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Hybridization scheme for the quasiparticle spectrum of superfluid 4He
Physica B 263—264 (1999) 367—369
Hybridization scheme for the quasiparticle spectrum of superfluid He N. Gov*, E. Akkermans Department of Physics, Technion, 32000 Haifa, Israel
Abstract We present a new description of the quasiparticle spectrum of superfluid He based on the assumption that in addition to the density excitations proposed by Feynman, there exist localized modes that represent elementary vortex-cores. The energy spectrum which results from the hybridization of these two kinds of excitations is compared to the experimental data. A second branch of excitations interpreted as a quantized vortex-loop is obtained whose energy agrees with critical velocity experiments. 1999 Elsevier Science B.V. All rights reserved. Keywords: Superfluid Helium 4; Elementary excitations; Hybridization
In this letter, we present a new theoretical description of the quasiparticle spectrum of superfluid He. Better quantitative agreement with the experimental results was achieved by refinements of the wave function proposed by Feynman for a review see Ref. [1], using more complicated variational wave functions at the expense of a simple interpretation as a pure density fluctuation. We propose a different approach [2] by assuming the existence of two kinds of excitations in the system: Delocalized density fluctuations (Feynman) and localized excitations associated with exchange rings of atoms see for instance Refs. [3,11] (vortexcores). These two kinds of excitations are not independent but instead are hybridized in a way
* Corresponding author. Fax: #972-4-8221514; e-mail: [email protected].
reminiscent of the case of excitons in dielectric crystals [4]. To write down the Hamiltonian describing the interacting localized excitations, we follow the approach of Hopfield and Anderson [4,5] and assume that the role of the density fluctuations is to induce an effective dipolar interaction between the localized modes H " ( u #X(k))(bRb #) I I I # X(k)(bRbR #b b ), (1) I \I I \I I where X(k) is a (real and negative) matrix element which depends on the microscopic details of the dipolar interaction, u is the bare excitation en ergy of the localized-mode, and the operators b obey bosonic commutation relations. The coupling between the phonons and the localized modes
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 3 8 6 - 6
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N. Gov, E. Akkermans / Physica B 263—264 (1999) 367—369
is described by the Hamiltonian [4] H " (j(k, u )b #k(k, u )a )(aR#a )#CC, I I I \I I
(2)
where the two functions j and k are given by j(k)"i u (!3X(k)/2e(k)) and k(k)" ! u 3X(k)/2e(k). The energy e(k)" k/2mS(k), m is the mass of the helium atoms and S(k) the structure factor of the liquid, and the operators a obey bosonic commutation relations. H is diag onalized using the Bogoliubov transformation b "u(k)b #v(k)bR . which gives the spectrum I I \I E"( u ( u #2X(k)). (3) The two functions u(k) and v(k) are given by u(k)"1/2(( u #X(k)/E(k))#1) and v(k)"1/ 2(( u #X(k)/E(k))!1). The Hamiltonian H # H #H is quadratic and is diagonalized by means of the canonical transformation a "Aa # I I Bb #CaR #DbR and a "Ba #Ab # I \I \I I I I DaR #CbR . The corresponding dispersion rela\I \I tion is 6 e(k) X(k) "1! . (4)
u 1!(E(k)/ u ) E(k) We choose for the energy u the highest value of the phonon—roton spectrum, i.e. where it terminates, namely u "2D and the energy D cor responds to the roton minimum. To solve the dispersion relation and to obtain the phonon part of the spectrum, we use Eq. (3) into Eq. (4). This gives E(k)"(1/2)e(k) i.e. an expression independent of the matrix element X(k). Using now this latter expression into Eq. (4), we obtain the other branch describing the hybridized local modes at E" 2 u "4D. We compare our results for the energy spectrum with the experimental data. Using the experimental [6,7] structure factor S(k) we calculate the spectrum: E(k)" k/4mS(k) for the lower branch, which we compare with the experimental results [8,9] obtained at two different pressures (Fig. 1). The agreement is good between the maxon momentum and the termination point of the phonon—roton branch, where our approach be-
Fig. 1. Comparison between the experimental energy spectrum [9,8] (points) and the theoretical expression (solid line) where the structure factor S(k) is obtained from independent measurements [6,7]. (a) and (b) correspond, respectively, to the saturation vapor pressure and to P"24 atm. The dashed line at 4D indicates the position of the branch of the vortex-loop excitations.
