The interaction-sensitive states of a trapped two-component ideal Fermi gas and application to the virial expansion of the unitary Fermi gas
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b The interaction-sensitive states of a trappedtwo-component ideal Fermi gas and application tothe virial expansion of the unitary Fermi gas
Shimpei Endo and Yvan Castin ∗ Laboratoire Kastler Brossel, ENS-PSL, CNRS, UPMC-Sorbonne Universit´es andColl`ege de France, Paris, France ∗ [email protected] Abstract.
We consider a two-component ideal Fermi gas in an isotropic harmonicpotential. Some eigenstates have a wavefunction that vanishes when twodistinguishable fermions are at the same location, and would be unaffected by s -wavecontact interactions between the two components. We determine the other, interaction-sensitive eigenstates, using a Faddeev ansatz. This problem is nontrivial, due todegeneracies and to the existence of unphysical Faddeev solutions. As an application wepresent a new conjecture for the fourth-order cluster or virial coefficient of the unitaryFermi gas, in good agreement with the numerical results of Blume and coworkers.PACS numbers: 03.75.Ss - Degenerate Fermi gases
1. Introduction and motivations
We consider a three-dimensional trapped two-component ideal Fermi gas. The twocomponents, noted as ↑ and ↓ , correspond to two spin components of a single fermionicatomic species, or to two different fully polarised fermionic atomic species. The singleparticle masses m ↑ and m ↓ in each component may thus differ. There is no coherentcoupling between the two states ↑ and ↓ so the total particle numbers in each component N ↑ and N ↓ are fixed, not simply the total particle number N = N ↑ + N ↓ . One can thentake as reference spin configurations the N ↑ + N ↓ configurations ↑ . . . ↑↓ . . . ↓ , wherethe wavefunction ψ ( r , . . . , r N ) is antisymmetric under the exchange of the positions ofthe first N ↑ particles, and under the exchange of the positions of the last N ↓ particles.The particles are trapped in the isotropic harmonic potential U σ ( r ) = m σ ω r / σ = ↑ , ↓ in such a way that the angular oscillation frequency ω is σ -independent. In the experiments on cold atoms, where the interaction strength canbe tuned via a Feshbach resonance [1, 2], our system is not a pure theoretical perspectiveand can be realised.Imagine now that one turns on arbitrarily weak binary contact interactions betweenopposite spin particles. As the interaction has a zero range, it acts only among pairs ofparticles that approach in the s -wave. If one treats the interaction as a Dirac delta to deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion interaction-sensitive states, and the unshifted ones to interaction-insensitive states.This criterion can be implemented experimentally, by measuring the energy levels in thetrap [3]. Interestingly, the interaction-insensitive states have a vanishing wavefunctionwhen any pair of particles converge to the same location; they thus remain unaffectedby the interaction whatever its strength, provided that it remains zero range.Even if this is an ideal gas problem, it is to our knowledge not treated in theclassic literature. The interactions are usually of nonzero range, in nuclear physics or inquantum chemistry, and are not restricted to the s -wave channel; in this traditionalcontext, our problem totally lacks physical motivation. This is probably why thisproblem was not mentioned in the classic book of Avery on hyperspherical harmonics[4], although the wavefunctions we are looking for are particular cases of hypersphericalharmonics, as we shall see. Actually, specifically determining the interaction-sensitivestates, and not simply all the eigenstates of trapped non-interacting fermions, isnontrivial due to the occurrence of large degeneracies of the unperturbed spectrumin an isotropic harmonic trap, so one faces the diagonalisation of large matricesin the degenerate perturbation theory, even if the problem can be first analyticallyreduced by the explicit construction of hyperspherical harmonics in Jacobi coordinatesthat are invariant (up to a global sign) under the exchange of identical fermions[5]. This degeneracy issue is reminiscent of the Fractional Quantum Hall Effect forcontact interactions between cold atoms in an artificial magnetic field, where themacroscopic degeneracy of the Lowest Landau Level makes it nontrivial, even to firstorder perturbation theory, to determine the gapped phases induced by the interactions[6]. The famous Laughlin wavefunction, when transposed to spinless bosons, is actuallyan interaction-insensitive state, which is thus automatically separated in energy spacefrom the other, interaction-sensitive states when a repulsive contact interaction isturned on. This is why, in reference [7], the interaction-insensitive states were termedlaughlinian states.Another physical motivation is the calculation of the cluster or virial coefficients ofthe spatially homogeneous spin-1 / s -wave scattering length. It is indeed nowpossible to measure the equation of state of the unitary gas with cold atoms [8, 9, 10],from which one can extract the cluster coefficients up to fourth order [8, 10]. Werecall that the cluster coefficients b N ↑ ,N ↓ are, up to a factor, the coefficients of theexpansion of the pressure of the thermal equilibrium gas of temperature T in powersof the small fugacities z σ = exp( µ σ /k B T ), that is in the low-density, non-degeneratelimit where the chemical potential µ σ of each spin component σ tends to −∞ [11].For the unitary gas, it is efficient to use the harmonic regulator technique of reference[12], that is to determine the cluster coefficients B N ↑ ,N ↓ ( ω ) for the trapped system, inorder to use its SO(2,1) dynamical symmetry [13, 14, 15]; then one takes the ω → b N ↑ ,N ↓ . It only remains to solve trapped few-body problems, since deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion B N ↑ ,N ↓ can be expressed in terms of the energy spectrum of all the n ↑ + n ↓ systems, with n σ ≤ N σ . For the third cluster coefficient, this procedure was implemented numericallyin reference [16], and then analytically in reference [17] by a generalisation to fermions ofthe inverse residue formula used for bosons in reference [18]. The predictions agree withthe experimental results. For the fourth virial coefficient, its numerical implementationby a direct calculation of the first few energy levels of four trapped fermions could notbe pushed to low enough values of ~ ω/k B T to allow for a successful comparison withexperiment [19], and its analytical implementation is still an open problem [20].In all these calculations, what is actually computed is the difference ∆ B N ↑ ,N ↓ ( ω )between the cluster coefficients of the unitary gas and of the ideal gas, so as to get ridof the contributions of the interaction-insensitive states, which are common to the twosystems and exactly cancel. So for the ideal gas, one must determine the energy levelsof the interaction-sensitive states. For the 2 + 1 fermionic systems (or equivalently for 3bosons) this was done analytically in references [16, 18]. For the 3+1 and 2+2 fermionicsystems, this was done numerically for the first few energy levels in reference [19]. Inthis work, we obtain from a Faddeev ansatz an analytical prediction for all values of N ↑ and N ↓ . We then face a subtlety of the problem, that was already known for 2 + 1fermions [7]: some of the energy levels predicted by our Faddeev ansatz are unphysicaland must be disregarded, since the corresponding wavefunction is zero. We solve thisissue for 3 + 1 and 2 + 2 fermions, with a general analytical reasoning completed for2 + 2 fermions by a case by case analysis.The paper is organised as follows. In section 2, we introduce the basic theory toolsalready available in the literature [21], allowing us to reduce the problem to a zero energyfree space problem with a wavefunction of the Faddeev form, each free space solution,characterised by a scaling exponent s , giving rise in the trapped system to a semi-infiniteladder of interaction-sensitive energy levels equispaced by 2 ~ ω . In section 3, we givethe corresponding scaling exponents s for an arbitrary N ↑ + N ↓ spin configuration. Insection 4, we investigate for N = 4 the unphysical values of s , that are artifacts of theFaddeev ansatz. In section 5 we present some applications to the cluster expansion ofthe unitary gas, with a new conjecture for the fourth cluster coefficient and a comparisonto the numerical results of [19]. We conclude in section 6.
2. The theoretical building blocks
In this section, we remind the reader how, due to scale invariance, the energy levelsof the trapped system can be deduced from the zero-energy free space solutions, moreprecisely from their scaling exponents (for a review, see reference [21]). We also explain,building on a footnote of reference [18], how the interaction-sensitive states of the idealgas can be singled out from the interaction-insensitive ones using a Faddeev ansatz forthe N -body wavefunction. deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion In free space, the ideal gas Hamiltonian H free = N ↑ X i =1 − ~ m ↑ ∆ r i + N X i = N ↑ +1 − ~ m ↓ ∆ r i (1)is scale invariant. Therefore, if ψ free ( r , . . . , r N ) is an eigenstate of H free with theeigenvalue zero, H free ψ free = 0 (2)so is ψ λ free ( r , . . . , r N ) ≡ ψ free ( λ r , . . . , λ r N ), where all coordinates are multiplied bythe same arbitrary scaling factor λ >
0. An elementary consequence is that one canchoose ψ free to be scale invariant, which means that the wavefunctions ψ λ free and ψ free areproportional. The corresponding scaling exponent s of ψ free is then conveniently definedas follows: ψ free ( λ r , . . . , λ r N ) = λ s − N − ψ free ( r , . . . , r N ) ∀ λ > ψ free ( r , . . . , r N ) is a positively homogeneous function of the coordinatesof degree s − (3 N − /
