Dynamic many-body theory: Dynamic structure factor of two-dimensional liquid 4 He
aa r X i v : . [ c ond - m a t . o t h e r] J un Dynamic many-body theory:Dynamic structure factor of two-dimensional liquid He E. Krotscheck , and T. Lichtenegger , Department of Physics, University at Buffalo, SUNY Buffalo NY 14260 and Institut f¨ur Theoretische Physik, Johannes Kepler Universit¨at, A 4040 Linz, Austria (Dated: September 16, 2018)We calculate the dynamic structure function of two-dimensional liquid He at zero temperature employing aquantitative multi-particle fluctuations approach up to infinite order. We observe a behavior that is qualitativelysimilar to the phonon-maxon-roton-curve in 3D, including a Pitaevskii plateau (L. P. Pitaevskii, Sov. Phys.JETP , 830 (1959)). Slightly below the liquid-solid phase transition, a second weak roton-like excitationevolves below the plateau. PACS numbers: 67.30.em, 67.30.H, 67.10.-j
INTRODUCTION
The static and dynamic structure of few-layer films of liquid helium absorbed on solid substrates at low temperatures has beenstudied experimentally, e.g. within neutron scattering measurements [1–4], and theoretically [5–12]. The earliest investigationsof excitations [5–8, 11] were based on generalizations of Feynman’s theory of excitations in the bulk liquid [13] and thereforeonly qualitative. Later work [10, 12, 14] employed correlated basis functions (CBF) theory [15–17]. These methods are simpleenough for the application to non-uniform geometries including the inhomogeneity of the substrates and the non-trivial densityprofile of the films. Agreement with measurements of the dynamic structure was either semi-quantitative, or required somephenomenological input for a quantitative description of the various excitation types seen in the experiments [18] such as “layer-phonons”, “layer-rotons”, or “ripplons”.Since then the development of theoretical tools for describing the dynamics of bulk quantum liquids has made significantprogress, providing a quantitative description in the experimentally accessible density range for low and intermediate momenta,probing the short-range structure of the system [19–22]. Due to the increasingly complicated form of more elaborate methods,application to inhomogeneous geometries is less straightforward. Building on the success of our method for both bulk He [23]and He [21, 22, 24], we here investigate mono-layer films of He which can be treated as strictly two-dimensional liquids.Recently, novel numerical methods [25–27] have appeared that give access to dynamic properties of quantum fluids. Theseare algorithmically very important developments that will ultimately aide in the demanding elimination of background andmultiple-scattering events from the raw data. However, it is generally agreed upon that the model of static pair potentials like theAziz interaction describes the helium liquids accurately. Hence, given sufficiently elaborate algorithms and sufficient computingpower, such calculations must reproduce the experimental data. The aim of our work is somewhat different: The identification ofphysical effects like phonon-phonon, phonon-roton, roton-roton, maxon-roton . . . couplings that lead to observable features inthe dynamic structure function is, from simulation data, only possible a-posteriori whereas the semi-analytic methods pursuedhere permit a direct identification of these effects, their physical mechanisms, and their relationship to the ground state structuredirectly from the theory.
