Atomic Fermi gas at the unitary limit by quantum Monte Carlo methods: Effects ofthe interaction range

Xin Li,1 Jindrich Kolorenc,1,2 and Lubos Mitas1

1Department of Physics, North Carolina State University, Raleigh, NC 27695, USA2 I. Institut fur Theoretische Physik, Universitat Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany

(Dated: May 10, 2011)

We calculate the ground-state properties of unpolarized two-component Fermi gas by the diffusionquantum Monte Carlo (DMC) methods. Using an extrapolation to the zero effective range of theattractive two-particle interaction, we find E/Efree to be 0.212(2), 0.407(2), 0.409(3) and 0.398(3) for4, 14, 38 and 66 atoms, respectively. The obtained energies are lower than those from the previouslyreported fixed-node DMC studies. In order to test the quality of nodal surfaces and to estimate theimpact of the fixed-node approximation we perform released-node DMC calculations for 4 and 14atoms. Analysis of the released-node and the fixed-node results suggests that the main sources ofthe fixed-node errors are long-range correlations which are difficult to sample in the released-nodeapproaches due to the fast growth of the bosonic noise. Besides energies, we evaluate the two-bodydensity matrix and the condensate fraction. We find that the condensate fraction for the 66 atomsystem converges to 0.56(1) after the extrapolation to the zero interaction range.

I. INTRODUCTION

In recent years, the homogeneous Fermi gas with at-tractive interactions has been studied extensively boththeoretically and experimentally due to the success incooling atoms into ultracold dilute condensates.1–3 Bytuning the interaction strength through the Feshbachresonance,4–7 the system can cross from the Bardeen-Cooper-Schrieffer (BCS) superfluid phase, where thes-wave scattering length as is negative, to the Bose-Einstein condensate (BEC), where as is positive. Sincethere is no symmetry change of the quantum state in-volved, the system exhibits the well-known BCS-BECcrossover.

In the special case corresponding to the diverging scat-tering length, as →∞, the system is in a strongly inter-acting regime called the unitary limit. In this regime theinterparticle spacing rs is the only relevant scale, and therest of the quantities are universal and system indepen-dent. The total energy of this system can be convenientlywritten as E = ξEfree, where Efree is the energy of thenon-interacting atomic gas and ξ is a system independentparameter. Experimental measurements of ξ have beenperformed using 6Li and 40K atoms by investigating theexpansion rate of the atomic cloud and the sound prop-agation in it.8–12 Simultaneously, a number of theoreti-cal and numerical estimations of ξ have been reported,including diffusion Monte Carlo (DMC)13–18 as well aspath integral Monte Carlo, lattice simulations and ana-lytical methods.18–27 The resulting estimates fall between≈ 0.25–0.45 showing that the actual value has not beensettled yet and is still of significant interest due to theuniversal nature of the unitary limit.

One of the most interesting properties of the unitarygas is the robust presence of the pairing condensate whichinvolves a large fraction of the system. The study of pair-ing effects is thus much more straightforward than, say,in superconducting materials, where only a sliver of thefermions around the Fermi level forms the condensate

and the attractive interaction is much more complicated.The quantum Monte Carlo (QMC) methods have the ad-vantage that the condensate can be detected directly, byevaluating the off-diagonal two-particle density matrixand by monitoring its behavior at large distances.16,18

The goal of our study is twofold. First, any actual sim-ulation involves only a finite system, and the quantitiesrelevant for the thermodynamic limit have to be obtainedfrom appropriate extrapolations. The unitary system isnot trivial in this respect, since pairing with infinite scat-tering length is described by a function with a slow fall-offat large distances. It is necessary to analyze the finite-sizescaling of the quantities of interest and to test whetherthe actual limit of infinite dilution, or, equivalently, ofpoint-like character of the interaction, has indeed beenreached. Second, the impact of the fixed-node approxi-mation in the quantum Monte Carlo method is not verywell understood for this system since there is nothing tocompare with: so far the fixed-node formulation of theQMC methods appears to be the only approach that isable to provide an upper bound for the total energy. Thishas motivated us to probe the accuracy of the nodes byreleased-node QMC simulations and by improvements inthe variational flexibility of the employed wave functions.

