The unitary gas in an isotropic harmonic trap: symmetry properties and applications

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The unitary gas in an isotropic harmonic trap:

symmetry properties and applications

Félix Werner, Yvan Castin

To cite this version:

Félix Werner, Yvan Castin. The unitary gas in an isotropic harmonic trap: symmetry properties and

applications. Physical Review A, American Physical Society, 2006, 74, pp.053604. �10.1103/Phys-

RevA.74.053604�. �hal-00088162�

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ccsd-00088162, version 1 - 31 Jul 2006

F´elix Werner and Yvan Castin

Laboratoire Kastler Brossel, ´Ecole Normale Sup´erieure, 24 rue Lhomond, 75231 Paris Cedex 05, France

(Dated: July 31, 2006)

We consider N atoms trapped in an isotropic harmonic potential, with s-wave interactions of infinite scattering length. In the zero-range limit, we obtain several exact analytical results: mapping between the trapped problem and the free-space zero-energy problem, separability in hyperspherical coordinates,SO(2,1) hidden symmetry, and relations between the moments of the trapping potential energy and the moments of the total energy.

PACS numbers: 03.75.Ss, 05.30.Jp

Strongly interacting degenerate Fermi gases with two spin components are studied in present experiments with ultra-cold atoms [1]: by tuning the interaction strength between the atoms of different spin statesviaa Feshbach resonance, one can even reach the so-called unitary limit [2] where the interaction strength in the s-wave chan- nel reaches the maximal amplitude allowed by quantum mechanics in a gas. More precisely, this means that thes- wave scattering amplitude between two particles reaches the value

fk=−1

ik (1)

for the relative momentak that are relevant in the gas, in particular for kof the order of the Fermi momentum kF of the particles. This implies that thes-wave scattering length ais set to infinity (which is done in practice by tuning an external magnetic field). This also implies that k|re| ≪1, where re is the effective range of the interaction potential, a condition well satisfied in present experiments on broad Feshbach resonances.

The maximally-interacting gas defined by these conditions is called the unitary quantum gas [2]. It has universal properties since all the details of the interaction have dropped out of the problem. Theoretically, for spin 1/2 fermions with equal populations in the two spin states, equilibrium properties have been calculated in the thermodynamical limit in the spatially homogeneous case using Monte-Carlo methods; at finite temperature [3, 4], and at zero temperature with a fixed node approximation [5]. In practice, the unitary gases produced experimentally are stored in essentially harmonic traps, which rises the question of the effect of such an external potential.

In this paper, we consider a specific aspect of this question: restricting to perfectly isotropic harmonic traps, but with no constraint on the relative spin populations, we show that the unitary quantum gas admits interesting symmetry properties that have measurable consequences on its spectrum and on the many-body wavefunctions.

These properties imply that there is a mapping between theN-body eigenfunctions in a trap and the zero-energy N-body eigenfunctions in free space; theN-body problem is separable in hyperspherical coordinates; and there exist relations between the moments of the trapping po-

tential energy and those of the total energy at thermal equilibrium.

A unitaryBose gas was not produced yet. This is related to the Efimov effect [6]: when three bosons interact with a short range potential of infinite scattering length, an effective three-body attraction takes place, leading in free space to the existence of weakly bound trimers. This effective attraction generates high values ofkso that the unitarity condition Eq.(1) is violated. It also gives a short lifetime to the gas by activating three-body losses due to the formation of deeply bound molecules [7, 8, 9]. In an isotropic harmonic trap, for three bosons, there exist efimovian states [10, 11], but there also exist eigenstates not experiencing the Efimov effect [10, 12]. These last states are universal (in the sense that they depend only on ¯h, the massmand the trapping frequencyω) and they are predicted to be long-lived [12]. The results of the present paper apply to all universal states, fermionic or bosonic, but do not apply to the efimovian states.

