The pairing Hamiltonian with two pairs in many levels

Transcription

1 arxiv:nucl-th/3662v 23 Jun 23 The pairing Hamiltonian with two pairs in many levels M. Barbaro, R. Cenni 2, A. Molinari and M. R. Quaglia 2 Dipartimento di Fisica Teorica Università di Torino Istituto Nazionale di Fisica Nucleare Sez. di Torino Torino Italy 2 Dipartimento di Fisica Università di Genova Istituto Nazionale di Fisica Nucleare Sez. di Genova Genova Italy Abstract We address the pairing problem for two pairs of fermions in many levels of a potential well. We introduce a new number, referred to as like-seniority, which provides a measure of the degree of collectivity of the states. We point out that two critical values of the coupling constant might exist and we explore the behaviour of the energy, the wave functions and the transition amplitudes, in particular around these critical points. It is worth noticing that, in some definite cases, including the one of metal bands, the pairs can escape the trapping mechanism. Introduction In a previous paper [] we examined in detail the problem of the fermionic pairing Hamiltonian, a useful model for the atomic nuclei actually, more generally, for any many-body Fermi system in the simple situation of one pair of nucleons coupled to an angular momentum J = living in a set of L single particle levels (s.p.l.), e.g. in a major shell. This problem, solved long ago for the case of a fully degenerate major shell hosting any number

2 of pairs, still deserves some attention when the s.p.l. are split in energy. Actually this is the situation encountered in both nuclei and metals, with the important difference that in a nucleus a major shell is typically split into five or six s.p.l. with different angular momenta, whereas in a metal the number of non-degenerate levels entering into a band corresponds to a significant fraction of the Avogadro number. Moreover in a heavy nucleus the number of pairs living in a level may be as large as, say, eight while in a metal is one. In the above quoted paper two main results were achieved:. the energy position of the collective mode E coll was related to the statistical features of the s.p.l. distribution. In particular we showed that only few moments of the distribution, beyond of course the strength of the pairing interaction, are sufficient to accurately predict E coll ; 2. the eigenvalues trapped in between the unperturbed single particle energies (s.p.e.) e i (i =, L) were shown to belong to two different regimes, depending upon the strength of the interaction and the degeneracy of the s.p.l.. Specifically in the weak coupling regime the lowest lying eigenvalues were found to almost coincide with the s.p.e. of the corresponding unperturbed levels, the more so the smaller their degeneracy; instead for a strong enough interaction (provided the levels degeneracy increases with the s.p.e.) the low-lying trapped eigenvalues turned out to be pushed down close to the energy of the below lying unperturbed s.p.l. (namely, to the lowest allowed bound). Notably this transition occurs sharply, in correspondence of a precise value of the coupling constant. In the present paper we extend the previous analysis considering two pairs of fermions living in many levels. We shall see that although the findings of the one pair case are reflected in some of the states now encountered, the presence of two pairs leads to new important features. First the classification of the eigenstates of the pairing Hamiltonian in terms of the seniority quantum number v, which characterises the one level case, becomes insufficient and we argue that a new number v l should be introduced to organise the states in a scheme more in touch with physics. Indeed v l turns out to provide a measure of the degree of collectivity of the states and, as such, has an important bearing on the transition matrix elements and hence on 2

3 the physics of the stripping and pick-up pair reactions. Second the eigenvalues, unlike the one pair case, can get away from the trapping mechanism, in some definite cases, including the one of relevance for the metal bands. This occurrence, of importance for the possible onset of pairs condensation in a finite system, will be analysed in detail. In fact, although in the end our aim is to deal with the situation with many active pairs, in pursuing this scope we hope to gain insight on more involved situations already from the study of the two pairs case. In particular in this instance it may happen that two critical values of the coupling constant might exist, corresponding to the cross over between different regimes. This phenomenon, although reminiscent of a true phase transition which might eventually occur when the number of pairs is large, if the system is sizable enough not to incur in the Anderson s conjecture [2], actually is not such (the system s energy and its derivative turn out to be continuous functions of the coupling constant). Yet it might be asked, when m > 2 pairs are active and hence m critical values might exist, whether a number of different regimes occur, leading eventually to a true phase transition: in this connection the study of the case of three or four pairs is expected to shed light on how a super-fluid phase is eventually reached. This topic will be confronted in a forthcoming paper: presently we limit ourselves to investigate in depth the above mentioned new features arising in the two pairs system, including the Pauli blocking effect, which yields a correction to the simple doubling of the eigenvalues of the one pair case, the magnitude of the correction depending upon the degeneracy of the single particle levels. 2 Classification of the states As well-known the eigenvalues and the eigenvectors of the pairing Hamiltonian are found by solving the Richardson equations [3]. These, for the two pair case, reduce to the following system of two equations in the unknown E and E 2 (referred to as pair energies, although they not always carry a direct physical meaning): G L = Ω + 2G = (a) 2e E E 2 E 3

