Parity effect in a mesoscopic Fermi gas
PParity effect in a mesoscopic Fermi gas
Johannes Hofmann, Alejandro M. Lobos,
1, 2 and Victor Galitski
1, 3 Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics,University of Maryland, College Park, Maryland 20742-4111 USA Instituto de F´ısica Rosario, CCT-CONICET, Blvd. 27 de Febrero 210 bis, Rosario, Santa Fe C.P. 2000, Argentina School of Physics, Monash University, Melbourne, Victoria 3800, Australia (Dated: April 1, 2018)We develop a quantitative analytic theory that accurately describes the odd-even effect observedexperimentally in a one-dimensional, trapped Fermi gas with a small number of particles [G. Z¨urn et al. , Phys. Rev. Lett. , 175302 (2013)]. We find that the underlying physics is similar tothe parity effect known to exist in ultrasmall mesoscopic superconducting grains and atomic nuclei.However, in contrast to superconducting nanograins, the density (Hartree) correction dominates overthe superconducting pairing fluctuations and leads to a much more pronounced odd-even effect in themesoscopic, trapped Fermi gas. We calculate the corresponding parity parameter and separationenergy using both perturbation theory and a path integral framework in the mesoscopic limit,generalized to account for the effects of the trap, pairing fluctuations, and Hartree corrections. Ourresults are in an excellent quantitative agreement with experimental data and exact diagonalization.Finally, we discuss a few-to-many particle crossover between the perturbative mesoscopic regime andnon-perturbative many-body physics that the system approaches in the thermodynamic limit.
PACS numbers: 67.85.Lm,68.65.-k
Our understanding of quantum systems is usuallyfirmly rooted in either a few-body picture, where exactsolutions of a few-particle Schr¨odinger equation exist, ora many-body picture, where the system can be describedin a (quantum) statistical framework. In between thesetwo limits lies the mesoscopic regime, where finite par-ticle number and confinement have a strong effect onthe system’s properties. Mesoscopic systems occur nat-urally, for example, in nuclear physics, where a finitenumber of protons and neutrons form an atomic nucleus,or they can be engineered, such as in semiconductingquantum dots [1, 2] or superconducting nanograins [3–6]. For attractively interacting mesoscopic Fermi sys-tems, a key effect is that the ground-state energy is nota strictly convex function of the particle number, butthe interaction can cause some configurations to havelower binding energy (and thus enhanced stability) rel-ative to others [7, 8]. For example, this implies an en-hanced stability of nuclei with a “magic number” of con-stituents. A related effect exists for superconductingnanograins [7, 9, 10]: the binding energy of systems withan even number of spin-up and spin-down fermions (even-number parity) is enhanced compared to odd particlenumber systems with an unpaired fermion (odd-numberparity). This parity effect is a hallmark of mesoscopicsuperconductor systems and can be quantified by the so-called parity or “Matveev-Larkin” parameter [7, 9, 10],which denotes the excess energy of an odd parity staterelative to the mean of the neighboring fully paired evenparity states:∆ P = E l +1 −
12 ( E l + E l +2 ) , (1)where E l +1 denotes the ground-state energy of a fermionsystem with odd total particle number 2 l + 1. For non-interacting systems, the parity parameter (1) vanishes, and it is positive if there is a parity effect.In this Rapid Communication, we study mesoscopicone-dimensional Fermi quantum gases, and establish arigorous connection with well-known mesoscopic super-conducting systems. Our work is motivated by recentprogress in quantum gas experiments which can deter-ministically prepare systems of few fermions in a har-monic one-dimensional trap [11]. These systems werestudied for repulsive [12] and attractive [13] interactionsand spin-balanced [13] and spin-imbalanced configura-tions [14], and used to simulate models of quantum mag-netism [15–17]. Motivated by a recent experiment [13],here we study a spin-balanced few-fermion system withattractive interaction in a harmonic trap, i.e., ensembleswhich contain an equal number of spin-up and spin-downfermions for a total even particle number, and a singleunpaired fermion for an odd total particle number. InRef. [13], following the preparation of an ensemble witha definite particle number, the trapping potential wastilted for a variable time, allowing fermions to tunnel outof the trap. From the tunneling times obtained in the ex-periment, a separation energy was extracted [13, 18–20],which is defined as E sep ( N ) = ( E N − E N ) − ( E N − − E N − ) , (2)where E N ( E N ) is the ground-state energy of the inter-acting (noninteracting) system with N particles. At zerotemperature, E N is obtained by filling the lowest har-monic oscillator levels up to the Fermi level: for eventotal particle number 2 l , the states j = 0 , . . . , l − l + 1, the level l contains anadditional unpaired fermion. The parity effect is mani-fested in the separation energy in the form of an odd-evenoscillation, where the separation energy of an odd par- a r X i v : . [ c ond - m a t . qu a n t - g a s ] J un ticle number state is smaller than the separation energyof an even particle number state. The experiment [13]has been analyzed theoretically using exact diagonaliza-tion for small particle number [21–23]. However, forlarger numbers of particles, exact diagonalization is be-yond computational reach and different theoretical ap-proaches are necessary. Recent numerical works computeground-state properties using Monte Carlo methods foreven fermion numbers up to N = 20 [24] and coupled-cluster methods for up to N = 80 [25, 26]. In this pa-per, we employ analytical methods, which allow a directphysical interpretations of the experimental results andprovide complementary information to numerical works.Pairing in higher dimensions has been considered in [27–30].In the following, we analyze the mesoscopic pairingproblem, focusing on the weak-interaction limit whichcorresponds to the experimental situation [13]. The par-ity parameter takes a fundamentally distinct form in thefew-body and the many-body limits, interpolating be-tween a simple perturbative form and a manifestly non-perturbative many-body expression. We estimate a crit-ical particle number which marks the crossover betweenthe mesoscopic and the macroscopic regime, finding thatthis quantity scales exponentially with the interactionstrength, which suggests that the mesoscopic descriptionpersists over a wide range of particle number. Our the-ory is in accurate quantitative agreement with the ex-periment [13] and provides a theoretical framework tostudy the mesoscopic regime where N (cid:29)
1, which is offundamental interest to understand the emergence of su-perfluidity and superconductivity in physical systems.The Hamiltonian of a two-component Fermi gas in onedimension is (we set (cid:126) = 1) H = (cid:90) dx (cid:20)(cid:88) σ ψ † σ (cid:16) − ∂ x m + V ( x ) (cid:17) ψ σ − g ψ †↑ ψ †↓ ψ ↓ ψ ↑ (cid:21) . (3)Here, ψ σ ( x ) annihilates a fermion at x with mass m andspin σ , V ( x ) = mω x / ω , and g > a via g = 2 /ma . Wewrite the continuum model (3) in an oscillator basis byexpanding the fermion operators in terms of simple har-monic oscillator states ψ σ ( x ) = (cid:80) ∞ j =0 c jσ φ j ( x ), where φ j ( x ) is a normalized harmonic oscillator wavefunctionwith energy ε j = ω ( j + 1 /
2) and the operator c jσ annihi-lates a fermion with spin σ in state j . The Hamiltonianin oscillator space is H = (cid:88) jσ ε j c † jσ c jσ − g l − (cid:88) ijkl w ijkl c † i ↑ c † j ↓ c k ↓ c l ↑ , (4)where l ho = (cid:112) /mω denotes the harmonic oscillatorlength. The coupling is now state-dependent with aneffective interaction strength set by the overlap integral w ijkl = l ho (cid:82) dx φ i φ j φ k φ l .The theory in Eq. (3) can be solved in the absence of atrapping potential [31–33]. In the thermodynamic limit of a large system size L and large particle number N withconstant density n = N/L , the parity parameter corre-sponds to half the spin gap, which for small interactionstrength is ∆ P = π ε F (cid:112) γ hom /πe − π / γ hom [31], where γ hom = mg /n (cid:28) n = 2 √ N /πl ho . This gives aparity parameter [25]˜∆ = ∆ P ( N → ∞ ) = 4 N ωπ (cid:114) γπ e − π / γ , (5)where the dimensionless interaction strength is γ = πg √ N ωl ho . (6)Equation (5) is a manifestly nonperturbative expression.Note that despite the exponential suppression with γ , theparity parameter ∆ P scales with the Fermi energy. Themacroscopic limit is therefore characterized by ˜∆ (cid:29) ω .By contrast, in the mesoscopic limit of small particlenumber where ˜∆ (cid:28) ω , we expect simple perturbationtheory to hold. This is reminiscent of the Anderson cri-terion that marks the vanishing of superconductivity ifthe level spacing of a grain is larger than the bulk super-conducting gap [34]. Clearly, the crossover from a few tomany particles is manifest in the parity parameter. Theexpression (5) is extensive with particle number for con-stant γ , indicating that the crossover should be studiedwhile keeping γ fixed, i.e., imposing g ∼ √ N . In thefollowing, we consider the regime where γ (cid:28) g , ap-plicable to the mesoscopic regime ˜∆ (cid:28) ω . To this leadingorder, the ground-state energy is given by the expecta-tion value of Eq. (4) with respect to the noninteractingground state. The separation energy in Eq. (2) is thus E sep (2 l + 1) = − g l − l − (cid:88) j =0 w jl (7) E sep (2 l ) = − g l − l − (cid:88) j =0 w j ( l − , (8)which corresponds to the interaction energy of a sin-gle fermion in the outermost level interacting with thefermions in the lower shells. In Eqs. (7) and (8), wedefine the diagonal coupling w ij = w ijji , which can bedetermined in closed analytical form [35]. Note that theperturbative interaction correction is due to a mean-fieldshift of the single-particle energies. In Fig. 1, we show theresults of Eqs. (7) and (8) for the separation energy alongwith the experimental measurement [13] (black errorbars) for an interaction strength g /ωl ho = 0 .