2. Further using the free space translational invariance, oneimposes that the centre of mass of the system is at rest: ψ free ( r + u , . . . , r N + u ) = ψ free ( r , . . . , r N ) ∀ u ∈ R (4)A more elaborate consequence is that one can generate from ψ free a semi-infiniteladder of exact eigenstates of the Hamiltonian H of the trapped system, H = H free + H trap , H trap = N ↑ X i =1 m ↑ ω r i + N X i = N ↑ +1 m ↓ ω r i (5)Each rung of the ladder is indexed by a quantum number q ∈ N . The correspondingunnormalised wavefunction is [15] ψ q ( r , . . . , r N ) = L ( s ) q ( R /a ) e − P Ni =1 m i ωr i / (2 ~ ) ψ free ( r , . . . , r N ) (6)where m i is the mass of particle i , R is the internal hyperradius of the N particles R ≡ " m u N X i =1 m i ( r i − C ) / (7)involving the position of the centre of mass C = (cid:16)P Ni =1 m i r i (cid:17) / (cid:16)P Ni =1 m i (cid:17) of the systemand some arbitrary mass reference m u , a ho = [ ~ / ( m u ω )] / is the corresponding harmonicoscillator length and L ( s ) q ( X ) is the generalised Laguerre polynomial of degree q : L ( s ) q ( X ) ≡ X − s e X q ! d q d X q ( X q + s e − X ) (8)In a harmonic potential, the centre of mass motion and the relative motion are separable.Since the wavefunction ψ free and the internal variable R are translationally invariant,the wavefunction ψ q corresponds to the centre of mass motion in its ground state with deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion ~ ω/
2. The eigenenergy of ψ q is thus E q = ~ ω + E rel q , where E rel q is therelative or internal eigenenergy, given by [15] E rel q = ( s + 1 + 2 q ) ~ ω ∀ q ∈ N (9)Physically, this ladder structure reflects the fact that scale invariant systems acquire ina harmonic trap an exact breathing mode of angular frequency 2 ω [13, 14]. This mode,when quantised, is a bosonic mode of Hamiltonian 2 ~ ω ˆ b † ˆ b , where the creation operatorˆ b † and the annihilation operator ˆ b , obeying the usual commutation relation [ˆ b, ˆ b † ] = 1,are raising and lowering operators in each semi-infinite ladder, exciting and deexcitingthe breathing mode by one quantum [15]. Mathematically, this reflects the SO(2,1)dynamical symmetry of the trapped system, H being part of a SO(2,1) Lie algebra.One can show that the mapping (6) is complete, meaning that all eigenstates inthe trap with a ground state centre of mass are obtained if one uses all possible ψ free [15]. The trapped problem is thus reduced to a zero energy free space problem in therest frame and we only need in practice to determine the scaling exponents s of thecorresponding interaction-sensitive eigenstates ψ free . To filter out the interaction-sensitive states of the ideal gas, we use the techniqueproposed in a footnote of reference [18]. We introduce a zero-range interaction betweenthe opposite spin fermions, with a finite s -wave scattering length a , in the form ofWigner-Bethe-Peierls contact conditions on the N -body wavefunction [22, 23]: for all ↑↓ pairs, that is for all particle indices i and j , with 1 ≤ i ≤ N ↑ and N ↑ + 1 ≤ j ≤ N ,there exists a function A ij , called the regular part , such that ψ free ( r , . . . , r N ) r ij → = (cid:18) r ij − a (cid:19) A ij (( r k − R ij ) k = i,j ) + O ( r ij ) (10)Here, the relative coordinates r ij = r i − r j of particles i and j tend to zero at a fixedposition R ij = ( m i r i + m j r j ) / ( m i + m j ) of their centre of mass, different from thepositions r k , 1 ≤ k ≤ N and k = i, j , of the other particles. Due to the assumedtranslational invariance (4) of the wavefunction in free space, we have directly consideredhere A ij as a function of the relative positions r k − R ij . The idea now is that theinteraction-insensitive states have identically zero regular parts, A ij ≡
0, for all i and j .The interacting states, on the contrary, have nonzero regular parts, and they converge,when a →
0, to the desired interaction-sensitive states of the ideal gas.To solve Schr¨odinger’s equation in the presence of the contact conditions (10), oneformulates it in the framework of distributions [24, 25]. Due to the 1 /r ij singularities,to the identity ∆ r (1 /r ) = − πδ ( r ) and to the rewriting − ~ m i ∆ r i − ~ m j ∆ r j = − ~ M ↑↓ ∆ R ij − ~ µ ↑↓ ∆ r ij (11)with M ↑↓ = m ↑ + m ↓ the total mass and µ ↑↓ = m ↑ m ↓ /M ↑↓ the reduced mass of twoopposite spin particles, equation (2) acquires three-dimensional Dirac delta terms in the deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion H free ψ free ( r , . . . , r N ) = N ↑ X i =1 N X j = N ↑ +1 π ~ µ ↑↓ A ij (( r k − R ij ) k = i,j ) δ ( r ij ) (12)Multiplying formally equation (12) by the inverse of the operator H free , that is expressingits solution in terms of the Green’s function of a 3 N dimensional Laplacian, we obtain ψ free as a sum over i and j of Faddeev components, ψ free = N ↑ X i =1 N X j = N ↑ +1 F ij , with F ij ≡ H free π ~ µ ↑↓ A ij (( r k − R ij ) k = i,j ) δ ( r ij ) (13)Let us review the symmetry properties of the Faddeev components. First, the( i, j ) source term in equation (12) is translationally invariant, as well as H free , and sois F ij . Second, the ( i, j ) source term is invariant by rotation of r ij at fixed R ij , andso is F ij because the i and j Laplacians in H free can be rewritten as in equation (11);as a consequence, F ij depends on r ij only through its modulus r ij . Third, due to thefermionic exchange symmetry, the regular parts A ij are not functionally independent andcoincide with A N ↑ +1 up to a sign, which is the signature of the permutation that maps(1 , . . . , i, . . . , N ↑ , N ↑ + 1 , . . . , j, . . . , N ) to ( i, , . . . , i − , i + 1 , . . . , N ↑ , j, N ↑ + 1 , . . . , j − , j + 1 , . . . , N ): A ij (( x k ) k = i,j ) = ( − i − ( − j − ( N ↑ +1) A N ↑ +1 (( x k ) k = i,j ) (14)Similarly, the Faddeev components can all be expressed in terms of the first Faddeevcomponent F N ↑ +1 , noted as F for concision. Fourth, at fixed ( i, j ) = (1 , N ↑ + 1), thefermionic exchange symmetry among the last N ↑ − ↑ particles and among the last N ↓ − ↓ particles imposes that F ( r ; ( x k ) k =1 ,N ↑ +1 ) is a fermionic function of its first N ↑ − N ↓ − F ( r ; ( x σ ( k ) ) ≤ k ≤ N ↑ , ( x k ) N ↑ +2 ≤ k ≤ N ) = ǫ ( σ ) F ( r ; ( x k ) k =1 ,N ↑ +1 ) (15) F ( r ; ( x k ) ≤ k ≤ N ↑ , ( x σ ( k ) ) N ↑ +2 ≤ k ≤ N ) = ǫ ( σ ) F ( r ; ( x k ) k =1 ,N ↑ +1 ) (16)where σ , of signature ǫ ( σ ), is any permutation of N ↑ − N ↓ − a → ψ free ( r , . . . , r N ) = N ↑ X i =1 N X j = N ↑ +1 ( − i − j − ( N ↑ +1) F ( r ij ; ( r k − R ij ) k = i,j ) (17)The key point is that, in the ( i, j ) component, particles i and j approach in a purely s -wave relative motion, which is a necessary condition for them to be sensitive to s -wavecontact interactions.It will be shown in section 4 that this is not always sufficient to make ψ free interaction-sensitive, because the Faddeev ansatz leads in some cases to ψ free ≡
0, thatis to unphysical solutions. To investigate this point, the momentum space version of deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion π ~ µ ↑↓ A N ↑ +1 (( x k ) k ∈ I ) = 1(2 π ) Z Y j ∈ I d k j (2 π ) D (( k j ) j ∈ I ) e i P j ∈ I k j · x j (18)where all indices run over the set I of integers from 2 to N different from N ↑ + 1, I = { , . . . , N } \ { , N ↑ + 1 } (19)we obtain the Fourier space representation of the Faddeev component F ( r ; ( x k ) k ∈ I ) = Z d q (2 π ) Y j ∈ I d k j (2 π ) D (( k j ) j ∈ I ) e i P j ∈ I k j · x j e i q · r ~ q µ ↑↓ + ~ ( P j ∈ I k j ) M ↑↓ + P j ∈ I ~ k j m j (20)where r is any vector of modulus r and, physically, q is the relative wave vector ofparticles 1 and N ↑ + 1 and − P j ∈ I k j their total wave vector. This corresponds to thefollowing ansatz for the Fourier transform of the N -body wavefunction:˜ ψ free ( k , . . . , k N ) = δ ( P Ni =1 k i ) P Ni =1 ~ k i m i N ↑ X i =1 N X j = N ↑ +1 ( − i − j − ( N ↑ +1) D (( k n ) n = i,j )(21)in agreement with reference [28]. Obviously, D (( k j ) j ∈ I ) is fermionic with respect to itsfirst N ↑ − N ↓ − A N ↑ +1 and F . Also, its scaling exponent in the unitary limit canbe expressed in terms of the scaling exponent s of the wavefunction through the usualpower-counting argument for the Fourier transform: D ( λ ( k n ) n ∈ I ) = λ − ( s + N − ) D (( k n ) n ∈ I ) ∀ λ >
3. Scaling exponents of the interaction-sensitive states of the ideal gas
It is well known from the one-body case that all eigenstates of the trapped systemHamiltonian H are products of polynomials in the 3 N coordinates of the particles andof the Gaussian factor appearing in equation (6), and so are the ψ q . Taking q = 0 inthat equation, so that L ( s ) q ≡
1, one sees that the free space eigenstate ψ free ( r , . . . , r N ) isnecessarily such a polynomial, and so is the Faddeev component F ‡ . As F depends on ‡ Up to an appropriate coordinate rescaling to account for a possible mass difference m ↑ = m ↓ , ψ free ( r , . . . , r N ) is a harmonic polynomial of degree d , since it is homogeneous and of zero Laplacian,and is translationally invariant, so it can be written as R d Y d (Ω), where R is the internal hyperradius (7),Ω is a set of hyperangles and Y d is a so-called hyperspherical harmonic. We are however only interestedin the specific case of interaction-sensitive states, not discussed in the extensive book of Avery onhyperspherical harmonics [4]. The reference [5] implemented the formalism of Avery with cleverlychosen Jacobi coordinates ρ i , that are invariant (up to a global sign) under the exchange of identical deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion r (and not on the direction) of the relative coordinates of two ↑↓ particles,only even powers of r can contribute to its expansion, hence the specific ansatz: F ( r ; ( x i ) i ∈ I ) = X k ≥ r k P k (( x i ) i ∈ I ) (23)where the set I is given by equation (19). Since ψ free has a well defined scaling exponent s , see equation (3), F is a homogeneous polynomial of degree d = s − N −