We have carried out calculations of the ground-stateproperties of the dilute unitary Fermi gas by the fixed-node DMC (FN-DMC)28 method for 4, 14, 38 and 66atoms. By using a more versatile construction of the pairorbital in the BCS wave function and by extrapolatingthe effective range of the two-particle interactions Reff

to zero, we were able to obtain lower ξ than the previ-ously reported DMC calculations. These results suggestthat the extrapolation of Reff is important, especiallyfor smaller systems. In order to test the quality of thenodal surface of the BCS wave function, we have per-formed released-node DMC (RN-DMC)29,30 calculationsfor 4 and 14 atoms. This procedure has been carried outstarting from two types of nodal constraints: from theBCS nodes and from the Hartree-Fock (HF) nodes. Our

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RN-DMC results indicate that the nodal corrections aredriven mainly by long-range correlations which are diffi-cult to sample in the released-node framework due to therapid growth of the bosonic noise. We have calculatedalso the two-body density matrix and the condensatefraction for the 66 atom system, and we have estimatedthe corrections from the effective-range extrapolation onthese quantities.

II. METHOD

A. Hamiltonian

We consider a two-component Fermi gas with Hamil-tonian

H = −1

2

N/2∑i=1

∇2i −

1

2

N/2∑i′=1

∇2i′ +

∑i,i′

V (rii′) , (1)

where N is the total number of atoms, i and i′ correspondto the spin-up and spin-down atoms, and rii′ denotes thedistance |ri − ri′ |. The atoms are located in a cubic boxwith the side L and we impose the periodic boundaryconditions. The two-particle potential V (rii′) is taken inthe Poschl-Teller form

V (rii′) = − 2µ2

cosh2(µrii′), (2)

whose effective range is Reff = 2/µ. The s-wave scatter-ing length as is infinite for all values of µ 6= 0.

B. Trial wave functions

In the majority of our calculations we employ trialwave functions of the BCS form multiplied with the Jas-trow factor (BCS-Jastrow) as given by

ΨT (R) = ΨBCS(R)eJ(R) , (3)

where

ΨBCS(R) = A[ N/2∏i,i′=1

φ(ri, ri′)

]= det[φ(ri, ri′)] . (4)

Here A represents the antisymmetrization operator andφ(ri, ri′) is the pair orbital. The vector R encom-passes all atomic coordinates ri and ri′ . Additionally, wehave carried out a subset of calculations also with theHartree-Fock-Jastrow (HF-Jastrow) trial functions, inwhich ΨBCS is replaced with a product of two Slater de-terminants of one-particle orbitals (simple plane waves).The HF-Jastrow wave function reads as

ΨSJ(R) = det[ϕa(ri)] det[ϕa(ri′)]eJ(R). (5)

The pair orbital φ(ri, ri′) in ΨBCS is written as a linearcombination of Gaussian functions

φ(ri, ri′) =

1∑l,m,n=−1

∑k

dke−αk(xi−x′

i+lL)2

× e−αk(yi−y′i+mL)2e−αk(zi−z′i+nL)2 , (6)

where dk are expansion coefficients, and ri = (xi, yi, zi)and ri′ = (xi′ , yi′ , zi′) are coordinates of i and i′ atomsinside the simulation box of linear size L. We choose suffi-ciently large exponents αk so that only the first neighborshell of periodic images contributes to the sum, that is,the Gaussian functions are negligible at distances largerthan 3L/2. The pair orbital is smooth with zero deriva-tive at the boundary of the simulation cell. The Jastrowfactor J(R) is constructed in a similar way as the pairorbital φ(ri, ri′) for both different spin atoms and samespin atoms.

A typical trial wave function includes around 30 to40 variational parameters that are optimized by mini-mizing a linear combination of the total energy and itsvariance.31 Although the Jastrow factor does not changethe nodal surface, accurate description of the pair cor-relations makes the variational optimization much moreefficient and robust. When the effective range of the po-tential approaches zero, more Gaussian functions withlarger exponents αk are included in the Jastrow factor inorder to keep the accuracy of the trial function consis-tently high. On the other hand, and somewhat surpris-ingly, we find that similar adjustment of the pair orbitalwith the changing effective range is relatively minor.