comes meaningless, i.e. for kK2.5 As \, where the spectrum becomes that of a free particle. From the experimental dynamic structure factor the intensity of the single quasi-particle scattering intensity Z(k) is obtained. Since in our description the ground-state has a finite population of localized mode pairs, we expect Z(k) for the excitation of a single quasi-particle to be proportional to 1bRbR 2"u v , where the expectation value is calI \I I I culated in the ground state: Z(k)"pkI
u E(k) !1 , E(k) u
(5)
here I is a normalization constant. From the two independent measures of E(k) and S(k) we obtain
N. Gov, E. Akkermans / Physica B 263—264 (1999) 367—369
369
excitations: One is the delocalized density fluctuations (Feynman variational ansatz), and a second kind which is a localized excitation of microscopic size. We have assumed that these excitations are coupled in the same way as for excitons in dielectric systems. By writing a phenomenological Hamiltonian together with the dipolar approximation for the coupling, we obtained an excitation spectrum which involves two hybridized modes. One corresponds to the phonon—roton quasiparticle spectrum which is shifted by a factor of two towards lower energies relative to the Feynman result. The second is interpreted as vortex-loop modes. This picture describes quantitatively a large range of experimental results. Fig. 2. Comparison between the experimental scattering crosssection [8] Z(k) of single quasi-particle excitations (points) at 1.1 K and the theoretical expression (5). The two curves are obtained using, respectively, in Eq. (5) the experimental results for S(k) [6] (dashed line) and for the energy E(k) [9] (solid line).
using Eq. (5) a theoretical expression of the differential cross-section which fits well the experimental results as shown in Fig. 2. Moreover, in the low momentum limit, we obtain the proportionality between S(k) and Z(k). We found a second branch of excitations at the constant energy E"4D. It describes localized excitations of energies twice the bare vortex core energy so it is suggestive to interpret this mode as a single vortex loop. Experimental evidence in favor of this interpretation is provided by critical velocity experiments [10]. In phase-slippage studies of the critical velocity through an orifice, the critical velocity is driven by the thermal nucleation of vortexloops. The corresponding energy E is determined by the nucleation rate and is found to be E K 33$5 K, which is indeed very close to our result 4D"34.4 K. In conclusion, we have described the quasiparticle spectrum of superfluid He using two kinds of
Acknowledgements This work is supported in part by a grant from the Israel Academy of Sciences and by the fund for promotion of research at the Technion.
References [1] P. Nozieres, D. Pines, The Theory of Quantum Liquids, vol. II: Superfluid Bose Liquids, Addison-Wesley, Reading, MA, 1990. [2] N. Gov, E. Akkermans, cond-mat/9805372. [3] D.M. Ceperley, Rev. Mod. Phys. 67 (1995) 279. [4] J.J. Hopfield, Phys. Rev. 112 (1958) 1555. [5] P.W. Anderson, Concepts in Solids, Benjamin, New York, 1963. [6] E.C. Svensson, V.F. Sears, A.D.B. Woods, P. Martel, Phys. Rev. 21 (1980) 3638. [7] D.G. Henshaw, Phys. Rev. 119 (1960) 9. [8] A.D.B. Woods, R.A. Cowley, Rep. Prog. Phys. 36 (1973) 1135. [9] R.J. Donnelly, J.A. Donnelly, R.N. Hills, J. Low Temp. Phys. 44 (1981) 471. [10] E. Varoquaux, W. Zimmermann, O. Avenel, in: A.G.F. Wyatt, H.J. Lauter (Eds.), Excitations in Two-Dimensional and Three-Dimensional Quantum Fluids, Plenum Press, New York, 1991. [11] V.M. Gvozdikov, I.I. Fal’ko, Sov. Phys. J. 17 (1974) 802.
