Low-energy spectrum and finite temperature properties of quantum rings

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1 Eur. Phys. J. B 8, () DOI:.4/epjb/e-5-5 THE EUROPEAN PHYSICAL JOURNAL B Low-enery spectrum and finite temperature properties of quantum rins P. Koskinen, M. Koskinen, and M. Manninen a Department of Physics, University of Jyväskylä, 4 Jyväskylä, Finland Received 8 January Published online Auust c EDP Sciences, Società Italiana di Fisica, Spriner-Verla Abstract. Recently it was demonstrated that the rotational and vibrational spectra of quantum rins containin few electrons can be described quantitatively by an effective spin-hamiltonian combined with riid center-of-mass rotation and internal vibrations of localized electrons. We use this model Hamiltonian to study the quantum rins at finite temperatures and in presence of a nonzero manetic field. Total spin, anular momentum and pair correlation show similar phase diaram which can be understood with help of the rotational spectrum of the rin. PACS. 7..Hb Quantum wires 7..La Quantum dots 7..Pm Fermions in reduced dimensions Introduction Nowadays very sophisticated techniques allows one to construct a lare variety semiconductor heterostructures, which have turned out to be interestin both theoretically and practically and are often referred as future components of nanoelectronics. One class of structures form quantum dots, or artificial atoms, which have characteristic discrete enery spectrum like atoms (for a review see []). The principle of a quantum dot is to confine a bunch of conduction electrons of a semiconductor heterostructure in a small reion of space. Usually the electrons are confined from a two-dimensional (D) as. Similarly, in a quantum rin the electrons are confined to move in a rin-shaped, circularly symmetric, potential. Lorke et al. [] and Fuhrer et al. [] have shown that rins containin only a few electrons can be studied experimentally. A rin can be considered to be cut out from a strictly D electron as. However, in the D plane the rin will have a finite thickness, bein quasi-one-dimensional. Theoretical interest in quantum rins has been concentrated on the possibility of persistent currents [4 ]. Most of the work is based on studyin strictly D systems usin Hubbard-type model Hamiltonians (for a review on D systems see Ref. []). While these strictly D-systems are exactly solvable in some limitin cases and have revealed interestin results for stronly correlated system (e.. Luttiner liquid and chare-spin separation, for a review see Ref. []), it is not easy to draw their relation to a realistic quansi-d rin with lon-rane Coulomb interactions. a Matti.Manninen@phys.jyu.fi The pioneerin work on ab initio electronic structure calculations for quasi-d rins with a few electrons were done by Chakraborty et al. [, 4] usin confiuration interaction (CI) technique. The exact many-body wave function of the CI calculation has a circular symmetry and the internal structure is only seen in correlation functions, which reveal the tendency to antiferromanetic orderin of the electron spins, in areement with the results of the Hubbard models. The formation of an antiferromanetic chain of localized electrons is seen more clearly from the results of the the density functional mean field theory which shows the internal symmetry of the manybody state [5]. In a many-body formalism the internal antiferromanetic order has been observed in studyin the pair-correlation function [6] and the excitation spectrum [7, 8]. Also in the density functional theory the excitation spectrum indicates the existence of antiferromanetic spin density wave [9]. The study of the excitation spectrum has shown that narrow rins can be accurately described by mappin the exact Hamiltonian to a model of localized electrons [8]. Similar idea have been used earlier for noncircular quantum dots by Jefferson and Häusler []. In this paper we will study the quantum mechanics and statistical physics of small quantum rins usin the model Hamiltonian suested earlier [8]. We extend the model for finite manetic fields and find an interestin variation of the spin-state as a function of the manetic field. We show that the parameters of the model Hamiltonian are smooth functions of the thickness and radius of the rin. Consequently, the qualitative properties of the rin are insensitive to the parameters of the rin. Due to the exact solution of the model Hamiltonian the statistical physics of the rins can be solved.

