An exact solution to the Bertsch problem and the non-universality of the Unitary Fermi Gas
AAn exact solution to the Bertsch problemand the non-universality of the Unitary Fermi Gas ∗ E. Ruiz Arriola,
1, †
S. Szpigel,
2, ‡ and V. S. Tim´oteo
3, § Departamento de F´ısica At´omica, Molecular y Nuclear and Instituto Carlos I deFisica Te´orica y Computacional, Universidad de Granada, E-18071 Granada, Spain Centro de R´adio-Astronomia e Astrof´ısica Mackenzie,Escola de Engenharia, Universidade Presbiteriana Mackenzie, Brazil Grupo de ´Optica e Modelagem Num´erica (GOMNI), Faculdade de Tecnologia,Universidade Estadual de Campinas - UNICAMP, Brazil (Dated: November 7, 2018)We analyze the universality of the Unitary Fermi Gas in its ground state from a Wilsonian renormalizationpoint of view and compute the effective range dependence of the Bertsch parameter ξ exactly. To this end weconstruct an effective block-diagonal two-body separable interaction with the Fermi momentum as a cut-offwhich reduces the calculation to the mean field level. The interaction is separable in momentum space and isdetermined by Tabakin’s inverse scattering formula. For a vanishing effective range we get ξ = π − = . . ≥ ξ ≥ − / Keywords: Unitary Fermi Gas, Inverse Scattering, Separable potential
During the 10th Conference on Advances in Many- BodyTheory, that took place in Seattle in 1999, G. F. Bertsch posedthe following challenge (see [1]):
What are the ground state properties of the many-bodysystem composed of spin-1/2 fermions interacting via a zerorange, infinite scattering-length contact interaction?. It maybe assumed that the interaction has no two-body bound states.Also, the zero range is approached with finite ranged forcesand finite particle number by first taking the range to zero andthen the particle number to infinity.
The Unitary Fermi Gas was proposed as a simple, universaland scale invariant dilute system where interactions at low en-ergies are strong ranging from atoms in the ultracold regimeto nuclear systems such as neutron stars [2, 3]. More specifi-cally, if α is the scattering length, r the effective range and k F the Fermi momentum, the unitary limit is meant to describesystems in the range 1 / α (cid:28) k F (cid:28) / r . The interaction ischaracterized by an isotropic scattering amplitude given by f = ∞ ∑ l = ( l + ) e i δ l sin δ l k P l ( cos θ ) ≡ − α + r k − ik , (1)with P l ( cos θ ) Legendre polynomials and δ l ( k ) the phaseshifts, δ ( k ) = cot − (cid:18) − α k + r k (cid:19) , δ l ( k ) = l ≥ . (2)For α → − ∞ and r → δ ( k ) = π /
2. In this limit,for a Fermi system with two (spin) species the Fermi momen-tum k F is the only dimensionful quantity and hence the totalenergy per particle should be proportional to the energy of thefree Fermi Gas: EN = ξ k F M , (3) where ξ is the Bertsch parameter which is expected to be a universal number , and in the r (cid:54) = universal function of the combination r k F .This apparently simple problem provides an example of astrongly correlated fermion system and has been a major the-oretical and experimental challenge over the last two decades.Experimental measurements on ultracold atomic gases for avanishing effective range, r =
0, yield [4] ( ξ = . ( ) ), [5]( ξ = . ( ) ), [6] ( ξ = . ( ) ) and [7] ( ξ = . ( ) ). InRef. [2] over 50 different calculations based on different manybody simulations and experiments are listed and, with a fewexceptions ξ ∼ . − .
