The work by T. K. Ghosh and K. Machida [cond-mat/0510160 and Phys. Rev. A 73,
013613 (2006)] on the sound velocity in a cylindrically confined Fermi
superfluid obeying a power-law equation of state is shown to make use of an
improper projection of the sound wave equation. This inaccuracy fully accounts
for the difference between their results and those previously reported by
Capuzzi et al. [cond-mat/0509323 and Phys. Rev. A 73, 021603(R) (2006)]. In
this Comment we show that both approaches lead exactly to the same result when
the correct weight function is used in the projection. Plots of the correct
behavior of the phonon and monopole-mode spectra in the BCS, unitary, and BEC
limits are also shown.Comment: Comment on cond-mat/051016

F. Federici

M. Abramowitz

M. P. Tosi

P. Capuzzi

P. M. Morse

P. Vignolo

Physical Review A

English

arXiv

Fil: Capuzzi, Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Vignolo, P.. No especifica;Fil: Federici, F.. No especifica;Fil: Tosi, M. P.. No especifica

Capuzzi, Pablo

Vignolo, P.

Federici, F.

Tosi, M. P.

CONICET Digital

Comment on “Sound velocity and multibranch Bogoliubov spectrum of an elongated Fermi superfluid in the BEC-BCS crossover”

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Comment on “Sound velocity and multibranch Bogoliubov spectrumof an elongated Fermi superfluid in the BEC-BCS crossover”P. Capuzzi,* P. Vignolo,† F. Federici,‡ and M. P. Tosi§NEST-CNR-INFM, Scuola Normale Superiore, I-56126 Pisa, ItalyReceived 20 February 2006; published 9 November 2006The work by T. K. Ghosh and K. Machida Phys. Rev. A 73, 013613 2006 and cond-mat/0510160 on thesound velocity in a cylindrically confined Fermi superfluid obeying a power-law equation of state is shown tomake use of an improper projection of the sound wave equation. This inaccuracy fully accounts for thedifference between their results and those previously reported by Capuzzi et al. Phys. Rev. A 73, 021603R2006 and cond-mat/0509323. In this Comment, we show that both approaches lead exactly to the same resultwhen the correct weight function is used in the projection. Plots of the correct behavior of the phonon andmonopole-mode spectra in the BCS and unitary limits and in the BEC regime are also shown.DOI: 10.1103/PhysRevA.74.057601 PACS numbers: 03.75.Kk, 03.75.Ss, 47.37.qIn their recent study on sound propagation in an elongatedFermi superfluid in the BEC-BCS crossover, Ghosh andMachida 1 have reported a calculation of the sound veloc-ity u1 for the case of a power-law equation of state EOS aspreviously analyzed for bosons by Zaremba 2 and for fer-mions by Capuzzi et al. 3. Their result is found to be in-correct due to the use of an improper projection of the soundwave equation.The eigenvalue equation for small-amplitude densitymodes nqreiqz is obtained by linearization of the hydro-dynamic equations around equilibrium and readsM2nq = q2n0/nn=n0nq−  · n0/nn=n0nq , 1where n0 is the equilibrium density profile,  the frequencyof the perturbation, and q its wave vector along z. Equation1 reduces to Eq. 8 in Ref. 1 for a power-law EOSn=Cn.To obtain the dispersion relation q for any value of q,one must resort to the numerical solution of the eigenvalueequation 1. A possible method to solve such an equationconsists of expanding the eigenmodes nq in a complete setof basis functions see, e.g., Zaremba 2 asnr = bnr , 2where = nr ,m labels the basis functions, with nr the radialnumber and m the number for the azimuthal angular momen-tum. By inserting this expression into Eq. 1 and projectingthe result onto an element of the basis, a matrix representa-tion of the eigenvalue equation is found, which is suitable fora numerical solution. This procedure allows some freedom inthe choice of the basis and of the projection, as long as thesesatisfy the boundary conditions. However, in order to obtaina standard eigenvalue equation of the formv = A · v 3one must choose a projection in which the basis is orthogo-nal 4. For the basis functions adopted in Ref. 1, cf. Eqs.10 and 11, the projection must be performed with aweight function wr 1− r̃2−0 5, where r̃=r /R and0=1/−1, R being the radius of the density profile. Theorthogonality condition thus reads wr̃n*r̃nr̃d2r̃  . 