Hybridization scheme for the quasiparticle spectrum of superfluid He N. Gov*, E. Akkermans Department of Physics, Technion, 32000 Haifa, Israel
Abstract We present a new description of the quasiparticle spectrum of superfluid He based on the assumption that in addition to the density excitations proposed by Feynman, there exist localized modes that represent elementary vortex-cores. The energy spectrum which results from the hybridization of these two kinds of excitations is compared to the experimental data. A second branch of excitations interpreted as a quantized vortex-loop is obtained whose energy agrees with critical velocity experiments. 1999 Elsevier Science B.V. All rights reserved. Keywords: Superfluid Helium 4; Elementary excitations; Hybridization
In this letter, we present a new theoretical description of the quasiparticle spectrum of superfluid He. Better quantitative agreement with the experimental results was achieved by refinements of the wave function proposed by Feynman for a review see Ref. [1], using more complicated variational wave functions at the expense of a simple interpretation as a pure density fluctuation. We propose a different approach [2] by assuming the existence of two kinds of excitations in the system: Delocalized density fluctuations (Feynman) and localized excitations associated with exchange rings of atoms see for instance Refs. [3,11] (vortexcores). These two kinds of excitations are not independent but instead are hybridized in a way
* Corresponding author. Fax: #972-4-8221514; e-mail: [email protected].
reminiscent of the case of excitons in dielectric crystals [4]. To write down the Hamiltonian describing the interacting localized excitations, we follow the approach of Hopfield and Anderson [4,5] and assume that the role of the density fluctuations is to induce an effective dipolar interaction between the localized modes H " ( u #X(k))(bRb #) I I I # X(k)(bRbR #b b ), (1) I \I I \I I where X(k) is a (real and negative) matrix element which depends on the microscopic details of the dipolar interaction, u is the bare excitation en ergy of the localized-mode, and the operators b obey bosonic commutation relations. The coupling between the phonons and the localized modes
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 3 8 6 - 6
368
N. Gov, E. Akkermans / Physica B 263—264 (1999) 367—369
is described by the Hamiltonian [4] H " (j(k, u )b #k(k, u )a )(aR#a )#CC, I I I \I I
(2)
where the two functions j and k are given by j(k)"i u (!3X(k)/2e(k)) and k(k)" ! u 3X(k)/2e(k). The energy e(k)" k/2mS(k), m is the mass of the helium atoms and S(k) the structure factor of the liquid, and the operators a obey bosonic commutation relations. H is diag onalized using the Bogoliubov transformation b "u(k)b #v(k)bR . which gives the spectrum I I \I E"( u ( u #2X(k)). (3) The two functions u(k) and v(k) are given by u(k)"1/2(( u #X(k)/E(k))#1) and v(k)"1/ 2(( u #X(k)/E(k))!1). The Hamiltonian H # H #H is quadratic and is diagonalized by means of the canonical transformation a "Aa # I I Bb #CaR #DbR and a "Ba #Ab # I \I \I I I I DaR #CbR . The corresponding dispersion rela\I \I tion is 6 e(k) X(k) "1! . (4)
u 1!(E(k)/ u ) E(k) We choose for the energy u the highest value of the phonon—roton spectrum, i.e. where it terminates, namely u "2D and the energy D cor responds to the roton minimum. To solve the dispersion relation and to obtain the phonon part of the spectrum, we use Eq. (3) into Eq. (4). This gives E(k)"(1/2)e(k) i.e. an expression independent of the matrix element X(k). Using now this latter expression into Eq. (4), we obtain the other branch describing the hybridized local modes at E" 2 u "4D. We compare our results for the energy spectrum with the experimental data. Using the experimental [6,7] structure factor S(k) we calculate the spectrum: E(k)" k/4mS(k) for the lower branch, which we compare with the experimental results [8,9] obtained at two different pressures (Fig. 1). The agreement is good between the maxon momentum and the termination point of the phonon—roton branch, where our approach be-
Fig. 1. Comparison between the experimental energy spectrum [9,8] (points) and the theoretical expression (solid line) where the structure factor S(k) is obtained from independent measurements [6,7]. (a) and (b) correspond, respectively, to the saturation vapor pressure and to P"24 atm. The dashed line at 4D indicates the position of the branch of the vortex-loop excitations.
