaleparamerizing the model, with respect to which all quan-tities are measured. It appears here, rather surprisingly,unrenormalized by the interactions which are present inthe system and independent of the raw cut-off, unlikethe case for a dot placed on the boundary [30]. We willcomment on this further in the next section but for nowwe examine the expression (51). First we can check thatupon inserting K = 1 in the above expression we recoverthe non interacting result n d = 12 − π arctan (cid:16) (cid:15) Γ (cid:17) . (52)For other values we may evaluate (51) by contour integra-tion and obtain an expansion of n d for (cid:15) < Γ or (cid:15) > Γ K = K = K = ϵ Γ n d K = K = K = Γ ϵ n d FIG. 5: (Color Online). The dot occupation at small (left)and large (right) dot energy, (cid:15) / Γ, for different values of
K >
1. The effect of attractive interactions is to suppress the dotoccupation as compared to the non interacting case (dashedline). This effect becomes stronger for increasing K . giving n d = − (cid:104)(cid:80) ∞ n =0 a n (cid:0) (cid:15) Γ (cid:1) n +1 + b n (cid:0) (cid:15) Γ (cid:1) (2 n +1) / ( K − (cid:105)(cid:80) ∞ n =0 c n (cid:16) Γ (cid:15) (cid:17) n +1 for Γ < (cid:15) (53)where a n , b n and c n are constants. Furthermore the ca-pacitance of the dot is χ = ∂n d ∂(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 = 1 π ( K − . (54)We see that at low energy, (cid:15) < Γ the system is stronglycoupled with the dot becoming hybridized with the bulk.At the low energy fixed point ( (cid:15) = 0) the dot is fullyhybridized and has n d = 1 /
2. The leading term in theexpansion about this is (cid:15) / Γ which indicates that theleading irrelevant operator has dimension 2. We iden-tify it as the stress energy tensor [39]. The next orderterm ( (cid:15) / Γ) / ( K − is due to the backscattering which isgenerated at low energies but is irrelevant for K > (cid:15) > Γ, the system becomes weaklycoupled with the fixed point ( (cid:15) → ∞ ) describing a de-coupled empty dot, n d = 0. The expansion about thisfixed point is in terms of integer powers indicating thatthe tunnelling operator d † ψ (0) has dimension 1 /
2. Thefirst few terms of the expansion are plotted in Fig. 5from which we see that the dot occupation is suppressedas a function of (cid:15) for K > B. K < The ground state takes a different form in the re-gion φ ∈ [ − π,
0] which corresponds to K ∈ [0 ,
1] . Itis constructed by taking the chiral parameters λ j ∈ R to be real and the rapidities placed on the 2 πi line i.e.Im( z α ) = 2 π . Inserting these values into the Bethe equa-tions and then passing to the continuous form we obtaina set of coupled integral equations for the distributionsof the charge, ρ − ( z j ) = 1 /L ( z j − z j − ) and chiral vari-ables σ ( λ j ) = 1 /L ( λ j − λ j − ) which we can again splitinto bulk and dot contributions. The bulk contributions K = .55 K = .66K = ϵ Γ n d K = .55 K = .66K = Γ ϵ n d FIG. 6: (Color Online).The dot occupation at small (left) andlarge (right) dot energy for different values of K . The effect ofrepulsive interactions K < K decreases. ρ b − ( z ) and σ b1 ( λ ) are governed by the continuous Betheequations, D e z/ π = ρ b − ( z ) − (cid:90) ∞− B (cid:48) a ( z − y ) σ b1 ( y ) (cid:90) ∞− B (cid:48) a ( λ − y ) ρ b − ( y ) = σ ( λ ) + (cid:90) ∞−∞ a ( λ − y ) σ b1 ( y )(55)where the rapidities are bounded by − B (cid:48) ( (cid:15) ). When thedot energy is set to zero we have that B (cid:48) (0) = ∞ and thebulk ground state distributions are found to be, ρ b − ( z ) = D e z/ π , (56) σ b1 ( λ ) = D e z/ π cos ( φ/ . (57)The fundamental excitations above this ground stateconsist of adding holes to either of these distributions.The energy of these are ε h ( z ) = 4 πρ b − ( z ) and ε h ( λ ) =4 πσ b1 ( λ ) for a charge hole and chiral hole respectively.As in the previous section these are used to determine B (cid:48) which gives the same relation as (48). The dot occu-pation is subsequently obtained by integrating over thedot part of the charge distribution n d = (cid:82) ρ d − ( z )d z whichis determined by, g ( λ − c ) = ρ d − ( λ ) + (cid:90) ∞− B (cid:48) g ( λ − y ) ρ d − ( y ) , (58) g n ( x ) = 12 π (cid:90) ∞−∞ e iωx sinh ( π − φ ) ω φω ) sinh ( nπω ) . (59)The solution is again determined using the Wiener-Hopfmethod with the result that the dot occupation for K < ω = i ( K − n +1) / n d = − (cid:80) ∞ n =0 a n (cid:0) (cid:15) Γ (cid:1) n +1 (cid:80) ∞ n =0 c n (cid:16) Γ (cid:15) (cid:17) n +1 + b n (cid:16) Γ (cid:15) (cid:17) (2 n +1) / (1 − K ) (60)with the capacitance being given by (54). As in the K > φ < φ > πi -2 πi ν -string 1 FIG. 7: (Color Online) At finite temperature the rapidity andchiral variables may form z − λ strings where n λ s and 2 n z sform a set given by (62). On the left we show how a 2-string,4-string and the negative parity 2 ν -string are arranged for φ <
0. On the right we depict the same for φ >
0. Noteonly the z positions are changed when going from left to rightwhich results in a change in sign of the energy from the strings. and weakly coupled at high energy with the same leadingterms in the expansion about these points however theterm generated by the backscattering now appears in theexpansion about the high energy fixed point. This stemsfrom the fact that backscattering is relevant for K < K = 1 case, see Fig. 6.The dot occupation for the embedded system is simplyobtained from (51) by using the mapping K → /K . V. RG FLOW
In the previous section we derived exact expressions forthe dot occupation for the side-coupled model as a func-tion of (cid:15) measured with respect to the strong couplingscale. This strong coupling scale is given by Γ, the levelwidth. It does not depend on K as might be expected foran interacting model and in fact coincides with the freemodel. To understand why the level width is not renor-malised by K we can make use of the mapping to the em-bedded model. The strong coupling scale in the embed-ded model should behave similarly to the single lead case,where a dot is placed at a Luttinger liquid edge [8]. Foran arbitrary Coulomb interaction, U this is D (Γ / D ) /α where α = 1 + 2 [arctan ( g ) − arctan ( U )] /π [30]. Taking U = g , as required by the mapping (see (5)), reducesthis to Γ, the free value. The non-renormalization of thelevel width suggests that the tunnelling operator d † ψ ± (0)should have the same dimension as the free model whichis confirmed by the high energy expansions of the dotoccupation. This is in stark contrast to the the fact thatfermions in a Luttinger liquid (away from the edge) havedimension ( K +1 /K ) /
4. Thus the remarkably simple ex-pression for the strong coupling scale and critical expo-nents present here stand in contrast to a quite substantialmodification of the fermions in the vicinity of the dot.We now have the following picture of the side-coupledsystem. For all K ∈ [0 ,
2] the system flows from weakcoupling at high energy to strong coupling at low en-ergy. The low energy fixed point describes a dot whichis fully hybridized with the bulk and has the fixed pointoccupation n d = 1 /
2. The hybridized dot then acts as abackscattering potential via co-tunnelling. The leadingirrelevant operator which perturbs away from the fixedpoint is the stress energy tensor and results in odd inte-ger powers of (cid:15) / Γ in the dot occupation. For
K > (cid:15) / Γ) / ( K − resulting in a suppression of the dotoccupation at (cid:15) >
0. For
K < n d = 0 for (cid:15) → ∞ or n d = 1 for (cid:15) → −∞ .By reducing the energy scale we flow away from the fixedpoint with the tunnelling operator d † ψ ± (0) which is theleading relevant operator and has dimension 1 / /(cid:15) in n d .Additionally when K < /(cid:15) ) / (1 − K ) to appear resultingin an enhancement of the dot occupation . VI. THERMODYNAMICSA. K = ν − ν In this section we study the finite temperature prop-erties of the dot by calculating the free energy. To doso we use the methods developed by Yang and Yang [40]and later extended by Takahashi [37] based on the stringhypothesis. This states that in the thermodynamic limitthe solutions of the Bethe equations take complex valuesorganised into strings. The form of the strings dependupon the model and the values of the parameters therein.To simplify matters we take φ = ± π/ν