40. The numerical resemblance sug-gests, as it was tacitly expected in those studies, that this is auniversal quantity which is uniquely determined by the condi-tions spelled out originally by Bertsch’s challenge. Here, weprovide an exact solution of the problem and show, contraryto this general and widespread belief, that more informationis needed than assumed hitherto.From a theoretical point of view the strategy to deal withthe Unitary Fermi Gas has been a two-step approach: onefirst tunes a two-body interaction to fulfill the Bertsch scat-tering condition, Eq. (1), and then uses it to solve the many-body problem. These two stages are usually discussed sepa-rately in the literature. A prototype calculation is the one pur-sued recently by Conduit and Schonberg [8] where, for any k F , they have tuned a potential to Eqs. (2) and considered aMonte Carlo simulation up to N =
294 particles and verifythe expected O ( N − ) trend to the extrapolated N = ∞ ther-modynamic limit. They find ξ = . ( ) , the most precisedetermination to date, and consider also finite effective rangevalues in the interval, r k F ∈ [ − , ] .In the present letter we advocate for a different strategy.Indeed, there is a well-known and inherent ambiguity associ-ated to this approach; one can undertake a phase-equivalent a r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec unitary transformation of the potential [9] (see e.g. [10] for areview). Here we propose to take advantage of this arbitrari-ness by building an interaction in block-diagonal form wherethe separation scale in the momentum, Λ , is actually chosen tocoincide with the Fermi momentum, Λ = k F . This has the im-portant consequence that the mean field result already yieldsthe exact many-body solution and a suitable choice permitsto construct an analytic solution without violating any of theconditions originally spelled out by Bertsch.For our purposes it is convenient to formulate the two-bodyscattering problem for two identical particles of mass M inmomentum space for the kinematics ( (cid:126) P + (cid:126) p / ,(cid:126) P − (cid:126) p / ) → ( (cid:126) P + (cid:126) p (cid:48) / ,(cid:126) P − (cid:126) p (cid:48) / ) where (cid:126) P is the (conserved) CM momentum and (cid:126) p and (cid:126) p (cid:48) therelative momenta before and after the collision respectively.The Lippmann-Schwinger equation, which in operator formreads T ( E ) = V + V G ( E ) T ( E ) with G ( E ) = ( E − H ) − ,becomes [11], (cid:104) (cid:126) p (cid:48) | T ( E ) | (cid:126) p (cid:105) = (cid:104) (cid:126) p (cid:48) | V | (cid:126) p (cid:105) + (cid:90) d q ( π ) (cid:104) (cid:126) p (cid:48) | V | (cid:126) q (cid:105)(cid:104) (cid:126) q | T ( E ) | (cid:126) p (cid:105) E − q / µ + i ε , (4)where µ = M / (cid:104) (cid:126) p (cid:48) | T ( E ) | (cid:126) p (cid:105) = π µ ∑ lm l Y lm l ( ˆ p ) Y lm l ( ˆ p (cid:48) ) ∗ T l ( p (cid:48) , p , E ) , (5)with Y lm l ( ˆ p ) spherical harmonics and similarly for the po-tential V l . In terms of the means of the K − matrix fulfilling, T l = K l / ( + i √ µ EK l ) which half-off-shell fulfills K l ( p (cid:48) , p ) = V l ( p (cid:48) , p ) + π − (cid:90) ∞ dq q V l ( p (cid:48) , q ) p − q K l ( q , p ) . (6)where − (cid:82) stands for the principal value integral and K l ( p (cid:48) , p ) ≡ K l ( p (cid:48) , p , E = p / µ ) . The relation with the phase-shifts inEq. (1) follows from − π f = (cid:104) (cid:126) p (cid:48) | T ( E ) | (cid:126) p (cid:105) and is given bytan δ l ( p ) p = − K l ( p , p ) . (7)Clearly, the conditions in Eq. (2), are fulfilled by taking V l ( p , p (cid:48) ) = , for l ≥ . (8)The problem is then to find the