THEORETICAL FRAMEWORK
The behavior of N identical, non-relativistic particles in an external field U ext ( r ) , interacting via a pair potential V int ( r , r ′ ) , isgoverned by a microscopic Hamiltonian H = − N (cid:229) i = ¯ h m (cid:209) i + N (cid:229) i = U ext ( r i ) + N (cid:229) i , j = i < j V int ( r i , r j ) . (1)The ground state is written in the Feenberg form [28] | Y i = e U | f i , (2)where | f i is a non- or weakly-interacting model wave function containing the appropriate symmetry and statistics of the system,and U ( { r k } ) = N (cid:229) i = u ( r i ) + N (cid:229) i , j = i < j u ( r i , r j ) + N (cid:229) i , j , k = i < j < k u ( r i , r j , r k ) + . . . (3)is the correlation operator consisting of n -particle correlation functions u n .For homogeneous Bose systems such as three- and two-dimensional He, | f i can be chosen to be 1 and the wave function(2,3) is in principle exact. The empirical Aziz potential [29] as interaction between the helium atoms has turned out to leadto results in quantitative agreement with experiments, see Ref. 30 for a review. With minimal phenomenological input, thesame accuracy can be obtained with integral equation methods [31, 32]. In that case, the correlation functions are optimized byminimizing the ground state energy E , viz. d E d u n = dd u n h Y | H | Y ih Y | Y i = . (4)Dynamics is treated along basically the same lines. In the presence of a time-dependent external perturbation d H ( { r k } ; t ) = (cid:229) i d U ext ( r i ; t ) , (5)the time-dependent generalization of the ground state wave function (2) is | Y ( t ) i = e − i E t / ¯ h e d U ( t ) | Y i (cid:2)(cid:10) Y (cid:12)(cid:12) e R e d U ( t ) (cid:12)(cid:12) Y (cid:11)(cid:3) / , (6)where d U ( { r k } ; t ) = (cid:229) i d u ( r i ; t ) + (cid:229) i < j d u ( r i , r j ; t ) + · · · (7)is the complex excitation operator . Its components, the fluctuations d u n ( r , . . . , r n ; t ) of the correlation functions, are determinedby the least action principle [33, 34] d Z dt (cid:28) Y ( t ) (cid:12)(cid:12)(cid:12)(cid:12) H + d H ( t ) − i¯ h ¶¶ t (cid:12)(cid:12)(cid:12)(cid:12) Y ( t ) (cid:29) = , (8)which generalizes the Euler-Lagrange Eq. (4) to the time-dependent case. MULTI-PARTICLE FLUCTUATIONS AND DENSITY-DENSITY RESPONSE
For weak external perturbations, the relationship between the perturbing external field and the induced density fluctuation dr ( r ; t ) = Z d r ′ dt ′ r ( r ) c ( r , r ′ ; t , t ′ ) r ( r ′ ) d U ext ( r ′ ; t ′ ) + O (cid:0) d U (cid:1) (9)is linear and defines the density-density response function c ( r , r ′ ; t , t ′ ) . In homogeneous, isotropic geometries where the groundstate density r ( r ) = r is constant, the density-density response function is most conveniently formulated in momentum andenergy space and defines the dynamic structure function S ( k , ¯ h w ) = − p I m c ( k , ¯ h w ) , (10)spelled out here for zero temperature and consequently ¯ h w > d U F ( t ) = (cid:229) i d u ( r i ; t ) , (11)for the fluctuations leads to the time-honored Feynman dispersion relation [13] e F ( k ) ≡ ¯ h k mS ( k ) . (12)Here, S ( k ) is the static structure function which can be obtained from experiments or ground state calculations. In this approxi-mation, S ( k , ¯ h w ) is described by a single mode located at the Feynman spectrum.The importance of including at least two-particle fluctuations d u ( r , r ′ ; t ) was first pointed out by Feynman and Cohen in theirseminal work on “backflow” correlations [35]. A somewhat more formal approach was taken by Feenberg and collaboratorswho derived a Brillouin-Wigner perturbation theory in a basis of correlated wave functions [15–17, 36, 37]. These approachesdetermine, rigorously speaking, only the energy of the lowest-lying mode. The equations of motion method (8) employed hereprovides access to the full density-density response function c ( k , ¯ h w ) = S ( k ) ¯ h w − S ( k , ¯ h w ) + i h + S ( k ) − ¯ h w − S ∗ ( k , − ¯ h w ) + i h , (13)where S ( k , ¯ h w ) is the phonon self-energy. In practically all applications, the excitation operator (7) has been truncated at thetwo-body level and the convolution approximation was used [31, 38] which is simple enough to be employed in non-uniformgeometries [12, 18]. Then, the self-energy has the form S ( k , ¯ h w ) = e F ( k ) + Z d d p d d p ( p ) d r d ( k + p + p ) | V ( ) ( k ; p , p ) | ¯ h w − e F ( p ) − e F ( p ) + i h , (14)where d is the dimension of the system. The three-body vertex V ( ) ( k ; p , p ) describes the decay of a density fluctuation withwave vector k into two waves with wave vectors p and p . It can be calculated in terms of ground state quantities, its generalform