Condensation of Cooper Triples
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Condensation of Cooper Triples
Sora Akagami, Hiroyuki Tajima, and Kei Iida
Department of Mathematics and Physics, Kochi University, Kochi 780-8520, Japan (Dated: February 12, 2021)The condensation of Cooper pairs, originating from the Fermi-surface instability due to a weaklyattractive interaction between two fermions, opened a new frontier for exploring many-body physicsin interdisciplinary contexts. In this work, we discuss the possible condensation of Cooper triples,which are three-body counterparts of Cooper pairs for three-component fermions with a three-bodyattraction. Although each composite trimer-like state obeys the Fermi-Dirac statistics, its aggregatecan form a condensate at zero center-of-mass momentum in the presence of the internal degrees offreedom associated with the relative momenta of constituent particles of momenta close to theFermi surface. Such condensation can be regarded as bosonization in infinite -component fermions.We propose a variational wave function for the condensate of Cooper triples in analogy with theBardeen-Cooper-Schrieffer ground state and obtain the ground-state energy.
PACS numbers: 03.75.Ss
Introduction — A quantum many-body problem has beenencountered not only in condensed matter physics butalso in nuclear and particle physics. On top of interac-tions between constituents, the degrees of freedom suchas spin, isospin, and flavor play an important role incharacterizing a variety of strongly correlated systems.One of the most striking examples is conventional su-perconductivity, which is triggered by the formation ofa Cooper pair, i.e., a pairing state of two electrons withspin up and down in the presence of the Fermi sphere.The Bardeen-Cooper-Schrieffer (BCS) theory [1], whichsuccessfully explains the microscopic mechanism of su-perconducting phase transition in terms of condensationof Cooper pairs, has developed a fundamental basis forthe description of superfluid states in cold atoms [2, 3] aswell as dense matter [4, 5].Most of quantum many-body effects in spin-1 / (a) (b) k k' k π / π / π / R k ^ R k ^ FIG. 1: (a) Momentum-space configurations of three fermionsof different species on a spherical surface that have zerocenter-of-mass momentum. The momenta k and k ′ of oneof the three fermions can take on various directions. (b) Thecorresponding in-plane configuration of the three fermions ofmomenta k , ˆ R k , and ˆ R k . The angle between any pair ofmomenta is 2 π/ this perspective, ultracold atoms provide a valuable op-portunity to manipulate multi-body interactions in a sys-tematic manner [24]. In particular, by reducing the two-body interaction to zero, one can realize a unique sys-tem that is governed by the three-body interaction hid-den behind the two-body interaction [25–34]. Effects ofthe three-body interaction have been studied extensivelyin theories of, e.g., pair superfluidity of bosons [35–38],three-component Fermi gases [39–41], quantum boundstates [42], and Efimov states [13, 43].In this work, we theoretically predict that Coopertriples can condense in a three-component Fermi system.Although each Cooper triple obeys the Fermi-Diracstatistics, a macroscopic number of triple states canoccupy an effective single state of zero center-of-massmomentum in the presence of various sets of threeparticles having different species and different momentaon a spherical surface close to the Fermi surface as shownin Fig. 1. This exotic condensation can be regarded asbosonization in infinite -component fermions associatedwith relative momenta. For an attractive three-bodyforce which has extensively been discussed in cold atomphysics [6], we propose a variational wave function forthe Cooper triple condensation in analogy with the BCSground state. We obtain the ground-state energy in athree-component fermionic system with a three-bodyinterspecies attraction to show that the Cooper triplestate is energetically more favorable than the normalstate. In what follows, we take the system volume to beunity and units in which ¯ h = k B = 1. Hamiltonian — We consider a Hamiltonian ˆ H for non-relativistic three-component (r,g,b) fermions of equalmass m , which is given byˆ H = X γ =r , g , b Z d r ˆ ψ † γ ( r ) (cid:18) −∇ m − µ γ (cid:19) ˆ ψ γ ( r )+ Z d r d r X γ = γ ′ ˆ ψ † γ ( r ) ˆ ψ † γ ′ ( r ) ˆ V ( r − r ) ˆ ψ γ ′ ( r ) ˆ ψ γ ( r )+ Z d r d r d r ˆ ψ † r ( r ) ˆ ψ † g ( r ) ˆ ψ † b ( r ) × ˆ V ( r , r , r ) ˆ ψ b ( r ) ˆ ψ g ( r ) ˆ ψ r ( r ) , (1)where ˆ ψ γ ( r ) and µ γ are the field operator and the chem-ical potential for fermions of component γ , respectively,and ˆ V ( r − r ) and ˆ V ( r , r , r ) are two- and three-body