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Kondo effect in charm/bottom nuclei Shigehiro Yasui ∗ Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan WediscusstheKondoeffectforisospin-exchangeinteractionbetweenaD¯,Bmesonandavalence nucleonincharm/bottomatomicnucleiincludingthediscreteenergy-levelsforvalencenucleons. To investigate the binding energy by the Kondo effect, we introduce the mean-field approach for the boundstateoftheD¯,B mesonincharm/bottomnuclei. Assumingasimplemodel,weexaminethe validityofthemean-fieldapproximationbycomparingtheresultswiththeexactsolutions. Wealso estimate the effect of the quantum fluctuation beyond the mean-field approximation. We discuss thecompetitionbetweentheKondoeffectandtheothercorrelations invalencenucleon,theisospin symmetry breakingand thenucleon pairings. 6 1 PACSnumbers: 12.39.Hg,14.40.Lb,14.40.Nd,21.85.+d 0 2 I. INTRODUCTION becomesenhancedduetothenon-cancellationofloopef- n fects from multiple number of particle-hole pairs around a J In recent hadron and nuclear physics, the flavors of the Fermi surface, and the system becomes a strongly 1 nuclear systems are extended to multi-flavor direction; coupledoneregardlesstothesmallcouplingbetweenthe 3 strangeness flavor for hypernuclei and K¯-mesic nuclei. valence fermion and the impurity particle. As results, the impurity particle with spin-exchange interaction can Inthesedays,nuclearsystemswithcharm/bottomflavor ] changethe transportproperties of the condensedmatter h (charm/bottomnuclei)arealsoinvestigatedinmanythe- systems. This is called the Kondo effect. The conditions p oreticalstudies. Oneofthemostinterestingpropertiesin - charm/bottomnucleiisthatthemassesofcharm/bottom for emergence of the Kondo effect are: (i) heavy impu- p rity, (ii) existence ofFermi surface (or degenerate state), hadronsaremuchheavierthanthenucleonmass. Forex- e h ample,themassofthelightestcharmmeson,aD¯ meson, (iii) loop contribution as quantum effect and (iv) non- [ is1870MeV,whichisabouttwiceaslargeasthenucleon Abelianinteraction,suchasthespin-dependentforce[7]. As far as those conditions are satisfied, the Kondo ef- mass. Themassofthelightestbottommeson,aBmeson, 1 fect can emerge in any quantum systems for any kind of is 5280 MeV, which is even about 5.6 times as large as v constituent particle and energy scale. 7 the nucleon mass. To investigate the behavior of such a Recently, the Kondoeffect is discussedfor the isospin- 2 heavyparticleinnuclearsystemisaninterestingproblem 2 as the impurity physics. This is important, not only for exchange interaction with SU(2) isospin symmetry be- 0 understandingthehadrondynamics(hadroninteraction, tween a charm/bottom hadron and a nucleon in nu- 0 clear matter, and for the color-exchange interaction change of hadron in medium) and the nuclear structure, 2. but also for unveiling the fundamental properties of the with SU(3) color symmetry between a light (up, down, 0 strong interaction, such as the spontaneous breaking of strange)quarkandaheavy(charm,bottom)quark[8,9]. 6 The Kondo effect in strong magnetic fields is also dis- chiral symmetry and the color confinement in vacuum, 1 cussed [10]. Those can be studied in experimental stud- in Quantum Chromodynamics (QCD). In fact, it is dis- : v cussedthatthe“heavy-quarkspinsymmetry”asthefun- ies in high energy accelerator facilities. To research the i damental symmetry in QCD is essentially important in Kondoeffectincharm/bottomatomicnuclei,itisimpor- X tantto considerfinite-volume effects anddiscrete energy the heavy hadron interactionand the mass spectroscopy r levels of valence nucleons. Here we mean that a valence a of heavy hadrons [1–4]. nucleon is the nucleon which is an active degree of free- Existenceofimpurities canaffectthe propertiesofthe dom in a model space in shell-structure in atomic nu- matter systems. As the famous and important impurity clei. Inthepresentwork,focusingonthediscreteenergy- physics in condensed matter systems, the Kondo effect levels of valence nucleons, we study the Kondo effect in has been investigated for a long time [5–7]. We consider charm/bottom nuclei. that the valence fermion (the quasi-particle forming the Asmentioned,thenon-Abelianinteractionisoneofthe Fermi surface in medium; electrons in metal) and the essentialconditionsforthe Kondoeffect. Ingeneral,itis impurity particle (atom with finite spin) has the spin- dependent force (~s S~ type interaction with spin oper- known that there are severalkinds of non-Abelian