a r X i v :c o n d -m a t /0410578v 1 [c o n d -m a t .s u p r -c o n ] 22 O c t 2004
Comparison between a diagrammatic theory for the BCS-BEC crossover and
Quantum Monte Carlo results
P.Pieri,L.Pisani,and G.C.Strinati
Dipartimento di Fisica,Universit`a di Camerino,I-62032Camerino,Italy
(Dated:February 2,2008)
Predictions for the chemical potential and the excitation gap recently obtained by our diagram-matic theory for the BCS-BEC crossover in the superfluid pha are compared with novel Quantum Monte Carlo results at zero temperature now available in the literature.A remarkable agreement is found between the results obtained by the two approaches.
PACS numbers:03.75.Ss,03.75.Hh,05.30.Jp
The recent experimental realization of the BCS-BEC crossover with ultracold trapped Fermi atoms 1ha
s given impetus to theoretical investigations of this crossover.In a recent paper 2,the t-matrix lf-energy approach (orig-inally conceived for the normal pha 3)was extended to the superfluid pha,aiming at improving the description of the BCS-BEC crossover by including pairing fluctua-tions on top of the BCS mean-field approach considered in Refs.4and 5.
In this theory,the effects of the collective Bogoliubov-Anderson mode is explicitly included in the fermionic lf-energy,thus generalizing the theory due to Popov for a weakly-interacting (dilute)superfluid Fermi gas 6.The theory is bad on a judicious choice of the fermionic lf-energy,such that it reproduces the fermionic mean-field BCS behavior plus pairing fluctuations in the weak-coupling limit as well as the Bogoliubov description for the composite bosons which form in the strong-coupling limit.In the intermediate-coupling region of interest about the unitarity limit,where no small parameter ex-ists to control the many-body approximations,the theory is able to capture the esntial physics of the problem,as the excellent agreement with a previously available QMC calculation 7at the unitarity point (k F a F )−1=0has al-ready shown 8,and as more extensively demonstrated by the prent comparison with more recent QMC data 9,10spanning the whole crossover region.The theory of Ref.2is completely ab initio and it contains no adjustable pa-rameter.Although the comparison with QMC data is here limited to the zero-temperature limit where t
hey are available,the predictions of the theory of Ref.2ex-tend as well to finite temperature and across the critical temperature.
Purpo of this Brief Report is to compare the the-oretical predictions obtained from the theory of Ref.2with novel Quantum Monte Carlo (QMC)data 9,10,which were published after completion of Ref.2.A quantitative comparison between the results for the density profiles obtained from a local density version 8to the theory of Ref.2and the experimental data was already prented in Ref.11.
Both our calculations and the QMC calculations of Refs.9and 10are bad on a model Hamiltonian de-scribing a system of fermions mutually interacting via
-1
-0.5
0.5
1
-4
-3
-2
-10
1
2
µ/εF f o r µ>0 a n d µ/ε0/2 f o r µ<0
(k F a F )
-1
t-matrix-I t-matrix-II
BCS FNQMC Galitskii 0.6 a F FIG.1:Chemical potential at zero temperature vs the cou-pling parameter (k F a F )−1.The results of the prent theory (t-matrix-I)and of its version without the inclusion of the lf-energy shift Σ0(t-matrix-II)are compared with the BCS mean field (BCS),the Fixe
d-node QMC data from Ref.10(FNQMC),the Galitskii’s expression for the dilute Fermi gas (Galitskii),and the asymptotic expression for strong coupling using the result a B =0.6a F .
an attractive contact potential.In Ref.12this Hamilto-nian was proved appropriate to describe the BCS-BEC crossover with trapped Fermi gas.In this model,the only dimensionless parameter reprenting the effective coupling strength is the (inver of the)product k F a F between the Fermi wave vector k F and the fermionic scattering length a F .For the homogeneous gas here con-sidered,k F =(3π2n )1/3where n is the particle density.Comparison will be made at zero temperature only,since finite-temperature QMC calculations for the BCS-BEC crossover are not yet available.
The overall agreement between the two alternative (di-agrammatic and QMC)calculations turns out to be quite good,expecially in the most interesting intermediate-coupling regime about (k F a F )−1=0.Figure 1shows the comparison for the chemical potential at zero tem-perature,as obtained by our calculation 2and by the Fixed Node QMC (FNQMC)calculations of Ref.10.As discusd in Ref.2,on the weak-coupling side we find
2
0.5
1
1.52
-1-0.500.51
∆m /εF
(k F a F )
-1
FIG.2:Excitation gap ∆m at zero temperature vs the cou-pling parameter (k F a F )−1.The results of the prent theory (t-matrix-I)are compared with the Green’s function QMC data of Refs.7and 9(GFQMC)as well as with the BCS mean field (BCS).
it appropriate to introduce a constant shift Σ0in the bare Green’s function entering the lf-energy.This shift needs to be included only for coupling values (k F a F )−1≤−0.5,such that the lf-energy can be considered to be approximatively constant.The curve obtained by this procedure is reported in Fig.1with the label t-matrix-I and corresponds to the data reported in Fig.6of Ref.2.For completeness,we also report in Fig.1the curve ob-tained without the inclusion of the lf-energy shift Σ0[with the label t-matrix-II](by definition,the two curves I and II coincide when (k F a F )−1≥−0.5).Our results are in excellent agreement with the FMQMC data in the
range −0.5<∼(k F a F )
−1<∼0.5spanning the crossover region.