2 484 The European Physical Journal B The oranization of the paper is as follows. In Section we will describe the model Hamiltonian and its relation to the exact Hamiltonian, in Section we present results for zero external manetic field. In Section 4 the rins in the presence of a manetic field are studied and Section 5 concludes the paper. Theoretical model The model potential for a quantum rin can be considered to be the parabolic V (r) = m ω (r R),whereR is the radius of the rin. The parameter ω describes physically the strenth of the confinin radial potential: The larer ω, the more strictly electrons are confined to the distance R and deviations in the radial direction are small. For a iven number of electrons the rin becomes more onedimensional if either ω or R or both are increased. The normal many-body Hamiltonian if the quantum rin is H = i= ] [ m i + V (r i ) + i<j e 4πɛɛ r i r j () The eienstates of () were calculated usin the confiuration-interaction (CI) method in reference [8]. We have extended these calculations for several values of R and ω. In the case of a relative narrow rin we observed earlier [8] that the enery spectrum could be accurately described with a model Hamiltonian consistin of antiferromanetic Heisenber term combined with free rotations of the rin and vibrational states of the localized electrons. The effective Hamiltonian of the system can then be written as H eff = J i,j S i S j + I M + a ω a n a, () where the first term is the normal Heisenber spin- Hamiltonian and the second term describes the rotation of the molecule with moment of inertia I. Note that the rotation axis is fixed. The last term describes the eneries of the vibrational excitations of the system, ω a bein the enery of the vibrational mode a and n a the number of quanta associated with that specific vibrational mode. In a narrow rin the vibrational states are clearly separated from the rotational states like in molecules. However, in the case of very shallow rins (and in quantum dots []) the vibrational levels are close to the rotational levels and the model Hamiltonian becomes less accurate. Note that the Heisenber model is a limitin case of many models of stronly correlated D-systems [,]. Here it can be understood as comin from the tiht-bindin couplin (see e.. Ref. [] between the electron localized in pocket states []). The eienstates of the model Hamiltonian () can be solved as follows. First, solvin the pure Heisenber model for N localized electrons in a rin is straihtforward. The Table. Symmetry properties of all the eienstates of the Heisenber model for a rin of six electrons. L i shows the anular momenta which the state can et in vibrational mode i (i = is the vibrational round state). In case of deenerate states C i shows the trace of the rotation operation. Enery Spin C C C 6 σ d σ v L L L L.8,5,4.8,4,5.5,4,5.8,5,,4,,5,4.,4,,5,,4,5.5,5,4.5,4,,5,,4,5.,5,,4,,5,4.8,4,5.5,5,4.78,5,,4,,5,4.8,5,4.,4,,5,,4,5.5,4,5 basis consists of N Fock states φ i = σ σ σ N (N localized spin- electrons have N possible combinations of their spins, σ i =, ). By direct diaonalization the solved eienstates can be chosen to be eienstates of the Heisenber Hamiltonian, S and S z. Obtainin the anular momenta of these states is not that simple, but one can determine them in the followin way. The eneral rotation operator actin on states is of the form e iθm, where M is the anular momentum operator and θ is the rotation anle (with direction). We define the rotation operator R actin on basis states φ i as R σ σ σ N = σ N σ σ N, so that R is equivalent to a rotation of an anle θ =π/n. Because Ψ j = c i φ i,soe iθm Ψ j = c i R n φ i = c i φ i, when θ = nπ/n. If Ψ j is nondeenerate eienstate of M, the riht hand side becomes e i πn N M Ψ j.inthecase of deenerate states (state Ψ j is not an eienstate of R) one has to inspect the trace of R, which will ive us the possibility to determine the anular momentum with help of the character table of the C Nv roup. Followin the precedin method M ets the values...n, but the spectrum can be obtained for all possible anular momenta. Because e iθm =e θ ) and θ = kπ/n, states with anular momentum M = M +nn =, n = ±, ±... have similar symmetry properties. Table shows explicitly the symmetry properties of a rin for six electrons. In reference [8] it was shown that the exact rotational spectrum calculated with the CI method consists of separated vibrational bands each of which havin spin structure consistent with the Heisenber model. The rotational enery increases simply as M. Fiure shows two examples of the enery spectra of the model Hamiltonian for six electrons toether with the spectrum of the noninteractin electrons in an infinitely narrow rin. (For comparison of the spectra of the model Hamiltonian to those of the true Hamiltonian we refer to our earlier paper [8].) In the lowest panel the separation of the spectrum to rotational and vibrational states is clearly seen. Usin the term iθ(m+ πn