I. OUR MODEL FOR THE UNITARY GAS The physical system considered in this paper is a set of N particles of equal mass m (an extension to different masses is given in Appendix A). The particles are of arbitrary spin and follow arbitrary statistics; the Hamiltonian is supposed to be spin-independent so that theN-body wavefunctionψthat we shall consider corresponds to a given spin configuration [13]. The particles are trapped by the same isotropic harmonic potential of frequencyω. We collect all the positions~ri of the particles in a single 3N component vector:

X~ ≡(~r1, . . . , ~rN). (2) Its norm

X =kXk~ = v u u t

N

X

i=1

r²_i (3)

is called the hyperradius. We will also use the unit vector

~n≡X/X~ (4)

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2 (which may be parametrized by 3N−1 hyperangles). The

coordinates (X, ~n) are called hyperspherical coordinates [14]. The total trapping potential energy simply writes

Htrap= 1

2mω²X². (5)

The interaction between the particles is assumed to be at the unitary limit defined in Eq.(1); one can then replace the interaction by contact conditions on the N- body wavefunction (see e. g. [15, 16] and references therein): when the distancerij =k~rj−~rik between par- ticlesiandjtends to zero, there exists a functionAsuch that

ψ(X~ ) =

r_ij→0

A(R~ij,{~rk:k6=i, j}) rij

+O(rij) (6) whereR~ij = (~ri+~rj)/2 is the fixed center-of-mass position of particlesi and j, and{~rk : k6=i, j}are the positions of the other particles. When none of the particle positions coincide, the stationary wavefunctionψ solves Schr¨odinger’s equation,Hψ=Eψ, with the Hamiltonian

H=−¯h²

2m∆X~ +1

2mω²X². (7) This model is expected to be exact for universal states in the limit of a zero range of the interaction potential [15].

II. SCALING PROPERTIES OF THE TRAPPED UNITARY GAS

A. What is scale invariance ?

A fundamental property of the contact conditions Eq.(6) is their invariance by a rescaling of the spatial coordinates. More precisely, we define a rescaled wave- functionψλ by

ψλ(X~ )≡ψ(X/λ)~ (8) where λ > 0 is the scaling factor. Then, if ψ obeys the contact conditions, so doesψλ for anyλ. Note that this property holds only because the scattering length is infinite (for a finite value ofa, 1/rij in Eq.(6) would be replaced by 1/rij−1/a, which breaks scale invariance).

In free space (that is for ω = 0), the scale invariance implies the following property: if ψ is an eigenstate of energyE, thenψλis an eigenstate of energyE/λ²for any λ[17]. This implies the absence of bound states in free space: otherwise the scaling transform would generate a continuum of states which are square integrable (after elimination of the center of mass variables), and this is forbidden for a Hermitian problem [18].

WhenE= 0, one finds (see Appendix B) that the free space eigenstates can be assumed to be scale-invariant, i.e. there exists an exponentν such that

ψλ(X~ ) =λ⁻^νψ(X~). (9)

Taking the derivative of this relation with respect toλin λ= 1, this shows thatψis an eigenstate of the dilatation operator,

Dˆ ≡X~ ·∂X~, (10) with the eigenvalueν. This result is interesting for section III.

The presence of a harmonic trap introduces the harmonic oscillator length scaleaho=p

¯

h/mω, so that the eigenstates cannot be scale-invariant as in Eq.(9). How- ever, ifψobeys the contact condition, so do theψλ’s: as we shall see, this allows to identify general properties of the eigenstates in the trap.

B. Scaling solution in a time dependent trap We now assume that the trap frequencyω, while keep- ing a fixed valueω(0) for all timest <0, has an arbitrary time dependence at positive times. Let us assume that, at t ≤ 0, the system is in a stationary state of energy E. Then at positive times the wavefunction of the system will be deduced from thet= 0 wavefunction by the combination of gauge and scaling transform [19]:

ψ(X, t) =~ e⁻^iEτ(t)/¯^h

λ(t)^3N/2 e^imX²^λ(t)/2¯^˙ ^hλ(t)ψ(X/λ(t),~ 0) (11) where the time dependent scaling parameter obeys the Newton-like equation

λ¨=ω²(0)

λ³ −ω²(t)λ (12) with the initial conditions λ(0) = 1,λ(0) = 0. We also˙ introduced an effective timeτ given by

τ(t) = Z t

0

dt^′

λ²(t^′). (13) This result may be extended to an arbitrary initial state as follows:

ψ(X, t) =~ 1

λ(t)^3N/2e^imX²^λ(t)/2¯^˙ ^hλ(t)ψ(˜X/λ(t), τ~ (t)).

(14) where ˜ψ evolves with thet <0 Hamiltonian (i.e. in the unperturbed trap of frequencyω(0)).

