Order parameter, correlation functions, and fidelity susceptibility for the BCS model in the thermodynamic limit

Transcription

1 PHYSICA REVIEW B 89, 3452 (204) Order parameter, correlation functions, and fidelity susceptibility for the BCS model in the thermodynamic limit Omar El Araby,2 and Dionys Baeriswyl,3 Department of Physics, University of Fribour, Chemin du Musée 3, CH-700 Fribour, Switzerland 2 Institute for Theoretical Physics, University of Amsterdam, Science Park 904, N-090 Amsterdam, the Netherlands 3 International Institute of Physics, Natal-RN, Brazil (Received 9 April 203; revised manuscript received 2 April 204; published 29 April 204) The exact round state of the reduced BCS Hamiltonian is investiated numerically for lare system sizes and compared with the BCS ansatz. A canonical order parameter is found to be equal to the larest eienvalue of Yan s reduced density matrix in the thermodynamic limit. Moreover, the limitin values of the exact analysis aree with those obtained for the BCS round state. Exact results for the round-state enery, level occupations, and a pseudospin-pseudospin correlation function are also found to convere to the BCS values already for relatively small system sizes. However, discrepancies persist for a pair-pair correlation function, for interlevel correlations of occupancies and for the fidelity susceptibility, even for lare system sizes where these quantities have visibly convered to well-defined limits. Our results indicate that there exist nonperturbative corrections to the BCS predictions in the thermodynamic limit. DOI: 0.03/PhysRevB PACS number(s): F I. INTRODUCTION The microscopic theory of Bardeen, Cooper, and Schrieffer (BCS) [] represents aruably the central paradim of superconductivity, but it plays also a crucial role for superfluid helium-3 [2], ultracold ases of fermionic atoms [3,4], atomic nuclei [5], and neutron stars [6]. The theory involves two elements: on the one hand the so-called reduced BCS Hamiltonian, where only scatterin processes between zeromomentum pairs of fermions are taken into account, on the other hand a variational ansatz for the round state of this Hamiltonian, a coherent superposition of products of pair wave functions. In this paper we address the question whether the BCS ansatz is the exact round state of the reduced BCS Hamiltonian in the limit of an infinitely lare system size. An early arument for the asymptotic validity of the BCS ansatz was iven by Anderson [7], who pointed out that BCS theory should be nearly valid because in the limit of lare numbers quantum fluctuations die out. ater explicit calculations showed that indeed the round-state enery, level occupation, and the free enery were exactly predicted by BCS theory in the thermodynamic limit [8 0]. Moreover, for a specific sinle-particle spectrum ( step model ) Mattis and ieb concluded that the BCS wave function was exact in this limit []. Our results, based on Richardson s exact solution of the reduced BCS Hamiltonian [2,3], confirm that many quantities, for instance level occupancies or the roundstate enery, are predicted accurately by BCS theory in the thermodynamic limit. This is also true for an order parameter, defined accordin to Yan s concept of off-diaonal lonrane order (ODRO) [4]. However, for other quantities, such as a pair-pair correlation function, interlevel occupancy fluctuations, and the fidelity susceptibility, BCS predictions are found to differ from the (numerically) exact results, even for very lare system sizes. The paper is oranized as follows. Section II describes Richardson s exact solution for the eienstates of a simplified form of the reduced BCS Hamiltonian. Section III deals with the round state, on the one hand in BCS approximation, on the other hand by evaluatin the exact solution numerically. The exact round-state enery is shown to approach rapidly the BCS prediction as a function of system size. In Sec. IV it is shown that Yan s ODRO is encoded in a canonical order parameter, which is found to convere to the BCS result in the thermodynamic limit. Correlation functions involvin hihest occupied molecular orbitals (HOMOs) and lowest unoccupied molecular orbitals (UMOs) are calculated in Sec. V. While for