45, which
FIG. 1. (color online) (a) Separation energy as a function ofparticle number N for interaction strength g /l ho ω = 0 . et al. [13], which are shown as greenerror bars. In addition, we compare with results from exactdiagonalization [21], indicated by gray bars. (b) Orange line:separation energy including only pairing interactions. TheHartree density interaction is essential to fit the experimentaldata. corresponds to the value used in the experiment [13]. Re-markably, the perturbative result provides a very accu-rate description of the experimental data and is also invery good agreement with results from a numerical exactdiagonalization of the Hamiltonian (4) (gray bars) [21].The parity parameter is∆ P ( ˜∆ (cid:28) ω ) = g w ll ωl ho ω ∼ ˜∆ (cid:114) ω ˜∆ , (9)where ˜∆ was defined in Eq. (5).To gain insight into the physical mechanisms con-tributing to the separation energy and the parity param-eter, we assume that for weak interactions, pairing takesplace predominantly within a harmonic oscillator shell,i.e., that the ground-state properties can be described asexcitations of paired levels: levels are either occupied bya pair of fermions or empty. This implies that only theinteraction terms that connect two levels are retained: w ij = w iijj = w ijij = w ijji [36]. The effective Hamilto- nian takes the form: H eff = (cid:88) nσ ε n c † nσ c nσ − g l − (cid:88) ij w ij c † i ↑ c † i ↓ c j ↓ c j ↑ − g l − (cid:88) i (cid:54) = j w ij c † i ↑ c † j ↓ c j ↓ c i ↑ . (10)There are two interaction terms: The first, which we callthe pairing term , destroys a pair of spin-up and spin-down fermions in one oscillator level and creates a pairin a different one. The second, which we refer to as the Hartree term , does not create excitations but providesa density-dependent energy shift to the single-particlelevels (note that the perturbative result is due to thistype of interaction). A third possible interaction termwhich exchanges the spin between two simply occupiedlevels (
Fock term ) does not contribute to the balancedsystem that we consider. Compared to pairing modelsused for superconducting nanograins, the pairing interac-tion takes a more complicated level-dependent form andinvolves an additional Hartree term, which is in fact es-sential to describing the experimental data of Ref. [13].Figure 1(b) shows the leading order prediction for theseparation energy only taking into account the pairing in-teraction. As is apparent from the figure, this predictionis in complete disagreement with the experiment. Notethat the while the Hartree term is crucial for a correctdescription of the separation energy, it does not affectthe parity parameter of Eq. (9) to leading order.We obtain the ground-state energy for fixed particlenumber from the limit [7]lim β →∞ Ω eff = min N ( E N − µN ) , (11)where Ω eff is the free energy of the system obtained fromthe grand canonical partition function Z = e − β Ω eff =Tr e − β ( H eff − µN ) (where Tr denotes the trace over allmany-particle eigenstates), i.e., the grand canonical en-semble projects onto a sector with definite particle num-ber. However, because of the parity effect of Eq. (1), theprescription (11) only allows us to study configurationswith even particle number. Nevertheless, we can relatethe ground-state energy of a system with an odd number2 l + 1 of fermions to the ground-state energy of a sys-tem with an even number 2 l : since the Hamiltonian (10)only couples fully occupied or empty levels, the unpairedorbital of an odd-particle number state does not partic-ipate in the interaction and decouples; i.e., it