As shown by Rosch and Pitaevskii [20], the existence of such a scaling and gauge time dependent solution is related to a SO(2,1) hidden symmetry of the problem.

This we rederive in the next subsection.

C. Raising and lowering operators, and SO(2,1) hidden symmetry

We consider the following gedanken experiment: one perturbs the gas in an infinitesimal way by modifying the

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trap frequency in a time interval 0 < t < tf. After the excitation period (t > tf), the trap frequency assumes its initial value ω(0). The scaling parameter then slightly deviates from unity, λ(t) = 1 +δλ(t) with |δλ| ≪ 1.

Linearizing the equation of motion Eq.(12) in δλ, one finds thatδλoscillates as

δλ(t) =ǫe⁻^2iωt+ǫ^∗e^2iωt, (15) where we set ω = ω(0) to simplify the notation. The gedanken experiment has therefore excited an undamped breathing mode of frequency 2ω [20].

We now interpret this undamped oscillation in terms of a property of the N-body spectrum of the system.

Expanding Eq.(11) to first order inδλ(t) leads to ψ(X, t) =~ e^iαh

e^−iEt/¯^h−ǫe^−i(E+2¯^hω)t/¯^hL+

+ ǫ^∗e^−i(E−2¯^hω)t/¯^hL−

iψ(X,~ 0) +O(ǫ²)(16) (the phaseαdepends on the details of the excitation procedure). This reveals that the initial stationary state E was coupled by the excitation procedure to other stationary states of energies E±2¯hω. Remarkably, the wavefunction of these other states can be obtained from the initial one by the action of raising and lowering operators:

L+ = +3N

2 + ˆD+ H

¯

hω −mωX²/¯h (17) L− = −3N

2 −Dˆ + H

¯

hω −mωX²/¯h. (18) Repeated action ofL+andL−will thus generate a ladder of eigenstates with regular energy spacing 2¯hω.

The hidden SO(2,1) symmetry of the problem then results from the fact thatH,L+andL−have commutation relations equal (up to numerical factors) to the ones of the Lie algebra of the SO(2,1) group, as was checked in [20]:

[H, L+] = 2¯hωL+ (19) [H, L−] = −2¯hωL− (20) [L+, L−] = −4H

¯

hω (21)

From the general theory of Lie algebras, one may form the so-called Casimir operator which commutes with all the elements of the algebra, that is withH andL±; it is given by [20]:

Cˆ=H²−1

2(¯hω)²(L+L−+L−L+). (22) Consider a ladder of eigenstates; as we will show later, the hermiticity ofH implies that this ladder has a ground energy step, of valueE0. Within this ladder, the Casimir invariant assumes a constant value,

C=E0(E0−2¯hω). (23)

D. Virial theorem

Another application of the existence of raising and lowering operators is the virial theorem for the unitary gas. For a given eigenstate of H of energy E and real wavefunctionψ, L−|ψiis either zero (if ψis the ground step of a ladder) or an eigenstate of H with a different energy. Assuming thatH is hermitian, this implies hψ|L−|ψi= 0, and leads to [21]:

hψ|H|ψi= 2hψ|Htrap|ψi. (24) At thermodynamical equilibrium, one thus has

hHi= 2hHtrapi, (25) that is the total energy is twice the mean trapping potential energy.

This virial theorem is actually also valid for an anisotropic harmonic trap (this result is due to Fr´ed´eric Chevy). One uses the Ritz theorem, stating that an eigenstate of a hermitian Hamiltonian is a stationary point of the mean energy. As a consequence, the function ofλ

E(λ) ≡ hψλ|H|ψλi hψλ|ψλi

= λ⁻²hψ|H−Htrap|ψi+λ²hψ|Htrap|ψi(26) satisfies (dE/dλ)(λ = 1) = 0, which leads to the virial theorem. This relies simply on the scaling properties of the harmonic potential, irrespective of its isotropy.

The proportionality betweenhHiandhHtrapiresulting from the virial theorem was checked experimentally [22].

III. MAPPING TO ZERO-ENERGY FREE-SPACE EIGENSTATES

Usually, the presence of a harmonic trap in the experiment makes the theoretical analysis more difficult than in homogenenous systems. Here we show that, remarkably, the case of an isotropic trap for the unitary gas can be mapped exactly to the zero-energy free-space problem (which remains, of course, an unsolved many-body problem).