pseudospin operators BCS theory is aain found to aree with the limit of the exact solution, this is not true for pair operators or for level occupancies. A similar discrepancy is found for the round-state fidelity susceptibility, as shown in Sec. VI. The results are summarized in Sec. VII. II. HAMITONIAN AND ITS EXACT EIGENSTATES The reduced BCS Hamiltonian in the form introduced by Richardson [2] for describin nucleons coupled by pairin forces is H = ε c σ c σ c μ c μ c c, () σ μ,,μ where c σ and c σ are, respectively, creation and annihilation operators for fermions in level with spin σ and ε = W 2 + W (2 ), =,...,. We use the width W as 2 unit of enery, W =, and of course assume >0. The Hamiltonian has particle-hole symmetry, and therefore the chemical potential vanishes if the number of fermions equals (half fillin), the case considered in this paper. The calculations presented below can readily be performed for other forms of the sinle-particle spectrum, for instance for the tiht-bindin spectrum of the square lattice, but to discuss the eneric lare behavior it is advantaeous to choose a spectrum exhibitin neither deeneracies nor Van Hove sinularities. In the reduced BCS Hamiltonian () all levels are coupled equally, i.e., the interaction has infinite rane for in the space of quantum numbers (in k space for Bloch electrons) /204/89(3)/3452(0) American Physical Society

2 OMAR E ARABY AND DIONYS BAERISWY PHYSICA REVIEW B 89, 3452 (204) In classical statistical mechanics infinitely lon-rane interactions are enerally believed to be treated exactly by mean-field theory. This suests that the mean-field description of BCS for the Hamiltonian () is also exact in the thermodynamic limit. There is however a loophole in this arument. A quantum system in d dimensions corresponds to a classical system in d + dimensions. On the additional axis representin time the interaction does not have to be lon-raned. Therefore it is worthwhile to investiate the lare limit of Richardson s exact solution in detail. The eienstates of the Hamiltonian () can be classified accordin to the number of sinly occupied levels. The round state belons to the subspace where all levels are either doubly occupied or empty ( even). Within this subspace the operators c σ c σ are identical to b b, where b = c c and b = c c (2) create and annihilate pairs, respectively. Therefore the level occupancy can be written as n := c σ c σ = 2b b σ (3) and the Hamiltonian () is equivalent to H = 2 ε b b b μ b (4) μ,,μ in the subspace where sinle occupancy is forbidden. The operators b,b, and n can be combined to pseudospin operators s [7] with components s x = 2 (b + b ), s y = i 2 (b b ), (5) s z = 2 (n ). In terms of these operators the Hamiltonian (4) reads H = 2 ɛ s z s μ+ s, (6) μ, μ where s μ± = s x ± is y, and represents an XY ferromanet with lon-rane interaction in an inhomoeneous transverse field. This Hamiltonian is part of a larer family of interable models, for which eienstates and eienvalues were found by Gaudin [5]. Interability means that there exist operators R, =,...,, which commute amon themselves and with the Hamiltonian. For our model the R operators are [6] R = s z + μ,μ s μ s ɛ μ ɛ. (7) One readily verifies that for the case considered here (N = ) the Hamiltonian (6) can be written as H = 2 ɛ R, (8) = which therefore also commutes with all operators R. The exact eienstates of the Hamiltonian () form = 2 pairs have the form [2,3] R = M B i 0, B i = i= = 2ε λ i b, (9) where 0 is the vacuum state, c σ 0 =0, and the rapidities λ i satisfy the Richardson (or Bethe) equations = 2ε λ k M i,i k 2 λ k λ i = 0. (0) The systems for which these equations can be directly solved are rather small, but recent alorithmic proress [7] allows us to study much larer sizes than before. Analytical insiht has been provided by Gaudin [8] in the continuum limit ( ), usin an analoy to electrostatics. His result was used to show [9,20] that the BCS equations for the ap, the chemical potential, and the round-state enery are reproduced in the thermodynamic limit. The low-enery excitations have also been obtained by solvin Richardson s equations analytically in the stron-couplin limit [2]. III. GROUND STATE AND GROUND-STATE ENERGY The