is [39] V ( ) ( k ; p , p ) = ¯ h m s S ( p ) S ( p ) S ( k ) (cid:2) k · p ˜ X ( p ) + k · p ˜ X ( p ) − k ˜ X ( k , p , p ) (cid:3) (15)where ˜ X ( p ) = − / S ( p ) is the “direct correlation function” and ˜ X ( k , p , p ) is the irreducible part of the three-body distributionfunction, see appendix .The lowest excitation branch is obtained by solving e ( k ) = R e S ( k , e ( k )) . (16)When consistent approximations are used, the solution of Eq. (16) is identical to what was obtained by CBF perturbation theory.Eqs. (13), (14) and (15) give the correct physics up to and somewhat beyond the roton minimum, the solution of Eqs. (14),(16) bridges about about 80 percent of the discrepancy between the Feynman spectrum e F ( k ) and the experiment. The mostprominent shortcoming of the approximation is that it misses the energy of the plateau. The reason for this shortcoming is thatthe energy denominator in the self-energy (14) contains the Feynman energies.There are several ways to improve upon this: Brillouin-Wigner perturbation theory has been worked out by Lee and Lee [37]up to fourth order from which the general scheme can be seen. Some low-order processes contributing to the self-consistentself-energy are shown in Fig. 1.Unfortunately that work did not utilize the fact that the ground state should be optimized and, therefore, obtained also spuriousdiagrams. The most complete derivation within the equations of motion scheme includes time-dependent triplet correlations[19, 20]. The theory reproduces, for the lowest mode, the first diagrams of CBF perturbation theory. The expected result is thatthe self-energy in Eq. (14) should be replaced by the self-consistent form e F ( p ) + e F ( p ) = ⇒ S ( p , ¯ h w − e F ( p )) + S ( p , ¯ h w − e F ( p )) , (17)which leads to quantitative agreement between the theoretical excitation spectrum and the experimental phonon-roton spectrum.It still contains the Feynman energy as argument which should also be calculated self-consistently. We have simplified this partof the calculation by using the calculated phonon-roton spectrum in the energy arguments of the self-energy. This provides aslight improvement of the description in the momentum regime of the plateau. (a) (b) (c) (d) FIG. 1. Leading-order Feynman diagrams for the dynamic response function. (a) represents a single Feynman density wave, (b) shows thesplitting into and recombination of two intermediate waves as described by pair fluctuations, whereas (c) and (d) are three-phonon excitationsindicating the beginning of the self-consistent summation of (b). Processes of more complicated structure like one-to-three transitions havebeen neglected in the present calculation.
DYNAMIC STRUCTURE FUNCTION OF TWO-DIMENSIONAL HE The only quantity needed for the calculation of the self-energy is the ground state distribution function g ( r ) and/or the staticstructure function S ( k ) . These quantities have been calculated in the past and are available in pedagogical and review-typeliterature, see Refs. 30 and 40. We have here used the HNC-EL method including four and five-body elementary diagrams andtriplet correlation functions as described in Ref. 32.We have calculated the dynamic structure function in the regime between the equilibrium density of the system of r = .
042 ˚A − and the solidification density of r = .
064 ˚A − [30] in steps of D r = .
002 ˚A − . Compared to earlier work we haveused an improved method for calculating the three-body vertex V ( ) ( k ; p , p ) as described in appendix . This leads to a slightlowering of the roton minimum by about 0 . . . . . D , roton wave number k D and roton ”effective mass” m , e ( k ) = D + ¯ h m ( k − k D ) . (18)Fig. 4 shows the density dependence of the roton energy and wave number. For that purpose, we have fitted the spectra in aregime of k = k D ± .
15 ˚A − by the form Eq. (18). The values of D and k D are somewhat sensitive to the choice of the momentumrange used for the the fit, especially at the lower densities where the roton minimum is not very pronounced. Because of this werefrain from showing a comparison with the simulation data in Fig. 4, Figs. 2 contain the same information but include error barsand are more informative. These pieces of information are the standard quantities that characterize the phonon-roton spectrum.Let us now focus on those features of the dynamic structure function where the 2D case differs visibly from the 3D system: • It was already noted in Ref. 30 that the speed of sound is low compared to the same quantity in 3D. The consequence is astrong anomalous dispersion which has, in turn, the consequence that long wavelength phonons can decay up to a densityof about 0 .