3 P. Koskinen et al.: Low-enery spectrum and finite temperature properties of quantum rins 485,,,,,,,,,,,,,,,, (a) (b) Fi.. Enery spectra of six-electron rins for the anular momenta 6. The numbers above the enery levels indicate the spin. (a) noninteractin electrons, (b) model Hamiltonian with IJ = 4, (c) model Hamiltonian with IJ =.7. The enery scales are in arbitrary units. used in nuclear physics we call the lowest enery state for each anular momentum the yrast state, and the lowest vibrational band as yrast band. In the case of six electrons there are only three vibrational modes with enery relations ω : ω : ω =: 7/ :. The hiher vibrational states can be constructed with the help of the yrast band by roup-theoretical methods. Table shows how these vibrational states are constructed for the case of six electrons. The parameters of the model Hamiltonian can be fitted to the results of the exact Hamiltonian. However, knowin that the model Hamiltonian describes narrow quantum rins well, it is interestin to study its properties more systematically in the parameter space determined by J, I, and ω a at zero and at finite temperatures. Since one of the parameters determines the enery scale and in most cases the vibrational states do not matter (as will be shown below) we have in fact only one parameters J/(/I) =IJ. The accuracy of the model Hamiltonian depends on the one-dimensionality and on the radius of the rin. The (c) Table. Ratio of the enery differences 6 and (defined in text) calculated with exact diaonalization method. I/I is the ratio of the exact moment of inertia to that of a riid rin of classical electrons. r s C f 6/ I/I one-dimensionality can be expressed in terms of a parameter C F [] which is essentially the excitation enery of a radial mode. The rin radius can be related to the onedimensional density parameter r s. The resultin relations are R = Nr s /π and ω = C F π /(mrs). The Heisenber couplin enery of the model Hamiltonian can be fitted to the splittin of the lowest band (vibrational round state) at a iven anular momentum. For example, for six electrons J can be determined from the enery difference of the lowest two S =,M = states, see Fiure. In the Heisenber model this enery difference is.66j as seen from Table. Similarly, the moment of inertia can be fitted by requirin that the enery difference between the lowest M =6andM =statesisn/i. For narrow rins the fitted moment of inertia is very close to the classical estimate NmR. The accuracy of the model Hamiltonian can then be estimated by studyin how well the other enery states are described. The model Hamiltonian predicts that the spectrum for M = is identical to that for M =6. In Table we compare the enery difference of the lowest two S =statesform = 6 (denoted by 6 )tothatfor M =( ). Table also ives the ratio of the moment of inertia determined from the exact spectrum (I) tothe classical estimate (I = NmR ). Note however, that the moment of inertia can also be fitted to the exact results. It should be emphasised that usin only one parameter all the excited states of the yrast band with their correct spin assinment (shown in Fi. b) are reproduced with the model Hamiltonian with the same or better accuracy than the example of the enery ratio shown in Table. In the limit of a strictly D case (ω ) the model Hamiltonian results (numerically) exactly the same spectrum as the true Hamiltonian. Within the rane r s 6 and C F 5 the product IJ can be accurately described (for six electrons) as IJ =[.7 +.4(r sc F )+.4 6 (r sc F ) ]. () In the limit of r s orc F this fit will ive IJ =.64 which is surprisinly close to the similar fit to the noninteractin spectrum, which would ive IJ = 5. The effect of a manetic field B perpendicular to the plane of the quantum rin can be taken into account