conventional BCS round state is defined as where u = BCS = (u + v b ) 0, () E + ε E ε, v =, (2) 2E 2E with E = (ε ) /2. The ap parameter is determined by minimizin the enery expectation value. For the present model we obtain = ( 2sinh ) (3) in the limit. The BCS state can also be written as BCS e B 0, B = v b u. (4) Its projection on a subspace with a definitive number of M pairs, (M) BCS (B ) M 0, resembles the Richardson solution (9), but, as emphasized by Combescot and collaborators [22], in the BCS state all pair operators are equal (B ), while they are all different in the Richardson solution (B i ). In the thermodynamic limit the conventional and numberprojected BCS round states are expected to be equivalent, but for finite they differ, especially for weak couplins. Thus conventional BCS theory predicts a phase transition at a critical couplin strenth c (), below which the ap parameter vanishes, while there exists only a crossover for the number-projected BCS round state [23,24]. The behavior is completely smooth for the exact solution [24]. Here we concentrate on the reion > c (), where we can expect the different round states to mere. For lare the critical couplin strenth is approximately iven by

3 ORDER PARAMETER, CORREATION FUNCTIONS, AND... PHYSICA REVIEW B 89, 3452 (204) E BCS FIG.. (Color online) Ground-state enery of the reduced BCS Hamiltonian. Symbols represent the exact result for different system sizes while the dashed line stands for the BCS result in the thermodynamic limit. c () [ln(2) + γ ], where γ is Euler s constant (γ = ). The exact round state can be analyzed numerically by scannin the Richardson equations from = 0uptosome finite value >0. The computations are reatly simplified by introducin the variables =, (5) 2ɛ i λ i which satisfy the substituted Bethe equations [7] 2 μ = 0. (6) 2 ɛ μ ɛ μ,μ These quadratic equations can be readily solved for much larer system sizes than the oriinal Richardson equations (0). Some quantities are simple functions of. Thus the roundstate enery E 0 () is iven by the formula M E 0 = λ i + 2 = 2ɛ M 2. (7) i= = Results for different sizes are shown in Fi.. As expected, the curves convere very rapidly towards the asymptotic limit of BCS theory, E0 BCS 4 coth,. (8) IV. ORDER PARAMETER The order parameter of conventional BCS theory [25], F = 0 b 0 =, (9) vanishes for a definitive number of particles and one has to search for alternatives. A canonical pairin parameter has been proposed by von Delft et al. [26] and adopted in other studies [27,28], can = ( b b c c c c ) /2, (20) where we have used the notation O := 0 O 0. Numerical calculations for the exact round state [24] indicate that can F for. However, can does not probe phase coherence and therefore the quantity can = ( b b c c c c ) (2) was juded to be more adequate [29]. Within BCS theory one has can BCS = ( ) 2 2E = 2 arctan for. (22) 2 This expression reaches a finite limitin value (/4) for, while both F and can increase indefinitely with and represent asymptotically a pair bindin enery rather than a measure of order. The pseudospin operators (5) can be used to rewrite the order parameter can. First we notice the eneral relation b b = 2 3 s ( n ). (23) It is easy to see that both the BCS ansatz () and the exact round state (9) are eienstates of s 2 with eienvalue 3 4 and we may write can = n (2 n ). (24) 4 can = 0 for the filled Fermi sea, where n vanishes for ε >ε F and is equal to 2 for ε <ε F. Therefore this order parameter measures deviations from the level distribution of the filled Fermi sea. This is very satisfactory, at the same time there exist other Fermi surface instabilities leadin to similar level redistributions as superconductivity. Hence can is somewhat less specific than Gorkov s order parameter F, which is based on the breakin of aue symmetry. In another proposal, inspired by Yan s ODRO, the correlation functions C μ = 0 b μ b 0 (25) are summed up to yield the parameter [27] OD = C μ, (26) μ, which is equal to F 2 for the BCS round state in the limit. However, Yan s concept of ODRO is based on the larest eienvalue of the matrix C rather than on the sum of its matrix elements. Thus ODRO exists if (and only if) C has an eienvalue of the order of the particle number, i.e., of the order in the present case. This is indeed true for the conventional BCS round state, for which the correlation functions are iven by (half fillin) { E ɛ 2E C μ =,μ=, (27) 2 4E μ E,μ. To find the eienvalues of the matrix C we have to calculate the determinant C ωi where I is the unit matrix. Introducin the quantities f = 2E, = E ɛ 2E (28)