inter- · actionin(charm/bottom)nuclei;(i)interactionchanging ators ~s and S~ for the valence fermion and the impu- totalangularmomentum(sumofspinandorbitalangular rity particle, respectively) with SU(2) spin symmetry. momentum) of valence nucleon, (ii) interactionchanging Then, the effective interaction in low-energy scattering heavy-quark spin, and (iii) interaction changing isospin of heavy hadron and valence nucleon. (i) The first induces the Kondo effect in deformed nu- cleiwhoseshapeisdifferentfromthesphericalone. This ∗ [email protected] may be a phenomena irrelevant to heavy impurity. In- 2 stead, the coupling of nucleon spin to quantum rota- [39, 40]. This has been applied also to the quantum dot tion of the deformed nucleus is important. In Ref. [43], systems with the Kondo effect [41]. The mean-field ap- Sugawara-Tanabe and Tanabe argued that the Coriolis proach provides us with a useful method for theoreti- force in deformed nucleus plays the interesting role of cal analysis and gives an intuitive understanding of the non-Abelian interaction. In this case, the Coriolis force Kondo effect. We recall that the idea of the mean-field compels the spins of valence nucleons aligned along the approximation, or the Hartree-Fock approximation, has spin of the deformed nuclei in the same direction (anti- beenknowntobe veryusefulinnuclearphysics[42]. We pairing force), and hence the Kondo effect reduces the expectthatthemean-fieldapproachfortheKondoeffect strengthofthe effective couplinginthelow-energylimit. in charm/bottom nuclei will give us a straightforward (ii) The second is the interaction in the heavy-quark extension towardthe impurity physics in nuclear theory. effective theory based on QCD, which is given by 1/m The paper is organized as the followings. In Sec- Q expansion for the heavy quark mass m [1, 2]. It is tion2,weintroducetheeffectiveinteractionforexchang- Q known that the heavy-quarkspin is the conservedquan- ingisospinbetweenaD¯,B mesonandavalencenucleon. tity in the heavy-quark limit (m ), regardless to Oneofthe purposesinthis paperisto studythe validity Q → ∞ the non-perturbativeinteractionto lightquarksandglu- of the mean-field approximationfor many-body problem ons. Incharm/bottomnucleiwithaboundΛ ,Λ baryon in the Kondo effect, when the valence nucleons occupy c b (isospin 1/2, spin-parity 1/2+) [44, 45], the heavy-quark thediscreteenergy-levels. InSection3,weinvestigatethe spiniscarriedbythe Λ ,Λ baryon,whenonlythelead- approximate solution for the Kondo effect in the mean- c b ing order in the 1/m expansion is considered. In the field approximation. We introduce the auxiliary fermion Q heavy-quarklimit,theheavy-quarkspin,namelythespin fieldforisospinofD¯, B meson,andapplythe mean-field of Λ , Λ baryon, cannot flip, and hence cannot induce approximation in the extended Fock space. We consider c b the Kondo effect in charm/bottom nuclei. the quantum fluctuation by the random-phase approxi- (iii) The third gives the Kondo effect by isospin ex- mation (RPA) and show that the approximate solution change between a D¯, B meson (isospin 1/2, spin-parity becomesclosertothe exactone. InSection4,we discuss 0 ) and a valence nucleon. The isospin exchange is still the competition between the Kondo effect and the cor- − available in the heavy-quark limit, because the isospin- relations of valence nucleons. We investigate the isospin degrees of freedom remain for the D¯, B meson in this breaking of the valence nucleons, and the nucleon pair- limit. It has been discussed by many theoretical studies ings. The final section is devoted for conclusion. that the D¯, B meson can be bound in nuclear matter1; the quark-meson coupling model [11, 12], the QCD sum rules [13–16], the hadron-interaction model [17–28], and II. HAMILTONIAN FOR KONDO EFFECT the quark-interaction model [29]. Some of them suggest WITH DISCRETE ENERGY-LEVELS thataD¯,B mesonisboundbyattractivepotentialinnu- clear matter. It is interesting that the pion-exchange in- A. Model setup teractionbetweenaD¯,B mesonandanucleoncanbeat- tractiveenoughtoformsomebound/resonantstates[30– We consider a D¯, B meson (isospin 1/2) as a heavy 34]2. Inthepresentstudy,weinvestigatetheD¯,B meson impurity particle in atomic nuclei. In order to treat the astheimpurityfortheKondoeffectincharm/bottomnu- isospin-exchange interaction between a D¯, B meson and clei. WenotethataΛc,Λb hasnoisospinandhencedoes a valence nucleon in a simple form as far as possible, we not induce Kondoeffect by isospin-exchangeinteraction. consider