More precisely, all the universal N-body eigenstates can be put in the unnormalized form:

|ψν,qi= (L+)^q e⁻^X^ˆ²^/2a²^ho|ψ⁰_νi (27) and have an energy

Eν,q = (ν+ 2q+ 3N/2)¯hω (28) whereqis a non-negative integer,L+is the raising operator defined in Eq.(17), andψ⁰_ν is a zero-energy eigenstate of the free-space problem which is scale-invariant:

ψ⁰_ν(X/λ) =~ ψ_ν⁰(X~ )/λ^ν (29)

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4 for all real scaling parameterλ,ν being the real scaling

exponent [23].

We also show that the reciprocal is true, that is each zero-energy free-space eigenstate which is scale-invariant with a real exponentν generates a semi-infinite ladder of eigenstates in the trap, according to Eq.(27,28).

We note that Eq.(28) generalizes to excited states a relation obtained in [24] for the many-body ground state.

A. From a trap eigenstate to a free-space eigenstate

We start with an arbitrary eigenstate in the trap. By repeated action of L− on this eigenstate, we produce a sequence of eigenstates of decreasing energies. According to the virial theorem Eq.(24), the total energy of a universal state is positive, since the trapping potential energy is positive. This means that the sequence produced above terminates. We call ψ the last non-zero wavefunction of the sequence, an eigenstate ofH with energyE that satisfies L−|ψi= 0. To integrate this equation, we use the hyperspherical coordinates (X, ~n) defined in Eq.(3,4).

Noting that the dilatation operator is simply ˆD=X·∂X

in hyperspherical coordinates, we obtain:

ψ(X~ ) =e⁻^X²^/2a²^hoX^E/(¯^hω)⁻^3N/2f(~n). (30) Then one defines

ψ⁰(X~)≡e^X²^/2a²^hoψ(X~ ). (31) One checks that this wavefunction obeys the contact conditions Eq.(6), since X² varies quadratically with rij at fixed R~ij and {~rk, k 6= i, j}. ψ⁰ is then found to be a zero-energy eigenstate in free space, by direct inser- tion into Schr¨odinger’s equation. But one has also from Eq.(30,31):

ψ⁰(X~) =X^E/(¯^hω)−3N/2f(~n), (32) so that ψ⁰ is scale-invariant, with a real exponentν related to the energy E by Eq.(28). This demonstrates Eq.(27,28) forq= 0, that is for the ground step of each ladder.

One just has to apply a repeated action of the raising operatorL+on the ground step wavefunction to generate a semi-infinite ladder of eigenstates: this corresponds to q > 0 in Eq.(27,28). Note that the repeated action of L+ cannot terminate since L+|ψi= 0 for a non-zeroψ implies thatψis not square-integrable.

B. From a free-space eigenstate to a trap eigenstate The reciprocal of the previous subsection is also true:

starting from an arbitrary zero-energy free-space eigenstate that is scale-invariant, one multiplies it by the Gaussian factor exp(−X²/2a²_ho), and one checks that the

resulting wavefunction is an eigenstate of the Hamilto- nian of the trapped system, obeying the contact conditions [25]. Applying L+ then generates the other trap eigenstates.

C. Separability in hyperspherical coordinates Let us reformulate the previous mapping using the hyperspherical coordinates (X, ~n) defined in Eq.(3,4). A free-space scale-invariant zero-energy eigenstate takes the formψ⁰(X~ ) =X^νfν(~n), and the universal eigenstates in the trap have an unnormalized wavefunction

ψν,q(X~ ) =X^νe^−X²^/2a²^hoL^{(ν−1+3N/2)}_q (X²/a²_ho)fν(~n), (33) whereL^(.)q is the generalized Laguerre polynomial of de- greeq. This is obtained from the repeated action ofL+

in Eq.(27) and from the recurrence relation obeyed by the Laguerre polynomials:

(q+1)L^(s)_q+1(u)−(2q+s+1−u)L^(s)_q (u)+(q+s)L^(s)_q−1(u) = 0.

(34) We have thus separated out the hyperradiusX and the hyperangles ~n. The hyperangular wavefunctions fν(~n) and the exponentsν are not known forN ≥4. However, we have obtained the hyperradial wavefunctions, i.e. the X dependent part of the many-body wavefunction. A more refined version of these separability results can be obtained by first separating out the center of mass (see Appendix C), but this is not useful for the next Section.