4 486 The European Physical Journal B by addin the followin terms in the model Hamiltonian (these can be obtained with the normal minimal substitution) H B = e 8mc B R N + e mc BL z + µ B BS Z. (4) By considerin the manetic field as a flux penetratin the rin φ = πr B and approximatin the moment of inertia by I = NmR, we et in atomic units M (a) H = J S i S j (5) i,j [ + R N M + N ( φ φ ) ( ) ] φ + (M + S z ), φ (6).5... (b) where φ is the flux quantum. In addition to the flux φ we have (by nelectin the vibrations) only two parameters in the Hamiltonian (6), the Heisenber couplin J (or R J) and the effective Lande factor. In a quantum rin the manetic flux can be constructed usin a homoeneous manetic field, in which case the electron spins also couple with the field ( = ), or the field can be restricted inside the rin, in which case the electrons are in a zero manetic field ( = ). Also a partial penetration of the field in the rin reion can be described with nonzero -values. In describin real quantum rins in semiconductors, our parameter depend thus on the effective Lande factor of the material in question and on the relative strenth of the manetic field on the rin perimeter as compared in the center. Results at zero field. Hund s first rule In quantum dots the electronic structure follows the Hund s first rule, that is the spin of the round state of an open shell is at maximum [,4]. The same is true also for quantum rins, althouh in this case the maximum spin can be only S = since each shell consists of only two states correspondin to sinle particle anular momenta m and m. The total spin of the round state of the Heisenber Hamiltonian is S = for all even number of electrons. However, combined with the rotational states the round state of the Heisenber model is not necessarily the round state of the total Hamiltonian. The lowest S = state belons to the M = round state of the riid rotation only if N =(k + ), where k is an inteer. In this case the total round state has S =.On the other hand, in the case N =4k the lowest S = state of the Heisenber Hamiltonian belons to the anular momentum M = N/. Due to the M term in the Hamiltonian the total enery of this state will be pushed hiher than the S = state belonin to M = rotational state. This means that the model Hamiltonian results the. Specific heat k B T [J] k B T [J] 7. (c) 7. Fi.. (a) Enery levels for different anular momentum values (M) in units of J. (b) The pair correlation function at different nearest neihbours (denoted by numbers) as a function of the temperature. (c) The specific heat as a function of the temperature. The rin has six electrons, vibrational states are nelected and IJ =6. Hund s first rule. (Strictly speakin this requires that IJ is small enouh, but, this condition is always valid whenever the model Hamiltonian is a ood approximation to the real system.). Heat capacity At finite temperatures the excited states will be populated accordin to the probabilities determined by the Boltzmann factor. At hih temperatures the total spin (the quantum mechanical quantity bein S ) approaches to the averae determined by the Heisenber Hamiltonian, since all states will be populated with equal probability. The total enery has an expected monotonic behaviour with increasin temperature, but the heat capacity shows a sharp peak at low temperatures due to the discreteness of the enery levels. This is demonstrated in Fiure where we show the positions of the enery levels and the heat capacity. Note that the heat capacity peak appears at a much lower temperature than the enery of the first

