A new effective interaction for the trapped Fermi gas
aa r X i v : . [ c ond - m a t . o t h e r] J un A new effective interaction for the trapped Fermi gas
Y. Alhassid, G.F. Bertsch, and L. Fang Center for Theoretical Physics, Sloane Physics Laboratory,Yale University, New Haven, CT 06520 Department of Physics and Institute of Nuclear Theory, Box 351560University of Washington, Seattle, WA 98915 (Dated: June 27, 2007)We apply the configuration-interaction method to calculate the spectra of two-component Fermisystems in a harmonic trap, studying the convergence of the method at the unitary interaction limit.We find that for a fixed regularization of the two-body interaction the convergence is exponentialor better in the truncation parameter of the many-body space. However, the conventional regu-larization is found to have poor convergence in the regularization parameter, with an error thatscales as a low negative power of this parameter. We propose a new regularization of the two-bodyinteraction that produces exponential convergence for systems of three and four particles. From thesystematics, we estimate the ground-state energy of the four-particle system to be (5 . ± . ~ ω . PACS numbers: 31.15.-p, 03.75.Ss, 21.60.Cs, 71.15.Nc
The study of cold trapped atomic condensates has be-come a rich field experimentally. By providing a stronglyinteracting system that is well defined, it also offers physi-cists an unprecedented opportunity to assess theoreticaltechniques that cross the boundaries of disciplines. In theso-called unitary limit, the only dimensional scale of theproblem is fixed by the harmonic trap frequency. System-atic studies have begun on small systems using fixed-nodeMonte Carlo [1, 2] and density functional methods [3].Remarkably, the exact wave functions and energies of the A = 3 system are known, calculated by solving a singletranscendental equation [4]. Our work here is in the con-text of the configuration-interaction (CI) method, widelyused in atomic, molecular, and nuclear spectroscopy. Westudy the convergence of the CI method with respect toa regularization parameter of the two-body interactionand find that a simple regularization scheme that renor-malizes the interaction produces slow convergence of thethree- and four-particle spectra. We introduce a new ef-fective interaction that gives exponential convergence, atleast in small systems. Hamiltonian.
The cold trapped atom system is modeledby the Hamiltonian H = − A X i =1 ~ m ∇ i + A X i =1 mω r i + X i The two particle system ( A =2) is separable in center of mass and relative coordinates r = r − r . The center of mass Hamiltonian describes an harmonic oscillator with frequency ω and mass 2 m , whilethe relative-coordinate Hamiltonian is H rel = − ~ µ ∇ r + µω r + V δ ( r ) with reduced mass µ = m/ 2. The two-particle energies are given by E = (2 N + L + 3 / ~ ω + ε nl , (2)where N , L and n, l are the radial quantum number andangular momentum of the center of mass and relativemotion, respectively. The energies ε nl are the eigenvaluesof H rel , and may be derived from the boundary conditionat the origin imposed by the unitary interaction [5]. Thecontact interaction affects only the l = 0 partial waves,and shifts each s -wave oscillator energy down by one unitof ~ ω [6]. Thus we have ε nl = (2 n + l + 3 / − δ l, ) ~ ω ; n = 0 , , , . . . (3) The renormalized contact interaction. In the CI method,the contact interaction in Eq. (1) must be treated explic-itly. However, a δ -function interaction cannot be used inthree dimensions without a regularization. We shall dothis by truncating the space of relative-coordinate wavefunctions to a q subspace defined by the lowest q + 1 os-cillator l = 0 wave functions (see also Ref. [7]). Withinthe truncated space the relative-coordinate Hamiltoniancan be written as( H rel ) ( q ) n,n ′ = (2 n + 3 / ~ ωδ n,n ′ + V ( q ) n,n ′ (0 ≤ n, n ′ ≤ q ) (4)where V ( q ) n,n ′ = ~ ωχ q ψ n (0) ψ n ′ (0) , (5)and ψ n (0) = π − / p (2 n + 1)!! / (2 n n !) is the ( n, l = 0)oscillator wave function at r = 0 for an oscillator of radius1. The parameter χ q is a dimensionless normalizationconstant related to V by χ q = ( ~ /µ ) − / ( ~ ω ) / V .We determine the normalization constant χ q by requir-ing the ground-state energy of the truncated Hamiltonianto equal the exact value for the unitary contact interac-tion, ε = ~ ω/ 2. The separable form of (5) permitsan algebraic diagonalization of the Hamiltonian. Eacheigenvalue ε of (4) satisfies the dispersion formula χ − q = − q X n =0 ψ n (0)(2 n + 3 / − ε/ ~ ω . (6)Requiring ε = ε = ~ ω/ χ q = − π / q X n =0 (2 n − n n ! ! − . (7)We note that the sum in (7) diverges as q / for large q [8].Thus, the strength of the δ -function goes to zero as q →∞ , showing the need for a renormalization procedure. Asimilar relation between the strength of the interactionand the cutoff can be derived for a plane-wave basis. Inthat case the relation is V = − π ~ /µ Λ where Λ is amomentum cutoff [9]. This value of V agrees with theasymptotic expression of Eq. (7) [8] once we equate thecorresponding cutoff energies as ~ Λ / µ = (2 q +3 / ~ ω .The excited states of the q -truncated Hamiltonian (4)have energies ε ( q ) n that differ from the exact unitary spec-trum (3). Using the dispersion relation (6), we find thatthe error in the energy δε ( q ) n = ε ( q ) n − ε n goes to zero atlarge q , but only at a rather slow rate, δε ( q ) n ∼ q − / . Wepresent evidence below that this slow convergence is alsopresent in the q -renormalized energies for the A = 3 and A = 4 systems. This makes it problematic to extrapolatethe q series to estimate the true q → ∞ energies. A new effective interaction. We have considerably morefreedom to construct the q -space interaction than we haveexploited so far. The only requirement on the q -spaceHamiltonian is that it converge to the unitary limit forlarge q . For example, in effective field theory one mayintroduce derivatives of the contact interaction to fit cer-tain properties of the two-particle Hamiltonian. Here wepropose the following prescription to improve the q -spaceinteraction: simply require that the relative-coordinateHamiltonian reproduce all q + 1 s -wave eigenvalues ofEq. (3). We can do this and still keep the separable formfor the interaction, V eff( q ) n,n ′ = − ~ ωf n f n ′ . (8)A motivation for preserving the separable form is givenin the discussion below. There are q + 1 independentvariables f n in the interaction (8) and the same numberof eigenvalue equations having the form of Eq. (6) with f n replacing p | χ q | ψ n (0). Using the conditions that all q + 1 lowest l = 0 unitary eigenvalues (3) ( n = 0 , . . . , q )are reproduced, we find the following q + 1 equations for f n q X n =0 f n n − r ) + 1 = 1 ( r = 0 , . . . , q ) . (9)Eqs. (9) determine a unique solution for f n ( n =0 , . . . , q ) [10]. We choose the sign of the real numbers f n to coincide with the sign of ψ n (0). Using the con-vention that the harmonic oscillator wave functions bepositive at the origin, the unique solution for f n is f n = s (2 n + 1)!!(2 n )!! [2( q − n ) − q − n )]!! . (10)The interaction defined by (8) and (10) is different fromthe renormalized contact interaction for any q . However,its eigenfunction components (in the 3-D oscillator ba-sis) converge to the corresponding unitary eigenfunctioncomponents in the limit of large q with an error of ∼ q − .In comparison, the eigenvector components of the renor-malized contact interaction converge to the same unitaryeigenvector components but at a slower rate of ∼ q − / . CI method and truncation of many-particle space. Inthe CI approach, one uses a single-particle basis in thelaboratory frame and constructs a many-particle basisof Slater determinants for A fermions. In our prob-lem, a natural choice for the single-particle basis arethe eigenstates of the three-dimensional harmonic oscil-lator. These states are labeled by orbital quantum num-bers a = ( n a , l a ), the orbital magnetic quantum number m a , and an additional two-valued quantum number (e.g.spin) to distinguish the two species of fermions.A way to truncate the many-particle space must bespecified, because there is no natural truncation associ-ated with the interaction except in the trivial cases q = 0or A = 2. There are a number of truncation schemesin the literature; here we will define a truncated single-particle orbital basis and construct the A -particle wavefunction allowing all possible anti-symmetrized productstates. In particular, we shall use all single-particle statesin the oscillator shells N = 0 , . . . , N max with N = 2 n a + l a to construct the many-particle states. There will betwo limiting processes necessary to calculate the many-particle energies. The first is N max → ∞ , which we willinvestigate for fixed q . Then, with converged q -regulatedenergies we estimate the q → ∞ limit.Two technical aspects of our calculations should bementioned. The