4 G L = Ω + 2G = (b) 2e E 2 E E 2 or, equivalently, by adding and subtracting the above, G L = 2 G L = Ω 2e E + G Ω 2e E G L = L = Ω 2e E 2 = (2a) Ω + 4G =. (2b) 2e E 2 E 2 E In (,2) the e are the unperturbed s.p.e., whose degeneracy is Ω, and G is the strength of the pairing force. Since the Richardson equations deal with pairs of fermions (nucleons) coupled to an angular momentum J =, their eigenvalues are those of the zero-seniority states (v = ) and are simply given by E = E + E 2. Importantly, these eigenvalues display different degrees of collectivity: hence they are conveniently classified in terms of the latter. For this purpose, as mentioned in the introduction, we introduce a number v l, to be referred to as like-seniority, that counts in a given state the number of particles prevented to take part into the collectivity, not because they are blind to the pairing interaction (indeed they are coupled to J = ), but because of the trapping mechanism, even in the strong coupling regime. Specifically we shall ascribe the value v l = to the fully collective state, v l =2 to a state set up with a trapped pair energy while the other displays a collective behaviour and v l =4 to the state with two trapped pair energies. Clearly a detailed knowledge of the behaviour of the solutions of the Richardson s equations with the coupling constant G is required to understand where the concept of like-seniority is actually useful. This will be discussed in the sequel. However it is natural to anticipate that the classification of the states in terms of v l becomes more neat in the strong coupling domain. Since, however, in the weak coupling regime the pattern of the states is simpler we shall commence to examine this domain in the next section. 4

5 3 The weak coupling domain The weak coupling limit is of course best suited for a perturbative treatment. Accordingly, adopting for the pair energies E, E 2 the expressions (where Gx i is a perturbation), one has E i = 2e i + Gx i (3) Ω 2e E i = Ω i Gx i + i Ω 2(e e i ) Gx i. (4) Hence, expanding in G, the system () becomes + Ω x G Ω 2(e e ) + 2G 2(e 2 e ) + O(G2 ) = (5a) + Ω 2 x 2 G 2 Ω 2(e e 2 ) + 2G 2(e e 2 ) + O(G2 ) =, (5b) where the indices and 2 select one configuration out of the unperturbed ones. At the lowest order in G, if 2, we get x i = Ω i and E i = 2e i GΩ i. Thus the energy eigenvalue E = E + E 2 becomes E = 2(e + e 2 ) G(Ω + Ω 2 ). (6) Instead, if = 2, we have to solve the system + Ω x G Ω 2(e e ) + 2 x 2 x + O(G 2 ) = (7a) + Ω 2 x 2 G 2 Ω 2(e e 2 ) + 2 x x 2 + O(G 2 ) =. A generic solution reads (7b) x,2 = (Ω ) ± i Ω (8) (note that in this case, when Ω >, E and E 2 are always complex conjugate). Hence the energy becomes E = 4e 2G(Ω ), (9) 5

6 which is of course real. In comparing (6) and (9) with the lowest energy of one pair, which, in the weak coupling regime, is [] E = 2e GΩ, () one sees that, while (6) corresponds to the sum of two contributions like () (the two pairs ignore each other), in (9) a Pauli blocking effect appears. 4 The strong coupling domain 4. Like-seniority In this Section we study the v l = (and hence lowest) eigenvalue in the strong coupling limit. In this connection we already know that in the degenerate case (when all the s.p.e. coincide) only the collective eigenvalue E = 4ē 2G(Ω ) () exists, ē being the energy and Ω the degeneracy of the level. The above yields the leading order contribution to the energy in the non-degenerate case and represents a good estimate when the spreading of the s.p.e. levels is small with respect to GΩ, as it is natural to expect. To show this occurrence, following [], we introduce the new variables where now ē = Ω x i = E i 2ē GΩ L Ω e, Ω = = (2) L Ω. (3) We further define the variance σ = Ω ν (e ν ē) Ω 2 (4) ν = and the skewness γ = σ 3 Ω Ω ν (e ν ē) 3 (5) ν 6

7 of the level distribution; moreover we introduce the expansion parameter Then the system of eqs. (2) becomes 2 + ( ) + α2 + α3 γ + + ( x2 + α2 x x 2 x 3 x 2 2 x ( + α2 x 2 ) + α3 γ + ( x2 + α2 x 3 x 2 2 α = 2σ GΩ. (6) ) + α3 γ + x 3 2 ) + α3 γ + x 3 2 = (7a) + 4 =. Ω x 2 x (7b) At the leading order the system (7) is easily solved, yielding x () (Ω ) Ω,2 = ± i ; (8) Ω Ω at second order, setting and linearising in d, d 2, we obtain x (2) (Ω ),2 = α 2(Ω 2) Ω Ω (Ω ) ± i Ω Likewise, at third order, we get x (3) x (2),2 = x (),2 + d,2 (9) i Ω 2 α2. (2) (Ω ) 3/2 (Ω ),2 = α 2(Ω 2) Ω (Ω ) + (Ω 4) α3 γ (Ω ) Ω ± i i Ω Ω 2 α2 (Ω ) ± Ω 3/2 iα3 γ. (2) (Ω ) 3/2 So, at this order, the total energy of the system turns out to read E 4ē GΩ = 2(Ω ) Ω 2α 2(Ω 2) (Ω ) + 2α3 γ (Ω 4) (Ω ) + O(α4 ). (22) Since the collective energy of one pair of nucleons living in L levels in the strong coupling regime is [] E 2ē GΩ = α2 + γα 3 + O(α 4 ), (23) 7

8 one sees that, when Ω, (22) becomes just twice the value (23) and, moreover, the imaginary part of x and x 2 goes to zero as / Ω. Thus, in this limit, the Pauli interaction between the two pairs vanishes, as expected: in other words the two pairs behave as two free quasi-bosons condensed in a level whose energy is given by (23). In Table we compare the result (22) with the exact one assuming that the two pairs live in the first L levels of a harmonic oscillator well with frequency ω. Actually the v l = collective state of concerns arises from an unperturbed configuration having a pair in the lowest and a pair in the next above s.p.l. (with the present units the energy of such a configuration is 8). We see that the difference between the two results never exceeds 5 %, even for G = G/ ω as low as.. Note also that the energy of the v l = state does not depend upon ω, i.e. it scales with the size of the well. G α E () E (2) E (3) E exact Table : Energies, in units of ω, of the state of zero like-seniority for different values of G = G/ ω at the order, 2 and 3 in the expansion parameter α compared with the exact ones. The s.p.e. levels (L = 4) and degeneracies are those of a 3-dimensional harmonic oscillator, ω being the harmonic oscillator constant. 4.2 Like-seniority 2 The Richardson equations () are coupled through a term that accounts for the Pauli principle. Would it be absent, the eigenvalues associated to v l = 2 8