comes meaningless, i.e. for kK2.5 As \, where the spectrum becomes that of a free particle. From the experimental dynamic structure factor the intensity of the single quasi-particle scattering intensity Z(k) is obtained. Since in our description the ground-state has a finite population of localized mode pairs, we expect Z(k) for the excitation of a single quasi-particle to be proportional to 1bRbR 2"u v , where the expectation value is calI \I I I culated in the ground state: Z(k)"pkI
u E(k) !1 , E(k) u
(5)
here I is a normalization constant. From the two independent measures of E(k) and S(k) we obtain
N. Gov, E. Akkermans / Physica B 263—264 (1999) 367—369
369
excitations: One is the delocalized density fluctuations (Feynman variational ansatz), and a second kind which is a localized excitation of microscopic size. We have assumed that these excitations are coupled in the same way as for excitons in dielectric systems. By writing a phenomenological Hamiltonian together with the dipolar approximation for the coupling, we obtained an excitation spectrum which involves two hybridized modes. One corresponds to the phonon—roton quasiparticle spectrum which is shifted by a factor of two towards lower energies relative to the Feynman result. The second is interpreted as vortex-loop modes. This picture describes quantitatively a large range of experimental results. Fig. 2. Comparison between the experimental scattering crosssection [8] Z(k) of single quasi-particle excitations (points) at 1.1 K and the theoretical expression (5). The two curves are obtained using, respectively, in Eq. (5) the experimental results for S(k) [6] (dashed line) and for the energy E(k) [9] (solid line).
using Eq. (5) a theoretical expression of the differential cross-section which fits well the experimental results as shown in Fig. 2. Moreover, in the low momentum limit, we obtain the proportionality between S(k) and Z(k). We found a second branch of excitations at the constant energy E"4D. It describes localized excitations of energies twice the bare vortex core energy so it is suggestive to interpret this mode as a single vortex loop. Experimental evidence in favor of this interpretation is provided by critical velocity experiments [10]. In phase-slippage studies of the critical velocity through an orifice, the critical velocity is driven by the thermal nucleation of vortexloops. The corresponding energy E is determined by the nucleation rate and is found to be E K 33$5 K, which is indeed very close to our result 4D"34.4 K. In conclusion, we have described the quasiparticle spectrum of superfluid He using two kinds of
Acknowledgements This work is supported in part by a grant from the Israel Academy of Sciences and by the fund for promotion of research at the Technion.
References [1] P. Nozieres, D. Pines, The Theory of Quantum Liquids, vol. II: Superfluid Bose Liquids, Addison-Wesley, Reading, MA, 1990. [2] N. Gov, E. Akkermans, cond-mat/9805372. [3] D.M. Ceperley, Rev. Mod. Phys. 67 (1995) 279. [4] J.J. Hopfield, Phys. Rev. 112 (1958) 1555. [5] P.W. Anderson, Concepts in Solids, Benjamin, New York, 1963. [6] E.C. Svensson, V.F. Sears, A.D.B. Woods, P. Martel, Phys. Rev. 21 (1980) 3638. [7] D.G. Henshaw, Phys. Rev. 119 (1960) 9. [8] A.D.B. Woods, R.A. Cowley, Rep. Prog. Phys. 36 (1973) 1135. [9] R.J. Donnelly, J.A. Donnelly, R.N. Hills, J. Low Temp. Phys. 44 (1981) 471. [10] E. Varoquaux, W. Zimmermann, O. Avenel, in: A.G.F. Wyatt, H.J. Lauter (Eds.), Excitations in Two-Dimensional and Three-Dimensional Quantum Fluids, Plenum Press, New York, 1991. [11] V.M. Gvozdikov, I.I. Fal’ko, Sov. Phys. J. 17 (1974) 802.