effectivelyblocks a level from the Hilbert space. Hence [7, 9, 10] E l +1 = ε l +1 + E (cid:48) l , (12)where the energy E (cid:48) l is computed without the blockedlevel l . To analyze the effective theory of Eq. (10), weeliminate the quartic interaction terms using a doubleHubbard-Stratonovich transformation in both the den-sity and pairing channel, which introduces three auxil-iary fields ∆ i , ∆ ∗ i , and K i . To this end, we define theoperators q i = ( c † i ↑ c i ↑ + c † i ↓ c i ↓ ), q + i = ( c † i ↑ c † i ↓ + c i ↓ c i ↑ ),and q − i = i ( c † i ↑ c † i ↓ − c i ↓ c i ↑ ). The Hamiltonian (10) takesthe form H − µN = T − V + − V − − V , where T = (cid:88) j ξ j n j + g l − (cid:88) i w ii , (13) V α = g l − (cid:88) ij w ij q αi q αj , (14)with ξ j = ε j − µ . The last constant term in Eq. (13) arisesfrom a fermion commutator. Since the symmetric ma-trix ( w − ) ij /g l − is positive definite, the four-fermioninteraction terms can be reduced using three standardHubbard-Stratonovich transformations [37] for the op-erators q αi introducing conjugate real fields x αi , where α = 0 , ± . Identifying K j = x j and ∆ j = x + j + ix − j , thepartition function reads: Z = (cid:90) (cid:20) N (cid:89) τ,i D ∆ i ( τ ) D ∆ ∗ i ( τ ) D K i ( τ ) (cid:21) Tr (cid:2) U ∆ ( β, (cid:3) × exp (cid:20) − βC − (cid:90) β dτ (cid:88) ij ( w − ) ij g l − (∆ ∗ i ∆ j + K i K j ) (cid:21) , (15)where C = (cid:80) i ( g l ho w ii + ξ i ) with ξ j = ξ j − K j , N is thepath integral normalization N = (cid:90) (cid:20)(cid:89) τ,i D ∆ i ( τ ) D ∆ ∗ i ( τ ) D K i ( τ ) (cid:21) × exp (cid:20) − (cid:90) β dτ (cid:88) ij ( w − ) ij gl − (∆ ∗ i ∆ j + K i K j ) (cid:21) , (16)and U ∆ ( β,
0) = T τ exp (cid:20) − (cid:90) β dτ (cid:88) j χ † j (cid:18) ξ j − ∆ j − ∆ ∗ j − ξ j (cid:19) χ j (cid:21) (17)with χ j = ( c j ↑ , c † j ↓ ) T [38].We first consider the saddle-point approximation andminimize the Euclidean action in Eq. (15) with respectto K i and ∆ i . To this end, we first integrate out thefermions in the partition function, which gives the effec-tive action S eff [ { ∆ j , K j } ] = − (cid:88) j tr ln[ − G − ,j ]+ (cid:90) β dτ (cid:26)(cid:88) ij ( w − ) ij g l − K i K j + (cid:88) i ξ i + g (cid:88) i w ii (cid:27) , (18)where the trace runs over Matsubara indices. Thematrix element of G ,j is given by (cid:104) iω n | G ,j | iω n (cid:48) (cid:105) = δ n,n (cid:48) G ,j ( iω n ) = δ n,n (cid:48) (cid:2) iω n − ξ j σ (cid:3) − = δ n,n (cid:48) iω n + ξ j σ ( iω n ) − ξ j .Varying the action S eff in Eq. (18) with respect to K j ,∆ j , and µ , we obtain the mean-field saddle-point solu-tion defined by K i = g l ho (cid:80) j w ij (cid:16) − ξ j E j (cid:17) and ∆ i = g ωl ho (cid:80) j w ij ∆ j E j , where E j = (cid:113) ξ j + ∆ j . The solution ofthese equations determines the value of the Hartree field K i and the gap ∆ i at the saddle point. Note that for aneven particle number, the saddle-point equations corre-spond to the solution of a BCS pairing ansatz [35, 39, 40]with coherence factors v i = 1 − u i = [1 − ( ξ i − K i ) /E i ] / γ , the only saddle-point solution for ∆ i corresponds to ∆ i = 0, which implies vanishing off-diagonal long-range order and absence of superfluidityin the weak-coupling regime. This is the fluctuation-dominated regime where ˜∆ defined in Eq. (5) is muchsmaller than the harmonic oscillator level spacing ω [9].The Hartree fields are given by K i = g l − (cid:80) l − j =0 w ij ,which correspond to the single particle energy shift com-puted to leading order in perturbation theory in g usingthe noninteracting ground state of the Fermi gas. Usingthe