5 P. Koskinen et al.: Low-enery spectrum and finite temperature properties of quantum rins 487 Table. Pair correlation function for the round state of the six electron rin. For comparison the results of noninteractin tiht bindin model are shown. neihbour TB TB excited state. This, as well as the sharpness of the peak, is caused by the lare deeneracy of the first excited state. At hiher temperatures the population of the new states increases smoothly and the heat capacity decrease to a constant value. The effect of the vibrational states is only marinal for the for the heat capacity at low temperatures. The reason is that the vibrational states are well separated from the lowest rotational band and a lare number of the rotational levels is already populated before the vibrational states start to et marked population.. Pair correlation function A natural way to analyse the internal structure of a manybody system is to study the pair correlation function. In the model Hamiltonian the correlation at zero temperature (round state) arises solely from the Heisenber model. We define the spin-pair-correlation function as follows: σσ (d) = Ψ j n i,σ n i+d,σ Ψ j, (7) where n i,σ is the number operator for an electron with spin σ at Heisenber chain site i. So (d) describes the probability for the electron at the site i to have an electron with parallel spin at dth nearest-neihbor site (the spin of the electron at the site i can be either or ) and similarly for opposite spins (d). The definition is independent of i by symmetry. Table ives as an example the pair correlation functions for the six-electron rin. For comparison the correlations for noninteractin electrons in tiht bindin (TB) model are shown. Incidentally, the correlations of the TB model aree with the values of the pair-correlation function of free electrons in a narrow rin, calculated at anles correspondin to the localized electrons in the TB model. Naturally, the Heisenber model has a much stroner correlation. In fact the result of the Heisenber model is in ood areement with the pair-correlation calculated from the exact Hamiltonian in the limit of a narrow rin. Nevertheless, the correlation decreases fast with the distance makin the pair correlation function not a clear sinature of the particle localization as discussed in reference [8]. In the infinite one-dimensional antiferromanetic Heisenber model the pair correlation decreases as /d, d bein the distance between the electrons [5]. Fiure shows an example of the temperature dependence of (d) for different d in the case of a six electron rin. Fiure shows that the antiferromanetic confiuration vanishes essentially at temperatures above the heat capacity peak, i.e. at temperatures smaller than the enery of the fist excited state. This orderdisorder-transition is similar to that observed by Borrmann et al. [6]. However, Fiure shows this behaviour more clearly, because the model Hamiltonian allows us to o the the zero-temperature limit which was not possible usin the diffusion Monte Carlo method of reference [6]. Especially, one should note that the stron correlation comin from the round state can only be seen at very low temperatures and requires that the Monte Carlo method is able to describe the round state essentially exactly. In the limit T the pair-correlation functions approaches to for all nearest-neihbors, so all spins are oriented at random. Aain, the vibrational modes exhibit almost no contribution whatsoever to the pair-correlation functions even with small ω. The reason is that when the vibrational states start to be populated all the possible Heisenber states of the lowest rotational band are already nearly evenly populated and the correlation has disappeared. 4 Rins at finite manetic field A finite manetic field chanes the anular momentum at which the minimum enery is reached and, in the case of nonzero, splits the deeneracy of different S z states. Fiure a shows the phase diaram of S z for six electrons at zero temperature as a function of manetic field and. In the limit of a lare field (and nonzero ) the spin is naturally, because the enery is minimized by alinin all the spins parallel to the field. On the other hand in the limit of small field, irrespective of, the spin has the same value as without the manetic field. As discussed above, for even number of particles this is either or and for odd number of electrons it is /. Between these extreme limits the spin exhibits interestin periodic behaviour as a function of manetic flux. This is a result of the periodicity of the yrast spectrum (Fi. ) as a function of M. Wecan clarify this by writin the relevant terms in (6) as r [ N ( M + φ ) ] N + φ S z. φ φ Minimizin the enery with specific φ means that M Nφ/φ (M has only discrete values). By increasin φ by one flux quantum M decreases by N, ivin ultimately the periodicity of φ. Dependin on J the periodicity can be also φ /N or φ /. The periodicity φ /N results if J is so small that all the yrast states for a iven M are almost deenerate and, consequently, the enery increases monotonously as a function of M. The periodicity φ / is seen, when J becomes so lare that the yrast states with L = kn and L = kn + N/ are lower in enery than the neihbourin states, see Fi. ). This behavior is not restricted to rins with six electrons, but is a eneral property of quantum rins as demonstrated in Fiure d,