4 OMAR E ARABY AND DIONYS BAERISWY PHYSICA REVIEW B 89, 3452 (204) ω max pairin strenth ω max can BCS FIG. 2. (Color online) arest eienvalue of the matrix C as a function of for different couplin strenths, = 0.,...,.5. we can write C ωi = FAF where F is diaonal with F = f and { a, μ =, A μ = (29), μ, with a = ( ω)/f 2. Thus the eienvalues of C are iven by the zeros of det A = (a ) +. (30) a μ Toether with = μ= a = 2 [ (E ɛ ) 2 4E 2 ω] (3) we arrive at the eienvalue equation 2 + = 0. (32) (E = ɛ ) 2 4E 2 ω For eienvalues ω of order the summand has to chane sin somewhere between = and =. In turn, if all the terms can BCS FIG. 3. (Color online) Order parameter can as a function of couplin strenth for different system sizes. The dashed line represents the BCS result in the thermodynamic limit FIG. 4. (Color online) Pairin strenth, as measured by can / (diamonds) and lim (ω max /) (dots). The full line retraces the BCS result. in the sum are neative, the eienvalue has to be of order, for which in the limit Eq. (32) implies ωmax BCS = ( = 2E ) 2 = BCS can. (33) To find out whether this remarkable equality of order parameter and larest eienvalue of the reduced density matrix remains valid beyond BCS theory, we have calculated both ω max and can for the exact round state. The matrix elements C μ can be expressed as sums of certain determinants which depend explicitly on the rapidities λ i [28] and are not simple functions of the quantities. Nevertheless it turns out to be advantaeous to solve first the quadratic equations for and then use the procedure outlined in Ref. [30] to extract the rapidities. Results for the larest eienvalue ω max of C are depicted in Fi. 2 as functions of for various couplin strenths. A linear behavior is clearly observed already for modest system sizes with slopes that aree perfectly well with BCS theory. Fiure 3 shows the exact results for the quantity can /, which also converes rapidly towards the BCS prediction as increases. Therefore the relation ω max = can is also found to hold for the exact round state and ω max can be used interchaneably as order parameter. The results are summarized in Fi. 4 where the exact values for ω max / and can / at lare are seen to aree both with each other and with BCS theory. We conclude that the natural canonical order parameter can be defined either by Eq. (2) or as the larest eienvalue of the reduced density matrixc. Both quantities are faithfully predicted by BCS theory. V. CORREATION FUNCTIONS We have shown above that the round-state enery E 0, the larest eienvalue ω max of the reduced density matrix C, and the order parameter can are correctly predicted by BCS theory as the system size tends to infinity. The same is true for the level occupancy n, i.e., for the diaonal elements of C. But what about nondiaonal matrix elements of C, i.e., correlation functions 0 b μ b 0 with μ? To answer this question we have studied the special case where μ is the

5 ORDER PARAMETER, CORREATION FUNCTIONS, AND... PHYSICA REVIEW B 89, 3452 (204) C H S H FIG. 5. (Color online) HOMO-UMO pair-pair correlation function. Symbols represent the exact solution, while the dashed lines stand for the BCS result (with increasin from riht to left). lowest unoccupied molecular orbital (UMO) and is the hihest occupied level (HOMO), i.e., ε μ = ε = (2). For the conventional BCS round state we find CH BCS = ( ) 2 /[ + (2 ) 2 ], where represents the ap parameter for levels (M = pairs). CBCS 2 H vanishes for < c() and is finite for > c (). Results for the exact round state are shown in Fi. 5 and compared to the BCS predictions. There is ood areement for lare couplin strenths but, in contrast to BCS, slihtly above c () there is a peak which does not decrease with increasin system size. We have extracted both the peak values C max and the locations of the maxima max by fittin the numerical data with