52 (24)so that each polynomial P k is homogeneous of degree d − k as long as d − k ≥ ψ free has a zeroeigenenergy with respect to the free space Hamiltonian, see equation (2): H free F ( r N ↑ +1 ; ( r i − R N ↑ +1 ) i ∈ I ) = 0 (25)From the explicit form (1) of H free , modified with equation (11) for the first ↑ and ↓ particles, and the chain rule of differential calculus, this is turned into a differentialequation for F :(∆ r + ˆ D ) F ( r ; ( x i ) i ∈ I ) = 0 (26)Here ∆ r , the usual three-dimensional Laplacian, can be restricted to its radial part r − ∂ r ( r · ) as far as the variable r is concerned, and the differential operator ˆ D , actingonly on the vectorial variables of the Faddeev component, is given byˆ D = (1 − t ) N ↑ X i =2 ∆ x i + t (2 − t ) N X j = N ↑ +2 ∆ x j + 2 t (1 − t ) N ↑ X i =2 N X j = N ↑ +2 ∇ x i · ∇ x j (27)with the mass ratio t = m ↑ m ↑ + m ↓ ∈ ]0 ,
1[ (28)When applied to the expansion (23), the equation (26) gives a recurrence relation onthe polynomials P k , P k +1 (( x i ) i ∈ I ) = − k + 2)(2 k + 3) ˆ DP k (( x i ) i ∈ I ) ∀ k ≥ D on thepolynomial P , the generating polynomial.In conclusion, to generate an arbitrary interaction-sensitive state ψ free of zero energyin free space, one simply has to arbitrarily choose a polynomial P (( x i ) i ∈ I ) which ishomogeneous of degree d ∈ N and antisymmetric under the exchange of its first N ↑ − N ↓ − fermions. For example, for equal mass ↑↑↓ fermions, it took ρ = r − r and ρ = ( r + r ) / − r .To express however the fact that, in an interaction-sensitive state, the opposite-spin particles 1 and 3approach in the s -wave, one must rather use a system of coordinates containing r = r − r , whichis not invariant by permutation of particles 1 and 2. This is why we introduced the extra ingredient ofthe Faddeev ansatz in equation (17), not relying on a specific choice of Jacobi coordinates. deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion F ( r ; ( x i ) i ∈ I ) = X k ≥ r k ( − ˆ D ) k (2 k + 1)! P (( x i ) i ∈ I ) (30)Then, one reconstructs the wavefunction ψ free from equation (17), and one generates asemi-infinite ladder of interaction-sensitive eigenstates of the trapped system using themapping (6).A natural choice, inspired by the rotational invariance, is to take as a basis of thepolynomials of a single vectorial variable x the set of homogeneous monomials x x n + ℓ Y m ℓ (ˆ x ) (31)where n ∈ N , ˆ x = x /x is the direction of x , parametrised by a polar angle andan azimuthal angle in spherical coordinates, and Y m ℓ is the corresponding sphericalharmonic of orbital quantum number ℓ ∈ N and azimuthal quantum number m (inroman style to avoid confusion with a mass). To construct P , one then puts one ↑ fermion in each ( n i , ℓ i , m i ) state for 2 ≤ i ≤ N ↑ , and one ↓ fermion in each ( n i , ℓ i , m i )state for N ↑ + 2 ≤ i ≤ N , where the monomial states are chosen freely, except forthe constraint that, within each spin manifold, they must be different and sorted inalphanumeric order to avoid multiple counting. This simple construction leads to atotal degree d = P i ∈ I (2 n i + ℓ i ) and to a scaling exponent s = 3 N −
52 + X i ∈ I (2 n i + ℓ i ) (32)According to the equation (9) the corresponding semi-infinite ladder of internal energiesof interaction-sensitive states is E rel q = (cid:18) q + 32 (cid:19) ~ ω + X i ∈ I (cid:18) n i + ℓ i + 32 (cid:19) ~ ω (33)This writing lends itself to a simple physical interpretation. The first term is an energylevel of a harmonically trapped fictitious particle with zero angular momentum; thisfictitious particle corresponds to the relative motion of two opposite spin fermions inthe trap, and its restriction to the zero angular momentum sector ensures that it issensitive to s -wave interactions. The second contribution in equation (33) is any energylevel of an ideal gas of N ↑ − ↑ fermions and N ↓ − ↓ fermions in the trap.As we shall see, the result (32) has to be refined for N >
2, as well as the transparentform (33): some scaling exponents are unphysical and must be disregarded. , and fermions For few-body systems, it is most convenient to take generating polynomials P witha well defined total angular momentum ℓ . As the r variable in equation (30) carriesa zero total angular momentum, the Faddeev component F and the correspondingwavefunction ψ free have a total angular momentum ℓ . This conclusion extends to the deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion ψ q since the variable R and the Gaussian factor in equation (6)are rotationally invariant (remember that the centre of mass of the gas is in its groundstate). Similarly, the eigenstates have the same parity as P .For 2 + 1 fermions, the sum in equation (32) contains a single term. The generatingexponents of the interaction-sensitive states are thus s ℓ,n = 2 n + ℓ + 2 , ∀ ( n, ℓ ) ∈ N (34)with a degeneracy 2 ℓ + 1 and a parity equal to the natural parity ( − ℓ . This agreeswith reference [16].For 3 + 1 fermions, the sum in equation (32) runs over the set I = { , } so itinvolves the principal n i and orbital ℓ i quantum numbers of particles 2 and 3. As theseare identical fermions, it is more convenient to use the principal ( n com , n rel ) ∈ N andorbital ( ℓ com , ℓ rel ) ∈ N quantum numbers of their centre of mass and relative motions,rewriting (32) as s = 72 + 2( n com + n rel ) + ℓ com + ℓ rel (35)and restricting to odd values of ℓ rel . From the composition of the two angular momenta ℓ com and ℓ rel , an angular momentum ℓ can be obtained if and only if ( ℓ rel , ℓ com , ℓ ) canbe the lengths of the sides of a triangle, that is | ℓ rel − ℓ com | ≤ ℓ ≤ ℓ rel + ℓ com , or moreconveniently | ℓ rel − ℓ | ≤ ℓ com ≤ ℓ rel + ℓ (36)The resulting parity ( − ℓ rel + ℓ com can now differ from the natural parity ( − ℓ . We writeit as σ ( − ℓ , where σ = ±
1. Equivalently, ℓ rel + ℓ com ≡ ℓ + (1 − σ ) / s ( σ ) ℓ,n = 2 n + ℓ + 1 − σ , ∀ ( n, ℓ ) ∈ N , ∀ σ ∈ {− , } (37)It remains to sum the natural degeneracy 2 ℓ + 1 over all values of ( n com , n rel ) and( ℓ com , ℓ rel ) to obtain the full degeneracy D ( σ ) ℓ,n = (2 ℓ + 1) X ℓ rel ∈ N +1 ℓ + ℓ rel X ℓ com = | ℓ − ℓ rel | X ( n rel ,n com ) ∈ N δ n rel + n com ) ,p − ℓ rel − ℓ com (38)where δ is the Kronecker symbol and p = ℓ + 2 n + − σ . The sum over ( n rel , n com ) isreadily performed using the variables n tot = n rel + n com ∈ N and n rel ranging from 0 to n tot , as the summand depends only on n tot . This sum is nonzero only if ℓ com ≤ p − ℓ rel and if p − ℓ rel − ℓ com is even, this second condition being taken care of by inclusion of afactor [1 + ( − p − ℓ rel − ℓ com ] /
2. Similarly, one introduces a factor [1 − ( − ℓ rel ] / ℓ rel due to the fermionic antisymmetry. This leads to D ( σ ) ℓ,n = (2 ℓ + 1) X ℓ rel ∈ N min ( ℓ + ℓ rel ,p − ℓ rel ) X ℓ com = | ℓ − ℓ rel | (cid:20) − ( − ℓ rel (cid:21) (cid:20) − p − ℓ rel − ℓ com (cid:21) × (cid:18) p − ℓ rel − ℓ com (cid:19) (39) deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion ≤ p − ℓ ≤ ℓ and ℓ < p − ℓ § andthe subcases σ = ±
1. We finally obtain for 3 + 1 fermions: D ( σ ) ℓ,n = (2 ℓ +1) n (2 ℓ +1+ σ )( n +1)( n +2) − [ σ +( − ℓ ] h n +1+ − n io I = { , } are nowdistinguishable. One reuses the last calculation, simply relaxing the parity condition on ℓ rel , that is removing the factor − ( − ℓ rel in equation (39). The scaling exponent of theinteraction-sensitive states, written as in equation (37), now has a degeneracy D ( σ ) ℓ,n = (2 ℓ + 1) (cid:18) ℓ + 1 + σ (cid:19) ( n + 1)( n + 2)2 (41)Both results (40) and (41) vanish at all n for ( ℓ, σ ) = (0 , −
1) as they should, sinceisotropic states of two particles (corresponding to the set I ) necessarily have the naturalparity +1. Both also include unphysical scaling exponents corresponding to a vanishingFaddeev ansatz wavefunction (17); this will be corrected in section 4.
4. Refining the theory: exclusion of the unphysical solutions
For
N >
2, some of the scaling exponents predicted in section 3 are unphysical, as theydo not correspond to any interaction-sensitive state of the ideal gas: the correspondingFaddeev ansatz wavefunction (17) vanishes, due to the destructive interference of itsindividually nonzero Faddeev components. This problem was already solved for N = 3:there is a single unphysical solution [7], corresponding to ( n, ℓ ) = (0 ,
0) in equation (34),that is to a generating polynomial P = 1 and a Faddeev component F = 1 obviouslygiving ψ free ≡ N >
3. Weinvestigate it explicitly for N = 4. An infinite number of unphysical solutions is easilypredicted by a formal reasoning in Fourier space with divergent integrals, in section 4.1.Then we perform a real space calculation on a case by case basis in section 4.2: for aspecific unphysical solution, taken as an example, we confirm the value of the generatingpolynomial P predicted by the general Fourier space reasoning, giving a meaning tothe divergent integrals by analytic continuation; we also show that some unphysicalsolutions are missed by the Fourier space reasoning. N = 4We start with the Faddeev ansatz (21) for the Fourier transform ˜ ψ free ( k , . . . , k N )of the wavefunction. It may happen that ˜ ψ free is identically zero, although theindividual contributions D (( k n ) n = i,j ) are not. The corresponding scaling exponent isthen unphysical and must be disregarded. § In the first case, ℓ com runs from ℓ − ℓ rel to ℓ + ℓ rel for 0 ≤ ℓ rel ≤ p − ℓ , from ℓ − ℓ rel to p − ℓ rel for p − ℓ < ℓ rel ≤ ℓ , and from ℓ rel − ℓ to p − ℓ rel for ℓ < ℓ rel ≤ p + ℓ . In the second case, ℓ com runs from ℓ − ℓ rel to ℓ + ℓ rel for 0 ≤ ℓ rel ≤ ℓ , from ℓ rel − ℓ to ℓ + ℓ rel for ℓ < ℓ rel ≤ p − ℓ , and from ℓ rel − ℓ to p − ℓ rel for p − ℓ < ℓ rel ≤ p + ℓ . In both cases, the sum over ℓ com is empty for ℓ rel > p + ℓ . deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion D is a non identically zero solutionof D ( k , k ) − D ( k , k ) + D ( k , k ) = 0 ∀ k , k , k (42)From equation (22), it is expected that D ( k , k ) has a finite limit when k → + ∞ :lim k → + ∞ D ( k , k ) = f ( k ) (43)Taking this limit in equation (42) leads to the correctly antisymmetrised form D ( k , k ) = f ( k ) − f ( k ) (44)More generally, differentiating (42) with respect to k and k , one sees that D ( k , k )has a vanishing crossed differential, which leads to the same ansatz (44). The value ofthe function f ( k ) is actually imposed, up to constant factor, by the rotational symmetryand the scaling invariance. For a total angular momentum ℓ and a scaling exponent s ,we get f ( k ) = k − ( s + ) Y m ℓ (ˆ k ) (45)where Y m ℓ is a spherical harmonic and ˆ k = k /k is the direction of k . Clearly f ( k ), D ( k , k ) and the final wavefunction have the natural parity ( − ℓ . Furthermore, aswe have seen, the Faddeev component F must be a homogeneous polynomial of degree d . From the usual power-counting argument in the Fourier transform, we find that s = d + 7 /