9 could be simply obtained by adding the collective energy E carried one pair and the trapped energy E 2 carried by the other pair. This situation is recovered in the very strong coupling limit, where the splitting between the s.p.e. can be disregarded (so that all the s.p.e. become essentially equal to ē) and both ē and E 2 are very small with respect to E. Indeed the first equation of () then becomes yielding G + Ω E 2 E = (24) E = G(Ω 2), (25) namely the energy of the state with two pairs and v = 2 in the one level problem. This result, setting a correspondence between states with v l = 2 and v = 2, helps to shed light on the relationship between the concepts of seniority and like-seniority (and, in parallel, between the physics of a broken and a trapped pair). Returning to the non degenerate case, we search for a solution of the Richardson equations (). For this purpose, denoting with E () and E (ν) 2 the first and the ν-th eigenvalues (ν ) of the equation G L = Ω 2e E =, (26) we observe that in the strong coupling limit the extra (Pauli) term can be approximated as follows 2G E 2 E 2G E (ν) 2 E () 2G 2e ν 2ē + GΩ 2 Ω. (27) Hence the equations of the system () turn out to be decoupled and can accordingly be recast as G + 2 E (ν) 2 E () G 2 E (ν) 2 E () Ω 2e E = Ω 2e E 2 =. (28a) (28b) 9

10 Then, defining G () eff G (2) eff G + 2 E (ν) 2 E () we rewrite the equations (28) in the form (29) G 2, (3) E (ν) 2 E () G () eff G (2) eff Ω 2e E = Ω 2e E 2 =. (3a) (3b) These show that in the strong coupling regime the Pauli principle just re-scales the coupling constant, the rescaling being, however, different for the collective and the trapped states. Assuming E (ν) 2 E () GΩ, which is justified in the strong coupling regime or when Ω is large enough, one immediately gets G () eff = G (2) eff = G + 2 Ω G 2 Ω (32), (33) namely the Pauli blocking quenches and enhances the strength of the pairing interaction for the collective and the trapped modes, respectively. We compare in Table 2 the exact results with the approximate ones (eq. (3)) in the h.o. case for L = 3, in the strong coupling regime ( G = 5). Note that in this example of the five existing states two have v l = 2, according to where the trapped pair is placed. In the Table we see that the estimated trapped energy E 2 satisfactorily reproduces the exact one, while the same does not happen for the collective energy E (the error being however 7%). To get a better estimate of E, we rewrite the first equation of the system

11 E exact E2 exact E E Table 2: Exact and approximate (eq. (3)) energies of the states with likeseniority 2 for G = 5. The s.p.e. (L = 3) and degeneracies are those of a 3-dimensional harmonic oscillator. () as follows G + 2 (E 2 E () L = ( ) n= Ω 2e E () E() E E 2 E () ( n= ) n E() E 2e E () ) n =. (34) Clearly, by truncating the expansion to the desired order, we get an algebraic equation for E. Assuming for E () the one pair result (26) and for E 2 the solution of (3), we obtain the values collected in Table 3, where a good agreement is seen to occur between the exact and the approximate solutions by keeping just a few terms of the expansion (34), which is thus seen to be fastly convergent. In concluding the discussion on the v l =2 states, it is worth noticing that the knowledge of the one pair dynamics - see eq. (3) - rules, to a large extent, the two pairs case as well. In particular, the same transition between two different regimes, already found in the one pair case and described in details in ref. [], occurs again for the trapped state, the only change being a rescaling of the critical value α cr which now is set by eq. (33). For the collective state, to get a more accurate agreement between exact and approximate results, it is necessary to account for more (but not many) terms in the expansion (34). Hence a compact expression of the collective energy in terms of the moments of the s.p.e. distribution no longer holds. 4.3 Like-seniority 4 We just mention now the class of states totally lacking in collectivity. Although one might distinguish two types of these, according to the unperturbed configuration from where they arise, namely two pairs living in the

12 G α E exact E2 exact E I E II E III E Table 3: Energies of the states with like-seniority v l =2 for various values of G in the strong coupling regime in the case L=3. The s.p.e. and degeneracies are those of a 3-dimensional harmonic oscillator. The approximate energies E, I E II, E III correspond to keep in the expansion (34),2 and 3 terms respectively. same level or in two different levels, no new physics appears to emerge in both instances. 5 The transition regime and the critical value of G In this Section we address a new phenomenon, not encoded in the physics of one pair. To study it we remind the occurrence of the following two situations in the weak coupling regime: if a state evolves, by switching on the interaction, from an unperturbed one having two pairs living in the same level, then the pair energies E and E 2 are always complex conjugate, while if the state evolves from an unperturbed one with two pairs living in two different unperturbed levels, then E and E 2 are real. This conclusion, valid in first order, holds as well for all the perturbative orders, since these are computed through a linearization procedure performed on the Richardson equations, which have real coefficients. On the contrary, in the strong coupling regime the pair energies E and E 2 of the v l = states are always complex conjugate. Since the collective state evolves from the lowest unperturbed state by switching on the interaction, it is clear that now two situations are possible, according to the degeneracy 2