with ν an integerso that K = ν ± ν . With this value fixed the hypothesisstates that the Bethe equations allow for the followingforms of the charge and chiral variables.The rapidities can be real or complex with Im( z ) =0 , π . These contribute bare energy ±D e z/ and we de-note the distributions of these ρ ± ( z ). The chiral variablescan take on complex values so that they arrange into n -strings with n < ν such that λ ( n ) l = λ ( n ) + iφ ( n − − l ) , l = 0 , . . . , n − λ on the iπ line which is sometimes called a negativeparity string. The λ n -strings have no bare energy andwe denote the distributions of their real part, called thestring centre σ n ( λ ) with n = ν denoting the negativeparity string. Also possible are z − λ n -strings consisting K = T Γ = .3 T Γ = .5 T Γ = T Γ = ϵ Γ n d K = T Γ = .3 T Γ = .5 T Γ = T Γ = ϵ Γ n d FIG. 8: (Color Online): The finite temperature dot occupa-tion is plotted as a function of (cid:15) / Γ for several values of thetemperature. Above we plot the dot occupation with K = (solid lines) and K = 1 (dashed lines). The repulsive bulkinteractions result in an enhancement of the dot occupationin comparison to the non interacting case. This is effect ismost pronounced for lower temperatures. At higher temper-ature the interacting and non interacting curves coincide ow-ing to the fact that the dot becomes decoupled. Below weplot the same for K = (solid lines) and plot again K = 1(dashed) for comparison. The dot occupation is suppresseddue to the attractive interactions wth the effect becomingmore pronounced for lower T / Γ. of 2 n z s and a λ n -string taking the values z ( n ) l +1 = λ ( n ) + iφ ( n − j ) + iπ + sgn( φ ) iπ (62) z ( n ) l + n +1 = λ ( n ) + iφ ( n − l ) + iπ − sgn( φ ) iπ (63)where j = 0 , . . . , n and l = 1 , . . . , n −
1. These contributebare energy E n = − φ ) cos ( nφ/ D e λ ( n ) / . In addi-tion there is also a negative parity z − λ string λ = λ ( ν ) + iπ, z , = λ ( ν ) ± i ( π − φ ) (64)which has energy 2 sin ( φ/ D e λ ( ν ) / . We denote the dis-tributions of the centres of the z − λ n -strings by ρ n ( z )with n = ν indicating the negative parity string.Severalstring type are depicted in Fig. 7 for both φ > φ < ϵ Γ = .1 ϵ Γ = .7 ϵ Γ = T Γ n d FIG. 9: (Color Online): The dot occupation for fixed (cid:15) o / Γas a function of temperature. The interaction is taken to be K = (dot-dashed lines), K = 1 (dashed lines) and K = (solid lines). We see the enhancement and suppression of thedot occupation for repulsive and attractive interaction withthe effect most pronounced as the temperature is lowered. following the procedure laid out in [37]. The approach iswell known and we just provide the main steps. The freeenergy F = E − T S , where E is the energy of an arbitraryconfiguration of strings and S is its associated Yang-Yang entropy, is minimized with respect to ρ ± , ρ n and σ n whichare solutions of the Bethe Ansatz equations. The result ofthis minimization gives the thermodynamic Bethe ansatz(TBA) equations which determine the minimum of F .Owing to the different string structures for K greaterthan or less than 1 we consider each region separately.We start with φ = − π/ν , corresponding to K = ν − ν <
1, describing repulsive interactions. In this region we findthe dot contribution to the free energy is F d = E d − T (cid:90) f ( x + 2 log (cid:18) T Γ (cid:19) ) log (1 + e ϕ − ( x ) ) − T (cid:90) f ∗ s ( x + 2 log (cid:18) T Γ (cid:19) ) log (1 + e κ ( x ) ) − T (cid:90) s ( x + 2 log (cid:18) T Γ (cid:19) ) log (1 + e κ ν − ( x ) ) (65)where E is the ground state energy due to the dot, s ( x ) = sech( πx/ φ ) / φ and ∗ denotes the convolution f ∗ g = (cid:82) f ( x − y ) g ( y )d y . The thermodynamic functions ϕ ± , ϕ n and κ n are related to the distributions ρ ± , ρ n and σ n respectively and are solutions of the TBA equationswhich in this case are ϕ + = s ∗ log (cid:18) e ϕ e κ (cid:19) , ϕ − = − e x/ + s ∗ log (cid:18) e ϕ e κ (cid:19) (66) ϕ n = s ∗ log (1 + e ϕ n − )(1 + e ϕ n +1 )(1 + e − ϕ ν ) δ n,ν − + δ n, s ∗ log (cid:18) e ϕ + e ϕ − (cid:19) (67) κ n = s ∗ log (1 + e κ n − )(1 + e κ n +1 ) δ n,ν − − δ n, (cid:20) e