identity in Eq. (11), the saddle-point contribution tothe ground-state energy of an system with even N = 2 l particle number is E l = 2 (cid:80) l − j =0 ε j − g (cid:80) l − i,j =0 w ij + g (cid:80) i w ii . Interestingly, this is not equal to the pertur-bative ground-state energy. The last term arises fromthe commutator term in Eq. (13). Despite the saddlepoint ∆ i being zero, fluctuations of the pairing field canmake an important contribution, and they are computedin the remainder of this paper. It turns out that thesepairing fluctuations contain a O ( g ) correction that can-cels the last term in the saddle-point contribution to E l and reproduces the perturbative result.To consider the effect of fluctuations around thesaddle-point solution. We write K i → K i + δK i and∆ i → δ ∆ i , and expand the action in Eq. (15) to sec-ond order in δK i and δ ∆ i . It turns out that thereis no correction due to fluctuations of the Hartreefields δK i . The partition function can be written as Z = Z sp Z δ ∆ , with Z sp the saddle-point contributionand Z δ ∆ = [det Γ] − the resulting quadratic functionalintegral in δ ∆ i , which can be exactly evaluated interms of the functional determinant [7, 37]. To de-rive this result, we expand the perturbation in Mat-subara space δK i ( τ ) = √ β (cid:80) iω n e − iω n τ δK i ( iω n ), and δ ∆ i ( τ ) = √ β (cid:80) iω n e − iω n τ δ ∆ i ( iω n ). The effective actionis S eff [ { θ j } ] = − (cid:88) j tr ln[ − G − ,j ] − (cid:88) j tr ln[1 − G ,j V j ]+ (cid:90) β dτ (cid:26)(cid:88) ij ( w − ) ij g l − (cid:104) δ ∆ ∗ i δ ∆ j + ( K i + δK i ) × ( K j + δK j ) (cid:105) + (cid:88) j ( ξ j − δK j ) (cid:27) , (19)where we separate a fluctuation part V j from the Green?sfunction G ,j . The matrix element of V j is given by (cid:104) iω n | V j | iω n (cid:48) (cid:105) = V j ( iω n − iω n (cid:48) )= − √ β (cid:104) δK j ( iω n − iω n (cid:48) ) σ + δ ∆ j ( iω n − iω n (cid:48) ) σ + + δ ∆ ∗ j ( iω n − iω n (cid:48) ) σ − (cid:105) . (20) Using − (cid:88) j tr ln[1 − G ,j V j ] = (cid:88) j ∞ (cid:88) n =0 j tr[ G ,j V j ] n , (21)we expand the effective action (19) to second order in V j .The functional integral in terms of δK j and δ ∆ j can thenbe performed exactly. The partition function involving δK j reads: Z δK = 1 N K (cid:90) (cid:20)(cid:89) k D δK k (cid:21) × exp (cid:20) − (cid:88) iω n (cid:88) ij ( w − ) ij g l − δK i ( − iω n ) δK j ( iω n ) − (cid:112) β (cid:88) i δK i ( iω m = 0) (cid:18) (cid:88) j ( w − ) ij g l − K j + ξ j | ξ j | − (cid:19)(cid:21) . (22)However, the zero-frequency contribution (second term) vanishes since K j is given by K j = g l − (cid:80) l − i =0 w ij . Theremaining quadratic term is irrelevant since it does not involve any single-particle energies. The partition functioninvolving δ ∆ j reads (discarding an irrelevant constant term) [7, 37]: Z δ ∆ = 1 N ∆ (cid:90) (cid:20)(cid:89) k D δ ∆ ∗ k D δ ∆ k (cid:21) exp (cid:20) − (cid:88) iω n (cid:88) ij δ ∆ ∗ i ( − iω n ) (cid:18) ( w − ) ij g l − + δ ij sgn ξ j iω n − ξ j (cid:19) δ ∆ j ( iω n ) (cid:21) = (cid:89) iω n det − (cid:20) δ ij + g l − w ij sgn ξ j iω n − ξ j (cid:21) = (cid:89) j sinh βξ j sinh β ˜ ξ j , (23)where by { ξ j } we denote the eigenvalues of A ij =2 ξ j δ ij − g w ij sgn ξ j , i.e., det(2 ˜ ξ k I − A ) = 0, or, respec-tively, det (cid:18) δ ij + g l − w ij sgn ξ j ˜ ξ k − ξ j (cid:19) = 1 + g l − (cid:88) i w ii sgn ξ i ˜ ξ k − ξ i = 0 , (24)where the second term holds for small corrections. Writ-ing ˜ ξ j = ξ j + δξ j and expanding in δξ j , the fluctuationcorrection to the free energy at zero temperature is: δ Ω = − β ln Z δ ∆ = (cid:88) j δξ j sgn ξ j , (25)where δξ i = − g w ii sgn ξ i l ho (cid:18) − g l ho (cid:88) j (cid:54) = i w jj sgn ξ j ξ j − ξ i (cid:19) − . (26)Hence, δE l = (cid:80) j δξ j sgn ξ j and δE l +1 = (cid:80) j (cid:54) = l δξ (cid:48) j sgn ξ (cid:48) j , where ξ (cid:48) and δξ (cid:48) are computed asin Eq. (26) with the l -th level excluded.The fluctuation correction to the separation energyand the parity parameter can be read off directly from thedefinitions (1) and (2). Note that the combined saddle-point and fluctuation correction contains the leading or-der perturbative result (see Fig. 1, where the separation energy is indicated by the red dashed line and the dia-mond symbol). There is a small quantitative correctionwhich improves the agreement with the exact diagonal-ization results by D’Amico et al. [21]. The fluctuationcorrection (26) is similar to the one encountered in su-perconducting nanograins in the limit where the super-conducting gap ∆ is much smaller than the level spacing δ(cid:15) [7].Interestingly, our analytical procedure also allows usto identify the critical particle number that marks thecrossover between the few-body regime ˜∆ (cid:28) ω and themany-body regime ˜∆ (cid:29) ω . The boundary of the meso-scopic regime is marked by a breakdown of the expan-sion (26). For large particle number, we can replace theharmonic oscillator matrix element by its semiclassicalexpression w jj ∼ / √ j and convert the summation to anintegral. This gives g l ho (cid:88) j (cid:54) = i w jj sgn ξ j ξ j − ξ i ∼ g ωl ho (cid:114) N ln 2 N ∼ γ ln 2 N, (27)indicating that by successively increasing particle num-ber, the few-body expansion loses validity at a criticalparticle number N c ∼ e /γ . In this case, the bulk par-ity parameter (5) is comparable to the level spacing ω ,which is a corresponding criterion as in superconductingnanograins [9]. Note that while the few-body to many-body crossover is manifested in the parity parameter atleading order, the ground-state energy is dominated bya Hartree mean-field term and will be less sensitive tothe crossover. From the perspective of superconductingnanograin systems, such a predominance of the Hartreecontribution over the fluctuation correction is an unex-pected effect [7, 9, 10]. Therefore, our findings prompta revision of both the theoretical modeling of nanograinsand the related experimental results [3–6].While the BCS pairing model can be solved ex-actly [41–48], this is not the case for the model (4) or thereduced pairing Hamiltonian (10). However, it would beinteresting to explore if the theory could be approximatedby a generalized Richardson-Gaudin model [49].In summary, we have computed the ground-state en-ergy, the separation energy, and the parity parameter fora trapped one-dimensional Fermi gas with weak attrac-tive interaction. We have used an insightful path-integralformalism which allows us to make useful connectionswith other physical systems (i.e., mesoscopic supercon-ductors). The parity parameter serves as an order pa- rameter that displays a fundamentally distinct behaviorin the mesoscopic and macroscopic regimes, and we es-tablish that the mesoscopic description persists for a widerange of particle number. Our results provide a quanti-tative description of the recent experiment [13]. A pathintegral treatment indicates that the ground-state energyand the parity effect are dominated by a Hartree meanfield contribution, with BCS pairing fluctuations provid-ing a subleading correction to this result. ACKNOWLEDGMENTS
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