C. Fixed-node and released-node DMC methods

The DMC method projects out the ground state froma given trial function ΨT by means of an auxiliary evolu-tion in the imaginary time, Φ(τ) ∼ exp(−τH)ΨT . By in-troducing importance sampling28 with the aid of a guid-ing function ΨG, we can write an integral equation forΦ(R, τ) in the form

ΨG(R)Φ(R, τ + ∆τ) =∫dR′

ΨG(R)

ΨG(R′)G(R,R′,∆τ)ΨG(R′)Φ(R′, τ) . (7)

For small ∆τ , the propagator G(R,R′,∆τ) can be ap-proximated using the Trotter-Suzuki formula as

ΨG(R)

ΨG(R′)G(R,R′,∆τ) ≈ G0(R,R′ + ∆τv(R′),∆τ)

× e−∆τ [EL(R)+EL(R′)−2ET ]/2, (8)

where v(R′) ≡ ∇ ln |ΨG(R′)| and G0(R,R′,∆τ) is theGreen’s function for non-interacting atoms that takes the

3

0.21 0.215

0.22 0.225

0.23 0.235

0.24 0.245

0.25

0 0.05 0.1 0.15

ξ 2,2

Reff/rs

rs=15.63

FN-DMC 0.405

0.41 0.415

0.42 0.425

0.43 0.435

0 0.05 0.1 0.15

ξ 7,7

Reff/rs

rs=10

FN-DMC

0.405 0.41

0.415 0.42

0.425 0.43

0.435

0 0.05 0.1 0.15

ξ 19

,19

Reff/rs

rs=10rs=10

FN-DMC 0.395

0.4 0.405

0.41 0.415

0.42 0.425

0.43 0.435

0.44

0 0.05 0.1 0.15 0.2

ξ 33

,33

Reff/rs

rs=10

FN-DMC

FIG. 1. The fixed-node energy for unpolarized unitary Fermigas as a function of the interaction range Reff/rs with linearextrapolation to Reff/rs = 0. The system sizes are 4, 14, 38and 66 atoms from the top left to the bottom right. Thestatistical error bars are smaller than the symbol size.

form of the diffusion kernel. The so-called local energyEL is given by

EL(R) =HΨG(R)

ΨG(R). (9)

The product ΨGΦ is represented by a set of sam-ples (also referred to as walkers) and this ensemble isevolved with the aid of a stochastic process simulat-ing Eqs. (7) and (8). In the fixed-node method we setΨG(R) = ΨT (R) and the fixed-node condition is im-posed by enforcing the sampling points to obey

ΨG(R)Φ(R, τ) ≥ 0 (10)

at all times. In the limit of long τ the solution con-verges towards the lowest-energy state consistent withthe boundary conditions given by the fixed nodes.

In the RN-DMC method the guiding function hasbosonic symmetry and its square should be close to thesquare of the fermionic ground state. We have used guid-ing functions in the form30,32

ΨG(R) =√

Ψ2T (R) + α 〈Ψ2

T 〉 , (11)

where⟨Ψ2T

⟩is the average value of Ψ2

T (R0) over all con-figurations, and R0 are the walker positions right afterthe nodal release. The tunable parameter α controls therate of walkers passing through the nodal region. Theguiding function is non-negative everywhere and there-fore the stochastic process propagates a mix of bosonic

and fermionic states. The fermionic component is fil-tered out by reweighting with the factor ΨT /ΨG so thatthe fermionic-state energy is given by

〈Φ0|H|ΨT 〉 =

∫dRΦ0(R)ΨG(R) ΨT (R)

ΨG(R)HΨT (R)ΨT (R)∫

dRΦ0(R)ΨG(R) ΨT (R)ΨG(R)

, (12)

where Φ0(R) denotes the exact fermionic ground state.Since this method is exponentially demanding both in

the projection time and in the number of atoms, it is im-portant to choose α so that the statistical information isrecovered as quickly as possible. If α is too large the fluc-tuations from the poor importance sampling overwhelmany useful signal very rapidly. On the other hand, atoo small value can bias the results. Since it is difficultto reach reliable error bars in this type of calculations,we have used the method mostly to identify the onsetand the amplitude of the energy decrease during the pro-jection period when the stochastic noise was acceptablysmall.