6 488 The European Physical Journal B. (a). (c) φ/φ 6. φ/φ 6.. (b). (d) φ/φ 6. 4 φ/φ 6. Fi.. Total spin (a) and total anular momentum (b) as a function of the manetic flux and effective Lande factor for a six electron rin (IJ = 6). Panel (c) shows the spin at a temperature of.5j. Panel (d) shows the spin for an eiht electron rin. showin the phase diaram for eiht electrons. These different periodicities have been observed also usin Hubbard models for the quantum rin []. The periodicity is shown also in Fiure b, which represents similar phase diaram for the anular momentum M as a function of manetic flux and Landé -factor. Because the state with S = has only one anular momentum value in the Heisenber model (see Tab. ), Fiure b shows only anular momentum values, 9,... for lare where the couplin to the spin is stron. When decreases also other anular momentum values start to exist, correspondin to different spin-values. Notice the resultin similarity of the phase diarams of S and M in Fiures a and b. Fiures a and d show that the reduction of spin from the maximum value is most easily obtained when the rin is penetrated with an inteer number of flux quanta. In this connection it is interestin to note that Maksym and Chakraborty [6] observed similar reduction of the spin from the maximum value at inteer numbers of flux quanta for a four electron quantum dot usin an exact CI calculation for the Hamiltonian () with R =.Inthecase of four electrons the yrast spectrum of a dot is very similar to that of a rin [] indicatin electron localization. Consequently, the model Hamiltonian ives a simple explanation to this phenomenon observed by Maksym and Chakraborty. The maximum spin belons to a different anular momentum than the low-spin state and the chane of the flux makes the anular momentum to jump with the period N/, until the field is so lare that only states with anular momenta with, 6,, 4 etc. (for 4 electrons) appear as round states. Fiure c shows the same phase diaram as Fiure a, but now at the temperature k B T =.5J. The periodicity vanishes above temperatures of the order of k B T J. At zero temperature the vibrational states have no meanin at all, but the remarkable fact is that even at finite temperatures the vibrational states do not have any effect on the phase diarams, not even even ω as small as J so that there is no ap between different vibrational bands. The pair-correlation functions have similar phase diaram as the spin and orbital anular momentum do. For example, the pair-correlation (d) = for all d, when the spin is three. Decreasin the spin the probability to have opposite spins as nearest neihbors increases with the same steps as in the phase diaram for spins. Let us now consider noninteractin electrons in an infinitely narrow rin. The one-particle states in a onedimensional rin can be solved easily with periodic boundary conditions. One obtains ψ(r) e imφ, and the enery E m m, where m is the anular momentum for the sinle-particle states. By settin N electrons in these states accordin to the Pauli principle, we obtain for the many-particle states M = i m i, E = i ɛ i and S z = i s z,i. The many-body spectrum of noninteractin electrons is shown in Fiure. The yrast states increase rouhly as M. The noninteractin spectrum differs in many ways from the spectra of the model (or exact) Hamiltonian shown in Fiure. For example, it has no different (vibrational) bands but rather a continuum of states above the yrast states. Nevertheless, the noninteractin spectrum has a couple of important connection points with the exact spectrum. As discussed above, the round state for N = 4k + has M = and S =, while for N =4kthe M = state is deenerate with the M = N/ state, the former havin spin or while the latter has spin. Moreover, the lowest S = N/ state corresponds to M = N/ in areement with the results of the model (or exact) Hamiltonian. Consequently, the dependence of the round state spin as function of the manetic field and the parameter is even quantitatively