s-wave interaction V ( p (cid:48) p ) from Eq. (2), as we will discuss shortly, after reviewing ourmany-body setup.Following the conventional strategy, once our effective in-teraction V ( p (cid:48) , p ) has been tuned to the Bertsch renormaliza-tion condition, Eq. (2), we turn now to the many body prob-lem. We will work first at lowest order in perturbation the-ory which corresponds to the mean field (Hartree-Fock) level,since this already provides an upper variational estimate for any V ( p (cid:48) , p ) . For a two-fermion species the energy per parti-cle at the Hartree-Fock level, is given by [12] EN = k F M + π M (cid:90) k F k dk (cid:18) − k k F + k k F (cid:19) V ( k , k )+ O ( V ) . (9) According to the standard variational argument, first orderperturbation theory provides an upper bound for the trueground state. If we have H = H + V and H ψ ( ) n = E ( ) n ψ ( ) n ,then for any normalized state ϕ we have E ≤ (cid:104) ϕ | H + V | ϕ (cid:105) so that taking ϕ = ψ ( ) n a Slater determinant leading to Eq. (9). E ≤ E ( ) n + (cid:104) ψ ( ) n | V | ψ ( ) n (cid:105) . The neglected higher order correc-tions correspond to transitions (cid:126) p → (cid:126) p (cid:48) above the Fermi-level, | (cid:126) P ± (cid:126) p / | ≤ k F ≤ | (cid:126) P ± (cid:126) p (cid:48) / | which requires p (cid:48) > k F > p . Note,however, that if we have V ( p (cid:48) , p ) = p (cid:48) > k F > p higherorder corrections vanish identically .The main ingredient in our construction is thus to separatethe two-body (relative) Hilbert space into two orthogonal (anddecoupled) subspaces H = H P ⊕ H Q which are below orabove some given Λ respectively. This can be denoted by pro-jection operators P and Q , fulfilling P = P and Q = Q and PQ = QP = P + Q =
1. This separation endows theHamiltonian H with a block structure, which can equivalentlybe transformed by a unitary transformation U into a block-diagonal form H = PHP PHQQHP QHQ = U PH Λ P Q ¯ H Λ Q U † (10)where H Λ and ¯ H Λ describe the low energy and high energydynamics respectively. Of course, we can split the Hamilto-nian as H = T + V , with T and V kinetic and potential en-ergies respectively. For the case when U commutes with thekinetic energy, [ U , T ] =
0, an equivalent decomposition holdsfor the potential V in terms of V Λ in the P-space and ¯ V Λ in theQ-space. Our idea is to assume already the former decompo-sition to the two-boby problem from the start and to considerthe following potential in the momentum P-space V Λ ( p (cid:48) , p ) = θ ( Λ − p (cid:48) ) θ ( Λ − p ) v ( p (cid:48) , p ) . (11)The effective interation V Λ ( p (cid:48) , p ) depends explicily on theseparation scale or cut-off Λ . It corresponds to a self-adjointoperator, V Λ ( p (cid:48) , p ) = V Λ ( p , p (cid:48) ) ∗ , acting in a reduced modelHilbert space with p , p (cid:48) ≤ Λ . Due to the fact that the transfor-mation is unitary we get that the phase-shift associated with V Λ ( p (cid:48) , p ) is just δ , Λ ( p ) = δ ( p ) Θ ( Λ − p ) , (12)in the P -model space. Note that for Λ = k F , the Q -space be-comes irrelevant in the many body problem, so that ξ = + π k F (cid:90) k F k dk (cid:18) − k k F + k k F (cid:19) V k F ( k , k ) . (13)We stress that for this choice of two body potential with onlythe s-wave contribution this equation is exact . We are only leftwith the determination of a suitable function v ( p (cid:48) , p ) . There are many inequivalent ways how this procedure can be carried outas a result of a finite number of steps. A particular implementation is toachieve Block-Diagonalization in a continuous way in terms of flow equa-tions [13]. We will exploit this freedom below.