IV. MOMENTS OF THE TRAPPING POTENTIAL ENERGY

A. Exact relations

As an application of the above results, we now obtain the following exact relations on the statistical properties of the trapping potential energy, relating its moments to the moments of the full energy, when the gas is at thermal equilibrium. For the definition of the trapping potential energy, see Eq.(5).

At zero temperature, its moments as a function of the ground state energyE0are given by:

h(Htrap)ⁿi=E0 E0+ ¯hω

. . . E0+ (n−1)¯hω

/2ⁿ. (35) At finite temperatureT, the first moment is given by the virial theorem

hHtrapi=hHi/2 (36) and the second moment by

h(Htrap)²i=

hH²i+hHi¯hω·cotanh ¯hω

kBT

/4.

(37)

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B. Derivation

The zero temperature result Eq.(35) follows directly from Eq.(33): forq= 0, the Laguerre polynomial is constant so that the probability distribution ofX is a power law times a Gaussian; the moments are then given by in- tegrals that can be expressed in terms of the Γ function.

For finiteT, the idea of our derivation is the following:

the hyperradial part of theN-body wavefunctionψν,q is known from Eq. (33); and thus the probability distribution of X in the state |ψν,qi is known, in terms of ν, q.

While the thermal distribution ofqis simple, the one of ν is not, butν is related to the total energy by Eq.(28).

We will need the intermediate quantities:

Bn,p≡ R^∞

0 du e^−uu^s+nL^(s)_q+p(u)L^(s)q (u) R^∞

0 du e⁻^uu^sh

L^(s)q (u)i2 , (38) where s≥0;n, q are non-negative integers; andpis an integer of arbitrary sign. These quantities can be calculated with the n= 0 ‘initial’ condition B0,p = δ0,p and the recurrence relation

Bn+1,p = −(q+p+ 1)Bn,p+1+ [2(q+p) +s+ 1]Bn,p

−(q+p+s)Bn,p−1 (39)

which follows from the recurrence relation Eq.(34) on La- guerre polynomials.

This allows to calculate the moments of the trapping energy in the step q of a ladder of exposant ν, using Eq.(33):

hψν,q|X²ⁿ|ψν,qi

hψν,q|ψν,qi =Bn,0a²ⁿ_ho. (40) Here we have set

s=ν−1 + 3N/2 (41)

in accordance with Eq.(33).

Assuming thermal equilibrium in the canonical ensem- ble, the thermal average can be performed over the sta- tistically independent variablesqands. The moments of q are easy to calculate, because of the ladder structure with equidistant steps:

hqⁿi= P+∞

q=0qⁿe⁻^q¯^hω/k^B^T P+∞

q=0e⁻^q¯^hω/k^B^T . (42) The moments ofs are not known exactly but they can be eliminated in terms of the moments of the total energy E and of the moments of q using the relation E = (s+ 1 + 2q)¯hω. This leads to the exact relations (36,37). This method in principle allows to calculate relations for moments of arbitrary given order, but the algebra becomes cumbersome.

V. CONCLUSION

In this paper we have derived several exact properties of the unitary gas in an isotropic harmonic trap. The spectrum is formed of ladders; the steps of a ladder are spaced by an energy 2¯hω, and linked by raising and lowering operators. This allows to map the trapped problem to the free-space one. The problem is separable in hyperspherical coordinates. This allows to derive exact relations between the moments of the trapping potential energy and the moments of the total energy. The relation between the first moments is the virial theorem; the relation between the second moments may be useful for thermometry, as will be studied elsewhere.

We thank F. Chevy, J. Dalibard, S. Nascimb`ene, L.

Pricoupenko, J. Thomas and C.-F. Vergu for very useful discussions. LKB is a Unité de Recherche de l’ENS et de l’Université Paris 6, associée au CNRS. Our research group is a member of IFRAF.

APPENDIX A: EXTENSION TO PARTICLES WITH DIFFERENT MASSES

All our results remain valid if the particles have different masses m1, . . . , mN; provided that the trapping frequencyω remains the same for all the particles. We define a mean mass:

m≡ m1+. . .+mN

N . (A1)

The definition ofX~ and X, given by Eq.(2,3) for equal masses, has to be generalized to:

X~ ≡

rm1

m~r1, . . . , rmN

m ~rN

. (A2)

X ≡ kX~k= v u u t

N

X

i=1

mi

mr²_i. (A3) With this new definition of X, the trapping potential energy is still given by Eq.(5).