050 ˚A − . • Similar to the 3D case we notice at low to moderate densities a feature which was tentatively called “ghost phonon”[20]. In contrast to the 3D system, where the ghost phonon disappears rapidly with increasing density, the feature is verypronounced even at a density of r = .
054 ˚A − . • At very high densities, slightly below the liquid-solid phase transition, we see a mode that is clearly separated from theplateau. The plateau itself is a threshold above which an induced density fluctuation of wave vector k and frequency w can decay, under energy and momentum conservation, into two rotons. This condition can be satisfied for all momenta −1 ) 0 2 4 6 8 10 12 − h w ( K ) − h w ( m e V ) r =0.044 Å −2 FeynmanCBFEOMArrigoni et al. 0.00.20.40.60.81.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5k (Å −1 ) 0 2 4 6 8 10 12 − h w ( K ) − h w ( m e V ) r =0.054 Å −2 FeynmanCBFEOMArrigoni et al. 0.00.20.40.60.81.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5k (Å −1 ) 0 2 4 6 8 10 12 − h w ( K ) − h w ( m e V ) r =0.062 Å −2 FeynmanCBFEOM 0.00.20.40.60.81.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5k (Å −1 ) 0 2 4 6 8 10 12 − h w ( K ) − h w ( m e V ) r =0.064 Å −2 FeynmanCBFEOMArrigoni et al. 0.00.20.40.60.81.0
FIG. 2. (Color online) The figure shows contour plots of the dynamic structure function for a sequence of densities as shown in the legends.The colors have been chosen to highlight the prominent features, darker colors correspond to higher values of S ( k , ¯ h w ) . The most strikingobservations are the appearance of a “ghost phonon” at low densities, and the presence of a secondary roton at high densities. For comparisonwe also show the Feynman spectrum, the spectrum obtained within CBF-BW perturbation theory, and the simulation data of Ref. 41. below twice the roton momentum. At high densities a signature of the resulting discontinuity in the imaginary part of theself-energy is visible not only beyond but also in the roton and even maxon regions.We have noted above that anomalous dispersion persists well beyond equilibrium density. This leads to the damping of long-wavelength phonons. Figs. 5 show cuts of S ( k , ¯ h w ) at long wavelengths. At the first glance, it appears that the phonon broadensat a wave number of k ≈ .
38 ˚A − . Closer inspection reveals, however, that a second, broad feature splits off the phonon andbecomes an isolated feature above k ≈ . − . Eventually the feature dissolves around k ≈ . − . The effect is also seenquite clearly in the two contour plots corresponding to the densities 0 .
044 ˚A − and 0 .
054 ˚A − shown in Figs. 2. On the otherhand, the broadening that should occur, due to anomalous dispersion, up to wave numbers of about 0.4 ˚A − , is hardly visible.The feature can be explained by examining the analytic structure of the self-energy in 2D. Specifically, we will show inappendix that the imaginary part of the self-energy has, in the limit ¯ h w → e ( k / ) , a discontinuity of the form I m S ( k , ¯ h w → e ( k / )) ∼− s k e ′ ( k / ) e ′′ ( k / ) q (cid:16) sign (cid:0) e ′′ ( k / ) (cid:1)(cid:0) ¯ h w − e ( k / ) (cid:1)(cid:17) , (19)which implies a logarithmic singularity of R e S ( k , ¯ h w → e ( k / )) . Eq. (19) is normally derived for the purpose of estimatingthe lifetime of phonons in the regime of anomalous dispersion [42]. However, it is also valid for normal dispersion e ′′ ( k / ) < | k e ′′ ( k / ) | ≪ e ′ ( k / ) , i.e. one should see the signature of the step function of the imaginary part of the self–energyup to about twice the wave number for which the dispersion relation e ( k ) is, to a good approximation, linear. This is exactlythe regime where the ghost phonon is seen in Figs. 2. We also note that the effect is stronger in 2D than in 3D becausethere the logarithmic singularity ln ( e ( k / ) − ¯ h w ) in the real part of the self-energy giving rise to Eq. (19) is replaced by e ( k ) ( K ) k (Å −1 ) r = 0.064 Å −2 r = 0.044 Å −2 FIG. 3. The figure shows the phonon-roton spectrum for the densities r = .