In the RN-DMC process, we can also pick up the sta-tistical signal from the walkers that have never crossedthe nodal surface, and in essence this provides the FN-DMC estimator. By monitoring these paths as well, wecan assess the consistency of the estimators and some-what better tune the parameter α for providing betterRN-DMC signal.

III. RESULTS

For benchmark purposes we first calculate perhaps thesmallest nontrivial system—four atoms. Our result isshown in Fig. 1, upper left panel. There is approximately10% energy drop when Reff/rs is reduced from 0.1279 to0.003998. We extrapolate Reff to zero using a linear fitand obtain ξ2,2 = 0.212(2). Here and in the rest of thepaper, the denominator Efree in the ratio ξ = E/Efree isevaluated in the same finite volume subject to the sameboundary conditions as the nominator E.

Our calculations with 14, 38 and 66 atoms are car-ried out analogously to the 4-atom case, and the extrap-olated values of ξ are 0.407(2), 0.409(3) and 0.398(3),respectively, as plotted in Fig. 1. In these calculations,the smallest effective range is Reff/rs = 0.003125. Itshould be noted that in the previous DMC calculations,the range of Reff/rs was between 0.1 to 0.2, and withinthis range our DMC results agree well with the previ-ously obtained values.14,15,18 Reduction of Reff decreasesthe energy in all cases although the decrease per atom issmaller in larger systems.

To test the quality of the nodal surfaces, we have car-ried out released-node calculations for 4-atom and 14-atom systems. The RN-DMC calculations for 4 atomswere done with Reff/rs = 0.06397. In a typical released-node run the number of walkers was about two millionso that the error bars were initially very small. In Fig. 2,the upper row shows the pair orbital along three distinct

4

0.24

0.25

0.26

0.27

0.28

0.29

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

ξ 2,2

EFt

FN RN

FN-DMC

FN-DMC α=0.5

RN-DMC α=0.5

0.225

0.23

0.235

0.24

0.245

0.25

0.255

0.26

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

EFt

FN RN

FN-DMC

FN-DMC α=0.5

α=0.2

α=0.05

RN-DMC α=0.5

α=0.2

α=0.05

0.22

0.222

0.224

0.226

0.228

0.23

0.232

0.234

0.236

0.238

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

EFt

FN RN

FN-DMC

FN-DMC α=0.2

α=0.05

α=0.01

RN-DMC α=0.2

α=0.05

α=0.01

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

φ(r)

r/L

φ 100

φ 110

φ 111

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

r/L

φ 100

φ 110

φ 111

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

r/L

φ 100

φ 110

φ 111

FIG. 2. The pair orbitals and FN-DMC and RN-DMC energies of the 4-atom unitary system with Reff/rs = 0.06397. Theupper row shows the pair orbitals with the lowest (left), intermediate (middle) and optimal (right) accuracy with regard to thevariational optimization. The lower row shows the corresponding DMC energies as functions of the projection time startingfrom the variational estimate. Note that the resolution of the left and right panels differs by an order of magnitude. Thevertical dotted lines indicate the instant of the nodal release.

directions (100, 110 and 111) of the interparticle dis-tance vector ri − rj . The lower row shows the FN-DMCand RN-DMC energies as they evolve with the projectiontime. The plots show convergence of the FN-DMC en-ergy followed by the nodal release. This is accomplishedby switching the guiding wave function from ΨT (R) tothe bosonic function ΨG(R) defined in Eq. (11).