2, in agreement with equation (24) specialised to N = 4. Finally, we takeas a particular case x = r = 0 and we isolate in equation (20) the contribution ofthe piece f ( k ) in D ( k , k ). We then perform the change of variables k = k k ′ and q = k q ′ and formally integrate over k and q the inverse of the energy denominator,which simply pulls out a factor k . We are left with an integral of the form Z d k Y m ℓ (ˆ k ) k − s e i k · x (46)This must be a homogeneous polynomial in x of angular momentum ℓ , of the form (31)with n any natural integer. Again using a power-counting argument or the changeof variable k = x k ′ , we arrive at the unphysical value of the scaling exponent s = 2 n + ℓ + , corresponding to the form (37) with σ = 1 and a degeneracy 2 ℓ + 1.There is however a little subtlety. In the particular case ( n, ℓ ) = (0 , d = 0 and s = 7 /
2, there cannot exist a nonzero fermionic polynomial P ( x , x ) of degree zero; the expression (46) is a constant, as the change of variable k = x k ′ shows, and so are the contributions to F (0; x , x ) of the pieces f ( k ) and f ( k ) of D ( k , k ), which thus exactly cancel. This was already taken into account inthe reasoning above equation (32) and there is no unphysical solution to disregard.As a consequence, we obtain a correction to the degeneracy of the scaling exponentsof the 3 + 1 interaction-sensitive states,¯ D ( σ ) ℓ,n = (2 ℓ + 1) 1 + σ − δ n, δ ℓ, ) (47)to be subtracted from the degeneracy D ( σ ) ℓ,n in equation (40). deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion ψ free ( k , . . . , k N ) is identically zero if D ( k , k ) − D ( k , k ) − D ( k , k ) + D ( k , k ) = 0 ∀ k , k , k , k (48)It is now expected thatlim k → + ∞ D ( k , k ) = f ( k ) and lim k → + ∞ D ( k , k ) = g ( k ) (49)As D is not subjected to any exchange symmetry, the functions f ( k ) and g ( k ) are ingeneral independent, but they both tend to zero at large k . Taking the limit k → + ∞ and k → + ∞ in equation (48), we obtain the ansatz D ( k , k ) = f ( k ) + g ( k ) (50)The more direct argument of cross-differentiation of equation (48) with respect to k and k , which kills all terms but the first one, also leads to the ansatz (50). The previous3 + 1 reasoning is readily adapted to this case. Due to the rotational symmetry and thescale invariance, f ( k ) = αk − ( s + ) Y m ℓ (ˆ k ) and g ( k ) = βk − ( s + ) Y m ℓ (ˆ k ) (51)where α and β are arbitrary constants. As the Faddeev components F (0; x , ) and F (0; , x ) must be of the form (31), with n any natural integer, we conclude that theunphysical scaling exponents are of the form (37) with a a parity σ = 1 relative to thenatural parity, and a degeneracy 2(2 ℓ + 1), the extra factor two reflecting the linearindependence of α and β .There is here again a little subtlety. In the particular case ( n, ℓ ) = (0 , d = 0 and the contributions to F (0; x , x ) of the pieces f ( k ) and g ( k ) in D ( k , k ) are constants proportional to α and β , so they are not linearly independent.No extra factor two is required.As a consequence, we obtain a correction to the degeneracy of the scaling exponentsof the 2 + 2 interaction-sensitive states,¯ D ( σ ) ℓ,n = (2 − δ n, δ ℓ, )(2 ℓ + 1) 1 + σ D ( σ ) ℓ,n in equation (41).The predictions (40), (41), (47), (52) can be tested against the results of reference[19], where the scaling exponents of the interaction-sensitive states of four trapped spin1 / s ≤ / k . As the table 1 shows, there is agreement for 3 + 1 fermions and forthe unnatural parity states of 2 + 2 fermions, but there is disagreement for the naturalparity states of 2 + 2 fermions. This means that some unphysical states are missed bythe above Fourier space reasoning. This is confirmed in section 4.2, where it is alsoexemplified that, surprisingly, the obviously sufficient conditions (42) and (48) to havean unphysical solution are not always necessary. k For the ( ℓ, n, σ ) = (0 , , +) channel of the 2 + 2 system, there is a typo in table I of the supplementalmaterial of reference [19], as kindly communicated to us by D¨orte Blume: the scaling exponent of theideal gas level labeled “st. no. 16” should be instead of . This is corrected here. deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion I , that is the spin ↑ particle i = 1 and the spin ↓ particle j = N ↑ +1, were in fact still there and both preparedin the mode ( n, ℓ, m) = (0 , , n i , ℓ i , m i ) i ∈ I that can be populated by fermions, a constraint not included in the reasoning aboveequation (32). This immediately leads to the occurrence of three types of unphysicalsolutions: • unphysical solutions of type ↑ : one puts one of the spin ↑ fermions of the set I , 2 ≤ i ≤ N ↑ , in the mode (0 , , ↓ fermions of the set I , N ↑ + 2 ≤ i ≤ N , are put in modes ( n i , ℓ i , m i ) = (0 , , • unphysical solutions of type ↓ : one puts one of the spin ↓ fermions of the set I , N ↑ + 2 ≤ i ≤ N , in the mode (0 , , ↑ of the set I , 2 ≤ i ≤ N ↑ , areput in modes ( n i , ℓ i , m i ) = (0 , , • unphysical solutions of type ↑↓ : one puts one of the spin ↑ fermions and one of thespin ↓ fermions of the set I in the mode (0 , , N . N = 4The previous reasoning in Fourier space, though elegant, is formal. It involves integralswith arbitrarily severe infrared divergences, see for example (46), since s can bearbitrarily large and positive. To believe in this reasoning, it is essential to extracta well defined prediction for the generating polynomial P (( x k ) k ∈ I ) of the unphysicalsolutions, and to check explicitly, by manipulating polynomials in real space, that thecorresponding Faddeev ansatz vanishes.We shall use two main recipes to obtain finite generating polynomials P from thediverging Fourier space integrals. First, we can pull out infinite constants, since P is defined up to a global factor. Second, we can use analytic continuation. Here, weexemplify the procedure for 3 + 1 fermions in the manifold ℓ = 1, n = 2 and σ = +1.According to the Fourier space reasoning, there should be a single unphysical solutionof azimuthal quantum number m = 0. The corresponding polynomial P ( x , x ) = F (0; x , x ), of degree d = 2 n + ℓ = 5, is given by P ( x , x ) = [ − i ∇ x φ ( x , x ) − ( x ↔ x )] · e z (53)where e z is the unit vector along the quantization axis z and where the function φ is φ ( x , x ) = Z d q d k d k (2 π ) k − ( d +8)2 e i( k · x + k · x ) ~ m ↑ [ k + k + t ( k + k ) + q − t ] (54)as it results from equation (20) and a differentiation with respect to x under the integralsign. First, we transform (54) only using scaling laws and scale invariances. At fixed k , we perform the change of variable k = k ′ − t t k to make the energy denominator deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion ℓ, n, σ ) s ( σ ) ℓ,n D ( σ ) ℓ,n ℓ +1 ¯ D ( σ ) ℓ,n ℓ +1 D Blume ℓ +1 D ( σ ) ℓ,n ℓ +1 ¯ D ( σ ) ℓ,n ℓ +1 D Ref . [ ] ℓ +1 (0 , , +) , , +) , , +) → , , +) → , , +) , , +) → , , +) → , , − ) , , − ) , , − ) , , +) , , +) → , , +) → , , − ) , , − ) Table 1.
For 3 + 1 fermions (left) and 2 + 2 fermions (right), values and degeneraciesof the scaling exponents of the interaction-sensitive states up to s = 19 /
2. The column D ( σ ) ℓ,n corresponds to the bare degeneracies (40) and (41). When subtractively correctedby the degeneracies of the unphysical solutions given in the column ¯ D ( σ ) ℓ,n , it agreeswith the numerical results of reference [19] reported in the column D Ref . [ ] (see ourfootnote k ). The values of ¯ D ( σ ) ℓ,n are given by the Fourier space predictions (47) and(52), corrected if necessary (and as indicated by an arrow) by the real space predictionsof section 4.2. The parity is σ ( − ℓ , σ = ± being relative to the natural parity ( − ℓ . rotationally invariant. Second we set k = (1+ t ) k ′ and k ′ = (1+2 t ) / k ′′ so that k ′ and k ′′ have identical coefficients in the energy denominator. This leads to the introductionof modified coordinates: X = (1 + t ) x − t x and Y = (1 + 2 t ) / x (55)We integrate over q using a scaling law, R d qq + Q ∝ Q for Q >