polynomials. The results shown in Fi. 6 confirm that the maximum saturates at a value of about 0.30 and its location max tends to a very small value, consistent with 0. While BCS theory predicts a simple step at = 0ofsize in the thermodynamic limit, our analysis 4 indicates that the exact solution exhibits a larer step at = 0, followed by a smooth decrease towards the asymptotic stron-couplin limit 4. As a second example we consider the pseudospinpseudospin correlation function max C max S μ = 0 s μ s 0, (34) max ln ln FIG. 6. (Color online) Maximum C max and its location max for the HOMO-UMO pair-pair correlation function FIG. 7. (Color online) S H as a function of couplin strenth for different system sizes. for which BCS theory predicts Sμ BCS = 2 δ μ + ɛ μɛ + 2. (35) 4E μ E For the HOMO-UMO levels we et SH BCS = 4 [(2 )2 ]/[(2 ) 2 + ], which tends to for. It is straihtforward to calculate S μ for the exact round state usin the R 4 operators defined by Eq. (7). The round state is an eienstate of these operators with eienvalues r = 4 μ,μ ɛ μ ɛ +, (36) where is iven by Eq. (5). Usin the Hellman-Feynman theorem for r = 0 R 0 we find (for μ ) r = ɛ μ 4 (ɛ μ ɛ ) + = 2 0 R 0 ɛ μ ɛ μ = (ɛ μ ɛ ) 0 s 2 μ s 0 (37) and therefore S μ = 4 (ɛ μ ɛ ) 2. (38) ɛ μ The pseudospin-pseudospin correlation function depends only on the quantities (and not explicitly on the rapidities λ i ) and therefore can be readily evaluated for lare system sizes. Fiure 7 shows results for the HOMO-UMO correlation function S H, in comparison with the BCS prediction. Clearly the exact results for S H approach the BCS prediction for > c (), and the curves mere more and more rapidly as increases. These results indicate that the pseudospinpseudospin correlation function is reproduced exactly by BCS theory for any value of in the thermodynamic limit. The pair-pair correlation function (25) can be written in pseudospin lanuae as C μ = 0 (s μx s x + s μy s y ) 0 = S μ 0 s μz s z 0, (39) where the (particle-hole) symmetry of C has been used

6 OMAR E ARABY AND DIONYS BAERISWY PHYSICA REVIEW B 89, 3452 (204) N H FIG. 8. (Color online) HOMO-UMO occupancy fluctuations for different system sizes. Accordin to Eq. (5) 0 s μz s z 0 measures correlations between level occupancies. It is illuminatin to consider fluctuations of these correlations, i.e., N μ := 0 (n μ n μ )(n n ) 0. (40) N μ vanishes accordin to BCS, but, in view of our previous findins for C μ and S μ it should differ from BCS and thus remain finite for the exact round state, even in the thermodynamic limit. This is indeed found by our numerical analysis, as shown in Fi. 8 for the HOMO-UMO occupancy fluctuations, which exhibit a pronounced minimum located slihtly above c (). While the location of the minimum min moves to the left as the system size increases its value N min remains essentially constant. This is clearly seen in Fi. 9 where N min and min are plotted aainst and /ln, respectively. For lare min.2/ln, in close areement with the correspondin behavior of the pair-pair correlations (Fi. 6). In order to understand better this behavior we have performed a perturbative analysis about the BCS mean-field round state. Details are iven in Appendix A. We also find clear minima, as shown in Fi. 0, but in contrast to the exact analysis not only do the locations of the minima decrease with but also their values. This can be seen explicitly from the min N min ln ln FIG. 9. (Color online) Minima N min of HOMO-UMO occupancy fluctuations (inset) and correspondin locations min as functions of (ln). N H FIG. 0. (Color online) HOMO-UMO occupancy fluctuations accordin to first-order perturbation theory about the BCS round state. The dashed line represents the asymptotic behavior N min () min. first-order result N () H = (2 )2 [ + 2( ) 2 ]. (4) [ + (2 ) 2 ] 5/2 For lare values of the dominant dependence of this function is throuh the ap parameter, with a minimum for (2 ) 2 = 5. Moreover the minimum value is simply proportional to its location, N min min.forlare and small we can use the relation e / and obtain N min () ln (42) We see that in first-order perturbation theory the minimum value of these fluctuations tends loarithmically to zero as a function of system size, while it