the Hamiltonian Theoretically, to obtain correctly the ground state H =H +H , (1) of the system with the Kondo effect is a highly non- 0 K perturbative problem, because the system becomes a where H is the kinetic term for the valence nucleon 0 strongly-coupledonedue totheenhancementofthecou- H = ǫ c c , (2) pling strength in the low-energy scattering. Several the- 0 k kσ† kσ oreticalapproacheshavebeendevelopedforthisproblem andH istheisospin-excXhange(Kondo)interactionterm K [6, 7]. One of the most effective methods is the numeri- calrenormalizationgroupinitiatedbyWilson[38]. Inthe HK =g ck′ †ck T++ck′ †ck T present analysis, we will adopt the mean-field approach ↓ ↑ ↑ ↓ − X(cid:0)+(ck′ †ck ck′ †ck )T3 , (3) ↑ ↑− ↓ ↓ with the coupling constant g. Here ckσ (ckσ†(cid:1)) is the annihilation (creation) operator for the valence nucleon in 1 The dynamics of a D¯, B meson (qQ¯; Q= c, b) is simpler than that of the antiparticle, a D, B¯ meson (q¯Q). Because the for- the kth single-state, where k = 1,...,N indicates the mer does not have qq¯annihilation in nuclear matter, while the single-state of the valence nucleon3, and σ = , is the ↑ ↓ latterhas. Thedifferenceoftheirbehaviorsisduetothecharge symmetrybreakingatfinitebaryonnumberdensity. 2 Thebound/resonant systems composed of aD¯, B mesonand a nucleonwereoriginallyinvestigatedbythebound-stateapproach 3 Forexample,N isgivenbyN =2j+1forj-shellinnuclearshell intheSkyrmemodelaspentaquarkstates [35–37]. model. 3 B. Exact energy eigenvalues TABLEI.Energy eigenvaluesoftheHamiltonian (1) forn= 1. Thenumbersinparenthesesarethenumbersofdegeneracy 1. Variational method for wave function factor. numberof valence nucleon n=1 For simplicity, we consider the single-particle states N I =0 I =1 with energy ǫk = ǫ for the valence nucleons. We use any N ǫ−3Ng (1), ǫ (N−1) ǫ+1Ng (1), ǫ (N−1) the representations imp and imp for the impurity 2 2 states with isospin |↑aind , res|p↓eictively. We also use ↑ ↓ the representation ψ(n) for the total state, composed | I,I3i ofanimpurity andvalencenucleons,withisospinI,its z component I and the number of valence nucleon n. We TABLEII.EnergyeigenvaluesoftheHamiltonian(1)forn= 3 consider as example the n = 1, 2, 3 cases in the follow- 2. The notations are the same as Table I. ings. numberof valence nucleon n=2 a. The n=1 case. We consider isospin I =0, 1. N I =1/2 I =3/2 For I =0, we assume the wave function 1 2ǫ (1) — ψ(1) = Γ c c , (7) 2 2ǫ−32Ng (1), 2ǫ (2), 2ǫ+21Ng (1) 2ǫ+21Ng (1) | 0,0i k k↑†|↓iimp− k↓†|↑iimp 3 2ǫ−32Ng (2), 2ǫ (5), 2ǫ+21Ng (2) 2ǫ+21Ng (2), 2ǫ (1) with unknown cXoefficie(cid:0)nts Γk = Γ1,...,ΓN(cid:1). By us- 4 2ǫ−32Ng (3), 2ǫ (10), 2ǫ+21Ng (3) 2ǫ+21Ng (3), 2ǫ (3) ing H|ψ0(1,0)i=E|ψ0(1,0)i, we o{bta}in { } 3 ǫΓ g Γ =EΓ , (8) k n k − 2 X up,downcomponentoftheisospin. WedefineT andT3 and hence the energy eigenvalues astheraising/loweringoperatorsandthez comp±onentof 3 the SU(2) isospin operator, and E =ǫ Ng (n.d.f.=1), ǫ (n.d.f. =N 1). (9) − 2 − 1 Thenumberinthe parenthesesarethe numberofdegen- T1 = (T++T ), (4) eracy factor (n.d.f.). 2 − For I = 1, considering I = 1, we assume the wave 1 3 T2 = (T+ T ). (5) function 2i − − (1) T , T and T satisfy the commutation relation of the |ψ1,1i= Γkck↑†|↑iimp, (10) 1 2 3 X SU(2) algebra with the unknowncoefficients Γ . By using H ψ(1) = { k} | 1,1i E ψ(1) , we obtain the relation | 1,1i [T ,T ]=iǫ T , (6) a b abc c 1 ǫΓ + g Γ =EΓ , (11) k n k 2 with a,b,c=1,2,3. X and hence the energy eigenvalues We note that, in the Hamiltonian(1), there is a quantum fluctuation of the impurity isospin, because the di- 1 E =ǫ+ Ng (n.d.f.=1), ǫ (n.d.f. =N 1). (12) rection of the impurity isospin is not fixed. Therefore, 2 − the dynamics of the valence nucleon is always affected We obtain the same values for I =0, 1. by the isospin fluctuation of the impurity, and hence it 3 − See Table I for summary. cannot be reduced to the one-body problem. This is one of the interesting properties of the Kondo effect. The b. The n=2 case We consider I =1/2, 3/2. purpose for us is to obtain the energy eigenvalues of the ForI =1/2,consideringI =1/2,weassumethewave 3 Hamiltonian (1) by considering the isospin fluctuation. function 2 1 ψ(2) = Γ0 [c c ] +Γ1 [c c ] [c c ] , (13) | 1/2,1/2i ( mn m†⊗ n† 00|↑iimp mn r3 m†⊗ n† 11|↓iimp− √3 m†⊗ n† 10|↑iimp!) X with unknown coefficients Γ0 , Γ1 having the properties Γ0 =Γ0 , Γ1 = Γ1 . Here we define mn mn mn nm mn − nm 1 [c c ] = c c c c , (14) m† n† 00 m † n † m † n † ⊗ √2 ↑ ↓ − ↓ ↑ (cid:0) (cid:1) 4 and with unknown coefficients Γ1 having the property mn Γ1 = Γ1 . By using H ψ(2) = E ψ(2) , we cm †cn †(I3=1) mn − nm | 3/2,3/2i | 3/2,3/2i ↑ ↑ obtain, through the relation [c c ] = 1 c c +c c (I =0) (15) m†⊗ n† 1I3 √2 m↑† n↓† m↓† n↑† 3 c c (I = 1) 1 m↓(cid:0)† n↓† 3 − (cid:1) 2ǫΓ1mn− 2g Γ1lm−Γ1ln =EΓ1mn, (18) forpairsofvalencenucleonswithisosingletandisotriplet, 1≤Xl≤N(cid:0) (cid:1) respectively, for short notation. By using H ψ(2) = | 1/2,1/2i with m<n, the energy eigenvalues E ψ(2) , we obtain the energy eigenvalues | 1/2,1/2i 1 3 E =ǫ n.d.f. = (N 2)(N 1) , E =ǫ Ng (n.d.f.