13 Ω of the lowest s.p.e. level. The first corresponds to Ω > : in this case the lowest unperturbed state has both pairs in the lowest level, E and E 2 are complex in both the weak and strong coupling limit and nothing unusual is expected to happen as a function of G. The second corresponds to Ω = : here, in the weak coupling limit, the two pairs must live on different levels, E and E 2 are necessarily real in a neighbourhood of the origin, but they become complex in the strong coupling regime. Thus a singularity in their behaviour as a function of the control parameter G is bound to occur. We shall focus on this second case and for this purpose we return to the Richardson equations. On the basis of the recognition that a transition from a regime where E and E 2 are real to one where they are complex conjugate should occur, it is natural to surmise that the transition takes place when E and E 2 coincide. In this case however the last term of the Richardson equations diverges, and, for the equations to remain meaningful, it must be compensated by another divergence in the sum appearing in the equations: this can only take place if E or E 2 coincides with an unperturbed eigenvalue, which indeed is what happens, as Fig. shows. In this figure we take L = 3 and the degeneracy and levels spacing of a 3-dimensional harmonic oscillator, which has Ω =. It is clearly apparent that, for small G, E and E 2 start decreasing from their unperturbed energies 3 and 5 (in units of ω ) respectively; then E increases, till it coincides with E 2 exactly at the s.p.e. 3. Note that the ground state energy E = E +E 2 still keeps decreasing with G, as it should. We now search for an analytic expression of the critical value of G where this transition occurs. To this aim we start from a generic solution with two pair energies E and E 2, which evolve with G till they come to coincide at an unperturbed energy 2e ν of a s.p.l. with a degeneracy Ω ν. In this situation eq. (2b), rewritten as (E 2 E ) 2 L = Ω (2e E )(2e E 2 ) + 4 =, (35) no longer depends upon G and allows us to express E 2 as a function of E. Indeed set E = 2e ν + x (36a) 3

14 8 6 E 4 E 2 2 E G ~ Figure : Behaviour of E and E 2 as functions of G (solid lines) for threedimensional harmonic oscillator well with 3 levels. The behaviour of E = E + E 2 is also reported (dashed line) E 2 = 2e ν + xϕ ν (x) (36b) with ϕ ν (x) assumed to be regular and non-vanishing in the origin. Insert then the above into (35) getting x 2 (ϕ ν (x) ) 2 ν Ω (2ǫ 2ǫ ν x)(2ǫ 2ǫ ν xϕ ν (x)) + + (ϕ ν(x) ) 2 Ω ν ϕ ν (x) which, in the limit of vanishing x, yields + 4 =, (37) ϕ ν () = Ω ν 2 ± 2 Ω ν Ω ν, 4

15 a real quantity when Ω ν =. This occurrence is crucial since in eq. (a), for x, two divergent terms appear, namely GΩ ν /(2e ν E ) = GΩ ν /x and 2G/(E 2 E ) = 2G/[x(ϕ ν (x) )]. Clearly they cannot cancel if the first term is real and the second is complex, whereas if Ω ν = then ϕ ν () = and the divergences exactly cancel. It follows that a critical point, where E = E 2 = 2ǫ ν, can only occur if Ω ν =. The same conclusion has been reached by Richardson [4] trough a different path. Assuming then, from now on, Ω ν = we search for a power expansion of ϕ ν (x). For this purpose note that Eq. (35) is automatically fulfilled, when expanded to first order in x, while at second order provides the constraint {ϕ ν()} 2 = 4 L =( ν) Ω (2e 2e ν ) 2, holding for any ν. Next we expand E 2 in powers of x and, introducing P (k)ν = to fourth order in x we obtain L =( ν) Ω (2e 2e ν ) k k, (38) E 2 = 2e ν x ± 2P (2)ν x 2 4P(2)νx 2 3 ( ) + ±9P 3 (2)ν + 2P3 (3)ν ± P4 (4)ν P (2)ν x 4 + O(x 5 ), (39) an equation linking E and E 2. The quantities (38) correspond to the inverse moments of the level distribution. To compute x we recast one of the equations of the system () as follows + G x G L =( ν) Ω + 2G = (4) 2e E E 2 E and notice that 2G G E 2 E x P (2)ν x = G 2 x P (2)νG. (4) 5

16 Hence the divergence in (4) is removed, the limit x can be taken and an equation for G is thus finally obtained. It admits the following two solutions G (ν)± cr = = P ()ν ± P (2)ν L =( ν) Ω 2e 2e ν ± L =( ν) Ω (2e 2e ν ) 2 (42) which represent two critical values for G. It is easily checked that < P (k+) < P (k), so that when ν = both the critical values (42) are always positive. For higher-lying s.p.l. (ν > ) whether they are positive or not should be checked case by case. However in the highest s.p.l. (ν = L) both values of G cr are negative. We explore now the behaviour of the energies E and E 2 around the critical value setting G = G cr (ν)± + δg (43) and expanding (4) up to the order x 2. We get x 2 = 2δGP (2)ν ( P()ν ± P (2)ν ) 2 3P 4 (2)ν + 4P (2)νP 3 (3)ν + P4 (4)ν + O ( δg 2), (44) namely the expression for x we need for our discussion. From the above indeed it follows that for ν =, where both G ()± cr exist and are positive, close to the lowest critical point G ()+ cr the solutions are real for δg < and imaginary for δg >. Close to the higher critical point G () cr the opposite occurs. The situation is illustrated in fig. 2, where the behaviour with G of E and E 2, again for a harmonic oscillator well and L = 3, is displayed. For the state arising from the unperturbed configuration with one pair in the first and the other in the second level, the pair energies E and E 2, real in the weak coupling limit, coincide at the critical point G ()+ cr (their common value being 2e ) and then become complex conjugate; the energy of the associated state evolves in the v l = collective mode. On the contrary, for the state arising from the configuration with two pairs living on the second level (provided Ω 2 > ), the two pair energies E and E 2 are complex conjugate in the weak coupling limit, coalesce into the energy 2e at the critical point G () cr and then become real. One of the two solutions remains trapped above 2e, while, 6