In the definition of the zero-range model, the contact conditions Eq.(6) remain unchanged, except that the fixed center of mass position of particlesi and j is now R~ij≡(mir~i+mjr~j)/(mi+mj).

In Appendix C, the center of mass position has to be redefined as

C~ =(m1r~1+. . .+mNr~N)

(m1+. . .+mN) (A4) and the internal hyperangular coordinates become:

R = v u u t

N

X

i=1

mi

m(~ri−C~)² (A5)

~Ω =

rm1

m

~r1−C~ R , . . . ,

rmN

m

~rN −C~ R

! . (A6)

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6 With these modified definitions, all the results of this

paper remain valid.

APPENDIX B: SCALE INVARIANCE OF THE ZERO-ENERGY FREE-SPACE EIGENSTATES In this appendix, we show that the zero-energy free- space eigenstates of the Hamiltonian may be chosen as being scale-invariant, that is as eigenstates of the dilatation operator ˆD, under conditions ensuring the hermiticity of the Hamiltonian.

Consider the zero-energy eigensubspace of the free- space Hamiltonian. This subspace is stable under the action of ˆD. If one assumes that ˆDis diagonalizable within this subspace, the corresponding eigenvectors form a complete family of scale invariant zero-energy states. If Dˆ is not diagonalizable, we introduce the Jordan normal form of ˆD.

Let us start with the case of a Jordan normal form of dimension two, written as

Mat( ˆD) = ν 1

0 ν

, (B1)

in the sub-basis|e1i,|e2i. The ket|e1iis an eigenstate of Dˆ with the eigenvalueν. We assume that the center of mass motion is at rest, with no loss of generality since it is separable in free space. Using the internal hyperspherical coordinates (R, ~Ω ) defined in Appendix C, we find that Dˆ reduces to the operatorR∂R. IntegratingR∂Re1=νe1

leads to

e1(X~) =R^νφ1(~Ω ). (B2) The ket|e2iis not an eigenstate of ˆDbut obeysR∂Re2= νe2+e1, which, after integration, gives

e2(X~ ) =R^νlogRφ1(Ω ) +~ R^νφ2(~Ω). (B3) One can assume that φ1 and φ2 are orthogonal on the unit sphere (by redifining e2 and φ2). It remains to use the fact that both e1 and e2 are zero-energy free- space eigenstates. From the form of the Laplacian in hyperspherical coordinates in d = 3N −3 dimensions, see Eq.(C5), the condition ∆X~e1= 0 leads to

TΩ~φ1=−ν(ν+d−2)φ1. (B4) The condition ∆X~e2 = 0 then givesT~Ωφ2=−ν(ν+d− 2)φ2−(2ν+d−2)φ1, which leads to the constraint [26]:

ν = 1−d/2. (B5)

At this stage, for this ’magic’ value of ν, it seems that there may exist non scale-invariant zero-energy eigenstates.

To proceed further, one has to check for the hermiticity of the free space Hamiltonian. This requires a reasoning

at arbitrary, non zero energy. We use the fact that the following wavefunction obeys the contact conditions,

ψ(X~ ) =u(R)R^νφ1(~n), (B6) where u(R) is a fonction with no singularity, except maybe in R = 0 [27]. Using again the expression of the Laplacian in internal hyperspherical coordinates, one finds thatψis an eigenstate of the free-space Hamiltonian ifu(R) is an eigenstate of

ˆh=−¯h²

2m(∂_R² +R⁻¹∂R). (B7) One checks that Hermiticity of the free-space Hamilto- nian for the wavefunctionψimplies hermiticity of ˆhfor the ‘wavefunction’u(R). Note that ˆhis simply the free- space Hamiltonian for 2D isotropic wavefunctions. It is hermitian over the domain of wavefunctionsu(R) with a non-infinite limit inR= 0. Including the ket|e2iin the domain of theN-body free-space Hamiltonian amounts to allowing for ‘wavefunctions’u(R) that diverge as logR for R → 0: this breaks the Hermiticity of ˆh, since this leads to a (negative energy) continuum of square integrable eigenstates of ˆh,

uκ(R) =K0(κR) (B8) with eigenenergy−¯h²κ²/2m, for allκ >0. HereK0(x) is a modified Bessel function of the second kind. Hermitic- ity may be restored by a filtering of this contiunuum [29], adding the extra contact conditionu(R) = log(R/l)+o(1) for R → 0, but the introduction of the fixed length l breaks the universality of the problem and is beyond the scope of this paper We thus excludee2from the domain of the Hamiltonian.