For couplings (k F a F )−1≤−0.5,the FNQMC results are extremely clo to both our curves,lying just in be-tween them.In the weak-coupling region (k F a F )−1<∼−2,our curves (as well as the FNQMC data)approach the asymptotic expression by Galitskii 13for the chem-ical potential of a dilute Fermi g
as.The BCS mean field (also reported in Fig.1)miss instead the Gal-itskii correction to the non-interacting chemical poten-tial.More specifically,we have verified that our the-ory with the inclusion of the lf-energy shift Σ0[t-matrix-I]recovers the complete Galitskii’s expression
µ/ǫF =1+415π2(11−2ln 2)(k F a F )2
including the cond-order correction in k F a F .The curve for the chemical potential obtained without the inclusion of the shift Σ0recovers instead only the leading order correc-tion linear in k F a F .[It can also be shown that neglect-ing the shift Σ0introduces a spurious additional term 33πk F a F )2
to the cond-order correction in the Galit-skii’s expression.]
On the strong-coupling side,for coupling values (k F a F )−1>∼0.5our results deviate somewhat from the
FNQMC data.This discrepancy is due to the fact that
in our approach the boson-boson scattering is treated at the level of the Born approximation,corresponding to the value a B =2a F of the bosonic scattering length a B .The import
ance of including the correct value of the bosonic scattering length (a B =0.6a F ,as calculated in Ref.14)in this region is clearly en from the agreement between the FNQMC data and the asymptotic expression µ=−ǫ0/2+µB /2,where ǫ0is the binding energy of the 2-body problem and µB =4πn B a B /m B ,with n B =n/2,m B =2m ,and a B =0.6a F .The asymptotic curve cor-responding to the value a B =2a F almost coincides with our curve in this region.[This curve is not reported in Fig.1for overall clarity.]It is,finally,interesting to mention that the inclusion of the next-order correction to the bosonic chemical potential,corresponding to the
expression µB =4πn B a B 3(n B a 3B /π)]
1/3
obtained in Ref.15,would worn appreciably the comparison be-tween the QMC data and the asymptotic curve in the
coupling region 0.2<∼(k F a F )−1<∼2.The inclusion of
this next-order term improves the comparison only in the truly asymptotic regime for (k F a F )−1>∼2(not re-ported in the figure),where the next-order correction to the bosonic chemical potential i
s,however,already quite small.This finding could (at least partially)explain the abnce of beyond-mean-field corrections on the bosonic side of the BCS-BEC crossover,recently reported in ex-periments with ultracold Fermi gas 16.
Quite generally,any theory of the BCS-BEC crossover connects the equation for the chemical potential µto the equation for the gap (order)parameter ∆in the super-fluid pha.The latter quantity is not directly accessible to the QMC simulations of Refs.9and 10.In Ref.9,however,the even-odd staggering of the ground-state en-ergy for a system with a finite number of particles was exploited to calculate the single-particle excitation gap ∆m .In a BCS-like framework (and for a sufficiently large number of particles)the gap ∆m is expected to coincide with the gap (order)parameter ∆when µis positive and with the quantity (∆2+µ2)1/2when µis negative.For a given coupling,this gap occurs at the wave vector |k |=
√
3
ure(full square).Even for the excitation gap,our results appear to be in remarkable agreement with QMC data
in the crossover region−1<∼(k F a F)−1<∼0.4.At larger couplings,the QMC results start instead to deviate from
our results,the discrepancy being mainly due to thefi-nite range of the interaction potential ud in the QMC calculations.In strong coupling,both our excitation gap and that calculated from QMC simulations tend,in fact, to half the value of the binding energyǫ0of the two-body problem.The binding energies for the contact potential and for thefinite-range potential ud in Ref.9are clo to each other only in a narrow range about(k F a F)−1=0. At the coupling value(k F a F)−1=1,the binding energy for thefinite-range potential of Ref.9is already larger by about40%than the contact-potential binding energy. This difference is responsible for the discrepancy between our values and the QMC data of Ref.9on the strong-coupling side,where the excitation gap is controlled by the binding energy of the two-body problem.
In conclusion,the theory of Ref.2for the BCS-BEC crossover in the broken-symmetry pha has been shown to compare extremely well with recent QMC data at zero temperature,especially in the intermediate-coupling (crossover)region which is the most interesting one both theoretically and experimentally.This agreement sug-gests that the choice of the fermionic lf-energy made in Ref.2captures the esntial physics of the problem,as soon as the fermionic degrees of freedom get
progressively quenched while forming composite bosons.
Acknowledgments
We thank S.Giorgini for discussions and for providing us with the datafile of Fig.2of Ref.10.
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