The released-node signal reflects the quality of thenodal surface of the trial wave function employed in theFN-DMC simulation. We have tested wave functionswith intentionally varied accuracy by employing subop-timal pair orbitals. The plot of the energy evolution inthe left panel of Fig. 2 shows a clear and pronounceddrop after the nodal release. As the quality of the pairorbital improves, this drop shrinks. For the fully op-timized BCS-Jastrow wave function (the right panel inFig. 2) the energy is reduced by less than 0.002 withinthe longest projection time we have tried. This fact aswell as comparison with other methods33 indicate thatour BCS wave functions are very accurate in this smallsystem and that the fixed-node error is marginal.

We observe an unexpectedly high sensitivity of thenodal quality to the details of the pair orbital at largedistances. This suggests an explanation for the rela-tively slow convergence of the released-node energy: thelong-range tails of the pair orbital affect the nodal hy-

persurfaces, although their contribution to the total en-ergy is relatively small. One can further deduce that thismakes the released-node method quite challenging to ap-ply since it requires sampling of long distances and thecorresponding correlations. This is, however, difficult toachieve because the diffusive motion of walkers is slow,proportional to t1/2, while the growth of the noise is fast,proportional to exp(∆BF t) where ∆BF is the differencebetween the bosonic and fermionic ground-state energies.

The time step ∆τ was set to 4 × 10−5r2s in all runs

and we have verified that the time-step bias of the RN-DMC results is negligible. The converged RN-DMC en-ergy should not depend on the parameter α in the bosonicguiding function ΨG provided one would be able to evolvethe stochastic process with the error bars under controluntil the full convergence. In the same time, the parame-ter α crucially affects the growth of the fluctuations withthe projection time as illustrated in Fig. 2.

The RN-DMC energy for 14 atoms with Reff/rs = 0.2is shown in Fig. 3. The error bars are estimated fromeight independent runs with two million walkers each. Inthe interval of EF t ≤ 0.2 after the nodal release the RN-DMC energy gain appears to be very small and the errorbars preclude to make any statistically sound estimationfor longer projection times. The rapid loss of resolutionis expected since the difference between the bosonic and

5

0.42

0.43

0.44

0.45

0.46

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

ξ 7,7

EFt

BCS

FN RN

FN-DMCFN-DMC α=5e-7

α=1e-7 α=5e-8

RN-DMC α=5e-7 α=1e-7 α=5e-8

FIG. 3. Evolution of the DMC energies for the 14-atom sys-tem with the best optimized BCS-Jastrow wave function. Theruns are for Reff/rs = 0.2. No statistically significant energydrop is observed after the nodal release that is indicated withthe vertical dotted line.

fermionic ground states grows with the number of atoms.Again, the RN-DMC signal exhibits little dependence onα we choose.

In order to make a comparison with a case displaying aclear fixed-node bias, we have carried out RN-DMC runsusing the Slater-Jastrow trial wave function, see Fig. 4.Since this wave function has the nodal surface of the non-interacting Fermi gas, the nodal surface is strongly dis-torted. As a result, we see a very pronounced released-node signal. However, within the projection time inter-val of EF t ≤ 0.2, the energy drops by only ≈ 0.015 forthe largest α we tested. This is very small consideringthat the true ground-state energy is at least an order ofmagnitude lower. This again illustrates the challengesof efficient application of the released-node method, atleast for present cases. Comparison of the Slater andBCS wave functions shows a significant effect of pairingas was demonstrated in earlier studies.13

In order to quantify the pairing effects we calculatethe two-body density matrix which enable us to evaluatethe condensate fraction. The projected two-body densitymatrix for spin-up and spin-down atoms is defined as

ρ(2)(r) =N2

4V 2

∫dRΦ(R)ΨT (R)ΨT (r1+r,r2+r)

ΨT (r1,r2)∫dRΦ(R)ΨT (R)

, (13)

where N is the total number of atoms and V is the vol-ume of the simulation cell. The density matrices havebeen calculated for the fixed-node wave functions andhence they correspond to the mixed estimators.28 Nev-ertheless, the mixed-estimator bias is negligible since thevariational Monte Carlo and DMC estimates of ρ(2) co-incide within error bars. This is a further evidence of thehigh accuracy of our trial wave functions.