7 P. Koskinen et al.: Low-enery spectrum and finite temperature properties of quantum rins 489 similar to that of the model Hamiltonian with a suitably chosen couplin constant IJ. In the limit of the extremely narrow rin, the model Hamiltonian describes not only the lowest rotational band exactly, but also all the vibrational states exactly (without any further parameters) [8]. It seems evident that in this stronly correlated limit the model Hamiltonian becomes an exact description of the true many-body Hamiltonian of interactin electrons. On the other hand many of the results of the model/true Hamiltonian are qualitatively similar to those obtained with the Hubbard-type models with contact interaction between the electrons [9 ]. Further work is needed to find relations between these models. 5 Conclusions We have studied the manetization and thermodynamic properties of quasi-one-dimensional quantum rins usin a model Hamiltonian previously shown to describe excellently the exact many-body spectra. The properties of the rins are overned by the symmetry of the quantum mechanical system, leadin to periodic properties as a function of the manetic flux. The spin of the round state has fluctuations obtained earlier usin exact diaonalization techniques. Similarly the observed maic anular momentum states which are seen as minima in the yrast line are a natural consequence of the model Hamiltonian. The temperature dependence of the pair correlation shows that the correlation disappears at low temperatures, correspondin an enery much lower than the lowest electronic excitation of the rin. A comparison to noninteractin electrons in a narrow rin shows a fundamental similarity which arises of the rotational symmetry and one-dimensionality of the system. The stronly correlated localized electrons result similar phase diaram (spin versus manetic field) to that of noninteractin electrons. We would like to thank Stephanie Reimann, Susanne Viefers and Prosenjit Sinha Deo for several useful discussions. This work has been supported by the Academy of Finland under the Finnish Centre of Excellence Proramme -5 (Project No , Nuclear and Condensed Matter Proramme at JYFL). References. T. Chakraborty, Quantum dots: A syrvey of the properties of artificial atoms (North-Holland, Amsterdam 999). A. Lorke, R.J. Lyuken, A.O. Govorov, J.P. Kotthaus, J.M. Garcia, P.M. Petroff, Phys. Rev. Lett. 84, (). A.Fuhrer,S.Lüscher, T. Ihn, T. Heinzel, K. Ensslin, W. Weschelder, M. Bichler, Nature 4, 8 () 4. F. Hund, Ann. Phys. (Leipzi), (98) 5. M. Büttiker, Y. Imry, R. Landauer, Phys. Lett. A 96, 65 (98) 6. L.P. Lévy, G. Dolan, J. Dunsmuir, H. Bouchiat, Phys. Rev. Lett. 64, 74 (99) 7. P. Sandström, I.V. Krive, Ann. Phys. 57, 8 (997) 8. S. Viefers, P.S. Deo, S.M. Reimann, M. Manninen, M. Koskinen, Phys. Rev. B 6, 668 () 9. F.V. Kusmartsev, J.F. Weisz, R. Kishore, M. Takahashi, Phys. Rev. B 49, 64 (994). F.V. Kusmartsev, Phys. Rev. B 5, 4445 (995). E.B. Kolomeisky, J.P. Straley, Rev. Mod. Phys. 68, 75 (996). H.J. Schulz, in Proceedins of Les Houches Summer School LXI, edited by E. Akkermans, G. Montambaux, J. Pichard, J. Zinn-Justin (Elsevier, Amsterdam, 995), p. 5. T. Chakraborty, P. Pietiläinen, Phys. Rev. B 5, 9 (995) 4. K. Niemelä, P. Pietiläinen, P. Hyvönen, T. Chakraborty, Europhys. Lett. 6, 5 (996) 5. S.M. Reimann, M. Koskinen, M. Manninen, Phys. Rev. B 59, 6 (999) 6. P. Borrmann, J. Hartin, Phys. Rev. Lett. 86, 8 () 7. L. Wendler, V.M. Fomin, A.V. Chaplik, A.O. Govorov, Phys. Rev. A 54, 4794 (996) 8. M. Koskinen, M. Manninen, B. Mottelson, S.M. Reimann, Phys. Rev. B 6, 5 () 9. A. Emperator, M. Barranco, E. Lipparini, M. Pi, Ll. Serra, Phys. Rev. B 59, 5 (999). J.H. Jefferson, W. Häusler, Phys. Rev. B 54, 496 (996). M.P. Marder, Condensed Matter Physics (Wiley, New York ). M. Manninen, M. Koskinen, S.M. Reimann, B. Mottelson, Eur. J. Phys. D 6, 8 (). S. Tarucha, D.G. Austin, T. Honda, R.J. van der Haae, L. Kouwenhoven, Phys. Rev. Lett. 77, 6 (996) 4. M. Koskinen, M. Manninen, S.M. Reimann, Phys. Rev. Lett. 79, 89 (997) 5. D. Vollhardt, in Perspectives in Many-Body Physics, edited by R.A. Brolia, J.R. Schrieffer, P.F. Bortinon (North- Holland, Amsterdam 994) 6. P.A. Maksym, T. Chakraborty, Phys. Rev. B 45, 947 (99)