Efimov states of fermionic species

J. H. Maceka

aDepartment of Physics and Astronomy, University of Tennessee, Knoxville, TN37996-1501 and Oak Ridge National Laboratory, Oak Ridge, TN

When three particles interact via zero-range potentials it is possible to find some exactsolutions of the three-body Schrodinger equation for infinite scattering lengths. For s-waveinteractions the exact solutions correspond to the famous Efimov states. Similar exactsolutions for two-body zero-range potentials with � = 1 are considered in this manuscript.Efimov states emerge in the limit as a1 → ∞. We also report exact solutions for � = 0with a0 �= 0.

1. INTRODUCTION

Zero-range potentials (ZRPs) emerge naturally in the coupled channel approach toreactions. For two body channels the complex interactions of particles are treated essen-tially exactly and are incorporated into the logarithmic derivative of the wave functionon a boundary rB. Outside of the boundary the potential vanishes and the motion isthat of free particles. To take this idea over to many-body channels with more than twofragments the hyperspherical close-coupling approach and a matching procedure is used[1]. In general, this is difficult to do except in the limit as R → ∞, since in that limitrB/R shrinks to zero and one may employ ZRPs, in first approximation. Furthermore theeffects of the two-body interaction are expressed in terms of the two-body effective rangeexpression

k2�+1 cot δ� = M�(k2) = − 1

a2�+1�

+1

2r2�−1�

k2 (1)

where a� is the scattering length, r� is the effective range corresponding to the two-bodyangular momentum quantum number �, and M�(k

2) is the M -matrix of Ross and Shaw[2,3].

The ZRPs that we use are those given by Stock et al [4] expressed as boundary condi-tions, namely,

limrk→0

[(2� + 1)!!

(2� − 1)!!a2�+1

�

∂2�+1

∂r2�+1k

− 1

]r�+1k Ψ = 0. (2)

In section II exact solutions for infinite scattering length for � = 0 and � = 1 ZRPs arediscussed.

Nuclear Physics A 790 (2007) 747c–751c

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2. HYPERSPHERICAL COORDINATES AND KNOWN SOLUTIONS

Hypersphrical coordinates for 3 identical particles ijk of mass m are defined in termsof the standard Jacobi coordinates rij = ri − rj and rij,k = rik − rij/2 according toμR2 = mijr

2ij + mij,kr

2ij,k where mij and mij,k are the reduced masses m/2 and 2m/3,

respectively, and μ is an arbitrary mass parameter. In mass-scaled coordinates uij and vij

that we employ here coordinates are scaled so that uij =√

mijrij, vij =√

mij,krij,k andμ = 1. The three pairs 12, 31, and 23 are labeled 1, 2 and 3 for notational convenience.The hyperradius now becomes R2 = u2

k + v2k, k = 1, 2, 3 while the hyperangles are any

one of the set R = uk, vk, αk where αk = arctan(uk/vk).With zero-range boundary conditions one simply employs the free-particle Hamiltonian

H = TR +Λ2

2R2(3)

which has the separable solutions [5]

Ψν(R) = S(ν, R)R−5/2(KR)1/2Zν(KR) (4)

where ν is a separation constant, Zν is a Bessel function, TR is the radial kinetic energyoperator, Λ2 the grand angular momentum operator and S(ν, R) is an eigenfunction ofΛ2. The boundary conditions in Eq. (2) are written using xk = uk/R, yk = vk/R and

∂

∂r=

1 − x2

R

∂

∂x+ x

∂

∂R(5)

In the case of � = 0 and � = 1, the boundary conditions depend upon R only through amultiplicative factor

limxk→0

[(2� + 1)!!

(2� − 1)!!

∂2�+1

∂x2�+1k

−(

R

a�

)2�+1]x�+1Ψ = 0 (6)

In the limit that a� → ∞ which is equivalent to M� → 0 the coordinate R no longer entersthe boundary conditions so that now the boundary conditions do separate, and Eq. (6)becomes an eigenvalue equation for the separation constant ν. Corresponding to eacheigenvalue νj, where j is a count number, there are effective hyperspherical potentialsgiven by

Veff(R) =ν2

j − 1/4

2R2, (7)

so that simple, separable, exact solutions are given by Eq. (4) with νj determined byEq. (6).