17 8 E, G ~ Figure 2: Behaviour with G of the solutions of the Richardson equations when the two critical points are real and positive for a harmonic oscillator well for L = 3: dashed lines denote the real part of the solutions (of course coincident) when the solutions are complex and solid lines the separate real parts. the other evolves into a collective state: the sum of the two yields the energy of a v l = 2 state. Note that the expansion of the energy E only embodies even powers of x δg, as it should, otherwise the total energy would become complex in some range of δg. In nuclear physics it may happen that a level with Ω = lies in the middle of a shell. Then it could still support the existence of two positive critical values of G, but the pair energies E and E 2 would be lowered only till they touch the neighbour lower lying s.p.l. then remaining trapped. Finally we consider the case of L, for simplicity equally-spaced, levels all 7

18 with degeneracy Ω ν =, a situation clearly of interest in solid state physics and in deformed nuclei. Here the existence of two positive critical points always occurs in the lowest level for L 3 (for L = 2 G () cr ). But the existence of G () cr implies complex pair energies E and E 2 for G < G () cr : however all the pair energies are real near G =, hence these can only originate from another transition occurring in the above lying s.p.l.. In short if there is a G cr (ν) > then also G (ν+)+ cr must exist, such that < G (ν+)+ cr < G cr (ν). Indeed the existence of G cr (ν) implies that the pair energies E and E 2 start real two levels above the level ν, merge at G (ν+)+ cr and then becomes complex. As such they come down to a lower level where they merge again at G cr (ν), thus becoming real again. Thus in this situation a solution can evade the trapping twice, as shown in fig. 3. We have numerically found, for equally spaced levels with Ω =, that two critical values of G indeed exist in the second s.p.l. when L 9 and in the third one when L 6, and so on. In conclusion we mention that it may occur that G (ν+)+ cr does not; in this case the two solutions in G (ν+)+ cr exists, but G cr (ν) become complex conjugate and keep their complex nature till they reach the strong coupling regime. In such a case the trapping mechanism is still working. 6 Wave functions In this Section we examine the wave functions of the states so far discussed. A generic v = eigenfunction of the pairing Hamiltonian is expressed in terms of the collective Grassmann variables Φ (referred to as s-quasibosons), according to [5] ψ n [Φ ](m) = n k= Bk (m) (45) where m actually denotes the set (m,...,m n ) labelling the unperturbed configuration of the state. In (45) B k(m) = L = β (k) (m)φ (46) can be viewed as the wave function of a pair. The β coefficients are related 8

19 E, G ~ Figure 3: The case of Ω = and nine equally spaced s.p.l. 9

20 to the eigenvalues E k according to: β (k) (m) = C k (m) 2e E k (m) (47) where the C k (m) are normalisation factors. When no confusion arises the index (m) will be dropped. Since (Φ, Φ ν) = Ω δ ν, it is convenient to replace the Φ s and the β s with Φ = Φ and β(k) = Ω β (k), (48) Ω respectively. Moreover it is natural to normalise a one pair state (p) > according to < (p) (p) >=, which entails C (p) = Ω (2e E(p)) 2. (49) We can still keep for a two pairs state C k (m) = Ω 2e E k (m) 2, k =, 2 (5) but, in this case, the state is no longer normalised to. Having already investigated [] the wave functions of a single pair living in many levels, we now compute the wave functions of two pairs, being mainly interested in studying the states undergoing a transition, in the sense discussed in Sec. 5. Assuming, for sake of illustration, L = 4 s.p.l. of a harmonic oscillator well, we focus on the v l = (m = (m, m 2 ) = (, 2)) and on the v l = 2 (m = (2, 2)) states. We display in fig. 4 and fig. 5 respectively, for different values of G, (k) their coefficients β as functions of the index. In the v l = case, for G ()+ (k) < G cr, the coefficients β (, 2) are real, while for G ()+ > G cr, they become complex conjugate (hence we show both their real and the imaginary part); the opposite occurs in the v l = 2 case. As it appears in the panel G =. of fig. 4, the vl = state initially has one pair in the first and one in the second level. As G approaches 2

21 G ()+ cr =.7, both pairs sit on the first level. Finally for G ()+ G cr, the weight of all the components of β () (, 2) and β (2) (, 2) are almost the same, thus displaying a collective behaviour. The (2,2) v l = 2 state (see fig. 5) instead starts up with two pairs on the second s.p.l. (see the case G =. in fig. 5). As G () approaches G cr =.287, the two pairs live on the lowest s.p.l.. Finally for G () G cr, all the components of β () (2, 2) have almost the same weight (like the components of the collective state of one pair) while only the first component of β (2) (2, 2) is significant (like for the trapped state of one pair). It is worth analysing the cases corresponding to G, G and G = G cr. () The weak coupling limit is immediate and yields β (m, m 2 ) = δ m and β (2) (m, m 2 ) = δ m2. The strong coupling limit is easily handled only when v l = (independently of Ω ), when the s.p.e. are very small with respect to E, E 2. In fact, using (2), one then easily gets β (,2) = Ω Ω ( Ω ± i ). (5) The structure of the wave function at the critical points (assuming the transition to occur on the = level) is more delicate. A tedious, although straightforward, calculation leads to } β () = { x2 2 P2 (2), (52) } β (2) = + { x2 2 P2 (2), (53) β () = { ( ) } 2 x x Ω + (54) 2ǫ 2ǫ 2ǫ 2ǫ β (2) = { ( ) 2 x x [ Ω ± 2P(2)(2ǫ 2ǫ ) ]} 2ǫ 2ǫ 2ǫ 2ǫ (55) (up to the order x 2 ). In the above x is connected to G by the relation (44) and in (55) the double sign refers to the two critical points G ()± cr. Although 2