two-particle matrix elements of the in-teraction in the oscillator basis are conveniently calcu-lated using the Talmi-Moshinsky brackets to transform torelative and center of mass coordinates [11]. The many-particle Hamiltonian is constructed and diagonalized us-ing the nuclear shell model code oxbash [12]. Unlikethe nuclear shell model, our orbitals are characterizedby integer angular momentum values. The two fermionspecies are distinguished in the same way as neutrons andprotons are distinguished in the nuclear application. max E ( ) max -3 -2 -1 ∆ E ( ) m a x N m a x N (a) (b) FIG. 1: Convergence in N max for the A = 3 ground-stateenergy. (a) E ( q ) N max versus N max for q = 3. Open circles corre-spond to the renormalized contact interaction and solid circlesto the interaction defined by (8) and (10). (b) ∆ E (3) N max versus N max in a logarithmic scale. All energies are in units of ~ ω . A=3 system. We now show the results for A = 3. Theground state of the A = 3 system is a negative-paritystate with total angular momentum L = 1 and energy4 . . . . ~ ω [4, 13]. In our CI convergence studieswe computed the ground-state energies E ( q ) N max for q =1 , , , N max = q, . . . , q , we find that E ( q ) N max converge exponen-tially or better in N max for both interactions. This isdemonstrated in Fig. 1. Panel (a) shows E ( q ) N max versus N max for q = 3. Both the renormalized contact interac-tion (open circles) and the new interaction (solid circles)are monotonically decreasing, as they must when thespace gets larger. The important point, seen in Fig. 1(b),is that the energy differences ∆ E ( q ) N max ≡ E ( q ) N max − − E ( q ) N max decrease rapidly on a logarithmic scale. In fact, the de-crease is steeper than linear on that scale, suggestingthat the convergence might be faster than exponential.The solid lines are quadratic fits to log(∆ E ( q ) N max ), usedto extrapolate to a value of E ( q ) ≡ E ( q ) ∞ . We observethe decrease rate of ∆ E ( q ) N max to be monotonically in-creasing with N max , so a conservative lower bound in E ( q ) is obtained using a fixed-rate extrapolation above N max = 7 with an average rate determined by the points N max = 5 , , 7. An upper bound for E ( q ) is given by E ( q )7 .Fig. 2(a) shows the converged or extrapolated energies E ( q ) versus q . These energies are monotonically increas-ing function of q . For the new interaction (solid circles),we observe a fast convergence to the known exact value(dotted line). Fig. 2(b) shows the absolute value of thedeviation δE ( q ) ≡ E ( q ) − E ( ∞ ) from the exact result ina logarithmic scale. The concavity of the curve for therenormalized contact interaction (open circles) indicatesthe convergence in q is slower than exponential. We findthis convergence to be consistent with a low negativepower law ∼ q − α with α in the range ∼ . − . A = 2 system it can be shown analytically q E ( q ) q -3 -2 -1 | δ E ( q ) | (a) (b) FIG. 2: Convergence of the q -regulated energies for the A = 3ground state. (a) E ( q ) versus q for both interactions (symbolsand units as in Fig. 1). The dotted line is the exact ground-state energy. (b) The error | δE ( q ) | in a logarithmic scale. that α = 1 / E ( ∞ ) . We calculated successive energy differ-ences ∆ E ( q ) ≡ E ( q − − E ( q ) and determined an averagerate of decrease λ of | ∆ E ( q ) | for q below a given q ′ . As-suming a fixed rate λ for q > q ′ , the extrapolated energyis [ λE ( q ′ ) − E ( q ′ − ] / ( λ − E ( ∞ ) , since the rate of decreaseof | ∆ E ( q ) | seems to be a monotonically non-decreasingfunction of q . Using q ′ = 3 and an average decrease rateof 3.28 (determined from ∆ E ( q ) at q = 1 , , E ( ∞ ) = (4 . ± . ~ ω , an accuracy of 0 . L π = 0 + firstexcited state at E ( ∞ ) = 4 . . . . ~ ω [4, 13]. Resultsare shown in Fig. 3(a). As in the ground-state case, weobserve a low negative power law convergence for therenormalized interaction and exponential convergence forour interaction. Using E (3) and an average decrease rateof 1.83 obtained from q = 1 , , 3, we estimate E ( ∞ ) =(4 . ± . ~ ω , an accuracy better than 0 . A=4 system. We also studied the L = 0 ground state ofthe A = 4 system with two particles of each species. Theresults for E ( q ) are shown in Fig. 3(b). Here the exactvalue E ( ∞ ) is unknown. An upper bound (using the newinteraction) is E (3)7 = 5 . ~ ω . A lower bound can beobtained as for the A = 3 system. The inset of Fig. 3(b)shows ∆ E ( q ) in a logarithmic scale