4Therefore, for Eq. 13 in Ref. 1 to be correct, the integralsdefining the matrix M must include the weight functionwr and thus readM = A2 d2r̃1 − r̃20r̃2+m+meim−m	Pnr0,m2r̃2 − 1Pnr0,m2r̃2 − 1 , 5where Pnr0,m are Jacobi polynomials. Equation 5 is whatshould have been used in Ref. 1, instead of Eq. 14, wherethe weight function wr is missing and the constant A takesa different value since nr has not been normalized withwr. A Fermi superfluid with =1 corresponds to a Bose-Einstein condensate BEC of molecules and has been previ-ously analyzed by Zaremba 2 for bosonic atoms. In thiscase wr=1, and Eq. 5 reduces to Eq. 14 in 1.To further analyze how the correct orthogonality condi-tion affects the results, we have numerically solved the ei-genvalue equation for sound propagation in a superfluidFermi gas in the BCS and unitary limits, and in the BECregime with y=1/kFa=0.25. Our results for the two lowestfrequency modes as functions of q are shown in Figs. 1 and2. The lowest mode, shown in Fig. 1, is soundlike and has aphononic dispersion relation at long wavelengths. We ob-serve that the slope of the dispersion relation, i.e., the soundvelocity, is lower than that found in Ref. 1 and bends downas q increases. The first excited state, displayed in Fig. 2,*Electronic address: capuzzi@sns.it†Electronic address: vignolo@sns.it‡Electronic address: fr.federici@sns.it§Electronic address: tosim@sns.itPHYSICAL REVIEW A 74, 057601 20061050-2947/2006/745/0576012 ©2006 The American Physical Society057601-1corresponds to a monopolar compressional mode that for q=0 is purely radial. Furthermore, its frequency is known ana-lytically at q=0 6 as 0=10/3 in the BCS and unitarylimits and as 0=2 in the BEC limit y→ +. Figure 2in the present work is to be compared with Fig. 3 in Ref. 1,where the monopolar mode is not correctly depicted. We alsonote that the effective mass associated with this mode is alsodifferent from that found by Ghosh and Machida.To obtain an analytical expression for the sound velocityu1	dq /dqq=0, one can use Eq. 13 in Ref. 1 for thelowest-frequency mode and expand it to first order in q2.This demonstrates that the off-diagonal terms in M do notenter the calculation of the sound velocity, as pointed out byZaremba 2 for bosons. From the definition 5, one obtainsM00= / +1 and thusu1 = 2 + 2vF, 6with vF=2̄ /M and ̄ the chemical potential, in agreementwith our result previously obtained in 3. This is the soundvelocity that also Ghosh and Machida should have obtainedif they had taken into account the weight function wr. Theirimproper projection of the eigenvalue equation leads them tothe pathological expression u1=2− /4vF, which pre-dicts no sound propagation for 2. The same problem af-fects the calculation of the effective mass mb for the mono-pole mode cf. Eq. 27 in Ref. 1, which once corrected ismb =M22 + 23/21 + 31 +  + 22. 7An alternative and more direct procedure to evaluate thesound velocity is the one that we have outlined in 3. Thespatial dependence is eliminated from Eq. 1 above by inte-grating in the x ,y plane. This yields the dispersion relationq = q 1M n0/nn=n0nqd2r nqd2r1/28for a perturbation with nqd2r0. Hence, the calculationof the sound velocity requires only the expression of nqcalculated at q=0. By using nq=0=  /nn=n0−1, we ob-tainu1 =  1M  n0d2r /nn=n0−1d2r1/2. 9This expression provides the exact velocity of sound propa-gation in cylindrically confined hydrodynamic gases withEOS n, and for nn leads to Eq. 6.1 T. K. Ghosh and K. Machida, Phys. Rev. A 73, 013613 2006.2 E. Zaremba, Phys. Rev. A 57, 518 1998.3 P. Capuzzi, P. Vignolo, F. Federici, and M. P. Tosi, Phys. Rev.A 73, 021603R 2006.4 P. M. Morse and H. Feshbach, Methods of Theoretical PhysicsMcGraw-Hill, New York, 1953, Vol. 1, Chap. 6.5 M. Abramowitz and I. A. Stegun, Handbook of MathematicalFunctions with Formulas, Graphs and Mathematical TablesDover, New York, 1965, Chap. 22.6 H. Heiselberg, Phys. Rev. Lett. 93, 040402 2004.FIG. 2. The same as in Fig. 1 for the monopole mode.FIG. 1. Dispersion relation q in units of  for the lowest-frequency sound mode as a function of qR with R the radius of thefermion density profile in the BCS limit. The dash-dotted, dashed,and solid lines correspond to fermions in the BCS and unitary lim-its, and in the BEC regime with y=0.25, respectively.COMMENTS PHYSICAL REVIEW A 74, 057601 2006057601-2

Comment on ``Sound velocity and multibranch Bogoliubov spectrum of an
  elongated Fermi superfluid in the BEC-BCS crossover"

Comment on ``Sound velocity and multibranch Bogoliubov spectrum of an elongated Fermi superfluid in the BEC-BCS crossover"

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