Poeschl-TellerUTP EXP λ / k F = ∞λ / k F = 0.9 ξ k F r FIG. 1. Exact Bertsch parameter ξ as a function of the effectiverange r in units of the Fermi momentum k F for α → − ∞ for aseparable potential from the Tabakin’s solution ( λ / k F = ∞ ) and itsSRG evolution to λ / k F = . r = ξ = . ( ) ), [5] ( ξ = . ( ) ), [6] ( ξ = . ( ) ) and[7] ( ξ = . ( ) ). Within the class of solutions given by Eq. (11) we can stillexploit the arbitrariness to choose an interaction which willprovide an analytical solution to the Bertsch’s problem. Here,we will search for a separable interaction solution of the form V ( p (cid:48) , p ) = ± g ( p (cid:48) ) g ( p ) . (14)The ± sign specifies a repulsive and an attractive interactionrespectively. This approach will work up to a value of p < Λ where the phase shift δ Λ ( p ) does reproduce Eq. (2). For aseparable potential of the form of Eq. (14) the solution of theLippmann-Schwinger equation reads reads [14, 15] p cot δ ( p ) = − V ( p , p ) (cid:20) − π − (cid:90) ∞ dq q p − q V ( q , q ) (cid:21) = − α + r p (15)For separable potentials the inverse scattering problem maybe solved in quadrature by the Tabakin’s formula devised in1969 [16] (for a review see e.g. [17]). In our case the attractivesolution without bound state will be the pertinent one, reading [ g ( k )] = sin δ ( k ) k exp (cid:20) −− (cid:90) ∞ − ∞ δ ( k (cid:48) ) k − k (cid:48) dk (cid:48) (cid:21) , (16)where δ ( − k ) = − δ ( k ) and the real and positive g ( k ) is takenprovided sin δ ( k ) > ≤ δ ( k ) ≤ π , a condition fulfilled by Eq. (2) for any value of α and r . In our case. accordingto Eq. (12) we have a limited integration interval − Λ ≤ k ≤ Λ . The case with α → − ∞ and r = p > g ( p ) = θ ( Λ − p ) (cid:112) Λ − p . (17)For Λ = k F , the potential satisfying exactly the conditions is V k F ( k (cid:48) , k ) = − θ ( k F − k (cid:48) ) (cid:113) k F − k (cid:48) θ ( k F − k ) (cid:113) k F − k , (18)which as expected depends on k F . It can be readily checkedthat for k > δ ( k ) = π θ ( k F − k ) . (19)A direct evaluation of the integral in Eq. (9) yields ξ = π − = . , (20)contrary to the “universal” value ξ = . − .
40 [2]. The case α → − ∞ and r (cid:54) = P − space of the potential V → U † VU which preserves the es-sential feature that the interaction does not allow transitionsabove the Fermi surface, but reshuffles the v ( p (cid:48) , p ) function.A simple way to generate a continuous one-parameter uni-tary transformation according to the previous requirementsis by means of the Similarity Renormalization Group (SRG)method introduced by Wilson and Glazek [18] (for a reviewsee e.g. [19]). Defining the Hamiltonian H s = T + V s at theoperator level the SRG equation with the Wilson, G s = T , gen-erator reads dV s ds = [[ T , V s ] , T + V s ] . (21)This evolution equation monotonously minimizes the Frobe-nius norm of the potential || V s || = tr ( V s ) , since d tr ( V s ) / ds < s → ∞ provides lim s → ∞ [ T , V s ] =
0, i.e. the potentialenergy operator becomes diagonal in momentum space andhence on-shell [20, 21] . Here, one has for v ( p , p (cid:48) ) in Eq. (11)and in our case ( Λ = k F ) the following equation, d v s ( p , p (cid:48) ) ds = − ( p − p (cid:48) ) v s ( p , p (cid:48) ) + π (cid:90) k F dq q × ( p + p (cid:48) − q ) v s ( p , q ) v s ( q , p (cid:48) ) , (22)where s = / λ and λ is the similarity cutoff. The flowequation generates a set of isospectral interactions that ap-proaches a diagonal form as s → ∞ (or λ → a fi - ¥ ; r = 0 ( N = 5 0 ) S R G - e v o l v e d s e p a r a b l e p o t e n t i a l o n - s h e l l l i m i t ( c o n t i n u u m ) : x = - 1 / 3 i n i t i a l s e p a r a b l e p o t e n t i a l : x = 0 . 5 6 u n i v e r s a l v a l u e : x = 0 . 3 7 x l / k F FIG. 2. Unitary Phase-equivalent evolution of the Bertsch parame-ter ξ as a function of the similarity cut-off λ ≡ / s / (squares) for adiscrete momentum grid with N =
50 points. For N = ∞ we mark theinitial separable potential corresponding to ξ = .
56 and the λ → ξ = − / = − .