This discussion may be extended to Jordan forms of higher order. For example, a Jordan form of dimension 3 generates a ket|e3isuch that ( ˆD−ν)e3 =e2. Bute2

must be excluded from the domain of the Hamiltonian by the above reasoning. Since we want the domain to be stable under ˆD,e3 must be excluded as well.

As a conclusion, to have a free-spaceN-body Hamil- tonian that is both hermitian and universal (i.e. with a scale-invariant domain) forces to reject the non scale- invariant zero-energy eigenstates, of the form Eq.(B3).

APPENDIX C: SEPARABILITY IN INTERNAL HYPERSPHERICAL COORDINATES We develop here a refined version of the separability introduced in subsection III C. First, we separate out the center of mass coordinates. Then we obtain the separability in hyperspherical coordinates relative to the internal variables of the gas, which allows to derive an effective repulsiveN −1 force and to get a lower bound on the energy slightly better than the one E >0 ensuing from the virial theorem.

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Let us introduce the following set of coordinates:

C~ =

N

X

i=1

~ri/N (C1)

is the position of the center of mass (CM);

R= v u u t

N

X

i=1

(~ri−C~)² (C2)

is the internal hyperradius; and

~Ω = ~r1−C~

R , . . . ,~rN −C~ R

!

(C3) is a set of dimensionless internal coordinates that can be parametrized by 3N−4 internal hyperangles. In these coordinates, the Hamiltonian decouples asH =HCM+ Hint with

HCM = − ¯h²

2N m∆C~ +1

2N mω²C² (C4) Hint = −¯h²

2m

∂_R² +3N−4

R ∂R+ 1 R²T~Ω

+1

2mω²R² (C5)

where T~Ωis the Laplacian on the unit sphere of dimension 3N−4. The contact conditions do not break the separability of the center of mass valid in a harmonic trap, so that the stationary state wavefunction may be taken of the form

ψ(X~ ) =ψCM(C)ψ~ int(R, ~Ω ). (C6) One can show [30] that there is separability in internal hyperspherical coordinates:

ψint(R, ~Ω ) = Φ(R)φ(Ω ).~ (C7) This form may be injected into the internal Schr¨odinger equation

Hintψint =Eintψint. (C8) One finds thatφ(~Ω ) is an eigenstate ofT~Ωwith an eigenvalue that we call−Λ. Note that the contact conditions Eq.(6) put a constraint onφ(~Ω ) only [27]. The equation for Φ(R) reads:

− ¯h² 2m

∂_R² +3N−4 R ∂R

Φ +

¯h²Λ 2mR² +1

2mω²R²

Φ

=EintΦ. (C9) A useful transformation of this equation is obtained by the change of variable:

Φ(R)≡R⁽⁵⁻^3N^)/2F(R), (C10)

resulting in

− ¯h² 2m

∂_R² + 1 R∂R

F +

¯h²s²_R 2mR² +1

2mω²R²

F

=EintF(R), (C11) wheresR is such that

s²_R= Λ +

3N−5 2

2

. (C12)

Formally, the equation for F is Schr¨odinger’s equation for a particle of zero angular momentum moving in 2D in a harmonic potential plus a potential∝s²_R/R².

For s²_R ≥ 0, one can choose sR ≥ 0. Assuming that there is noN-body resonance,F(R) is bounded for R→0 [31]. The eigenfunctions of Eq.(C11) can then be expressed in terms of the generalized Laguerre polynomials:

F(R) =R^s^RL^s_q^R[R²/a²_ho]e⁻^R²^/2a²^ho (C13) with the spectrum:

Eint = (sR+ 1 + 2q)¯hω. (C14) This gives a lower bound on the energy of any universal N-body eigenstate:

E≥5

2¯hω (C15)

forN >2 and in the absence of aN-body resonance.