044 ˚A − , 0 .
048 ˚A − ,. . . 0 .
064 ˚A − , the curve with the lowest rotonand the highest maxon corresponds to the highest density. Note that the dispersion for r = .
044 ˚A − is anomalous, we have in this case drawnthe peak of S ( k , ¯ h w ) . For a comparison with available simulation data, see Figs. 2. D ( K ) k D ( Å − ) r (Å −2 ) D(r) (K)k D (r) (Å −1) FIG. 4. The figure shows the roton energy D (left scale) and the roton wave number k D (right scale) as a function of density in the densityregime 0 .
050 ˚A − ≤ r ≤ .
064 ˚A − . For a comparison with available simulation data, see Figs. 2. k ( Å − ) S ( k , − h w ) ( K − ) , o ff s e t − h w (K) r =0.044Å −2 k ( Å − ) S ( k , − h w ) ( K − ) , o ff s e t − h w (K) r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 r =0.054Å −2 FIG. 5. The figures show cuts of S ( k , ¯ h w ) at long wavelengths at the densities r = .
044 ˚A − (left pane) and r = .
054 ˚A − (right pane). Thelong dashed line is the phonon dispersion relation, and the short-dashed line is the curve 2 e ( k / ) . At higher density, the phonon becomessharper but the ghost phonon is still visible. k ( Å − ) S ( k , − h w ) ( K − ) , o ff s e t − h w (K) r =0.064Å −2 FIG. 6. The figure shows cuts of S ( k , ¯ h w ) in the regime of the Pitaevskii plateau for the density r = .
064 ˚A − for a sequence of momenta2 . − ≤ k ≤ . − . p e ( k / ) − ¯ h w [42].A second striking feature is the appearance of a sharp mode below the plateau. We stress the difference: normally, the plateauis a threshold above which a wave of energy/momentum ( ¯ h w , k ) can decay into two rotons. This has the consequence that theimaginary part of the self-energy S ( k , ¯ h w ) is a step function and the real part has a logarithmic singularity [43]. A collectivemode is, on the other hand, characterized by a singularity of the S ( k , ¯ h w ) . Figs. 2 show, for the two highest densities, theappearance of a sharp discrete mode below the plateau. A close-up of the situation is shown in Fig. 6: Clearly the plateau startsat the same energy for all momenta. At a wave number of k ≈ . − , the collective mode is still merged into the continuum.With increasing wave number, we see, however, a clearly distinguishable mode about 0.3 K below the plateau. DISCUSSION
We have already made the essential points of our findings in the discussion of our results. Evidently, the difference betweentwo and three dimensions has quite visible effects on S ( k , ¯ h w ) , as mentioned above.Our findings about a secondary roton should shed some light on the discussion of the nature of the roton minimum. It hasbeen argued [44] that the roton is the “ghost of a vanished vortex ring” or [45, 46] “ghost of a Bragg Spot” due to the imminentliquid-solid phase transition. In this density region two-dimensional He already shows a strong signature of the triangular latticeinto which it eventually freezes [47–49] and can exhibit a so called hexatic phase [50].If the Bragg spot interpretation of the roton is correct, one should perhaps expect a second one and the ratio of the absolutevalues of the corresponding wave vectors should roughly satisfy k / k = ( p / ) ≈ .
73 because of the triangular lattice ofthe solid phase.Our results may indeed be interpreted as an indication that this is the case. It is certainly worth investigating this issue furtheralong the line of angular-dependent excitations [51]. A similar effect has been seen in cold dipolar gases [52] and the relationshipis worth examining.Finally a word about the comparison with simulation data [41]. Overall, the agreement appears satisfactory, most of ourresults are within the error bars of that calculation. The most visible discrepancy is seen at the highest density of r = .
064 ˚A − .At this density, the maxon energy is below twice the roton energy and the modes in this ( ¯ h w , k ) region can decay. One wouldexpect more strength at the decay threshold of 2 D as shown by our results whereas the Monte Carlo data indicate – despite largeerror bars – that the decay strength lies at higher energies. This point deserves further investigation, it might also explain whythe maxon energies at r = .