33. The hori-zontal line corresponds to the “universal” value ξ = .
37 obtainedby many calculations and experiments [2]. analytically [22]. Their numerical treatment requires intro-ducing a finite momentum grid, so that results in the con-tinuum are taken as a limiting procedure [14, 15]. Takingthe v s ( k , k ) into the mean field energy one obtains a phase-equivalent flow equation for the Bertsch parameter, ξ s . Defin-ing ϕ ( x ) = − x / + x /
2, Eq. (22) yields the inequality d ξ s ds = π (cid:18) π (cid:19) (cid:90) k F dq q (cid:90) k F dk k (cid:34)(cid:18) kk F (cid:19) − (cid:18) qk F (cid:19) (cid:35) × (cid:20) ϕ (cid:18) kk F (cid:19) − ϕ (cid:18) qk F (cid:19)(cid:21) | v s ( k , q ) | ≤ , (23)since ϕ ( x ) is a decreasing function, ϕ (cid:48) ( x ) = − ( − x ) / < ( x − y )[ ϕ ( x ) − ϕ ( y )] < < x , y <
1. This in-equality actually shows that the Bertsch parameter is not de-termined uniquely from the s-wave phase-shift and hence ξ isnot universal .In the on-shell limit, s → ∞ ( λ → v s ( p (cid:48) , p ) becomesdiagonal, and one has thus d ξ s / ds →
0. The limiting valueof Eq. (21) was determined in terms of the scattering phase-shifts [14, 15]. Adapted to our Eq. (22) and in the absence ofbound states, the limit becomes a fixed point. If k (cid:48) (cid:54) = k thenlim s → ∞ v s ( k , k ) = − δ ( k ) k , lim s → ∞ v s ( k (cid:48) , k ) = , (24)which is asymptotically stable [14, 15] and the solutions areattracted to this one. Hence, for δ ( k ) = π / s → ∞ ξ s = − = − . (25)This corresponds to a unstable system. Thus, the previousargument shows that regardless of the initial function v ( p (cid:48) , p ) at s = ξ , there is a phase-equivalentpotential where ξ < Λ = k F .In the case, k F < Λ , the mean field result is only an upperbound, ξ ≤ ξ MF . The flow equations for v s ( p (cid:48) , p ) and ξ MF readas Eq. (22) and Eq. (23) respectively with the replacement (cid:82) k F dq → (cid:82) Λ dq , so that the same inequality holds. Thus, onehas ξ s ≤ ξ MF → − /
3. In the general case, with finite α (cid:54) = r (cid:54) =
0, see Eq. (2), the sign of lim s → ∞ ξ s depends ontheir particular values. For instance for r =
0, the intercept ξ ∞ = α k F = − . α → − ∞ onehas r k F = . ξ . In particular, fromthe SRG equations on a momentum grid [14, 15] we can covercontinuously all values from the starting one to the final one .This is shown in Fig. 2 as a function of λ / k F .In particular, we could tune the SRG-scale λ to obtain fromthe potential V k F ( k (cid:48) , k ) given by Eq. (18) with ξ = .
558 the“universal” value ξ = . −
40 obtained in many calcula-tions and experiments [2]. We find that ξ = .
37 happensfor λ / k F = .
9, see Fig. 2. It is of course tempting to analyzethe effective range behaviour at the scale λ / k F = .
9. Thisis done in Fig. 1 and compared again with the recent MonteCarlo calculation of Conduit and Schonberg [8]. As we seethe lack of universality of ξ is reinforced for finite r evenafter tuning the r = local potentials. The solution of the inverse scatteringproblem exists [17] and will be discussed elsewhere. Sec-ondly, the potentials experienced between neutral atoms arevan der Waals-like and hence local. Thus, we conjecture thatlocality is the additional condition underlying the observeduniversality. Work along these lines is in progress. We take N =
50 points and Gauss-Legendre points. From the discretizedform of Eq. (13) we get ξ = . ξ = − . N ’sand small λ ’s [14, 15, 21]. We estimate an error in ξ about 0.03, which iscompatible with the expected flat behaviour at the fixed point. ∗ Supported by Spanish DGI (grant FIS2014-59386-P), Junta deAndalucia (grant FQM225), FAPESP (grant 2016/07061-3),CNPQ (grant 306195/2015-1) and FAEPEX (grant 3284/16). † [email protected] ‡ [email protected] §§