22 in the following higher orders in the x expansion will be needed, their expressions are heavy and not particularly enlightening: hence they will be not (,2) reported here. Notice that the β start linearly in x, i.e., in G G cr, thus displaying a branch point in the control parameter G at its critical value. 7 Transition amplitudes In this Section we study the transition amplitudes from the vacuum to a one pair state and from a one pair to a two pairs state, to explore the possible consequences of the transition between the different regimes previously discussed. These amplitudes, clearly relevant for the stripping or pick-up nuclear reactions, reflect the structure of the wave functions illustrated in the previous section. These, in turn, are strongly affected by the value of the coupling constant G and might carry a signal of the transition between different regimes, which is instead absent in the total energy behaviour versus G. We thus study the transition amplitude induced by the operator  =  j j m = 2 ( ) j m â j,m â j, m, (56) namely the one entering the pairing Hamiltonian, and examine the matrix elements < pair  > and < 2 pairs  pair >. (57) In the fully degenerate case a general expression for these transition amplitudes between states with any number n of pairs and seniority v, can be obtained by inserting into the pairing spectrum a complete set of states n, v >. Indeed one has  n +, v > < n +, v  n, v >< n, v n,v = < n +, v  n, v > 2 = ( n + v )( Ω n v ), 2 2 (58) thus getting for the transition amplitude ( < n +, v  n, v >= n + v )( Ω n v ), (59)

23 β ~ () G ~ = β ~ () G ~ = β ~ () G ~ = Re(β ~ () )=Re(β~ (2) ) G ~ = Re(β ~ () )=Re(β~ (2) ) G ~ = Re(β ~ () )=Re(β~ (2) ) G ~ = β ~ (2) G ~ = β ~ (2) G ~ = β ~ (2) G ~ = Im(β ~ () )=-Im(β~ (2) ) G ~ = Im(β ~ () )=-Im(β~ (2) ) G ~ = Im(β ~ () )=-Im(β~ (2) ) G ~ = Figure 4: In the L = 4 levels harmonic oscillator well, we display for different values of G (k) the coefficients β (or their real and imaginary part) as functions of for the v l = state. 23

24 Re(β ~ () )=Re(β~ (2) ) G ~ = Re(β ~ () )=Re(β~ (2) ) G ~ = Re(β ~ () )=Re(β~ (2) ) G ~ = β ~ () G ~ = β ~ () G ~ = β ~ () G ~ = Im(β ~ () )=-Im(β~ (2) ) G ~ = Im(β ~ () )=-Im(β~ (2) ) G ~ = Im(β ~ () )=-Im(β~ (2) ) G ~ = β ~ (2) G ~ = β ~ (2) G ~ = β ~ (2) G ~ = Figure 5: The same as in fig. 4, but for the v l = 2 state, made up of two pairs living on the second s.p.e. level. 24

25 which, while more transparent, coincides with the one of ref. [6]. Note that in (59) the seniority is conserved. For large G we could expect the matrix elements (57) to display an asymptotic behaviour coinciding with (59), namely <, Â > G < 2, v l Â, v l > G Ω (6) { 2(Ω ) for vl = Ω 2 for vl = 2 (6) if the like-seniority v l is conserved. However, like-seniority is not conserved at finite G: only in the strong coupling regime we expect a vanishing matrix element between states of different like-seniority. Actually even there the situation turns out to be somewhat more involved, as it will be seen in the following. We now compute explicitly the amplitude < (p) Â >. We immediately have < (p) Â >= Ω β (p), (62) which, using (47) and the Richardson s equations for one pair, namely can be rewritten as Ω 2e E = G, (63) < (p) Â >= C (p) Ω 2e E(p) = C (p) G. (64) To explore the strong coupling behaviour of the above, we first consider the v l = 2 case. Then the energy E(p) will assume, for G, some value trapped in between two s.p.l.: correspondingly the G-dependent normalisation constant C (p) will tend to a finite value C (p) and the asymptotic behaviour of the matrix element will read which goes to zero like G. < (p) Â > C (p) G, (65) 25

26 On the contrary, in the v l = case, where the unperturbed pair lives on the lowest level, E() GΩ and, for large G, we find C () G Ω and consequently < ()  > Ω, (66) in accord with (6). Thus at large G, for this matrix element, v l and v coalesce. Now we study the transition from one to two pairs, namely the matrix element of  between B (p) and B (m, m 2 )B 2(m, m 2 ): clearly the lack of normalisation of the latter should be accounted for. For this purpose a simple, although tedious, calculation yields < (m, m 2 )  (p) >= [ () ν [ β (m, m 2 ) β ν (2) (m, m 2 )] β (p) Ω ν + β ν (p) ] Ω 2 β (p)δ ν Ω the normalisation constant reading N, (67) N 2 = + ν [( β() ) (m, m 2 ) β ν (2) (m, m 2 ) β(2) (m, m 2 ) β ν () (m, m 2 ) 2 β() Ω (m, m 2 ) 2 ] β(2) (m, m 2 ) 2 δ ν. (68) Using the Richardson equations to get rid of the sums, (67) can be recast as follows < (m, m 2 )  (p) >= 2C (m, m 2 )C 2 (m, m 2 )C (p) N G[(E (m, m 2 ) E(p)][(E 2 (m, m 2 ) E(p)], (69) which is real since E and E 2 are either real or complex conjugate. The normalisation constant N will be computed in Appendix A. The amplitudes (69) are displayed in fig. 6 in the case of four s.p.l. of a harmonic oscillator well for the transitions from a one-pair system with p = (the ground collective state) or with p = 2 (the first excited trapped state) to a two pairs state with like-seniority (namely (m, n) = (, 2)) or to a two pairs state with like-seniority 2 and characterised by (m, n) = (2, 2). 26