=N 1), ǫ (n.d.f.=N2 2N+2), 2 − − − 2 − − (cid:18) (cid:19) 1 1 ǫ+ Ng (n.d.f.=N 1). (19) ǫ+ Ng (n.d.f.=N 1). (16) 2 − 2 − We obtain the same values for I3 =−1/2. We obtain the same values for I3 =1/2, 0, −1/2, −3/2. ForI =3/2,consideringI =3/2,weassumethewave See Table II for summary. 3 function c. The n=3 case We consider I =0, 1, 2. ψ(2) = Γ1 [c c ] , (17) For I =0, we assume the wave function | 3/2,3/2i mn m†⊗ n† 11|↑iimp X 1 1 ψ(3) = Γ c c c c + Γ c c c c | 0,0i lm l↑† l↓†√2 m↑†|↓iimp− m↓†|↑iimp ′lm m↑† m↓†√2 l↑†|↓iimp− l↓†|↑iimp 0 l<m N 0 l<m N ≤X≤ (cid:0) (cid:1) ≤X≤ (cid:0) (cid:1) 1 + Γ00 c c c c c c lmn2 l↑† m↓†− l↓† m↑† n↑†|↓iimp− n↓†|↑iimp 0 l<m<n N ≤ X ≤ (cid:0) (cid:1)(cid:0) (cid:1) 1 1 + Γ1lm1n√3 cl↑†cm↑†cn↓†|↓iimp−2 cl↑†cm↓†+cl↓†cm↑† cn↑†|↓iimp+cn↓†|↑iimp +cl↓†cm↓†cn↑†|↑iimp ,(20) 0 l<m<n N (cid:26) (cid:27) ≤ X ≤ (cid:0) (cid:1)(cid:0) (cid:1) withunknowncoefficientsΓ ,Γ ,Γ00 andΓ11 with For I = 1, considering I = 1, we assume the wave lm ′lm lmn lmn 3 l < m < n. By using H ψ(3) = E ψ(3) , we obtain the function | 0,0i | 0,0i energy eigenvalues as shown in Table III. (3) |ψ1,1i= Γlmcl↑†cl↓†cm↑†|↑iimp+ Γ′lmcm↑†cm↓†cl↑†|↑iimp 0 l<m N 0 l<m N ≤X≤ ≤X≤ 1 1 + Γ01 c c c c c + Γ10 c c c c lmn√2 l↑† m↓†− l↓† m↑† n↑†|↑iimp lmn l↑† m↑†√2 n↑†|↓iimp− n↓†|↑iimp 0 l<m<n N 0 l<m<n N ≤ X ≤ (cid:0) (cid:1) ≤ X ≤ (cid:0) (cid:1) 1 1 1 1 + Γ11 c c c +c c c +c c c , (21) lmn √2 l↑† m↑†√2 n↑†|↓iimp n↓†|↑iimp −√2√2 l↑† m↓† l↓† m↑† n↑†|↑iimp 0 l<m<n N (cid:26) (cid:27) ≤ X ≤ (cid:0) (cid:1) (cid:0) (cid:1) withunknowncoefficientsΓ ,Γ ,Γ10 andΓ11 with with unknown coefficients Γ11 with l < m < n. By lm ′lm lmn lmn lmn l < m < n. By using H ψ(3) = E ψ(3) , we obtain the using H ψ(3) = E ψ(3) , we obtain the energy eigenval- | 1,1i | 1,1i | 2,2i | 2,2i energy eigenvalues as shown in Table III. We obtain the ues as shownin Table III.We obtainthe same values for same values for I =0, 1. I =1, 0, 1, 2. 3 3 − − − For I = 2, considering I = 2, we assume the wave 3 function ψ(3) = Γ11 c c c , (22) | 2,2i lmn l↑† m↑† n↑†|↑iimp 0 l<m<n N ≤ X ≤ 5 TABLE III.Energy eigenvalues of theHamiltonian (1) for n=3. The notations are thesame as Table I. numberof valence nucleon n=3 N I =0 I =1 I =2 1 — — — 2 3ǫ−3Ng (1), 3ǫ (1) 3ǫ (1), 3ǫ+1Ng (1) — 2 2 3 3ǫ−3Ng (3), 3ǫ (4), 3ǫ+1Ng (1) 3ǫ−3Ng (1), 3ǫ (3), 3ǫ+1Ng (4) 3ǫ+ 1Ng (1) 2 2 2 2 2 4 3ǫ−3Ng (6), 3ǫ (11), 3ǫ+1Ng (3) 3ǫ−3Ng (3), 3ǫ (12), 3ǫ+1Ng (9) 3ǫ (1), 3ǫ+1Ng (3) 2 2 2 2 2 2. Method by pseudo-isospin SU(2) algebra energy (ǫk =ǫ)4. By using the identity For the single-particle states with ǫk = ǫ, we obtain TcT +TcT +2TcT =2T~c T~, (34) the energy eigenvalues specially by the simple method. + − − + 3 3 · For this purpose, we define the operator wefindthatthe interactionterm(3)canbeexpressedas H =2NgT~c T~. (35) K N · 1 CNσ = ckσ, (23) According to the compound isospin T~c+T~ =0, 1 (i.e. √N | | Xk=1 T~c T~ = 3/4,1/4, respectively), we obtain · − asacoherentsumofckσ. This satisfiesthe commutation 3Ng (T~c+T~ =0) relation for fermions HK =( −12Ng (T|~c+T~ |=1) . (36) 2 | | The original Hamiltonian (1) can be given as CNσ,CNσ′† =δσσ′. (24) n o H = ǫCkσ†Ckσ +2NgT~c T~, (37) · Defining the raising/lowering operators and the z com- X by the fermion operator C and the N 1 orthogonal ponent Nσ − fermion operators C (k = 1, ..., N 1). C with kσ kσ − k = 1,...,N 1 are linear combinations of c with T+c =CN↑†CN↓, (25) k = 1,...,N,−all of which are commutative witkhσCNσ, Tc =CN †CN , (26) and satisfy {Ckσ†,Ck′σ′} = δkk′δσσ′ (k,k′ = 1,...,N − − ↓ ↑ 1). From Eq. (37), we can indeed confirm the results in 1 T3c = 2 CN↑†CN↑−CN↓†CN↓ , (27) Tables I, II, III. (cid:16) (cid:17) and x, y components III. MEAN-FIELD APPROXIMATION IN KONDO EFFECT 1 Tc = Tc +Tc , (28) 1 2 + − In the previous section, we obtained the energy eigen- 1(cid:0) (cid:1) values by considering the simple case with ǫ =ǫ for the Tc = Tc Tc , (29) k 2 2i +− − Hamiltonian (1). In general cases, however, we need to (cid:0) (cid:1) perform diagonalization of large matrices with paying a we introduce the operator costtothenumericalcomputation. Moreover,suchdirect T~c =(Tc,Tc,Tc). (30) 1 2 3 4 WemaynotetheSU(2)algebraholdsformoregeneralcase, Those operators satisfy the SU(2) algebra; [Tc,Tc] = iǫabcTccwitha,b,c=1,2,3. HencewecallT~ctheapsebudo- T+c= N1 Xck′↑†Sk′kck↓, (31) iospoesrpainto.rWfoerdeiastcihngvuailsehnctehinsufrcolemont,hebeccoanuvseentT~iocn(aolrisCospin) T−c= N1 Xck′↓†Sk′kck↑, (32) Nσ 1 gives the coherent state of k = 1,...,N single-particle T3c= 2N X(cid:16)ck′↑†Sk′kck↑−ck′↓†Sk′kck↓(cid:17), (33) states. We emphasize that the pseudo-isospincanbe de- fined only when the single-particle states have the same withsymmetricSk′k (Sk′k =Skk′). 6 analysismay notbe useful for intuitive understandingof the Fock space for the boson fields has to be extended theresult. Inthissection,introducingthemean-fieldap- to infinite number of bosons in contrast to the fermion proachbasedonRef.[39–41],wediscusshowthisapprox- case,where fermion numbers are limited to two at most. imationbringsaneasywaytoobtainthe groundstateof Therefore, we consider that the fermion fields are more the Hamiltonian (1), and investigate the validity of the convenient than the boson fields in the present analysis. mean-field approach by comparing the results with the exact ones. We also consider the quantum fluctuation in RPA beyond the mean-field approximation. B. Isosinglet condensate (g>0) We note that the mean-field approach was applied to the cases with continuous number-density of valence Weconsidertheg >0case. Firstwediscussthemean- fermions in infinitely large system in condensed matter field approximation, and second we investigate the fluc- physics[39–41]. AsemphasizedinIntroduction,the pur- tuation by using RPA. poseinthepresentdiscussionistoinvestigatetheKondo effect in finite systems with discrete energy-levels of va- lencenucleonsincharm/bottomnuclei. Forthispurpose, 1. Mean-field approximation weapplythemean-filedapproachtothefinite-sizesystem with discrete energy-levels. As analogy, we remember We rewrite the Hamiltonian (1) as that the BCS theory, which is successful to describe the superconducting state with continuous number-density H = ǫkckσ†ckσ in infinitely large system, can be applied to pairings of valence nucleons in finite nuclei [42]. X 1 +g fσ†fσ′ck′σ′†ckσ ck′σ†ckσ − 2 (cid:18) (cid:19) X X A. Introducing auxiliary fermion fields +λ fσ†fσ 1 , (42) − (cid:16)X (cid:17) by using the relations (38)-(40) and the identity In order to describe the isospin of the impurity, we introduce the auxiliary fermion field f (σ = , ) [39– σ ↑ ↓ ck′ †ck T++ck′ †ck T +(ck′ †ck ck′ †ck )T3 41]. They satisfy the fermion commutation relation ↓ ↑ ↑ ↓ − ↑ ↑− ↓ ↓ 1 {fσ,fσ′†} = δσσ′ and {fσ,fσ′} = 0. We rewrite the = fσ†fσ′ck′σ′†ckσ − 2 ck′σ†ckσ, (43) isospin operators T , T , T of the impurity by using + 3 X X f as − where the constraintcondition(41) is used 5. In the last σ term in the right-hand side in Eq. (42), we consider the T+ =f †f , (38) constraint condition (41) by introducing the Lagrange ↑ ↓ T =f †f , (39) multiplier constant λ. Now we apply the mean-field ap- − ↓ ↑ 1 proximation. We introducethe mean-field fσ†ckσ asan T3 = 2(f↑†f↑−f↓†f↓). (40) expectation value of fσ†ckσ, sandwiched bhy the giround state, and define the isosinglet “gap” function [39–41] Because the number of the impurity should be always equalto one,we need to impose the constraintcondition [39–41] ∆=−g hfσ†ckσi. (44) X Using the relation fσ†fσ =1. (41) The Fock space satisfXying this condition is the physical g fσ†fσ′ck′σ′†ckσ Fock space which should be obtained. The Fock space =gXfσ†fσ′ ckσck′σ′†+δkk′δσσ′ with the other impurity numbers, fσ†fσ =0, 2, which − is indeedunphysical,needs to be excluded. Inthe mean- = Xg fσ†ck(cid:0)σck′σ′†fσ′ +Ng fσ(cid:1)†fσ P − field approximation, however, it will turn out that an X X extension of the Fock space to the multiple impurity- = g fσ†ckσ fσ†ckσ + fσ†ckσ − −h i h i numbers is useful to analyze the ground state of the Hamiltonian (1). In the followings, we consider sepa- × cXk′σ′(cid:16)†fσ′ −hck′σ′†fσ′i+hck′σ′†fσ′i(cid:17)+Ng rately the two cases of g > 0 and g < 0 in the Hamilto- (cid:0) (cid:1) nian (1). We note that the above decomposition of the operators T , T , T can be given by boson fields instead 5 The second term in the right-hand side in Eq. (43) does not of the f+ermi−on fi3elds. In the boson case, however, we include the flipping of the isospin of the valence nucleon, and hence could be neglected for the Kondo effect [41]. However, need to consider superposed fields of the bosons and the we keep this term throughout the analysis, because the present valence nucleons, fermions, in the mean-field approxi- discussionisdevotedtocomparisonoftheresultinthemean-field mation, which may lead to some difficulty. Moreover, approximationwiththeexactsolution. 