interactions, respectively. In this work, we assumethat the two-body interaction is vanishing for simplic-ity, while the weakly attractive three-body interaction ispresent. We also consider the system in which the chem-ical potentials are the same, i.e., µ r = µ g = µ b ≡ µ .At sufficiently low temperature and weak coupling, theFermi degeneracy is expected and therefore the availablemomentum space of fermions is restricted to around theFermi surface with the zero center-of-mass three-bodyscattering. In such a case, the system can be describedby the following effective Hamiltonian in the momentumspace,ˆ H eff = X γ X k ξ k ˆ n k ,γ + X k , ˆ R X k ′ , ˆ R ′ U k , k ′ ˆ F † k ′ , ˆ R ′ ˆ F k , ˆ R . (2)Here ξ k = | k | / (2 m ) − µ is the kinetic energy of a fermionwith momentum k measured from µ , and ˆ n k ,γ = c † k ,γ c k ,γ is the single-particle number operator with the annihila-tion (creation) operator c ( † ) k ,γ for fermions of momentum k and component γ . The second term in Eq. (2) denotesthe three-body interaction with a contact-type couplingconstant U , taken to be negative here. ˆ F ( † ) k , ˆ R is the trimerannihilation (creation) operator defined in terms of c k ,γ as ˆ F † k , ˆ R = ˆ c † k , r ˆ c † ˆ R k , g ˆ c † ˆ R k , b , (3)ˆ F k , ˆ R = ˆ c ˆ R k , b ˆ c ˆ R k , g ˆ c k , r , (4) where ˆ R is the operator that represents a 2 π/ k + ˆ R k + ˆ R k = ). These operatorssatisfy the anti-commutation relations { ˆ F k , ˆ R , ˆ F k ′ , ˆ R ′ } = { ˆ F † k , ˆ R , ˆ F † k ′ , ˆ R ′ } = ˆ0 , (5) { ˆ F k , ˆ R , ˆ F † k ′ , ˆ R ′ } = δ k , k ′ δ ˆ R k , ˆ R ′ k ′ δ ˆ R k , ˆ R ′ k ′ × h (ˆ1 − ˆ n k ,r )(ˆ1 − ˆ n ˆ R k ,g )(ˆ1 − ˆ n ˆ R k ,b )+ ˆ n k ,r ˆ n ˆ R k ,g ˆ n ˆ R k ,b i . (6) Possible condensation of Cooper triples — Let us pro-ceed to ask how condensation of Cooper triples, whichare three-body counterparts of Cooper pairs, can occurby considering a three-fermion configuration among dif-ferent components as shown in Fig. 1. For such threefermions near the Fermi surface, the absolute values ofthe respective momenta | k | , | ˆ R k | , and | ˆ R k | are aroundthe Fermi momentum k F , while the center-of-mass mo-mentum remains zero. In momentum space, as depictedin this figure, k , ˆ R k , and ˆ R k are located on the sameplane in such a way that the angles between two of themare 2 π/
3. When k and the plane are fixed, therefore, theother two, ˆ R k and ˆ R k , are automatically determined.From the anti-commutation relation associated with c k ,γ , one finds F † k , ˆ R F † k ′ , ˆ R ′ | i 6 = 0 ( k = k ′ , ˆ R k = ˆ R ′ k ′ , ˆ R k = ˆ R ′ k ′ ) , (7)where | i is the normalized vacuum state. Thisindicates that two trimers with zero center-of-massmomentum can coexist unless the two momenta of agiven component happen to be the same. Since k canbe taken in countless ways, one can find a countlessnumber of configurations of the three fermions withzero center-of-mass momentum in the presence of theFermi surface. It looks as if bosonization occurred in infinite -component fermions [44]. Indeed, it is similar tothe case of a three-dimensional gas of SU( N ) fermions inwhich the internal spin degrees of freedom of compositeparticles arise from the spin of constituent particles.Therefore, the Cooper triples can condense by occupyingthe zero-momentum state macroscopically regardless oftheir Fermi-Dirac statistics. Variational wave function for the Cooper triplecondensation — We propose a gauge fixed varia-tional wave function for the Cooper triples, in analogywith the BCS ground state, as | ψ T i = Y k ( u k + v k ˆ F † k , ˆ R ) | i , (8)where u k and v k are complex variational parameters asfunctions of k alone. Here, we set ˆ R in such a way thatif the momentum k of component r is in the direction of e = , the momentum ˆ R k of component g andthe momentum ˆ R k of component b are in the directionof e = √ / − / and e = −√ / − / , respectively;otherwise, k , ˆ R k , and ˆ R k are in the direction of V e , V e , and V e , respectively, with an appropriate rota-tion matrix V . This choice of ˆ R , leading to a specificorientation in momentum space, ensures that the state(8) can have any momentum of each component pickedup only once from the vacuum. Note that one can con-sider another ˆ R by setting e = √ θ/ √ θ/ − / and e = −√ θ/ −√ θ/ − / with θ = 0, i.e., by rotating themomentum plane on which a Cooper triple with k in thedirection of e resides by θ with respect to e . The re-sultant state is degenerate with the original state. Suchdegeneracy is similar to the case of different gauge ori-entations, which allows us to regard the variational pa-rameters as independent of the choice of ˆ R as well asthe gauge. Note also that in the