27 G ~ G ~ G ~ G ~ Figure 6: Transition matrix elements. Upper left panel: transition from p = (collective) state to (m, n) = (, 2) (v l = ) state; upper right panel: transition from p = (collective) state to (m, n) = (2, 2) (v l = 2) state; lower left panel: transition from p = 2 (trapped) state to (m, n) = (, 2) (v l = ) state; lower right panel: transition from p = 2 (trapped) state to (m, n) = (2, 2) (v l = 2) state. An asterisk denotes the position of the critical point. 27

28 Of these transitions we shall explore the behaviour around the critical values of G. First consider the weak coupling limit. Then eq. (67) simply provides < (m, n) Â (p) > G Ωm δ np + Ω n δ mp 2 δ mp δ np Ωm ( + 2 ), (7) δ mn Ω m which indeed is seen to occur in fig. 6, where one recognises for G the values 3 for the transition p = (m, n) = (, 2), for the transition p = (m, n) = (2, 2), for the transition p = 2 (m, n) = (, 2) and finally 2 for the transition p = 2 (m, n) = (2, 2). The behaviour around the critical points (in our cases G + cr =.7 and G cr =.287 for (m, n) = (, 2) and (m, n) = (2, 2), respectively) is quite delicate since both the numerator and the denominator are singular when E and E 2 tend, e.g., to 2e. In fact the numerator in (67) is vanishing like x 2 (see eqs. (36) and (39) for the definition of x), but so does the denominator. The transition (67) for small x thus becomes < (m, n) Â (p) > G G ± cr 2 G ± cr Ω β (p) 2ǫ 2ǫ ± β (p)p (2) 6P 4 (2) ± 8P (2)P 3 (3) + 2P4 (4). (7) The value of the transition matrix elements at the critical value of G is marked by an asterisk in fig. 6. Finally we comment the large G limit, which is ruled by eqs. (69) (and eqs.(74),(76)). First we observe that here the normalisation constant N (see eq.(76) in the Appendix) is always finite, as it should, and in fact, using the asymptotic expressions for E and E 2, turns out to read 2 2 for v Ω l = N 2 for v Ω l = 2 (72) not analytically available for v l = 4. Considering next in some details the transitions matrix element (69) from a one pair state with like-seniority or 2 to a two pairs state with likeseniority, 2 or 4, we find the following: 28

29 i) case of a one-pair initial state with v l =. As already observed this state, for large G, has E GΩ and N G Ω. The two-pairs final state may have: a) v l = : here, using the expressions for the pair energies E,2 and the normalisation constants C,2, one finds the asymptotic result 2(Ω ), which coincides with the degenerate one for zero v (not v l!). b) v l = 2: here one solution (say E ) behaves as GΩ, while the other (E 2 ) remains finite, as described in sec Specifically, one finds E G () eff Ω ΩG Ω + 2 and E 2 E, E. The normalisation constants are C = G () eff Ω and C2 (this one does not have a simple expression, but is G- independent and ). Using the above and the expression (72) for N, one gets for the matrix element the asymptotic value C 2 G Ω Ω 2, which is vanishing as G in the strong coupling limit, as it should, entailing a variation v l = 2 of like-seniority. c) v l = 4: no simple expressions for large G are available for C, C 2 and N, but their limiting values are. Furthermore, E and E 2, remaining trapped, are very small with respect to E. Thus the transition amplitude is deduced to be C C 2 N v l =4 2 G 2 Ω Ω. Note that this v l = 4 transition matrix element behaves as G 2. ii) Case of a one-pair initial state with v l = 2. For this state here the limiting values of C and E (denoted by C and E ) are finite: C is of order and E is of the order of the unperturbed levels. Again, the two-pairs final state may have: 29

30 a) v l = : in this case E is negligible with respect to E and E 2 and, since the asymptotic behaviour of these is known, the matrix element reads 2C G Ω(Ω ). Again a like-seniority transition v l = 2 entails a behaviour of the order G, but in this case the extra factor Ω induces a faster decrease of the transition matrix element, as clearly apparent in fig. 6 (lower left panel). b) v l = 2: this case is delicate. Defining E and E 2 (and likewise C and C 2 ) the limiting values of E and E 2 (C and C 2 ), we find for the asymptotic matrix element the expression 2C 2 C Ω 2G(E E 2 ). Now two different possibilities can occur: if both E and E2 are not confined in the range (2e, 2e ) then, in the strong coupling regime, their difference remains finite and the same occurs for the normalisation coefficients C and C2. Hence the matrix element is ruled by the factor /G and thus vanishing in this limit. Otherwise the difference E E2 tends to vanish. In fact, let E(G) be the solution of the one pair equation (63). Then E 2, in the strong coupling limit, solves eq. (3b), and, as a consequence, one has E 2 = E(G (2) eff ), with G(2) eff given by eq. (33). Thus, inserting E 2 = E y into eq. (3b) and expanding in y, one finds, using (63), y = 2 C 2 GΩ. Hence the G limit of the transition matrix element becomes C2 Ω. C Ω 2 To get the ratio C2 /C we derive (63) with respect to G and (3b) with respect to G (2) eff in order to pick out the normalisation constants. Taking then the ratios of lhs and rhs of the equations 3