7 = g fσ†ckσ fσ†ckσ ck′σ′†fσ′ ck′σ′†fσ′ for all k =1,...,N, because the diagonalizationof HMF − −h i −h i can be analytically performed. Such simplification does gX(cid:16) fσ†ckσ ck′σ′†fσ′ +(cid:17)(cid:0)ck′σ′†fσ′ fσ†ckσ (cid:1) not change the essence of the discussion. With the basis − h i h i c ,f (k =1,...,N,σ = , ),wegivethemean-field +gXh(cid:16)fσ†ckσihck′σ′†fσ′i+Ng, (cid:17) (45) H{akmσiltσo}nian HMF in terms o↑f t↓he 2(N +1)×2(N +1) matrix , wherethXeconstraintcondition(41)isusedagain,wesep- Hcf arate the Hamiltonian (1) into the mean-field part H MF ǫ 0 ∆∗ 0 0 0 and the fluctuation part H as ··· ··· fluc 0 ǫ ∆ 0 0 0  ··· ∗ ···  H =HMF+Hfluc, (46) ... ... ... ... ... ... ... ...   with cf =∆ ∆ ··· λ 0 0 ··· 0 , (51) H  0 0 0 ǫ 0 ∆∗  HMF = ǫkckσ†ckσ + ∆∗fσ†ckσ +∆ckσ†fσ  0 0 ··· 0 0 ǫ ··· ∆   ··· ··· ∗  X+λ fσ†fσ+ |X∆g|2(cid:16)−λ, (cid:17)(47)  ... ... ... ... ... ... ... ...   0 0 0 ∆ ∆ λ  X  ··· ···  and   as Hfluc= g fσ†ckσ fσ†ckσ ck′σ′†fσ′ ck′σ′†fσ′ ∆2 − −h i −h i 1X(cid:16) (cid:17)(cid:0) (cid:1) HMF =ψ†Hcfψ+ | g| −λ, (52) g ck′σ†ckσ+Ng. (48) with defining −2 X In the man-field approximation, we consider only the c 1 mean-field part HMF and neglect the fluctuation part ..↑ H [39–41]. We diagonalize H and introduce the  .  fluc MF Slater determinant by single-particle states. Then, we c  N  wpeitrhforrmesptehcet tvoarλiaatinodn∆forasthe expectation value hHMFi ψ = f↑↑ , (53)  c1  ∂  ..↓  H =0, (49)  .  ∂λh MFi    c  ∂  N↓  ∂∆hHMFi=0, (50)  f   ↓  and finally obtain λ and ∆. The ground-state energy is for short notation. It is worth to note that g >0 should given by substituting the λ and ∆ into H . be maintained, because the stability of the ground state MF In the following, to demonstrate the mhean-ifield calcu- is guaranteed by the positivity of ∆2/g in H . Then, MF | | lation explicitly, we consider the simple case of ǫ = ǫ we diagonalize analytically as k cf H 1 1 1 1 diag =diag ǫ,...,ǫ, (ǫ+λ D), (ǫ+λ+D),ǫ,...,ǫ, (ǫ+λ D), (ǫ+λ+D) Hcf 2 − 2 2 − 2 (cid:18) (cid:19) =diag(E ,...,E ,E ,E ,E ,...,E ,E ,E ), (54) 1 N 1 N N+1 1 N 1 N N+1 − − 1 with d = (c c ), (58) N 1σ 1σ Nσ − √2 − D = (ǫ λ)2+4N ∆2. (55) − | | 1 ǫ λ d = 1 − (c + +c ) Introducing the newpfields {dkσ} (k =1, ..., N) Nσ √2Nr − D 1σ ··· Nσ 1 1 ǫ λ d1σ = √2(c1σ−c2σ), (56) −√2r1+ −D fσ, (59) 1 1 ǫ λ d2σ = √2(c1σ−c3σ), (57) dN+1σ = √2Nr1+ −D (c1σ+···+cNσ) . . . 8 1 ǫ λ obtain the ground-state energy for the original Hamilto- + 1 − f , (60) √2 − D σ nian (1) r we represent the mean-field Hamiltonian HMF by EMF+shift =hψ0|HMF|ψ0i+hψ0|Hfluc|ψ0i ∆2 =EMF(ǫ,√Ng)+0 HMF =φ†Hcdfiagφ+ | g| −λ =ǫ Ng, (68) − ∆2 in the mean-field approximation. The binding energy = Ekdkσ†dkσ + | | λ, (61) g − Ng is different by about 33% in contrast to the exact − X value E =ǫ 3Ng/2in Section IIB. This difference with defining exact − originatesfromthelimitofthemean-fieldapproximation. We expect that the correction by the fluctuation, which d 1 .↑ is not included in the mean-field approximation, enables .  .  us to get the value close to the exact one. In the next d subsection,wewilldiscusstheenergycorrectionbyRPA.  N   d ↑  We furthermore discuss the result when the fluctuation φ= N+1↑ . (62) is completely included in Appendix A.  d1.↓  We leave a comment on the obtained wave function  ..  ψ0 . Representing ψ0 by the original fields ckσ,fσ ,   w| eifind that ψ is|a siuperposition of multipl{e numbe}r  ddN↓  ofimpurities,|i.e0.i fσ†fσ =0,1,2. However,weshould  N+1↓  remind us that only one impurity is allowedto exist due P We remark that the isospin components and for the to the condition(41). In fact,we confirmthis is satisfied ↑ ↓ valencenucleonsareseparatedinthematrix ,andthe as averageby cf H mixing part in the off-diagonal components is absorbed into the fluctuation part Hfluc. This separation indeed hψ0| fσ†fσ|ψ0i=1, (69) enables us to introduce the mean field for the valence X inthepresentmean-fieldapproximation[39–41]. Wealso nucleons. note that the groundstate ψ is a state superposedco- Now let us consider the variation of H with re- 0 MF | i h i herentlybymanystatesofvalencenucleonk =1,...,N. spect to λ and ∆. As