For three identical Bosons interacting through s−wave ZRPs and for total orbital an-gular momentum L = 0 the function S(ν, R) is given by

S(ν, R) =3∑

k=1

sin ν(π/2 − αk)

sin αk cos αk

. (8)

The boundary conditions then give

−RM0(k2) =

ν cos(νπ/2) − (8/√

3) sin νπ/6

sin νπ/2. (9)

J.H. Macek / Nuclear Physics A 790 (2007) 747c–751c748c

If M0(k2) = 0 then Eq. (6) is satisfied for an infinite number of eigenvalues νj. One of the

eigenvalues is complex with ν0 = it0, t0 = 1.00625 so that effective potential is attractivewith an infinite number of three-body bound states. These are the Efimov states [6] thathave attracted renewed attention owing to the magnetic tuning of scattering lengths foratom-atom collisions in the nanokelvin energy range [7,8].

The effective potential diverges with a strength −t20 − 1/4 such that the hyper-radialfunction also diverges as R → 0. This is the well-know Thomas effect [1] which requiresthat the solutions must be regularized in some way. Here I will suppose that they areregularized by introducing an infinite three-body potential barrier at some value R0. Fora perspective based upon effective field theory see the talk by H. W. Hammer at thisconference.

Recently, cold Fermionic atoms have been studied. In this case the spin wave functionχ(ijk) needs to be combined with the angular function S(ν, R) to obtain solutions withthe desired spin and symmetry. For spin 1/2 Fermions spins i and j are coupled to sij = 0,and the third spin k is coupled to give a total spin S = 1/2. In this case one finds thatthe eigenvalue equation [9] for ν is

RM0(k2) = 0 =

ν cos(νπ/2) − (4/√

3) sin νπ/6

sin νπ/2. (10)

This equation has only real roots, thus for this case there is no Efimov effect and noThomas effect.

The method used above may also be applied to � = 1 interactions. In that case the spinfunction has the coupling scheme (((1/2, 1/2)1, 1/2), S) and the 1+ symmetry has beenidentified by Esry [10] to be particularly important. In this case the function S(ν, R, σ)incorporates a solid bispherical harmonic proportional to the vector cross product xk×yk

and the function∑

k 2F1

(a, b, c, sin2 αk

)χ(kij) where 2F2(a, b; c; z) is a hypergeometric

function with a = −ν/2 + 2, b = ν/2 + 2 and c = 5/2. Upon using these functions in theboundary conditions for p−wave ZRPs we obtain [9]

(R

a1

)3 sin νπ/2

ν(ν2 − 4)− cos

νπ

2= C 2F1

(−ν

2+ 2,

ν

2+ 2;

5

2;1

4

), (11)

where C is given in terms of recoupling coefficients. For S = 3/2 one has that C = 2, andthat the eigenvalue equation for 1/a1 = 0 has only real roots. Thus for � = 1 interactionstwo-body ZRP and spin polarized Fermions there is neither a Thomas nor an Efimoveffect for the 1+ symmetry.

For S = 1/2 one finds that C = −1, and in this case there is one complex root ν = it0with t0 =0.6668. Accordingly if M1 = 0 there is both a Thomas effect and an Efimoveffect.

These are exact results for the special case that M�(0) = 0. When the first term inthe expansion of M�(k

2) vanishes the next term, i. e. the effective range term, is notsmall compared with the first, thus one needs some estimate of the effect of k2 terms forthree-body states. The effect of the k2 terms can be checked approximately by replacingk2 by a local energy k2 = (ν2 − lx(lx + 1)/3 − ly(ly + 1))/R2 obtained by expanding Λ2

through order α0k and comparing with the equation for the spherical Bessel functions. In

this case the boundary conditions become eigenvalue equations for νj(R) and there is a

J.H. Macek / Nuclear Physics A 790 (2007) 747c–751c 749c

corresponding adiabatic potential curve given by Eq. (6) which can be used in the radialequations to find bound and scattering states.

2 4 6 8 10

-1

-0.8

-0.6

-0.4

-0.2

0.2R/|r0|

Veff (R)E0

r0 < 0

r0 = 0

r0 > 0

Figure 1. Hyperspherical potential energycurves for s-wave ZRPs with 1/a0 = 0 andthree values of r0. Here E0 = 1/(2r0)

2,r0 �= 0.

2.5 5 7.5 10 12.5 15 17.5 20

-1

-0.8

-0.6

-0.4

-0.2

0.2

0.4

R/|r1|Veff (R)

E1

r1 < 0

1/r1 = 0

r1 > 0

Figure 2. Hyperspherical potential energycurves for p-wave ZRPs with 1/a1 = 0 andthree values of r1. Here E1 = 1/(2r2

1),1/r1 �= 0.