remains constant in a full treatment. This suests that first-order perturbation theory becomes more and more unreliable when approachin criticality, i.e., for, c () 0. We expect therefore that in the thermodynamic limit the critical behavior exhibits nonperturbative corrections beyond the BCS meanfield behavior. VI. FIDEITY SUSCEPTIBIITY A sensitive probe of fluctuations is the fidelity susceptibility χ F, which is often used in the context of quantum phase transitions [3,32] and can also characterize crossover phenomena [33,34]. For the reduced BCS Hamiltonian χ F may be defined as χ F () = 2 lim lnf (,δ), (43) δ 0 (δ) 2 where the fidelity F (,δ) is equal to the overlap 0 () 0 ( + δ) between round states associated with infinitesimally close-couplin constants. χ F () can be represented with respect to the eienstates n () of the Hamiltonian with couplin constant, by usin ordinary perturbation theory in powers of δ. One finds χ F () = n 0 μ,,μ 0 () b b μ n () 2 [E 0 () E n ()] 2. (44)

7 ORDER PARAMETER, CORREATION FUNCTIONS, AND... PHYSICA REVIEW B 89, 3452 (204) χ F χ F FIG.. (Color online) Fidelity susceptibility of BCS theory for = 0 2,0 3,0 4,0 5 (from riht to left) ln FIG. 3. (Color online) Fidelity susceptibility as a function of / ln for different couplin strenths. The BCS results are iven at / ln = 0. Therefore, in contrast to the correlation functions studied in Sec. V, the fidelity susceptibility probes all the eienstates of the Hamiltonian and not only the round state. The enery eienvalues E n () convere to the BCS values for, but this may not be true for all the matrix elements in the numerator. In conventional BCS theory the fidelity susceptibility can be obtained analytically. For the case studied here we obtain χ BCS F () = ( ) d 2 ε 2. (45) d 4 E 4 This function vanishes for < c () and diveres if approaches c () from above, as shown in Fi.. The size of the sinularity at c () decreases with increasin and disappears for, where χ F is iven by χf BCS () = [ ( )arctan ] , (46) with the asymptotic behavior χ BCS F () π 8 2 e / (47) for 0. There is no diverence at the critical point in the thermodynamic limit, instead there is a broad maximum for 0.26, representin a crossover from the small to the lare behavior. In the Bethe-ansatz framework the fidelity F (,δ)is iven by the determinant of an matrix [35], from which the fidelity susceptibility is calculated usin Eq. (43). Fiure 2 shows the exact results obtained in this way for different system sizes in comparison with the BCS result for.we observe a rapid converence to a limitin curve for > c (). This is confirmed by a detailed data analysis, illustrated in Fi. 3. Clearly the exact fidelity susceptibility levels off at a different value than the BCS prediction. The difference is larest around 0.26 (more than 50%), which is also the reion where both BCS and exact results exhibit a maximum NBCS 50 NBCS 00 NBCS 200 χ F BCS χ F NBCS 500 BCS FIG. 2. (Color online) Fidelity susceptibility as a function of couplin strenth. Symbols on the lower curves represent numerical results for the exact round state and various values of. The dashed line stands for the BCS result for FIG. 4. (Color online) Fidelity susceptibility as a function of couplin strenth for the number-projected BCS round state and different system sizes. The dashed line represents the conventional BCS result for

8 OMAR E ARABY AND DIONYS BAERISWY PHYSICA REVIEW B 89, 3452 (204) One may wonder whether the discrepancy between BCS and exact results for the fidelity susceptibility disappears if, instead of the conventional BCS ansatz, we use the numberprojected state (M) BCS. To deal with the number-projected BCS ansatz we have adapted a recursive scheme, used previously for calculatin the round-state enery [24]. Details are iven in Appendix B. The results shown in Fi. 4 indicate a clear converence between conventional and projected BCS states. Therefore the discrepancy between BCS and the exact solution cannot be removed by replacin the conventional BCS ansatz by the number-projected state. VII. DISCUSSION In this paper we have studied the exact round state of the reduced BCS Hamiltonian for lare system sizes. We have confirmed that both the round-state enery E 0 