a simple case, we consider the We also leavea comment about the gapfunction (44). systemwithone valencenucleon. The extensionto n va- In the mean-field approximation,we introduced the new lence nucleons is straightforward as discussed later. In fields d and considered the single-particle state for the presentcase,we have twodegrees offreedom; anim- kσ { } them. In this basis, the gap function gives the strength purityandavalencenucleon. Todescribethis systemby of the binding energy in the system. On the other hand, the fields d and d having the minimum energy E , N N N we consider t↑he groun↓d state intheoriginalfields ckσ,fσ ,the gapfunctiongivesthe { } strengthofthestatemixingbetweenthe valencenucleon ψ0 =dN †dN † 0 , (63) (ckσ) and the impurity (fσ) as seen in the matrix (51) | i ↑ ↓ | i (seealsoRefs.[39–41]). Although,thegapfunctiongives as the most stable state. Performing the variation for thedifferentphysicalmeaning(thebindingenergyorthe strength of the state mixing) according to the difference E (λ,∆)= ψ H ψ MF h 0| MF| 0i of the basis fields, they give essentially the same result. ∆2 =2E + | | λ (64) N g − 2. Fluctuation effect —RPA— with respect to λ and ∆, ∂ ∂ The mean-field approximation does not include the E =0, E =0, (65) ∂λ MF ∂∆ MF fluctuation effect. In this subsection, we investigate the fluctuation effect based on RPA [42, 46] (see also we obtain the values of λ and ∆ Refs. [47, 48] for application to the Hartree-Fock states λ=ǫ, ∆=√Ng. (66) and the BCS states in atomic nuclei). We rewrite the Hamiltonian(1)intermsof d insteadof c ,f as kσ kσ σ The ground-state energy for the mean-field Hamiltonian { } { } H is 3 MF H = ǫ Ng − 4 E (ǫ,√Ng)=ǫ Ng. (67) (cid:18) (cid:19) MF − (a a +a a )+(a a +a a ) 0 † 0 0 † 0 1 † 1 1 † 1 × ↑ ↑ ↓ ↓ ↑ ↑ ↓ ↓ Because we need to consider the energy shift 1 hψ0|Hfluc|ψ0i=0bythefluctuationpartHfluc,wefinally +4(cid:8)Ng (a0↑†a1↑+a0↓†a1↓)+(a1↑†a0↑+a1↓†(cid:9)a0↓) (cid:8) (cid:9) 9 1 gives the correction to the ground-state energy in the + Ng a0 †a0 † a1 †a1 † (a0 a0 a1 a1 ) 2 ↑ ↓ − ↑ ↓ ↑ ↓− ↑ ↓ mean-field approximation. Therefore, the ground-state +( 1)N(cid:0)g a a a a +(cid:1)a a a a energy in the mean-field approximation and the RPA is 0 † 1 † 0 1 0 † 1 † 0 1 − ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↓ 1 + N(cid:0)g a0 †a1 †+a0 †a1 † (a0 a1 +(cid:1)a0 a1 ) EMF+shift+RPA =EMF+shift+∆ERPA +((cid:18)−ǫ2+(cid:19)Ng)(cid:0) ↑ ↓ ↓ ↑ (cid:1) ↑ ↓ ↓ ↑ =ǫ 1(7 3√2)Ng − 2 − − N 1 ǫ 1.378Ng. (76) + − Ekdkσ†dkσ, (70) ≃ − This is the result for one valence nucleon. For n valence k=1 X nucleons(n 2N),onenucleonparticipatesinthe bind- where we define a = d and a = d for short ≤ 0σ Nσ 1σ N+1σ ingasdescribedaboveandtheleftn 1valencenucleons notation. Now we consider the RPA correlation energy − does not (see Eq. (54)). Therefore, the energy becomes byusingthegroundstate ψ =a a 0 inthemean- 0 0 † 0 † field approximation. | i ↑ ↓ | i E (n) nǫ 1.378Ng. (77) MF+shift+RPA Firstofall,wecalculateenergyeigenvaluesoftheRPA ≃ − modes. Wconsiderthefluctuationnearthegroundstate The binding energy 1.378Ng is about92%ofthe exact − ψ =a a 0 . We solve the RPA equation solution 3Ng/2 in Section IIB. Thus, by including the | 0i 0↑† 0↓†| i fluctuatio−n in the RPA, we get the energy close to the A B X X exact one. We expect that more closer value can be ob- =Ω , (71) B∗ A∗ ! Y ! ν Y ! tained when higher order fluctuations are taken into ac- − − count. In fact,we candiagonalizecompletely the Hamil- with tonian(70), dueto itssimplicity,andobtainthe ground- state energywhichis preciselythe same asthe exactone Aµνρσ =hψ0| a0ν†a1µ, H,a1ρ†a0σ |ψ0i, as presented in Appendix A. 1 = Ng(cid:2)δ δ (cid:2) (cid:3)(cid:3) µρ νσ 2 +Ng(δ δ δ δ +δ δ δ δ ) 3. Correspondence between exact solution and µ ν ρ σ µ ν ρ σ 1 ↑ ↓ ↑ ↓ ↓ ↑ ↓ ↑ mean-field+RPA solution + Ng(δ δ δ δ )(δ δ δ δ ),(72) µ ν µ ν ρ σ ρ σ 2 ↑ ↑− ↓ ↓ ↑ ↑− ↓ ↓ Let us see the correspondence between the mean- and field+RPA solution and the exact solution (Tables I, II, Bµνρσ = ψ0 a0ν†a1µ, H,a0σ†a1ρ ψ0 III;g >0). Concerningthegroundstate,wefindthatthe − h | | i former reproduces the latter within the approximation. 