The condensate fraction can be extracted from the two-body density matrix as

c =2V 2

Nlimr→∞

ρ(2)(r) . (14)

0.51

0.52

0.53

0.54

0.55

0.56

0 0.1 0.2 0.3 0.4 0.5

ξ 7,7

EFt

HF

FN RN

FN-DMCFN-DMC α=1e-3

α=5e-4 α=1e-4

RN-DMC α=1e-3 α=5e-4 α=1e-5

FIG. 4. DMC energies for 14 atoms obtained with the Slater-Jastrow wave function. The runs are for Reff/rs = 0.2. TheRN-DMC energy drops are significant when compared to theRN-DMC signal from the BCS-Jastrow wave function. Theparameter ξ7,7 drops by ≈ 0.015 within EF t ≈ 0.2 after thenodal release.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

2V2 /N

ρ(2

) (r)

kFr

c=0.56(1)

Reff/rs=0.2 Reff/rs=0.05 Reff/rs=0.0125

Reff/rs=3.125e-3

FIG. 5. The two-body density matrix for 66 atoms calculatedfrom the FN-DMC mixed estimator. The condensate fractionconverges to 0.56(1) for Reff/rs ≤ 0.05.

The calculated density matrices are shown in Fig. 5 withthe condensate fraction estimated from the long-rangelimit. The condensate fraction saturates for Reff ≤ 0.5at c = 0.56(1). This value is not too far from the resultsobtained previously.16,18

To illustrate the character of the nodal surfaces in theBCS-BEC systems, we present three-dimensional scansof the nodes for three wave functions corresponding tothe following scattering regimes: the free atomic gas withno pairing, our best unitary-limit wave function, and thewave function with enhanced pairing from the BEC sideof the BEC-BCS phase diagram (askF = 0.6592). Theleft column of Fig. 6 displays the nodal surface of the freeatomic Fermi gas. The delocalized nature of the systemis apparent. At the unitary limit, shown in the middlecolumn of Fig. 6, the shape of the nodal surface is sig-nificantly different as the pairing effects clearly dominateand lead to a localized character of the nodes from the

6

FIG. 6. Three-dimensional subsets of the nodal hypersurfacesfor three types of wave functions and corresponding phases inthe 14-atom system. The node is obtained by scanning thesimulation cell with a pair of spin-up and spin-down atomssitting on the top of each other while keeping the rest of theatoms at fixed positions (tiny spheres). From the left to theright, the columns show the nodal surfaces of the wave func-tions corresponding to the free Fermi gas, the unitary limitand the BEC side of the crossover. The lower row displaysthe same surfaces rotated by 45 degrees around the z-axis.

perspective of a pair of up and down spin atoms. Thenodes on the BEC side (the right column) do not differmuch from the unitary limit, except for a slightly more

pronounced localization.

IV. CONCLUSIONS

We have carried out QMC calculations of the zero-temperature, spin-unpolarized atomic Fermi gas in theunitary limit. We show that the interaction range im-pacts the resulting total energies significantly. By ex-trapolating the interaction range to zero we obtain theratio E/Efree for 4, 14, 38 and 66 atoms to be 0.212(2),0.407(2), 0.409(3) and 0.398(3), respectively. Our resultsagree well with the previous fixed-node DMC calculationswhen we employ similar simulation parameters such asthe atom density, the interaction range and the numberof atoms. From the released-node DMC calculations for4 and 14 atoms we have found that the convergence tothe correct and asymptotically exact ground-state ener-gies is unfavorably slow compared to the growth of thestatistical noise. We were able to identify only small en-ergy gains within the simulation times that allowed foracceptable signal to noise ratio. We have calculated thetwo-body density matrix and the condensate fraction inthe limit of zero interaction range, and we have foundonly small changes in these quantities when comparedwith the previous calculations. Our condensate fractionfrom the fixed-node DMC simulations is 0.56(1).

ACKNOWLEDGMENTS

This work is supported by ARO and by the NSF grantsDMR-0804549 and OCI-0904794.

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