and the level occupancies n aree with the BCS predictions in the thermodynamic limit. A canonical order parameter can, defined either throuh the concept of ODRO or by Eq. (2), was also found to tend asymptotically to the BCS value. The same turned out to be true for a pseudospin-pseudospin correlation function. These results support the conventional wisdom accordin to which the mean-field treatment of the reduced BCS Hamiltonian is exact in the thermodynamic limit. However, we did find counterexamples for which the exact results differ from those of BCS theory in this limit, namely the fidelity susceptibility, a pair-pair correlation function, and interlevel occupancy fluctuations. In this sense the BCS round state is not exact in all respects. The lare behavior of the two correlation functions for which discrepancies have been found suests that fluctuations produce nonperturbative corrections to mean-field critical behavior for, 0. It would be very interestin to explore this possibility in more depth, for instance usin field-theoretical techniques. Another direction of research could be the calculation of dynamic response or correlation functions, for which the discrepancies may be stroner and at the same time easier to measure than the quantities considered here. We have limited ourselves to s-wave pairin, but an interable model with p + ip pairin [36] could also be analyzed in a similar way. We do not expect any sinificant differences because for p + ip pairin the density of states around the Fermi enery is completely apped, as for s-wave pairin. An interestin case where the discrepancy between BCS and exact round state could be more severe than in these interable systems would be a pairin Hamiltonian where the ap parameter has nodes on the Fermi surface (as for d-wave pairin in two dimensions). ACKNOWEDGMENTS We are rateful to V. Gritsev for both his continuous interest in our work and helpful suestions. At the initial stae of our studies we have profited from the experience of B. Gut, who presented related issues in his Ph.D. thesis. We also thank A. Faribault, E. Yuzbashyan, and S. de Baerdemacker for stimulatin discussions, as well as W. Zwerer for insihtful comments. This work has been supported by the Swiss National Science Foundation. APPENDIX A: PERTURBATION THEORY Booliubov s version of BCS theory is based on the meanfield Hamiltonian H m = ε c σ c σ (c c + c c ), (A) σ which is diaonalized by a unitary transformation from fermion operators c σ to quasiparticle operators γ σ, i.e., H m = 0 + E γ σ γ σ, (A2) σ where E = ε and 0 = E. The mean-field round state m is the vacuum of quasiparticles, γ σ m = 0 for all,σ. The expectation value of the Richardson Hamiltonian () with respect to m ives the mean-field round-state enery E m = ( ) ( ) ε ε (A3) E 4E 2 2E The term of order / in the first sum is neliible in the thermodynamic limit, but for finite it has a small effect on the critical value c (), above which there is a finite ap, and on the value of for > c (). Without this term the minimization of E 0 with respect to yields the familiar ap equation =, (A4) 2 E which will be used in the followin. We have verified that this approximation has neliible effects for lare values of. We now set up a perturbative expansion around the meanfield solution. To do so, we introduce the bare Hamiltonian H 0 = H m + E m 0, (A5) and the perturbation H = c μ c μ c c + (c c + c c ) μ,,μ E m + 0. (A6) The Richardson Hamiltonian is then simply iven by H = H 0 + H (A7) and we may expand in powers of H. We note that the first-order correction to the round-state enery vanishes, m H m =0. The first-order correction to the round state is found to be = ε β 4 m 4 E 3 μ,,μ< (E μ E ε μ ε ) E μ E (E μ + E ) β μ β m, where β := γ γ creates pairs of quasiparticles. (A8)