31 thus obtained and recalling that E(G) has the same functional form of E 2 (G (2) eff ) we obtain C 2 eff ) C = G(2) G = 2 Ω. Therefore the asymptotic matrix element reads Ω 2, which coincides with the degenerate case. c) v l = 4: in this case all the quantities entering eq. (69) are finite and not coincident, hence the behaviour of the transition matrix element is ruled /G and vanishes in the strong coupling limit. 8 Conclusions The physics and the mathematics of the pairing Hamiltonian, viewed as a reduced version of the BCS model of superconductivity, has been recently remarkably well addressed by Sierra [7] and Dukelsky, Esebbag and Schuck [8]. These authors stress the importance and complement the finding of Cambiaggio, Rivas and Saraceno [9] concerning the integrability of the pairing Hamiltonian, essentially stemming from the classical Yang-Baxter equation. Moreover they shed light on the vast domain of connections linking the pairing Hamiltonian to the conformal field theory, to the renormalization group and to the Chern-Simons theory. In this paper we start from the Richardson equations, to which all the advanced approaches above referred to lead, to explore whether hidden aspects of the solutions of these equations still exist (and worth of finding). We believe this to be the case partly because we have indeed already found [] in the case of one pair living in many s.p.l. a transition between two different regimes occurring in correspondence of a critical value of the coupling constant G. Such a transition reflects the competition between the pairing force and the mean field and is neatly apparent when an infinite number of levels of the latter is accounted for (if only a finite number of s.p.l. are considered yet precursor phenomena of the transition show up). To our knowledge this transition was not previously brought to light. But also in the case of two pairs living in many s.p.l. new hidden physics is emerging, as the present paper shows. First we argue that a subtler classification of the eigenstates of the pairing Hamiltonian is useful. Thus the v = states ruled by the Richardson 3

32 equations are further classified in terms of the number (not a quantum one) v l, which accounts for their degree of collectivity. As our analysis shows, if G is small there is no point in introducing v l and, on the other hand, if G is very large, then v and v l coincide. If, however, G is intermediate, then the physics of the pairing Hamiltonian acting on a set of s.p.l. is quite different from the physics of the pairing Hamiltonian in the degenerate case. This difference is particularly marked in correspondence of a critical value of G (the lowest one of a pair): the hypothesis that this critical value of G coincides with or relates to the one found in [] is tempting and it will be explored in a forthcoming paper. Here we like to dwell on the origin of this pair of critical values of G which characterise each s.p.l., providing its degeneracy is the lowest possible one, namely Ω =. Their existence was already found by Richardson [4] in the n pairs problem, by means of a somehow more involved procedure. In our approach, that we believe more in touch with physics, we not only re-obtain the Richardson results, but we also explore the wave functions behaviour and the transition amplitudes, focusing our attention around the critical values of the pairing coupling constant. We find out that, while the pair energies and the wave functions components have a singularity actually a branch point for G = G cr, the physically relevant quantities, namely the total energy and the transition amplitudes, are regular around G cr, i.e. no phase transition occurs. The remarkable point about these critical values lies in their correspondence to the trapping mechanism, which, so rigid in the one pair case, can be circumvented in the two pairs case. It is thus natural to surmise that the Bose condensation phenomenon and, as a consequence, the related evenodd effect occur in nuclei through this mechanism. This issue also deserves further studies. A Appendix In this Appendix we deduce, whenever possible, analytic expressions for the normalisation constant. When both the pair energies are real there are no simple formulas for the normalisation coefficients C,2 (m, n) are available, whereas when E = E 2 one can use eq. (35) to get C,2 = E E (73)

33 Accordingly the expressions for the normalisation constant can be made simpler by distinguishing whether the solutions are real or complex conjugate. In the former case a tedious calculations provides N = 2[C2 (m, n) + C 2 2(m, n)] [E 2 (m, n) E (m, n)] 2, (74) still in terms of the unspecified normalisation constants C and C 2. In the latter case the normalisation is known, but, to get N, we need to define C k (m, n) = Ω (2e E k (m, n)) 2, k =, 2 (75) where, unlike in (5), no absolute values appear: thus the C k (m, n) are complex and C 2 = C. With the help of (75), and using (73), one then gets N = E E C 2( C 2 + C 2 2) C2 (E E 2 ). (76) 2 33

34 References [] M. Barbaro, R.Cenni, A. Molinari and M.R. Quaglia, Phys. Rev. C (22). [2] P.W. Anderson, J. Phys. Chem. Solids, 28 (959). [3] R.W. Richardson and N. Sherman, Nucl. Phys. 52, 22 (964). [4] R.W. Richardson, Jour. Math. Phys. 6, 34 (965). [5] M.B. Barbaro, A. Molinari, F. Palumbo and M.R. Quaglia, Phys. Lett. B476, 477 (2). [6] D. Bes and R.A. Broglia, Lezioni di Varenna International School of Physics Enrico Fermi, Course LXIX, edited by A. Bohr and R.A. Broglia, Varenna, 977, p.59. [7] G. Sierra arxiv:hep-th/4. [8] J. Dukelsky, C. Esebbag and P. Schuck, Phys. Rev. Lett. 87 (2) 6643 [arxiv:cond-mat/7477]. [9] M. C. Cambiaggio, A. M. Rivas and M. Saraceno, arxiv:nucl-th/