1 = Ng(cid:2)(δ δ (cid:2)δ δ )(δ(cid:3)(cid:3)δ δ δ ),(73) We consider the n = 1 case. For N single-particle µ ρ µ ρ ν σ ν σ 2 ↑ ↓− ↓ ↑ ↑ ↓− ↓ ↑ states of valence nucleon, we have one single-particle and obtain the RPA energy eigenvalues statewhichiscoupledtoimpurity(couplingorbital)and N 1 single-particle states which are not coupled (non- Ω = Ω ,Ω ,Ω ,Ω − { ν} { ±1 ±2 ±3 0} coupling orbital). In the ground state, one valence nu- = √2Ng, √2Ng, √2Ng,0 . (74) cleon occupies the coupling orbital, and forms the isos- ± ± ± inglet state as combined to the impurity isospin as the n o The zero-energy mode with Ω = 0 is due to the energy most stable state. Therefore, the number of degeneracy 0 degeneracy of the first term in the Hamiltonian (70) for factor is one. This corresponds to the ground state of ψ = a a 0 and ψ = a a 0 . This degener- I =0, 1 with energy ǫ 3Ng/2 in Table I. a|c0yiisspe0c↑†ial0↓i†n|tihemea|n-1fiielda1p↑p†ro1x↓†i|miation,andhence Weconsiderthen=−2case. Inthiscase,oneofthetwo should be regarded as the spurious one. Indeed, we will valencenucleonsoccupiesthecouplingorbital,andforms see such degeneracy will be resolved when higher order the isosinglet state combined with the impurity isospin. fluctuations are included in Appendix. A. The left valence nucleon occupies one of the N 1 non- − From the above result, we obtain the RPA correlation coupling orbitals. Becausethe fist valence nucleonforms energy [42, 46–48] the isosinglet state with the impurity, the addition of the second valence nucleon gives isodoublet state. The 1 1 ∆E = Ω TrA numberofdegeneracyfactorisN 1. Thisisthesameas RPA 2 ν − 2 the number of degeneracy factor−in the I = 1/2 ground ν>0 X 1 state with energy 2ǫ 3Ng/2 in Table II. = (3√2 5)Ng We consider the n−= 3 case. In this case, one valence 2 − nucleonoccupiesthe couplingorbitalandformsthe isos- 0.378Ng, (75) ≃− inglet state combined with the impurity isospin. The astheenergydifferencebetweenthemean-fieldstateand othertwovalencenucleonsoccupyoneortwooftheN 1 − the fluctuating state. Thus, the RPA correlation energy non-couplingorbitals,andformtheisosingletorisotriplet 10 state. For the isosingletstate, the number of degeneracy with factorisN(N 1)/2,becausethesecondtwovalencenu- − cleons can occupy the same single-particle states (N 1 H = ǫ c c − M′ F k kσ† kσ patterns)orcanoccupythedifferentsingle-particlestates ((N−1)(N−2)/2patterns). Forthe isotripletstate,the X+ ∆ickσ†(σσiρ)fρ+∆i∗fσ†(σσiρ)ckρ number of degeneracy factor is (N 1)(N 2)/2, be- − − 1X(cid:16) (cid:17) cause the two valence nucleons should occupy the differ- ∆i 2, (82) entsingle-particlestates((N 1)(N 2)/2patterns). We −g | | − − X confirmthose numbers of degeneracyfactor is consistent with those in the ground state of I = 0, 1 with energy and 3ǫ 3Ng/2 in Table III. − Hfl′uc =g fσ†(σi)σρckρck′σ′†(σi)σ′ρ′fρ′ C. Isotriplet condensate (g<0) −X ∆ickσ†(σσiρ)fρ+∆i∗fσ†(σσiρ)ckρ 1X(cid:16) 3 (cid:17) As mentioned previously, there is no stable isosinglet + ∆i 2+ ck′σ†ckσ 3Ng. (83) g | | 2 − condensate for g < 0. In this case, we need to consider We note that ∆Xi is given by tXhe matrix form the isotriplet condensate. ∆3 ∆1 i∆2 1. Mean-field approximation ∆i(σi)αβ = ∆1+i∆2 −∆3 ! . (84) − αβ We rewrite the Hamiltonian (1) as In the mean-field approximation,we consider only the mean-field part H and neglect the fluctuation part H = ǫkckσ†ckσ M′ F H , as performed in the isosinglet condensate in Sec- fl′uc X+g fσ†(σi)σρckρck′σ′†(σi)σ′ρ′fρ′ tion IIIB. We diagonalize HM′ F and introduce the Slater determinant by single-particle states. Then, we perform 3X +2g ck′σ†ckσ −3Ng, (78) tshpeecvtatroiaλtioanndfo∆rithase expectation value hHM′ Fi with re- X by using the identity ∂ ck′↓†ck↑T++ck′↑†ck↓T−+ ck′↑†ck↑−ck′↓†ck↓ T3 ∂λhHM′ Fi=0, (85) =X(cid:8)fσ†(σi)σρckρck′σ′†(σi)σ′ρ′fρ(cid:0)′ + 23 ck′σ†ckσ(cid:1) (cid:9) ∂∆∂ihHM′ Fi=0, (86) X3N, X (79) − and obtain λ and ∆i. The ground-state energy is given wherethe constraintcondition(41)isused. Defining the by substituting the obtained λ and ∆i into H . isotriplet “gap” function h M′ Fi In the followings, to demonstrate the mean-field cal- ∆i =g fα†(σi)αβckβ , (80) culation explicitly, we set ǫk =ǫ as a simple case, where h i the diagonalization of H can be performed analyti- M′ F X cally. Such simplification does not change the essence of weseparatetheHamiltonian(1)intothe mean-fieldpart the discussion. By using c ,f (k =1, ..., N, σ = , H and the fluctuation part H kσ σ M′ F fl′uc ), we write the mean-fie{ld Hami}ltonian H , with th↑e ↓ M′ F H =H +H , (81) 2(N +1) 2(N +1) matrix , M′ F fl′uc × Hc′f ǫ 0 ∆3 0 0 ∆1 i∆2 ∗ ∗ ∗ ··· ··· − 0 ǫ ∆3 0 0 ∆1 i∆2  ··· ∗ ··· ∗− ∗  ... ... ... ... ... ... ... ...    ∆3 ∆3 λ ∆1 i∆2 ∆1 i∆2 0  Hc′f = 0 0 ··· ∆1+i∆2 ∗−ǫ ∗ ∗−0 ∗ ··· ∆3∗ , (87)  ··· ··· −   0 0 ∆1+i∆2 0 ǫ ∆3   ··· ··· − ∗   ... ... ... ... ... ... ... ...     ∆1+i∆2 ∆1+i∆2 0 ∆3 ∆3 λ    ··· − − ···  