9 ORDER PARAMETER, CORREATION FUNCTIONS, AND... PHYSICA REVIEW B 89, 3452 (204) It is straihtforward to calculate correlation functions to first order in H. For the pair-pair correlation function (25)we obtain to first order in H C μ = m b μ b m + m (b μ b + b b μ) { ( ) = 2 ε μ + ε2 4E μ E 8E μ E Eμ 3 E 3 + (E μe ε μ ε ) 2 E μ E (E μ + E ) }. (A9) We consider now the special case where the two levels correspond, respectively, to the hihest occupied molecular orbital (HOMO) and to the lowest unoccupied molecular orbital (UMO), i.e., ε μ = ε = /(2). We find C H = ( )2 + 4( ) + + 4( ) 4. (A0) 2 2 [ + 4( ) 2 ] 5/2 Proceedin in the same way for the occupancy fluctuations (40) we find to first order in H N μ = 2 2 E μ E ε μ ε E 2 μ E2 (E μe ). (A) For the special case of HOMO-UMO levels we et N H = (2 )2 [ + 2( ) 2 ]. (A2) [ + (2 ) 2 ] 5/2 APPENDIX B: RECURSIVE METHOD FOR THE NUMBER-PROJECTED BCS STATE The BCS pair operator B = ( ) E ε /2 b (B) E + ε enerates the number-projected BCS round state (M) =(B ) M 0. (B2) Both the norm of the round state and the expectation value of the Hamiltonian can be calculated recursively [24]. We have used the recursive scheme for determinin the ap parameter for iven system sizes = 2M and couplin strenths. We show now how to adapt this method for calculatin the fidelity (M) F (, m m (M) ) = (M) m m (M) (M) m m (M), (B3) where (M) is the round state for the couplin strenth. The action of the operators b,n on (M) is iven by b (M) =Mf (M ) M(M )f 2 b (M 2), n (M) =2Mf b (M ), (B4) leadin to a recursion relation for the norm namely Z (M) = MZ (M ) where Z (M) := (M) (M), f 2 M(M ) f 3 S(M ) (B5), (B6) S (M) := (M) b (M ) (B7) is calculated throuh S (M) = Mf Z (M ) M(M )f 2 S(M ). (B8) Correspondin relations hold for Z (M) and S (M), while the overlap is obtained recursively as where V (M) =MV (M ) V (M) := (M) (M) f f M(M ) (B9) f 2 f W (M ), (B0) W (M) := (M) b (M ). (B) One also needs the quantity Y (M) := (M) b (M ). (B2) The system is closed by the recursion relations for W (M) and Y (M), W (M) = Mf V (M ) M(M )f 2 Y (M ), Y (M) = Mf V (M ) M(M )f 2 W (M ), toether with the initial conditions Z () = f 2, Z () = f 2, V () = S () = Y () = f, S () = W () = f. f f. (B3) (B4) [] J. Bardeen,. N. Cooper, and J. R. Schrieffer, Phys. Rev. 08, 75 (957). [2] A.J.eett,Rev. Mod. Phys. 47, 33 (975). [3] S. Giorini,. P. Pitaevskii, and S. Strinari, Rev. Mod. Phys. 80, 25 (2008). [4] I. Bloch, J. Dalibard, and W. Zwerer, Rev. Mod. Phys. 80, 885 (2008). [5] R.A.BroliaandV.Zelevinsky,50 Years of Nuclear BCS (World Scientific, Sinapore, 203). [6] K. Rajaopal and F. Wilczek, Handbook of QCD (World Scientific, Sinapore, 200), Chap. 35. [7] P. W. Anderson, Phys. Rev. 2, 900 (958). [8] N. N. Booliubov, D. N. Zubarev, and I. A. Tserkovnikov, Sov. Phys. JETP 2, 88 (96). [9] B. Mühlschleel, J. Math. Phys. 3, 522 (962). [0] R. J. Bursill and C. J. Thompson, J. Phys. A: Math. Gen. 26, 769 (993). [] D. Mattis and E. ieb, J. Math. Phys. 2, 602 (96)

10 OMAR E ARABY AND DIONYS BAERISWY PHYSICA REVIEW B 89, 3452 (204) [2] R. Richardson, Phys. ett. 3, 277 (963). [3] R. Richardson and N. Sherman, Nucl. Phys. 52, 22 (964). [4] C. N. Yan, Rev. Mod. Phys. 34, 694 (962). [5] M. Gaudin, J. Phys. (France) 37, 087 (976). [6] M. Cambiaio, A. Rivas, and M. Saraceno, Nucl. Phys. A 624, 57 (997). [7] A. Faribault, O. El Araby, C. Sträter, and V. Gritsev, Phys. Rev. B 83, (20). [8] M. Gaudin, Modèles Exactement Résolus (es Éditions de Physique, es Ulis, 995). [9] R. Richardson, J. Math. Phys. 8, 802 (977). [20] J. Román, G. Sierra, and J. Dukelsky, Nucl. Phys. B 634, 483 (2002). [2] E. A. Yuzbashyan, A. A. Baytin, and B.. Altshuler, Phys. Rev. B 68, (2003). [22] M. Combescot, W. V. Poosov, and O. Betbeder-Matibet, Physica C 485, 47 (203). [23] F. Braun and J. von Delft, Phys. Rev. ett. 8, 472 (998). [24] J. Dukelsky and G. Sierra, Phys. Rev. B 6, 2302 (2000). [25]. P. Gor kov, JETP 9, 364 (959). [26] J. von Delft, A. D. Zaikin, D. S. Golubev, and W. Tichy, Phys. Rev. ett. 77, 389 (996). [27] G.-S. Tian,.-H. Tan, and Q.-H. Chen, Europhys. ett. 50, 36 (2000). [28] A. Faribault, P. Calabrese, and J.-S. Caux, Phys. Rev. B 77, (2008). [29] J. von Delft and D. C. Ralph, Phys. Rep. 345, 6 (200). [30] O. El Araby, V. Gritsev, and A. Faribault, Phys. Rev. B 85, 530 (202). [3] P. Zanardi and N. Paunković, Phys. Rev. E 74, 0323 (2006). [32] S.-J. Gu, Int. J. Mod. Phys. B 24, 437 (200). [33] A. Khan and P. Pieri, Phys.Rev.A80, (2009). [34] B. J. Gut, Ph.D. thesis, University of Fribour, [35] A. Faribault and D. Schuricht, J. Phys. A: Math. Theor. 45, (202). [36] C. Dunnin, M. Ibanez, J. inks, G. Sierra, and S.-Y. Zhao, J. Stat. Mech. (200) P