versus q for the newinteraction. Again, the convergence seems to be at leastexponential. The straight line is a fit to log(∆ E ( q ) ) using q = 1 , , 3, and provides an average decrease rate of 2.14.Using the extrapolated E (3) = (5 . ± . ~ ω andthis average rate, we estimate E ( ∞ ) = (5 . ± . ~ ω .Our result agrees with fixed-node Monte Carlo estimatesof (5 . ± . ~ ω [1] and (5 . ± . ~ ω [2]. Discussion. There are a number of methodologies incurrent use to construct effective interactions for many-particle systems; among them, effective field theory q E ( q ) q -2 -1 δ E ( q ) q q -2 -1 ∆ E ( q ) (a) (b) FIG. 3: (a) E ( q ) versus q for the lowest L = 0 excited stateof the A = 3 system. The inset shows δE ( q ) versus q in alogarithmic scale. Symbols and units as in Fig. 2. (b) E ( q ) versus q for the L = 0 ground state of the A = 4 system.The inset shows ∆ E ( q ) versus q for the new interaction in alogarithmic scale. The solid line is a linear fit to q = 1 , , (EFT) and the unitary-transformation method have aconnection to the interactions discussed here. In EFT,the interaction is parameterized by contact terms (lead-ing order) and their derivatives. Our procedure to con-struct the q -renormalized contact interaction can thusbe considered as leading-order EFT. Its poor conver-gence suggests that EFT treatments will require deriva-tive terms to accurately model trapped fermion systems.Our improved interaction has some connection withSuzuki’s unitary regularization [14], a method widelyused in nuclear physics [15, 16, 17]. In Suzuki’s approach,an effective interaction is determined by a unitary trans-formation of the Hamiltonian that decouples a subspacefrom its complementary subspace. In practice, the trans-formation is performed on the two-particle Hamiltonian,giving a transformed Hamiltonian that is block diagonal.This block diagonal structure guarantees that the energyeigenvalues are reproduced in the truncated subspace.Our effective interaction also reproduces the exact two-particle spectrum in a truncated subspace but has theadvantage of being simple, i.e., separable.The unitary transformation of the two-particle Hamil-tonian cannot be carried out independently for all possi-ble pairings in the many-body Hamiltonian. When thistransformation is applied to the many-particle system, itgenerates higher-order many-body interactions that areusually simply neglected. For our Hamiltonian, addi-tional correction terms would be required if we were torelate it to a unitary-transformed Hamiltonian. Ratherthan attempting to compute these correction terms, wehave studied the convergence in the large q -limit, whereour effective interaction coincides with the contact inter-action. By studying the convergence, one can assess theusefulness of many of the specific details of the differentmethodologies. For example, there are other choices ofthe many-particle space truncation that might be more efficient. Non-unitary transformations might give fasterconvergence. The no-core-shell-model methodology [18]is an example where a particular choice was made.Our method can be applied for interaction strengthsaway from unitarity, at the slight cost of inverting nu-merically a ( q + 1)-dimensional matrix. It may also beinteresting to apply the method to uniform systems, us-ing the separability of the interaction in a plane-wavebasis. One caveat is that we have only examined three-and four-particle systems. It will be important to confirmthe exponential convergence when our interaction is usedto calculate the spectra of systems with more particles.We thank A. Bulgac, M. Forbes, M. Hjorth-Jensen andU. von Kolck for conversations, B.A. Brown and M. Horoifor their advice on the computer program oxbash , and S.Fujii in particular for his guidance on the unitary trans-formation models. This work was supported in part bythe U.S. DOE grants No. FG02-00ER41132, DE-FC02-07ER41457 and DE-FG02-91ER40608. [1] S.Y. Chang and G.F. Bertsch, arXiv: physics/0703190.[2] J. von Stecher, C.H. Greene, and D. Blume,arXiv:0705.0671.[3] A. Bulgac, arXiv:cond-mat/0703526.[4] F. Werner and Y. Castin, Phys. Rev. Lett. , 150401(2006).[5] The spectrum is most easily derived by noting that theboundary condition ( u ′ /u ) | r =0 = 0, imposed by the uni-tary interaction on the radial wave function u ( r ), effec-tively maps the 3-D s -wave problem onto the even solu-tions of the 1-D harmonic oscillator.[6] T. 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