0, as the change of variable q = Q q ′ shows; this amounts to extracting a diverging constant factor. Integrating overthe directions of k ′ and k ′′ and dropping the primes for simplicity, we are left with φ ( x , x ) ∝ Z + ∞ d k d k k ( k + k ) / XY k d +72 [cos( k X − k Y ) − cos( k X + k Y )](56)We move to polar coordinates, ( k , k ) = ( ρ cos θ, ρ sin θ ) to again take advantage of scaleinvariance: in the integral over ρ involving the first/second cosine term, we perform thechange of variable ρ = ρ ′ / | X sin θ ∓ Y cos θ | and we pull out a common infinite constantfactor R R + d ρ ′ ρ ′ d +4 cos ρ ′ . As d + 3 is even, we can remove the absolute values and we areleft with φ ( x , x ) ∝ Z π/ d θ cos θ [( X sin θ − Y cos θ ) d +3 − ( X sin θ + Y cos θ ) d +3 ] XY sin d +7 θ (57) deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion θ and of cos θ . Eliminating the cosine with cos θ = 1 − sin θ , we are left withintegrals over θ of the form f ( − n ), n ∈ N ∗ and f ( z ) ≡ Z π/ d θ sin z +1 θ (58)For ℜ z > − f ( z ) = π / z + 1)Γ( z + ) . By theusual analytic continuation of Euler’s Gamma function, one can extend f ( z ) to C \ R ,where it can also be written as f ( z ) = π / πz ) Γ( − z − )Γ( − z ) (59)thanks to Euler’s reflection formula Γ( z )Γ(1 − z ) = π/ sin( πz ). Unfortunately, this stillhas poles at the negative integers. As we are allowed to pull out from equation (57) aconstant diverging factor, we divide it by f ( − f . We now face A n ≡ lim ǫ → f ( − n + i ǫ ) f ( − ǫ ) (60)As the tangent function is periodic of period π , the troublesome first denominator inequation (59) is canceled out, the poles disappear and we obtain the recipe R π/ d θ sin − n +1 θ R π/ d θ sin − θ = A n = Γ( n − ) π / Γ( n ) ∀ n ∈ N ∗ (61)For d = 5 this leads to the finite prediction φ ( x , x ) ∝ [ A Y + A Y (7 X − Y )+ A Y (7 X − X Y +6 Y )+ A ( X − X Y +21 X Y − Y )+ A ( Y − X )( X − X Y + Y )] (62)Turning to the original variables x and x and calculating the gradient in equation (53),we obtain an explicit expression for P ( x , x ), and then from (30) an explicit expressionfor F ( r ; x , x ). We can then evaluate the four-body wavefunction when particles 1 and4 are at the same location, say at the origin of coordinates, from (17): ψ free ( , x , x , ) = F (0; x , x ) − F ( x ; − t x , x − t x ) (63)+ F ( x ; − t x , x − t x ) (64)After lengthy calculations, we find that it is zero at all x and x . While we have taken r = r = for simplicity in the above argument, we can also show, after lengthycalculations, that ψ free ( r , r , r , r ) is identically zero for all r i , ≤ i ≤
4. Thus, our P obtained from the Fourier space reasoning indeed generates an unphysical solution.Is this solution the only one, or is there some unphysical solution missed in section4.1? To answer this question, still in the manifold ℓ = 1, n = 2 and σ = +1 for 3 + 1fermions, we write P in the most general form P ( x , x ) = x · e z " X k =0 c k p k ( x , x ) − ( x ↔ x ) (65) deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion p k ( x , y ) is a basis of rotationally invariant homogeneous polynomials of degree4, for example p ( x , y ) = y , p ( x , y ) = x , p ( x , y ) = x y , p ( x , y ) = y ( x · y ), p ( x , y ) = x ( x · y ), p ( x , y ) = ( x · y ) , and the coefficients c k are unknown.Then, we calculate the Faddeev component and we expand the resulting polynomial ψ free ( , x , x , ) in the same basis, as in equation (65), with coefficients ( c ′ k ) ≤ k ≤ linearly related to the ( c k ) ≤ k ≤ via a six-by-six matrix A (too long to be given here).Then ψ free ( , x , x , ) is identically zero if and only if all the c ′ k are zero, that is A~c = ~ , (66)where the vector ~c collects the six unknowns ( c k ) ≤ k ≤ . For a mass ratio 0 < t < A is indeed of dimension one ¶ , and is spanned by the Fourierspace prediction discussed above.We have systematically searched for unphysical solutions missed by the Fourierspace reasoning for 2 + 2 fermions in natural parity states, for all the values of( ℓ, n ) appearing in the table 1. The strong motivation to do so is to recover thedegeneracies obtained numerically in reference [19], which by construction are exemptfrom unphysical solutions. We use the previous procedure, expanding P ( x , y ) overa basis of the homogeneous polynomials p k ( x , y ) of angular momentum ℓ and degree2 n + ℓ . We restrict to a zero angular momentum along e z , multiplying the obtaineddegeneracy by 2 ℓ + 1. As we have seen, for ℓ = 0, we take as a basis the set ofmonomials x n y n ( x · y ) n , with n + n + n = n . For ℓ = 1, we take the set( x · e z ) x n y n ( x · y ) n and ( y · e z ) x n y n ( x · y ) n , with n + n + n = n . For ℓ = 2, we take [3( x · e z ) − x ] x n y n ( x · y ) n , [3( y · e z ) − y ] x n y n ( x · y ) n and[3( x · e z )( y · e z ) − x · y ] x n y n ( x · y ) n , with n + n + n = n . From the generatingpolynomial with arbitrary coefficients c k in the basis, we calculate the polynomial ψ free ( , x , , y ) and expand it with coefficients c ′ k in the same basis. This gives thecoefficients of the matrix A relating the c ′ k to the c k : ~c ′ = A~c . The number of unphysicalsolutions is equal to the dimension of the null space of A . As indicated by an arrow inthe second ¯ D column of the table, this corrects the Fourier space prediction in six cases.We then obtain agreement with the numerical results of reference [19].In all cases, we have found that the unphysical solutions predicted by theFourier space reasoning (amenable to an explicit prediction for P ( x , y ) by analyticalcontinuation as explained in this section) are in the null space of the matrix A . As wenow show on a simple example, some elements of the null space are missed due to thefact that the conditions (42) and (48) are sufficient but not necessary. Let us considerthe case of 2 + 1 fermions and take D ( k ) = k − ( s +2) = k − , which corresponds to thealready known s = 2 unphysical solution. This does not satisfy the condition equivalentto (42) for 2 + 1 fermions, that is D ( k ) − D ( k ) = 0. Still the generating polynomial ¶ Interestingly, the null space of A is of dimension 2 for the infinite mass impurity t = 0 and ofdimension 4 for the zero mass impurity t = 1, leading to spurious unphysical solutions. deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion P is a constant, as a power-counting shows: P ( x ) ∝ Z d q d k D ( k ) e i k · x q + (1 − t ) k ∝ Z d k e i k · x k ∝ x (67)So is the Faddeev component and, due to the fermionic antisymmetry, the whole Faddeevansatz ψ free is zero + .
5. Implications for the cluster or virial expansion for the unitary gas
The cluster expansion is an expansion of the pressure of a thermal equilibrium systemin powers of the fugacity, that is at a low density or a high temperature relative to thequantum degeneracy threshold. It is a powerful tool, because it applies even for stronglyinteracting systems. Recently, the cluster coefficients were accessed experimentally inthe unitary spin 1 / v n of a generalisedEfimov transcendental function Λ( s ), while the interaction-sensitive energy levels ofthe unitary gas are related to roots u n of Λ( s ). In section 5.2 we obtain optimisedwritings of the third and fourth cluster coefficients in terms of sums P n ( e − ¯ ωu n − e − ¯ ωv n ),which allows us to extend the applicability of the numerical calculations of the fourthcluster coefficient of the reference [19] to lower values of ¯ ω ≡ ~ ω/ ( k B T ). In section 5.3,we produce some explicit results, showing that the conjecture of reference [20] for thefourth cluster clearly fails in the 2 + 2 fermion sector, and we construct on physicalgrounds a new, more successful conjecture. We consider now a zero energy E = 0 − solution of Schr¨odinger’s equation for two-component interacting fermions in free space, with ↑↓ contact interactions described + There is a s ↔ − s duality due to the evenness of Efimov’s transcendental function, see section 5.1.As D ( k ) ∝ k − ( s +2) , the dual of D ( k ) = k − is D ( k ) = 1. It corresponds to a negative value s = − D ( k ) − D ( k ) = 0 for a zero Faddeev ansatz. Forthree identical bosons, the unphysical solution in the sector ℓ = 1 , σ = 1 is s = 3 [7], correspondingto D ( k ) ∝ ˆ k · e z k , so its dual D ( k ) ∝ k · e z obeys the sufficient condition D ( k ) + D ( k ) + D ( k ) = 0 restricted to the subspace k + k + k = ; the unphysical solution in the sector ℓ = 0 is s = 4[7], corresponding to D ( k ) ∝ k − , so its dual D ( k ) ∝ k by no means obeys the sufficient condition D ( k ) + D ( k ) + D ( k ) = 0, but one can argue that D ( k ) + D ( k ) + D ( k ) ∝ k + k + k simplifieswith the energy denominator in the bosonic equivalent of equation (21), leading to ψ free ( r , r , r ) = 0except on a set of zero measure, if the Fourier transform is taken in the framework of distributions. deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion A (( x k ) k ∈ I ) = A N ↑ +1 (( r k − R N ↑ +1 ) k =1 ,N ↑ +1 ) then solves an integral equation [27, 28] M [ A ] = a − A (68)where the linear operator M does not depend on the scattering length a and the set I is given by the equation (19). In the unitary limit a − = 0 and in the ideal gas limit a − = ∞ , the gas is scale invariant and the function A has some scaling exponent s A ,conveniently defined by a shift of +1 in the exponent of equation (3): A ( λ ( x k ) k ∈ I ) = λ s A +1 − N − A (( x k ) k ∈ I ) ∀ λ > A (( x k ) k ∈ I ) = R s A +1 − N − A Φ(Ω A ) where R A is the hyperradius and Ω A are hyperangles parametrising the ( x k ) k ∈ I , one obtains animplicit equation for s A ,Λ ( σ ) ℓ ( s A ) a − =0 = 0 or Λ ( σ ) ℓ ( s A ) a − = ∞ = ∞ (70)where we could restrict to a subspace of fixed angular momentum ℓ ∈ N and parity σ ( − ℓ , σ = ±
1, due to the rotational invariance and the parity invariance. FormallyΦ(Ω A ) is the eigenvector of some linear s A -dependent operator M ( σ ) ℓ ( s A ) with a zero oran infinite eigenvalue, the function Λ ( σ ) ℓ ( s A ) is the determinant of that linear operator,Λ ( σ ) ℓ ( s A ) = det M ( σ ) ℓ ( s A ) (71)and is obviously independent of a . We call it Efimov’s transcendental function, because itwas calculated analytically by Efimov for N = 3 [29], see also references [7, 30, 31, 32, 33].For N = 4, it was evaluated numerically, for imaginary values of s A only [20, 26].Importantly, it is an even function of s A . In what follows, we assume that there is no N -body Efimov effect, which leads to known constraints on the mass ratio m ↑ /m ↓ for N = 3 [29, 33, 34] and for N = 4 [20, 26]. As a consequence, all the roots of Λ ( σ ) ℓ ( s A )are real. Considering (70) we call ( u ( σ ) ℓ,n ) n ∈ N the set of positive roots of Λ ( σ ) ℓ and ( v ( σ ) ℓ,n ) n ∈ N the set of positive poles of Λ ( σ ) ℓ , counted with a degeneracy 2 ℓ + 1, the negative rootsand poles being their opposites:Λ ( σ ) ℓ ( u ( σ ) ℓ,n >