For a complexs²_R, the effective 2D Hamiltonian is not hermitian and this case has to be discarded. Fors²_R<0, Whittaker functions are square integrable solutions of the effective 2D problem for all valuesEintso that, again, the problem is not hermitian. One may add extra boundary conditions to filter out an orthonormal discrete subset (as was done forN = 3 bosons [6, 10, 12, 33]) but this breaks the scaling invariance of the domain and generates non- universal states beyond the scope of the present paper.

To make the link with the approach of Section III, we note that

F(R) =R^s^R (C16)

is a solution of the effective 2D problem (C11) forω = 0, Eint = 0. Thus a solution of the internal problem Eq.(C8) at zero energy in free space is given by

ψint(R,Ω) =R^(5−3N^)/2+s^Rφ(Ω).~ (C17) Multiplying this expression byC^lY_l^m(C/C), one recovers~ theψ_ν⁰’s of Section III, with

ν= 5−3N

2 +sR+l. (C18)

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8

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[17] For three bosons, this holds only in the subspace of universal states.

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[23] There are in general several zero-energy free-space eigenstates with the same exponentν, so that our notationψν⁰

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[24] S. Tan, cond-mat/0412764.

[25] This holds also when the exponent ν of the free-space eigenstate is not real, in which case the Hamiltonian is not hermitian since the resulting eigenenergy in the trap is complex. This happens for the Efimov states of 3 bosons, which can be written as a sum of scale-invariant states with complex conjugate exponents. This exampli- fies the fact that the present algebra is meaningful for universal states only.

[26] To obtain Eq.(B5), consider ψ2 ≡ R^νu(R)φ2(~Ω). This wavefunction satisfies the contact conditions [27]. It thus belongs to the domain ofH (for an appropriate u(R)).

SinceHe1 = 0, hermiticity ofH implies he1, Hψ2i = 0;

which leads to the result.

[27] To establish this fact, one needs the following lemma: if ψ(X~) obeys the contact conditions, so does u(R)ψ(X~), where u(R) is a function with no singularity, except maybe inR= 0.This lemma results from the fact that R²=R²0+O(rij²), whereR²0is a non-zero constant in the rij → 0 limit [28], so thatu(R) =u(R0) +O(r²ij).

[28] We assume hereN≥3; in this case,R0 is different from zero since the contact conditions apply for R~ij 6= ~rk, whateverk6=i, j.

[29] P.M. Morse, H. Feshbach, Methods of Theoretical Physics, part II, p. 1665 (Mc Graw-Hill, 1953).

[30] The whole algebra of section III may be reproduced for the internal problem, with the modified raising and lowering operators: L±≡ ± ^3N−3₂ +R∂R

+^H_¯_hω^int−^mω_¯_h R². [31] The absence ofN-body resonance corresponds to the sim-

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ple boundary condition: (i)F(R) bounded for R → 0.

For 0 < sR < 1, N-body resonances can be taken into account by the modified boundary condition: (ii) F(R) = A R^s^R[R^−2s^R −ǫ l^−2s^R] +O(R^−s^R⁺²), where l∈]0,+∞] andǫ=±1 are fixed. If one is not exactly on theN-body resonance, one hasl <∞, which breaks scale invariance, thus invalidating the results of this paper. Ex- actly on theN-body resonance (e. g. when the energy of theN-body bound state in free space vanishes), one has l=∞, which preserves scale invariance: all the results of this paper remain valid; one must only replacesRby−sR

in Equations (C13,C14,C16,C17,C18), and Eq.(C15) be- comesE ≥3/2¯hω. For sR = 0, N-body resonances can be taken into account by the modified boundary condi-

tion: (iii)F(R) =Alog(R/l) +o(1), wherel∈]0,+∞[ is fixed. If one is not exactly on theN-body resonance, one hasl <∞, which breaks scale invariance, thus invalidating the results of this paper. Being exactly on theN-body resonance corresponds to taking the limitl→ ∞in (iii), which is equivalent to (i): the results of this paper then remain valid. ForsR≥1, the wavefunctionF(R) =R^−s^R is not square integrable nearR = 0 in 2 dimensions, so that the description ofN-body resonances becomes more complicated (see [32] for the caseN= 2).

[32] L. Pricoupenko, Phys. Rev. A73, 012701 (2006); L. Pri- coupenko, Phys. Rev. Lett.96, 050401 (2006).

[33] G. S. Danilov, Sov. Phys. JETP13, 349 (1961).