BCS case k just cor-responds to the relative momentum of a Cooper pair,whereas in the state (8) not only k but also ˆ R charac-terizes the relative momenta of two fermions in a Coopertriple: (1 − ˆ R ) k , ( ˆ R − ˆ R ) k , and ( ˆ R − k .Under h ψ T | ψ T i = 1, the normalization condition of u k and v k reads | u k | + | v k | = 1 , (9)where | u k | and | v k | physically represents the unoccu-pied and occupied probabilities of a Cooper triple with( k , r), ( ˆ R k , g), and ( ˆ R k , b), respectively. Then, one canexpress the ground-state energy of the three-componentsystem with three-body attraction considered here as E = h ψ T | ˆ H eff | ψ T i . After straightforward calculationsbased on such anti-commutation relations as Eqs. (5) and(6), we obtain E in terms of u k and v k as E = h ψ T | ˆ H eff | ψ T i = X k ξ k | v k | + X k X k ′ U k , k ′ v ∗ k v k ′ u ∗ k ′ u k . (10)Minimization of E with respect to the variational pa-rameters lead to u k = 1 √ ξ k p ξ k + ∆ k ! / , (11) v k = 1 √ − ξ k p ξ k + ∆ k ! / , (12) where ∆ k ≡ − X k ′ U k , k ′ u k ′ v k ′ = − X k ′ U k , k ′ h c † k ′ , r c † ˆ R k ′ , g c † ˆ R k ′ , b i (13)is the order parameter characterizing the Cooper triplecondensation. ∆ k is the expectation value of thefermionic operator, which is in a sharp contrast to theBCS superconducting gap [1].Before calculating the order parameter, we considerthe non-interacting case by setting ∆ k →
0. In such acase, we obtain u k = θ ( k F − | k | ) and v k = θ ( | k | − k F )from Eqs. (11) and (12), where θ ( x ) is the step function.This result indicates that Eq. (8) reproduces the wavefunction corresponding to the filled Fermi sphere, whichis given by | ψ FS i = Y | k |≤ k F ˆ F † k , ˆ R | i ≡ Y γ Y | k |≤ k F c † k ,γ | i . (14)Let us move on to the weakly interacting case. Substi-tution of Eqs. (11) and (12) into Eq. (10) leads to E = 32 X k ξ k − ξ k p ξ k + ∆ k ! + 14 X k X k ′ U k , k ′ ∆ k p ξ k + ∆ k ∆ k ′ p ξ k ′ + ∆ k ′ . (15)In the presence of the Fermi surface common to the threecomponents, we introduce an energy cutoff Λ such thatthe interaction works only for three fermions of momentaclose to the Fermi surface. We note that Λ corresponds tothe Debye frequency in a conventional BCS superconduc-tor with phonon-mediated interaction [1]. In the presentcase, we assume U k , k ′ = − U θ (Λ − | ξ k | ) θ (Λ − | ξ k ′ | ) , (16)where U is the positive constant. Substituting Eq. (16)to Eq. (13), we obtain∆ k = ∆ θ (Λ − | ξ k | ) , (17)where ∆ is the amplitude of the order parameter. In theweak coupling limit (∆ ≪ Λ), as in the BCS theory, ∆can be analytically obtained as∆ ≈
2Λ exp (cid:18) − ρ (0) U (cid:19) , (18)where ρ ( ω ) = 1(2 π ) (2 m ) / ( ω + µ ) / (19)is the density of state as a function of the single-particleenergy ω in an ideal Fermi gas. By combining Eqs. (16)and (17) with Eq. (15), we can express the ground-stateenergy as E ≃ E FS0 + 32 X | ξ k |≤ Λ | ξ k | − | ξ k | p ξ k + ∆ ! −
94 ∆ U , (20)where we split the summation in Eq. (15) as P k = P | ξ k | > Λ + P | ξ k |≤ Λ . While the former gives a large partof the Fermi-gas energy E FS0 since U k , k ′ = 0 there, thelatter is responsible for the interaction effect near theFermi surface. Under the assumption of ∆ ≪ Λ, the dif-ference in the ground-state energy between the Coopertriple and Fermi degenerate states is given by E − E FS0 ≈ − ρ (0)∆ ≈ − ρ (0)Λ exp (cid:18) − ρ (0) U (cid:19) < . (21)Whenever U is nonzero, therefore, the ground-stateenergy of the Cooper triple state is lower than that ofthe normal Fermi gas. In addition, Eq. (21) can beregarded as the condensation energy of the Cooper triplestate as in the BCS ground state with Cooper pairs [1]. Conclusion and Outlook — In conclusion, we have elu-cidated how the Cooper triple condensation can occurin a three-component Fermi gas within the variationalapproach inspired by the BCS ground state for two-component fermions. We have found that the Coopertriples can condense at zero center-of-mass momentumin the presence of a weak three-body attractive forceamong different components as well as the Fermi surfacecommon to the three components. The condensed stateis predicted to have nontrivial degeneracy associatedwith the gauge and momentum orientations, which mayinduce interesting topological properties. Moreover, itis interesting to examine properties of excited Coopertriples that have nonzero center-of-mass momenta,possible trimers in the strong coupling regime, lowerdimensions, the influence of population imbalance onthe Cooper triple condensation, and effects of two-bodyforces on possible competition or coexistence betweenCooper pairs and Cooper triples.
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