0) = 0 and Λ ( σ ) ℓ ( v ( σ ) ℓ,n >
0) = ∞ , ∀ n ∈ N (72)The last step is to relate the scaling exponent s A in equation (69) of the regularpart A to the scaling exponent s (3) of the wavefunction ψ free ( r , . . . , r N ). For theunitary gas, denoted by a diacritical sign, the term a vanishes in the Wigner-Bethe-Peierls contact condition, so ψ free ∼ r − A and, thanks to the shift of +1 of the exponentin the definition (69) one simply hasˇ s a − =0 = s A (73)For the ideal gas, the term a diverges in the Wigner-Bethe-Peierls contact condition, so ψ free ∼ a − A and s a − = ∞ = s A + 1 (74) deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion s are physical in the absence of N -body resonances [15, 35, 36, 37], we obtain the generalexpressions for the SO(2,1) ladders of internal energy levels of the unitary gas and of theideal gas in terms of the positive roots and the positive poles of Efimov’s transcendentalfunction, each energy level being counted with a degeneracy 2 ℓ + 1:ˇ E rel q a − =0 = ( u ( σ ) ℓ,n + 1 + 2 q ) ~ ω and E rel q a − = ∞ = ( v ( σ ) ℓ,n + 2 + 2 q ) ~ ω ∀ q ∈ N (75)This remarkable property was noticed and used in reference [18] for three bosons andin reference [17] for three fermions, but it was not physically interpreted. We havepresented here a general physical derivation of this fact, independently of the particlenumber. Note that (75) includes the unphysical solutions as defined in section 4 becauseit involves an integral equation (68) ultimately relying on the Faddeev ansatz; theseunphysical solutions are common to the ideal gas and the unitary gas, because theFaddeev ansatz, being zero, satisfies the Wigner-Bethe-Peierls contact conditions for allvalues of the scattering length a [18]. Whereas the u ( σ ) ℓ,n can probably not be determinedanalytically beyond N = 3, the v ( σ ) ℓ,n can be explicitly obtained from our results of section3. In the harmonic regulator method [12], one performs the cluster or virial expansionfor the thermal equilibrium harmonically trapped system. The grand potential of thetrapped two-component Fermi gas is by definitionΩ = − k B T ln + ∞ X N ↑ =0 + ∞ X N ↓ =0 Z N ↑ ,N ↓ z N ↑ ↑ z N ↓ ↓ (76)where Z N ↑ ,N ↓ is the canonical partition function of N ↑ + N ↓ fermions at temperature T in isotropic harmonic traps with a common angular frequency ω for the ↑ and ↓ components, and the fugacities z σ of the components are related to their chemicalpotentials µ σ by z σ = e βµ σ , with β = ( k B T ) − . In the low density, non-degeneratelimit µ σ → −∞ at fixed temperature, that is z σ →
0, one performs the so-called clusterexpansion of the grand potential [11]:Ω = − k B T Z ∞ X N ↑ =0 + ∞ X N ↓ =0 B N ↑ ,N ↓ z N ↑ ↑ z N ↓ ↓ (77)where Z = Z , = Z , , the single fermion partition function in the trap, is given by Z = 1[2 sinh(¯ ω/ with ¯ ω ≡ β ~ ω (78)It is convenient to restrict to the differences ∆ Z N ↑ ,N ↓ and ∆ B N ↑ ,N ↓ between theinteracting gas and the ideal gas values of Z N ↑ ,N ↓ and B N ↑ ,N ↓ : the ideal gas values deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion Z N ↑ ,N ↓ = Z ∆ Z rel N ↑ ,N ↓ (79)These internal or relative energies are what is ultimately calculated, see the equations(9), (33) and (75) and the references [16, 19]. Expanding the equation (76) in powers ofthe fugacities and equating the coefficients of z N ↑ ↑ z N ↓ ↓ to those of the equation (77), oneobtains up to fourth order:∆ B , = ∆ Z rel1 , (80)∆ B , = ∆ Z rel2 , − Z ∆ B , (81)∆ B , = ∆ Z rel3 , − Z Z rel2 , ∆ B , − Z ∆ B , (82)∆ B , = ∆ Z rel2 , − Z ∆ B , − Z (cid:18)
12 ∆ B , + ∆ B , + ∆ B , (cid:19) (83)At any given order, we have recursively used the relations obtained at a lower orderto eliminate partition functions ∆ Z rel in terms of cluster coefficients ∆ B . The clustercoefficients with N ↑ < N ↓ are obtained by exchanging the roles of ↑ and ↓ in the aboveexpressions. Note that ∆ B N ↑ , = ∆ B ,N ↓ = 0. Also the ideal gas values B a =0 N ↑ ,N ↓ are zeroexcept if N ↑ = 0 or N ↓ = 0. Last, from a use of the centre of mass and relative quantumnumbers of two ↑ fermions as explained around equation (35), one has Z rel2 , = X ℓ rel ∈ N +1 + ∞ X n rel =0 (2 ℓ rel + 1) e − (2 n rel + ℓ rel +3 / ω = e − ω/ (1 + 3 e ω )(2 sinh ¯ ω ) (84)From now on, the interacting two-component Fermi gas is taken in the unitarylimit a − = 0. For 1 + 1 fermions the scaling exponents in equations (3) and (9) arerespectively ˇ s = − / s = 1 / B , = X q ∈ N (cid:2) e − (2 q +1 / ω − e − (2 q +3 / ω (cid:3) = 12 cosh(¯ ω/
2) (85)For higher order cluster coefficients, the goal is to obtain optimized writings in terms ofthe following sums, S N ↑ ,N ↓ ≡ X n,ℓ,σ (2 ℓ + 1) h e − u ( σ ) ℓ,n ¯ ω − e − v ( σ ) ℓ,n ¯ ω i (86)where the roots u ( σ ) ℓ,n and poles v ( σ ) ℓ,n of Efimov’s transcendental function (71) for N ↑ + N ↓ fermions are defined in section 5.1, and since this sum only involves interaction-sensitivestates, the relative parity σ is +1 for N = 3 and ± N = 4. The motivation isthat these sums evoke the sums of residues resulting from the application of Cauchy’stheorem to a contour integration of the functions s A e − s A ¯ ω dd s A ln Λ ( σ ) ℓ ( s A ) [18], whichsuggests that they can be expressed in terms of an integral of these functions on the deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion N = 4 [20, 26]. Importantly, thesesums S N ↑ ,N ↓ include the unphysical solutions discussed in section 4. This is in markedcontrast with ∆ Z rel2 , , ∆ Z rel3 , and ∆ Z rel2 , where one can indifferently include or exclude theunphysical solutions for N = 3 and N = 4, since they are common to the ideal gas andthe unitary gas and cancel out in ∆ Z rel2 , , ∆ Z rel3 , and ∆ Z rel2 , .To express the ∆ B in terms of the sums S , we start from∆ Z rel N ↑ ,N ↓ = X n,ℓ,q,σ (2 ℓ + 1) h e − ( u ( σ ) ℓ,n +1+2 q )¯ ω − e − ( v ( σ ) ℓ,n +2+2 q )¯ ω i (87)then we use a plus-minus trick, writing exp[ − ( v ( σ ) ℓ,n + 2 + 2 q )¯ ω ] = exp[ − ( v ( σ ) ℓ,n + 1 + 2 q )¯ ω ] − (exp ¯ ω −
1) exp[ − ( v ( σ ) ℓ,n + 2 + 2 q )¯ ω ] to obtain∆ Z rel N ↑ ,N ↓ = S N ↑ ,N ↓ ω + ( e ¯ ω − Z rel , int . sens . with unphys . sol .N ↑ ,N ↓ (88)where Z rel , int . sens . with unphys . sol .N ↑ ,N ↓ = P n,ℓ,σ,q (2 ℓ + 1) exp[ − ( v ( σ ) ℓ,n + 2 + 2 q )¯ ω ] is the partitionfunction of the interaction-sensitive states of the relative motion of trapped 2 + 1fermions, 3 + 1 fermions or 2 + 2 fermions including the unphysical solutions. Thispartition function is known from the equation (33): it is equal to Z ℓ =01 times the partitionfunction of respectively one trapped particle Z , two trapped ↑↑ fermions Z , or twotrapped ↑↓ non-interacting particles Z a =01 , = Z . Here Z ℓ =01 = e − ¯ ω/ / [2 sinh ¯ ω ] is thesingle particle partition function restricted to the ℓ = 0 states and accounts for the term(2 q + ) ~ ω in the equation (33). Since( e ¯ ω − Z ℓ =01 − ∆ B , = 0 (89)this leads to the reduced forms ∗ ∆ B , = S , ω (90)∆ B , = S , ω − Z ∆ B , (91)∆ B , = S , ω − Z (cid:18)
12 ∆ B , + ∆ B , + ∆ B , (cid:19) (92)To obtain the cluster coefficients of the spatially homogeneous gas, one must calculatethe (finite) limit of the ∆ B when ¯ ω →
0. These forms are then optimised in the sensethat one has got rid in the ∆ B of the term ∝ Z diverging as 1 / ¯ ω for N = 3, and ofthe terms ∝ Z diverging as 1 / ¯ ω for N = 4. ∗ In evaluating P n,ℓ,σ,q (2 ℓ + 1) exp[ − ( v ( σ ) ℓ,n + 2 + 2 q )¯ ω ] we include unphysical solutions. Thus, thecorresponding sum S N ↑ ,N ↓ must also include unphysical solutions. An alternative choice would be toexclude the unphysical solutions in both sums; this would be inconvenient because we do not know allunphysical solutions for 2 + 2 fermions. deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion For 2 + 1 unitary fermions, an application of Cauchy’s theorem to the reduced form (90)as in reference [18] gives the exact result:∆ B , = X ℓ ∈ N (2 ℓ + 1) Z R d S π sin(¯ ωS )2 sinh ¯ ω dd S [ln Λ ℓ (i S )] (93)For 3 + 1 or 2 + 2 unitary fermions, it is not understood yet how the terms ∝ Z in equations (91) and (92), which diverge for ¯ ω → + contrarily to ∆ B N ↑ ,N ↓ , canbe compensated by poles of the logarithmic derivative of the corresponding Efimov’stranscendental function ♯ . Simply the following conjectured values were proposed inreference [20], by an abrupt generalisation of equation (93) with no justification:∆ B old conj N ↑ ,N ↓ = I N ↑ ,N ↓ (94)Here I N ↑ ,N ↓ is the following imaginary axis integral, shown to be finite in reference [20]: I N ↑ ,N ↓ ≡ X ℓ ∈ N ,σ = ± (2 ℓ + 1) Z R d S π sin(¯ ωS )2 sinh ¯ ω dd S h ln Λ ( σ ) ℓ (i S ) i (95)where Λ ( σ ) ℓ is given by the equation (71) and the corresponding linear operators M ( σ ) ℓ (i S )are given in reference [26] for 3 + 1 fermions and in reference [20] for 2 + 2 fermions.The resulting value of the fourth cluster coefficient ∆ b of the spatially homogeneoussystem for m ↑ = m ↓ is however in complete contradiction with the experimental results[8, 10], even for its sign.We make here a more detailed test of the conjecture (94). From the scalingexponents ˇ s ( σ ) ℓ,n and s ( σ ) ℓ,n of the interaction-sensitive states of the unitary and ideal four-particle systems calculated numerically up to the cut-off ˇ s = 19 / B , and ∆ B , for not too small ¯ ω . To evaluate ∆ B , , whichappears in the expressions of ∆ B , and ∆ B , , we do not use a numerically calculatedthree-body spectrum, but rather the exact expression (93).Various results for ∆ B , and for ∆ B , , corresponding to various expressions givenin the present paper, are plotted as functions of ¯ ω = β ~ ω in figure 1. The green dashedlines correspond to the original formulas (82) and (83). They start diverging at ¯ ω . . incorrect application ofthe optimised formulas (91) and (92), that is including in S , and S , only the physicalsolutions obtained by the reference [19]; the unphysical solutions are missing, and thered dotted curves start diverging at larger values of ¯ ω , ¯ ω ≃ .