For � = 0 and Bosons, the resulting potential curves are shown in Fig. (1). Closeexamination of the curves show that there is a 1/R2 behavior as R → ∞ so that thereis still the Efimov effect independently of the value of r0. For positive r0 the potentialcurves are unrealistic for R < 1.5r0 and need to be cut off. The upper portion of thecurve still exhibits the Efimov effect at large distances, but the lower part is unrealisticand corresponds to a spurious two-body bound state [11].

More significant effects are seen for � = 1 in Fig. 2 which shows the adiabatic hy-perspherical potential curves obtained for spin 1/2 Fermions in the 1+ state, i. e. thesymmetry where there is a Thomas effect divergence at small R. For r1 < 0 there isa spurious bound state and the lower curve exhibits the Thomas effect but neither thelowest state nor the first excited state show the Efimov effect. In contrast, if the effectiverange is taken to be infinite, then one obtains the dashed curve which shows both theThomas and Efimov effects. Such large effective ranges are not likely to occur for localpotentials. As similar absence of the Efimov effect is seen in when the effective range ispositive. There is a Thomas effect but no Efimov effect in this case also.

We have seen that there are exact, closed-form solutions when a0 → ∞ and r0 = 0.For finite a0, linear combinations of the separable solutions are needed to satisfy the ZRPboundary conditions. These solutions have the form

ψ(R) = R−5/2∫

cAνS(ν, R)(KR)1/2Zν(KR)νdν (12)

where c denotes a contour that must be chosen to satisfy physical boundary conditions.The ZRP boundary conditions give a three-term recurrence relation for A(ν);

Aν+1Xν+1 + Aν−1Xν−1 =2ν

Ka0

sinνπ

2Aν (13)

J.H. Macek / Nuclear Physics A 790 (2007) 747c–751c750c

where Xν = ν cos νπ/2 − (8/√

3) sin νπ/6.The case where a0 is positive is particularly interesting since three-body recombination

to a loosely bound dimer affects the lifetimes of Bose condensates. Using an analyticsolution for E = 0, we find the S− matrix element S00 corresponding to elastic scatteringof a Boson from the loosely bound dimer to be [5]

|S00(E)| = |(1 + 2iPeiΔr sin Δr)|, P =

(4π

3√

3− 1

)4(Ka0)

4

sinh2 πt0, (14)

Δr = Δ(R0) − arctan

(e−2πt0 sin 2Δ(R0)

1 + e−2πt0 cos 2Δ(R0)

), Δ(R0) = δ0 + t0 ln(R/a0), (15)

where δ0 is a phase factor determined from the phase of the wave function as R → 0. Thethree-body recombination coefficient K3 for E = 0 is given by

K3 = 2(2π)233/2 1 − |S00|2(Ka)4

h

ma4

0 = C3 sin2 Δrh

ma4

0 (16)

where C3 = 27π2(4π − 3√

3)/ sinh2 πt0 = 67.1177 . . .. The value of C3 agrees with theexpression obtained by Petrov [12] using the STM equation and by Bratten and Hammerusing a numerical solution of the same equation [7].

Support by the Chemical Science, Geosciences and Biosciences Division, Office of BasicEnergy Science, Office of Science, U.S. Department of Energy under Grant No. DE-FG02-02ER15283 is greatly appreciated.

REFERENCES

1. E. Nielsen et al, Phys. Rpts. 347 (2001) 373.2. M. H. Ross and G. L. Shaw, Ann. Phys. (NY), 12 (1961) 147.3. O. I. Kartavtsev and J. H. Macek, Few Body Systems 31 (2002) 249.4. R. Stock, et al, Phys. Rev. Lett. 94 (2005) 023202.5. J. H. Macek et al Phys. Rev. A 72 (2006) 032704.6. V. N. Efimov, Phys. Lett., 33B (1970) 563.7. E. Bratten and H.-W. Hammer, Phys. Reports 428 (2006) 261.8. T. Kraemer et al, cond-mat/0512394.9. J. H. Macek and J. Sternberg, Phys. Rev. Lett. 97 (2006) 023201.10. H. Suno, et al, New J. Phys. (2003) 53.11. R. G. Newton(1966), Scattering theory of waves and particles. (New York, McGraw-

Hiil Book Company, 1966).12. E. Bratten, private communication.

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