054 ˚A − differ more than expected. Otherwise the agreement is quite good, evidently the strengthshown at and above k = − follows indeed the kinetic energy branch in both calculations, whereas the plateau region hasrelatively little strength. Long-wavelength dispersion in 2D
In this appendix, we study the analytic structure of the self-energy as a function of an external energy ¯ h w in the limit¯ h w − e ( k / ) →
0. We assume that the solution e ( k ) of the implicit equation (16) has a negligible imaginary part.We look for processes where a state of wave vector k decays into two phonons of wave vectors p and p . In general oneexpects, for long wavelengths, a phonon dispersion relation of the form e ( k ) = ¯ hck + c k (20)where c is the speed of sound. In fact, it is easily shown that Eq. (16) leads to such a dispersion relation.The calculation is best carried out in relative and center of mass momenta, i.e. we set p = q − k p = − q − k . Then, it is clear that e ( | k / + q | ) + e ( | k / − q | ) (21)has, for all angles cos q ≡ x ≡ ˆ q · ˆ k , a relative extremum at q =
0. Expanding the energy denominator as e ( p ) + e ( p ) = e ( k / ) + (cid:20) e ′ ( k / ) k ( − x ) + e ′′ ( k / ) x (cid:21) q + O ( q ) , (22)we see that the the value 2 e ( k / ) is, at x =
1, a relative minimum if c > c < e ′ ≡ e ′ ( k / ) , e ′′ ≡ e ′′ ( k / ) . We do the calculation first for the case of anomalous dispersion. The three-body coupling matrix element assumes a finitevalue as q →
0, we therefore need to include only the leading term V (cid:18) k ; − k + q , − k − q (cid:19) ≈ V (cid:18) k ; − k , − k (cid:19) Then I m S ( k , ¯ h w ) ≈ (cid:12)(cid:12)(cid:12) V ( ) ( k ; − k , − k ) (cid:12)(cid:12)(cid:12) ( p ) r I m Z d q ¯ h w − e ( p ) − e ( p ) + i h ≈ (cid:12)(cid:12)(cid:12) V ( ) ( k ; − k , − k ) (cid:12)(cid:12)(cid:12) ( p ) r I m Z d qe ( q ) + ( e ( q ) − e ( q )) cos q + i h (23) = − (cid:12)(cid:12)(cid:12) V ( ) ( k ; − k , − k ) (cid:12)(cid:12)(cid:12) pr Z q + q − qdq p − e ( q ) e ( q ) (24)where e ( q ) ≡ ¯ h w − e ( k / ) − e ′ k q and e ( q ) ≡ ¯ h w − e ( k / ) − e ′′ q are the values of the energy denominator at x = x =
1. The integral is imaginary if the denoninator changes its sign for 0 ≤ cos q ≤
1. Since per assumption e ′′ ≪ e ′ / k wehave always e ( q ) < e ( q ) , therefore we need D E ≡ ¯ h w − e ( k / ) > e ′′ >
0, theimaginary part is picked up for q − < q < q + , where q − = s k D E e ′ q + = s D E e ′′ , (25)and, hence I m S ( k , ¯ h w ) = − (cid:12)(cid:12)(cid:12) V ( ) ( k ; − k , − k ) (cid:12)(cid:12)(cid:12) pr Z q + q − qdq p − e ( q ) e ( q ) q ( D E )= − s k e ′ e ′′ (cid:12)(cid:12) V ( k ; − k , − k ) (cid:12)(cid:12) r q ( D E ) . (26)For e ′′ <
0, there is no upper limit of the integration range of the internal momentum, but the integral converges becausethe three-phonon matrix element goes to zero for large momentum transfers, and the actual value depends on the details of theinteraction. However, we are interested only in the non-analytic behavior for D E →
0. To calculate this behavior, subtract andadd the matrix element at the position where the denominator has a second order node, i.e. we write (cid:12)(cid:12)(cid:12)(cid:12) V ( k ; q − k , − q − k ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) V ( k ; − k − k ) (cid:12)(cid:12)(cid:12)(cid:12) + D V ( k , q ) . (27) D V ( k , q ) still contributes to the imaginary part, but not to the non-analytic behavior. We must now distinguish between D E > D E <