2. The blue dashed-dottedlines correspond to the correct use of the optimised formulas (91) and (92): one includes ♯ Alternatively to such unexpected poles, one can invoke a nonzero contribution of the infinite-radiusquarters of circle used to connect the contour integration enclosing the poles on the real axis to theintegral along the imaginary axis. Due to the continuous spectrum of the operators M ( σ ) ℓ (i S ), onecan also suspect the existence of branch cuts for the function z dd z [ln Λ ( σ ) ℓ ( z )] in the complex plane;turning around those branch cuts in the contour integration would then lead to extra contributions.Our current knowledge of the function Λ ( σ ) ℓ ( z ), limited to z ∈ i R , does not allow to settle the problem. deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion S , and S , both the physical and the unphysical solutions up to the cut-off, wherethe degeneracies of the unphysical solutions are obtained as the difference of the baredegeneracies D ( σ ) ℓ,n given by the equations (40) and (41) with the degeneracies of thephysical solutions obtained numerically in reference [19]. As expected, the blue dashed-dotted lines start diverging at smaller values of ¯ ω , ¯ ω ≃
1. The black dashed line for∆ B , is an improvement over the blue dashed-dotted line: one sums over all unphysicalsolutions, without cut-off, assuming that their degeneracy ¯ D ( σ ) ℓ,n is given by the Fourierspace reasoning (47) of section 4.1; the divergence then starts at an even smaller value of¯ ω , ¯ ω . .
9. Such improvement can not be performed for ∆ B , , because the degeneracy¯ D ( σ ) ℓ,n predicted by the equation (52) is an underestimate, see the table 1 and the section4.2. The conjectured values (94) are plotted as black solid lines (lower black solidline on the right panel). For 3 + 1 fermions, they essentially agree with the bluedashed-dotted line up to the point of its cut-off induced divergence. This leaves thepossibility that the conjecture is correct for ∆ B , . Incidentally, its ¯ ω → + limit∆ B old conj3 , (0 + ) = 0 . .
025 of the approximatediagrammatic method of reference [38] (this value was communicated to us privately byJesper Levinsen). For 2 + 2 fermions, the conjectured values (94) clearly disagree withthe blue dashed-dotted lines even in the cut-off unaffected region ¯ ω ≥ B , (0 + ) = − .
036 of reference [38], by a factor close to 2. Theconjecture (94) is thus invalidated for ∆ B , .Let us now construct a less arbitrary conjecture than (94) for the fourth-ordercluster coefficients, building on our physical understanding. The imaginary axis integrals(95) have a finite limit when ω → + [20]. They must differ from the expected sum ofresidues S N ↑ ,N ↓ / (2 sinh ¯ ω ), that diverges when ¯ ω → + as we have seen, by counter-terms C N ↑ ,N ↓ of unelucidated mathematical origin: I N ↑ ,N ↓ = S N ↑ ,N ↓ ω − C N ↑ ,N ↓ (96)The new feature of the 3 + 1 and 2 + 2 fermion problem with respect to the 2 + 1one is that M ( σ ) ℓ (i S ) in equation (71), rather than being a finite size matrix of apurely discrete spectrum, is an operator with a continuous spectrum. We then expectthat this continuous spectrum is at the origin of the sought counter-terms. Crucially,the continuous spectrum can be interpreted in terms of decoupled asymptotic objects (DAOs), emerging for large amplitude oscillations of the four fermions in the trap orequivalently for eigenstates with large quantum numbers [20, 26]. For such asymptoticstates, one indeed expects that the N ↑ + N ↓ particles split into single particles or groupsof strongly correlated particles, that we call DAOs and that do not interact with theother groups or particles because they have high amplitude relative motions. Therelevant DAOs and their spectral properties are presented in the table 2. The partitionfunctions Z DAO of the DAOs in the trap, or more precisely the relative partition functions Z relDAO = Z DAO /Z after removal of the centre of mass, are easy to calculate since byessence the DAOs do not interact. They will be assumed to provide the counter-terms deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion Figure 1.
Fourth-order cluster coefficients ∆ B , (left panel) and ∆ B , (rightpanel) of a harmonically trapped two-component unitary Fermi gas with equal masses m ↑ = m ↓ , as functions of β ~ ω , where ω is the trapping angular frequency, β = 1 / ( k B T )and T is the temperature. Green dashed lines: from the original formulas (82) and(83) and the numerical four-body spectrum of reference [19]. Red dotted lines: froman incorrect use of the optimised formulas (91) and (92), only the physical solutionsbeing included. Blue dashed-dotted lines: from a correct use of (91) and (92), boththe physical and the unphysical solutions being included and subjected to a cut-off.Black dashed line (left panel only): idem, except that all unphysical solutions areincluded, with a degeneracy (47). All these lines diverge at low ¯ ω , because the four-body spectrum in reference [19] is calculated up to some cut-off [in our calculations,any three-body cut-off is avoided thanks to the exact expression (93)]. Black solid lines:the conjectured values; left panel: the old (94) and new (100) conjectures coincide andare in good agreement with the numerics; right panel: the old conjecture (94) (lowerblack solid line) disagrees with the numerics, whereas the new conjecture (105) (upperblack solid line) is in good agreement. in the new conjecture.The 3 + 1 fermion case is the simplest one. As a whole, the continuous spectra ofthe operators M ( ± ) ℓ (i S ) are composed of branches k Λ ↑↑↓ L (i k ) of degeneracy 2 L + 1, k ∈ R and L ∈ N . Each branch corresponds to two DAOs: a cluster of neighbouring,strongly correlated ↑↑↓ fermions of angular momentum L and a decoupled ↑ atom oforbital angular momentum compatible with the total spin ℓ . We call the ↑↑↓ clustera triplon ; its wavefunction has scaling exponents s given by the positive roots u ↑↑↓ L,n ofΛ ↑↑↓ L ( s ); in the trap, it moves as a free particle with an internal structure of energy levels( u ↑↑↓ L,n + 1 + 2 q ) ~ ω , ( L, n, q ) ∈ N . Its ideal gas counterpart and the difference, indicatedas above by the letter ∆, of their relative partition functions ∆ Z rel ↑↑↓ triplon , are given inthe table 2. We expect that the counter-term C , to be subtracted from S , / (2 sinh ¯ ω )is the difference of the relative atom+triplon partition functions in the unitary and idealgases, C , = ∆ Z rel ↑ atom+ ↑↑↓ triplon (97)hence the new conjecture I , = S new conj3 , ω − ∆ Z rel ↑ atom+ ↑↑↓ triplon (98) deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion ǫ unitint / ~ ω ǫ idealint / ~ ω ∆ Z relDAO ↑ or ↓ atom 0 0 0 ↑↓ pairon 2 q + q + ∆ Z rel1 pairon = ∆ B , ↑↑↓ triplon u ↑↑↓ L,n + 2 q + 1 v ↑↑↓ L,n + 2 q + 2 ∆ Z rel ↑↑↓ triplon = ∆ Z rel2 , − Z ∆ B , = ∆ B , ↑↓↓ triplon u ↑↓↓ L,n + 2 q + 1 v ↑↓↓ L,n + 2 q + 2 ∆ Z rel ↑↓↓ triplon = ∆ Z rel1 , − Z ∆ B , = ∆ B , Table 2.
For 3+1 or 2+2 trapped fermions, the relevant decoupled asymptotic objects (DAOs), their internal energy levels ǫ unitint and ǫ idealint and the difference (indicated by theletter ∆) of their relative partition functions in the unitary and ideal gas cases. EachDAO moves as a free particle in the trap, and internal or relative means after removalof this centre-of-mass motion. All integers q , L , n run over N . The pairon internalstates, which correspond to the interaction-sensitive states of the relative motion oftwo opposite-spin fermions, have a zero angular momentum and are not degenerate.The triplon internal energies of angular momentum L are related to the positive roots u L,n or poles v L,n of Efimov’s three-body transcendental function s Λ ↑↑↓ L ( s ) or s Λ ↑↓↓ L ( s ), and have a degeneracy 2 L + 1. The effective partition function of atriplon involves a subtraction of the partition function of two associated emergingDAOs (one atom and one pairon) to avoid double-counting. The equation (80) wasused, as well as the identity (81) and its ↑↔↓ counterpart. Considering the absence of atom-triplon interaction, one has∆ Z rel ↑ atom+ ↑↑↓ triplon = Z ∆ B , (99)where the factor Z is the partition function of the atom-triplon relative motion in thetrap. When combined with (91) and (94) this shows that the new conjecture coincideswith the old one for the 3 + 1 system:∆ B new conj3 , = ∆ B old conj3 , (100)This is partly accidental as the old conjecture was a guess. Two inspiring rewritingsof the above equations will be invaluable in what follows. First, we rewrite theconjecture (98) in a mathematically insightful way, that better reveals the key roleplayed by the continuous spectrum k Λ ↑↑↓ L (i k ), k ∈ R and L ∈ N , in the failure of thena¨ıve residue formula. Using (93) and (99) we get I , = S new conj3 , ω − Z X L ∈ N (2 L + 1) Z R d S π sin(¯ ωS )2 sinh ¯ ω dd S h ln Λ ↑↑↓ L (i S ) i (101)where the factor Z originates from the partition function of the DAOs relative motion.Second, we attribute the coincidence of the old and new conjectures to the absence ofatom-triplon correlations, hence the physically insightful rewriting of equation (100):∆ B new conj3 , = ∆ B old conj3 , + Z − (∆ Z ↑ atom+ ↑↑↓ triplon − Z ↑ atom ∆ Z ↑↑↓ triplon )(102)The 2 + 2 fermion case is richer. It leads to the expected continuous spectrumbranches k Λ ↑↑↓ L (i k ) and k Λ ↑↓↓ L (i k ) of degeneracies 2 L + 1, k ∈ R and L ∈ N [20],associated to the asymptotic splitting of the four fermions into one ↓ fermion plus one ↑↑↓ triplon, and one ↑ fermion plus one ↑↓↓ triplon. But the continuous spectrum of deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion M ( σ ) ℓ (i S ) for σ = +1 has an additional, nondegenerate branch k √ [1 − ( − ℓ cosh( kπ/ ],corresponding to two decoupled pairons , that is s -wave correlated pairs of neighbouring ↑↓ particles, with relative orbital angular momentum ℓ [20]. A pairon is a freelymoving bosonic particle in the trap, with an internal structure given in the table 2that reproduces the asymptotic three-body spectrum †† . The mathematical formulation(101) of the new conjecture immediately becomes I , = S new conj2 , ω − Z X L ∈ N (2 L + 1) Z R d S π sin(¯ ωS )2 sinh ¯ ω dd S h ln Λ ↑↑↓ L (i S ) + ln Λ ↑↓↓ L (i S ) i (103) − X n,ℓ (2 ℓ + 1) e − ¯ ω (2 n + ℓ + ) Z R d S π sin(¯ ωS )2 sinh ¯ ω dd S ln " − ( − ℓ cosh Sπ (104)Due to the identity (93) and its 1 + 2 counterpart, the second term reduces to − Z (∆ B , + ∆ B , ), which partially reconstructs the right-hand side of equation (92).In the third term, it is apparent that Z can no longer be factored out and that onemust keep a sum over the quantum numbers n and ℓ of the pairons relative motion,since the additional branch of the continuous spectrum depends on ℓ ; the integral over S can be calculated exactly by taking the sum and the difference of the odd and even ℓ integrals and using R R d S sin( xS )sinh( Sπ ) = tanh x , x ∈ R ; the result differs from the last term − Z ∆ B , / B new conj2 , = ∆ B old conj2 , + 132 1cosh ω cosh ¯ ω (105)To obtain the physical formulation (102) of the new conjecture, one must realize that,contrarily to the distinguishable atom and triplon, the pairons are identical bosons,which induces statistical correlations among them even if they do not interact, so∆ B new conj2 , = ∆ B old conj2 , + Z − (cid:20) ∆ Z −