x/ cos ( φ/ − s ∗ log (cid:18) e ϕ + e ϕ − (cid:19)(cid:21) (68)along with ϕ ν − = s ∗ log (1 + e ϕ ν − ) + ν(cid:15) T = − ϕ ν + ν(cid:15) T and κ ν − = s ∗ log (1 + e κ ν − ) = − κ ν . Just as in thecalculation of the dot occupation in the ground state theabove equations are independent of the cutoff which hasbeen removed while holding Γ fixed. These expressionsgive the exact dot free energy of the system in all temper-ature regimes. Their complicated nature precludes anyanalytic solution for the thermodynamic functions butare straightforwardly determined numerically through it-eration of the integral equations.Before doing this however we can examine them in thelimits of low and high temperature. The functions f ( x )and s ( x ) appearing in the free energy are sharply peakedabout zero meaning that for T → , ∞ the free energyis determined by the solutions of the TBA in the x →∞ , −∞ limits respectively. Setting (cid:15) = 0 and takingfirst the high temperature limit, x → −∞ we see that thedriving terms in the TBA vanish and the thermodynamic functions are constants e ϕ ± ( −∞ ) = 1, e ϕ j ( −∞ ) = e κ j ( −∞ ) = ( j + 1) − e ϕ ν − ( −∞ ) = e κ ν − ( −∞ ) = ν − . (70)Likewise in the opposite low temperature limit x → ∞ we get e ϕ − ( ∞ ) = 0 , e ϕ + ( ∞ ) = 3, e κ j ( ∞ ) = j − , e κ ν − ( ∞ ) = ν − e ϕ j ( ∞ ) = ( j + 2) − , e ϕ ν − ( ∞ ) = ν. (72)The free energy thus becomes linear in T in both the highand low temperature limit.Using these we can check the RG picture we arrived atearlier using the ground state dot occupation still holdstrue at finite temperature. Firstly note that the energyscale, the temperature in this case, is measured with re-spect to Γ which serves as both the strong coupling scaleand the level width for the model. Thus the system is1strongly coupled at low temperature T (cid:28) Γ and weaklycoupled at high temperature T (cid:29) Γ. Furthermore byinserting (71) (69) into (65) we obtain the g -function ofthe model, defined to be the difference in the UV and IRentropy of the impurity g = S UV − S IR = log 2 + 12 log (cid:18) K (cid:19) . (73)This is always positive for the range of values consid-ered in agreement with the requirement that as we movealong the RG flow by lowering the temperature, masslessdegrees of freedom are integrated out. The first termcomes from the charge degrees of freedom and corre-sponds to the entropy of a decoupled dot at high temper-ature which is fully hybridised at low temperature. Thesecond term comes from the chiral degrees of freedomand is the same as for the Kane-Fisher model of a backscattering impurity[41][29]. We see from this that at hightemperature the dot is decoupled and as T is lowered itbecomes hybridised with the dot whereupon it acts asa back scattering impurity. In the non interacting limitthe K → ϕ − )) ≈ exp (cid:0) − e x/ (cid:1) andlog (1 + exp ( κ )) ≈ exp (cid:0) − e x/ / cos ( φ/ (cid:1) for x (cid:29) C v ∼ T Γ (74)which agrees with the expectation that the irrelevant op- erator is the stress energy tensor.By numerically integrating the TBA and using them in(65) we can obtain the finite temperature dot occupationof the system. This is plotted in Fig. 8 for K = as afunction of (cid:15) / Γ at different values of the temperature,
T /
Γ. For the same value of K we plot the dot occupationat fixed (cid:15) / Γ as a function
T /
Γ in the Fig. 9. Comparingto the dashed lines which are the non interacting valueswe see that the dot occupation is enhanced just as it wasat zero T . This enhancement is strongest at low T and iswashed out at high temperature as the system becomesweakly coupled. B. K = ν +1 ν We turn now to the case of φ = π/ν or K = ν +1 ν > K > F d = E d − T (cid:90) f ( x + 2 log (cid:18) T Γ (cid:19) ) log (1 + e − ϕ + ( x ) ) − T (cid:90) f ∗ s ( x + 2 log (cid:18) T Γ (cid:19) ) log (1 + e ϕ ( x ) ) − T (cid:90) s ( x + 2 log (cid:18) T Γ (cid:19) ) log (1 + e κ ν − ( x ) )(75)with the TBA equations being ϕ + = 2 e x/ + s ∗ log (cid:18) e ϕ e κ (cid:19) , ϕ − = s ∗ log (cid:18) e ϕ e κ (cid:19) (76) ϕ n = s ∗ log (1 + e ϕ n − )(1 + e ϕ n +1 )(1 + e − ϕ ν ) δ n,ν − − δ n, (cid:20) s ∗ log (cid:18) e − ϕ + e − ϕ − (cid:19) + e x/ cos ( φ/ (cid:21) (77) κ n = s ∗ log (1 + e κ n − )(1 + e κ n +1 ) δ n,ν − − δ n, s ∗ log (cid:18) e − ϕ + e − ϕ − (cid:19) (78)and ϕ ν − = s ∗ log (1 + e ϕ ν − ) + ν(cid:15) T = − ϕ ν + ν(cid:15) T as wellas κ ν − = s ∗ log (1 + e κ ν − ) = − κ ν . Comparing to the K < e φ − and e − φ + havebeen exchanged and that the exponential driving termnow appears in the ϕ j equations rather than κ j ones.We gain insight to the K > x → −∞ remain unchanged and are given by(69), therefore as T → ∞ the system is the same re-gardless of K . In the low temperature limit however the solutions are different as should be the case giventhe ground state is of a different form. We get that e − ϕ + ( ∞ ) = 0 , e ϕ − ( ∞ ) = 3, e ϕ j ( ∞ ) = j − , e ϕ ν − ( ∞ ) = ν − e κ j ( ∞ ) = ( j + 2) − , e κ ν − ( ∞ ) = ν (80)Using these in the g function we obtain the same form as2before, g = log 2 + 12 log (cid:18) K (cid:19) . (81)Note however that although g >
0, the second term whichis due to the backscattering, is negative for
K > K → κ equation but in the ϕ equation instead and conse-quently we take log (1 + exp ( − ϕ + )) ≈ exp (cid:0) − e x/ (cid:1) and log (1 + exp ( ϕ )) ≈ exp (cid:0) − e x/ / cos ( φ/ (cid:1) for x (cid:29) C v ∼ T Γ + a (cid:18) T Γ (cid:19) α ‘ . (82)Again the leading order term coincides with the stresstensor being the leading irrelevant operator. The termscales as T α where α = 2 for K = ν +1 ν , ν >
2. Itis expected however that α becomes non integer whenincreasing K beyond this as is the case in the groundstate dot occupation.The finite temperature dot occupation can be obtainedby numerically integrating the TBA as in the previoussection and the results are plotted in Fig. 8 and Fig.9. We see that the dot occupation is suppressed as com-pared to K = 1 or K <
1, with the effect being mostpronounced at low temperature. At high T the dot be-comes decoupled and the occupation approaches that ofthe non interacting case. VII. CONCLUSION
In this article we have solved two related models ofquantum dots coupled to Luttinger liquids. The firstconsists of a dot side-coupled to the Luttinger liquidwhile in the second the dot is placed between two other-wise disconnected liquids. The latter also requires that a Coulomb interaction between the occupied dot and theend of the liquids is included and it is tuned to the samevalue as the bulk interaction. The side-coupled modelhowever, requires no such tuning.The solution shows that the two models are related bytaking K → /K which was shown previously throughbosonization [8]. We derived the Bethe equations forboth models and used them to construct the ground stateand derive exact expressions for the dot occupation in allparameter regimes. It was seen that the side-coupledsystem is strongly coupled at low energies so that thedot becomes fully hybridised with the bulk and acts as abackscattering potential. The effect of the backscatter-ing is to either suppress or enhance the dot occupationdepending on the sign of the interactions.The scaling dimensions of the leading relevant and ir-relevant operators about the UV and IR fixed points werefound to coincide with that of the free model. The sur-prising result that the fixed points appear, at least toleading order to be Fermi liquid is in start contrast tothe non-Fermi liquid nature of the bulk system.We then examined the finite temperature properties ofthe dot by deriving the Thermodynamic Bethe equationsand free energy of the system. It was seen that at lowtemperature dot is fully hybridised with the bulk and theinteractions resulting in a suppression or enhancementof the dot occupation. The effect of the interactions iswashed out at high temperature whereupon the dot de-couples.The lack of fine tuned parameters in the side-coupledmodel make it a good candidate for experimental realiza-tions. Such a system may be created placing a quantumdot near a carbon nanotube, the edge of a quantum Hallsample or a topological insulator. The dot occupationcan then be measured by means of a quantum point con-tact and compared to (51). Acknowledgments
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