0. For the former case we have I m S ( k , ¯ h w ) ≈ − (cid:12)(cid:12) V ( k ; − k , − k ) (cid:12)(cid:12) pr Z ... q − qdq p − e ( x ) e ( x ) . The momentum integral does not converge, but this is artificial because we have factored out the interaction since we are onlyinterested in the behavior due to the square-root singularity at q − . Therefore, write for D E > Z ... q − qdq q ( e ′ k q − D E )( D E + | e ′′ | q )= D E Z ... q − qdq r ( q q − − )( + | e ′′ | D E q )= q − D E Z ... xdx r ( x − )( + k | e ′′ | e ′ x ) = k e ′ × ( a number ) . (28)We can ignore the term k e ′′ / e ′ because, by assumption, | e ′′ | ≪ e ′ / k . The integral is then just a numerical value.For D E < Z ... q + dq q ( | D E | + e ′ k q )( | e ′′ | q − | D E | )= | D E | Z ... q + qdq r ( + e ′ k | D E | q )( | e ′′ || D E | q − )= q + | D E | Z ... xdx r ( x − )( + e ′ k | e ′′ | x ) = s k e ′ e ′′ × ( a number ) . (29)Here, the term e ′ k | e ′′ | dominates in the denominator. Since, by assumption, k | e ′′ | ≪ e ′ , the imaginary part has a discontinuity ofthe order of q k e ′ e ′′ at D E = r r r FIG. 7. The figure shows the leading order diagrams contributing to the irreducible three-body vertex X ( r , r , r ) . The usual diagrammaticconventions apply: circles correspond to particle coordinates, filled circles imply a density factor and integration over the associated coordinatespace. Solid lines represent correlation factors h ( r i , r j ) = g ( r i , r j ) − u ( r , r , r ) . Three-Body Vertex
Normally, the three-body vertex (15) is calculated in convolution approximation. An improvement can be achieved by sum-ming a set of three-body diagrams contributing to ˜ X ( k , p , q ) , which corresponds topologically to the hypernetted chain (HNC)summation. The first few diagrams are shown in Fig 7.The equations to be solved are best written in momentum space and relative and center of mass momenta, i.e. ˜ X ( p , p , p ) ≡ ˜ X ( q / + k , q / − k , q ) ≡ ˜ X q ( k ) . (30)The integral equation to be solved is ˜ X q ( k ) = Z d d p ( p ) d r ˜ h ( k − p ) ˜ N q ( p ) ˜ N q ( k ) = ˜ N ( CA ) q ( k ) + ˜ s q ( k ) d ˜ X q ( k ) , (31)where ˜ N q ( k ) is the set of nodal diagrams, and˜ N ( CA ) q ( k ) = ˜ h ( q + k ) ˜ h ( q − k ) + ˜ u ( q + k , q − k , q ) is the convolution approximation for this quantity. Also, we have abbreviated˜ s q ( k ) = [ S ( | p + q / | ) S ( | p − q / | ) − ] . (32)The equations can be easily solved by expanding all functions in terms of k , q , and the angle between the two vectors, e.g. ˜ h ( | k − k | ) = ¥ (cid:229) n = ˜ h n ( k , k ) cos ( n f ) This gives us the three-body vertex in the form˜ X q ( p ) = (cid:229) m cos ( m f ) X m ( q , p ) . We would like to thank C. E. Campbell, F. Gasparini and H. Godfrin for useful discussions. This work was supported, in part,by the Austrian Science Fund FWF under project I602. Additional support was provided by a grant from the Qatar NationalResearch Fund [1] B. Lambert, D. Salin, J. Joffrin, R. Scherm, Journal de Physique Lettres (18), 377 (1977)[2] W. Thomlinson, J.A. Tarvin, L. Passell, Phys. Rev. Lett. , 241 (1980)[3] H.J. Lauter, H. Godfrin, V.L.P. Frank, P. Leiderer, in Excitations in Two-Dimensional and Three-Dimensional Quantum Fluids , NATOAdvanced Study Institute, Series B: Physics , vol. 257, ed. by A.F.G. Wyatt, H.J. Lauter (Plenum, New York, 1991),
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