12 ∆( Z
21 pairon ) (cid:21) (106)where ∆ still represents the difference between the unitary gas and ideal gas values, forexample ∆( Z
21 pairon ) = ( Z unit1 pairon ) − ( Z ideal1 pairon ) . In the sum over the internal quantumnumbers q and q ′ of each pairon appearing in Z , the internal states ( q, q ′ ) and( q ′ , q ) are physically equivalent and shall not be double-counted. Also, when the relativeangular momentum ℓ of the pairons is odd, their internal state must be antisymmetric,which excludes the state ( q, q ). In the unitary gas case, this leads to the relative two-pairon partition function Z rel , unit2 pairons = ℓ even X n,ℓ (2 ℓ + 1) e − ¯ ω (2 n + ℓ + ) "X q e − ω (2 q + ) + 12 X q = q ′ e − ¯ ω (2 q + +2 q ′ + ) †† After removal of the centre of mass the energy levels of a pairon and a ↑ particle in the trap are(2 q + + 2 n + ℓ + ) ~ ω in the unitary limit. In the limit of large quantum numbers ℓ ≫ n ≫ u ↑↑↓ ℓ,n + 2 q + 1) ~ ω by o (1) ~ ω , which is confirmed by the exact asymptotic analysisof reference [7]. For the ideal gas there is an additional term ~ ω and one recovers the result (34). deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion ℓ odd X n,ℓ (2 ℓ + 1) e − ¯ ω (2 n + ℓ + ) " X q = q ′ e − ¯ ω (2 q + +2 q ′ + ) (107)We replace each term under the exponentials by to obtain Z rel , ideal2 pairons and form thedifference ∆ Z rel2 pairons = Z rel , unit2 pairons − Z rel , ideal2 pairons . This leads in equation (106) to exactly thesame result as in equation (105), which was not granted and is a good consistency check.To be complete, and make the new conjecture as physically transparent as possible,we note that it takes a posteriori a very simple form if, rather than using the four-bodysums S N ↑ ,N ↓ or cluster coefficients ∆ B N ↑ ,N ↓ , one turns back to the four-body partitionfunctions ∆ Z rel N ↑ ,N ↓ of equations (82,83): I , = ∆ Z rel , new conj3 , − Z , ∆ B , − Z ∆ B , (108) I , = ∆ Z rel , new conj2 , − Z ∆ B , − Z ∆ B , − ∆ Z rel2 pairons − Z ( Z − Z rel , ideal1 pairon )∆ B , (109)In the equation (108) the second and third terms in the right-hand side correspond to thetwo possible splittings of ↑↑↑↓ into DAOs, respectively ( ↑ ) + ( ↑ ) + ( ↑↓ ) and ( ↑ ) + ( ↑↑↓ );double-counting is avoided by the way the triplon partition function is calculated, seethe table 2. In the equation (109), the second, third, fourth and fifth terms in the right-hand side correspond to the four possible splittings of ↑↑↓↓ into DAOs, respectively( ↓ ) + ( ↑↑↓ ), ( ↑ ) + ( ↑↓↓ ), ( ↑↓ ) + ( ↑↓ ) and ( ↑ ) + ( ↓ ) + ( ↑↓ ). The only subtlety lies inthe fifth term: one must include only the interaction-insensitive states of the ( ↑ ) + ( ↓ )system, hence the subtraction of Z rel , ideal1 pairon from the partition function Z of their relativemotion, to avoid a double-counting with the first pairon contribution of the fourth term.The new conjectured value (105) for ∆ B , corresponds to the upper black solidline in the right panel of figure 1. It is now in good agreement with the numerics. Its¯ ω → + limit ∆ B new conj2 , (0 + ) = − . B , (0 + ) = − .
036 of reference [38]. There is therefore a good possibility thatthe new conjecture for ∆ B , is exact.In figure 2 we plot the total fourth-order cluster coefficient of the trappedunpolarised spin-1/2 unitary Fermi gas, or more precisely its deviation ∆ B ( ω ) = [∆ B , ( ω ) + ∆ B , ( ω ) + ∆ B , ( ω )] from the ideal gas value, as a function of ~ ω/k B T .The new (old) conjecture corresponds to the upper (lower) black solid line. Remarkably,it is not monotonic. Its ω → + limit is related to the homogeneous gas value ∆ b by ∆ B (0 + ) = ∆ b , where the homogeneous gas cluster expansion for the pressuredifference ∆ P = P unit − P ideal takes the form ∆ P λ /k B T = 2 P n ≥ ∆ b n z n with z ↑ = z ↓ = z and λ dB = (2 π ~ /mk B T ) / . It can be compared to the most recent experimentalresult ∆ b = 0 . B new conj4 (0 + ) = 0 . b new conj4 = 0 . z = 0 of data all having for accuracy reasons a fugacity z > deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion Figure 2.
Total fourth-order cluster coefficient ∆ B ( ω ) of the trapped unpolarisedspin-1 / β ~ ω as in figure 1. Upper (lower) blacksolid line: the new (old) conjecture. Green line with symbols: numerical resultsof reference [19] (disks connected by a solid line: actually calculated values; circlesconnected by a dashed line: values resulting from an extrapolation). Symbol with anerror bar: most recent experimental result [10]. ∆ b PIMC4 (0 + ) = 0 . Note added to the published version:
A precise calculation of the equation ofstate of the unpolarised unitary Fermi gas was performed with the diagrammatic MonteCarlo technique, see [K. Van Houcke, F. Werner, E. Kozik, N. Prokof’ev, B. Svistunov,M.J.H. Ku, A.T. Sommer, L.W. Cheuk, A. Schirotzek, and M.W. Zwierlein, NaturePhysics , 366 (2012)]. As shown in figure 3, the Monte Carlo data point to a value ofthe fourth cluster coefficient b of the uniform unitary Fermi gas in agreement with ournew conjecture (we recall that b n = ∆ b n + ( − n +1 n / so that b = ∆ b − ). They alsoexplain why a higher value of b is obtained if one extrapolates the data from an intervalof fugacity z > b can be accessed) but also in the weakly spin-polarisedcase where the spin susceptibility gives access to the cluster coefficients b , and b , separately.
6. Conclusion
We have determined the interaction-sensitive states of a harmonically trapped two-component ideal Fermi gas, using a Faddeev ansatz for the N -body wavefunction.We have found a simple rule to obtain the interaction-sensitive relative or internalenergy levels as follows. One removes one spin ↑ fermion and one spin ↓ fermion tobuild a pair of particles with a zero relative orbital momentum, which renders the stateinteraction-sensitive. One freely distributes the remaining N ↑ − ↑ fermions and the deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion z=exp( βµ ) [ n λ - ( b z + z + z )] / ( z ) Figure 3. (figure added to the published version) Disks: diagrammatic Monte Carloresults of reference [Nature Physics , 366 (2012)] for the unpolarised uniform unitaryFermi gas, as functions of the fugacity z . n is the total density of the gas and λ isthe thermal de Broglie wavelength. From nλ one subtracts the third order clusterexpansion so that the resulting function, when divided by 8 z , tends to the fourthcluster coefficient b when z tends to 0. Dotted line: linear extrapolation of the right-most two points to z = 0, giving a value of b within the error bars of the ENSexperimental value [8] (blue square with error bars). Dashed line: linear extrapolationof the left-most two points to z = 0, giving a value of b very close to our new conjecture(magenta star). remaining N ↓ − ↓ fermions among the single-particle energy levels of the harmonicoscillator, in a way compatible with Fermi statistics. One adds to the resulting energylevels the energy (2 q + ) ~ ω , where ~ ω may be interpreted as the internal energy ofthe subtracted ↑↓ pair and 2 q ~ ω , with q running over all natural integers, correspondsto the quantised excitation spectrum of the collective breathing mode of our SO(2,1)-symmetric system.This simple rule must be refined because some of its energy levels actuallycorrespond to a zero Faddeev ansatz for the wavefunction, by destructive interference ofindividually nonzero Faddeev components, and are unphysical solutions. This problemwas known for 2 + 1 fermions, where there is a single unphysical solution. We studiedit for 3 + 1 and 2 + 2 fermions and we found a countable infinite number of unphysicalsolutions. A Fourier space reasoning leads to a class of unphysical solutions that lendsitself to a simple picture: everything happens as if each particle of the subtracted ↑↓ pairwas prepared in the harmonic oscillator ground state | , , i ; the unphysical solutions inquestion then formally correspond to putting one of the N ↑ − ↑ particlesor one of the N ↓ − ↓ particles in the state | , , i , thus “violating” thePauli exclusion principle. In the case of 2 + 2 fermions, however, this is not the end ofthe story: there exist additional unphysical solutions, as a comparison to a numerically deal Fermi gas interaction-sensitive states and unitary Fermi gas virial expansion s ) such that the interaction-sensitiveenergy levels for the unitary gas can be expressed in terms of the positive roots u n ofΛ( s ). We then showed that the interaction-sensitive energy levels of the ideal gas canbe expressed in terms of the positive poles v n of Λ( s ). We reached an optimised writingof the third and fourth cluster coefficients in terms of the sums P n ( e − ¯ ωu n − e − ¯ ωv n ),with ¯ ω = ~ ω/ ( k B T ), which evoke the residues of Cauchy’s theorem applied to a contourintegration of s e − s ¯ ω dd s ln Λ( s ). This optimised writing allowed us to extend theapplicability domain of the numerical calculations of the cluster coefficients ∆ B , and∆ B , of the reference [19] to lower values of ¯ ω , before they diverge due to the energycut-off in the numerics. Over this range of values of ¯ ω , 1 . ¯ ω , it shows that theblind conjecture given in the reference [20] for ∆ B , is accurate, while the one for∆ B , disagrees by a factor ≃
2. Using a physical reasoning, we have constructeda new conjecture in terms of the decoupled asymptotic objects (DAOs) emerging inthe four-body interaction-sensitive spectrum at large excitation amplitudes or quantumnumbers, i.e. individual atoms, pairons and triplons, in the unitary and ideal gases.The new conjecture gives the same value for ∆ B , , and a new, now accurate value for∆ B , . The failure of the old conjecture for ∆ B , results a posteriori from the omissionof the statistical correlations between pairons induced by their bosonic nature. Themost recent numerical calculations, based on a dedicated path integral Monte Carloapproach, lead to a ¯ ω → Acknowledgements
S.E. thanks the Japan Society for the Promotion of Science (JSPS) for support. Thegroup of Y.C. is affiliated to IFRAF.
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