Physics Reports vol.176

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These propagators describe a gas of non-interacting quasiparticles in the Popov model. Correlations between these quasiparticles are taken into account in the B—P approximation.

Although it is explicitly second order in C, the B—P self-energies actually involve a selected set of higher-order diagrams, due to the fact that the first-order propagators G and G are also functions of C. We are now in the position to compare the Bogoliubov, HFB, Popov and many-body t-matrix approximation with the Beliaev—Popov approximation.

The following discussion is strongly influenced by the recent analysis of Proukakis et al. The Bogoliubov approximation In the Bogoliubov approximation, one retains in the interaction Hamiltonian see Eqs. In terms of vertex diagrams, this means that one keeps, in Fig. In other words, one only considers 0 14 57 collisions between condensate—condensate and condensate—excited atoms, but ignores those between two excited atoms.

Since the Hamiltonian now involves quadratic terms, it can be diagonalized by a canonical transformation, namely the Bogoliubov transformation, yielding a gas of non-interacting quasiparticles. The Bogoliubov approximation gives a good description of a weakly interacting dilute Bose gas at zero temperature, since the depletion of the condensate population of the excited states is small.

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However, as the gas becomes denser or the temperature becomes higher, it is not appropriate to neglect the interactions between excited atoms. Therefore, the Bogoliubov model lose its validity in these limits. In connection with the Beliaev—Popov approximation, the Bogoliubov self-energy diagrams only include a and b in Figs. Any average that includes three or more operators is neglected in the HFB approximation. With the above treatment, the Hamiltonian can again be diagonalized, defining a new set of quasiparticles.

The self-energy diagrams of the HFB approximation are shown in Fig.

The dashed lines in Fig. It has been generally assumed that the HFB diagrams can be generalized to include the effect of repeated collisions by simply replacing the bare interaction with a t-matrix. However, Proukakis et al. Their argument is equivalent to the following: the iteration of the selfconsistent anomalous propagator G in part f of Fig. Such a contact potential is equivalent to a t-matrix, approximated to first-order in a. Beliaev [34] was presumably the first to realize the problem of diagram over-counting when introducing a t-matrix. As we described earlier, he took care of this by adding strokes to the Fig.

The HFB self-energy diagrams in terms of the bare interatomic potential. As can be seen from 0 0 Eq. The precise cancellation 0 of those terms containing J in the final expressions of R and R given by Eqs. This new modified HFB approximation is not usually discussed in the literature and is not subject to the criticism made by Proukakis et al.

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As we mentioned in Section 3, the self-consistent HFB approximation has an energy gap in the excitation spectrum. It may not be clear at the first-order level of approximation why such problems arise in the HFB approximation but not the simpler first-order Popov approximation. In view of the above discussion, one can easily see that the gap arises because some of the diagrams are double-counted in R. The Popov approximation, on the other hand, is not only simpler but consistent to first 0 order.

It includes all self-energies computed to first order in the interaction [41].

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Finally, we would like to make an important comment concerning the use of a contact potential such as in Eq. Strictly speaking, the use of a contact potential makes sense only in a first-order approximation, in which the physical quantities are evaluated to first order in the interaction. In a higher-order approximation, explicit or implicit, one must take into account other higher-order terms contained in the t-matrix as shown in Eq.

In other words, it is not consistent to drop terms which are proportional to fI 2 in the t-matrix, but keep such terms in the 0 evaluation of other physical quantities. The problem of double-counting diagrams in the naive HFB approximation as discussed above is just one example. The only difference between the Popov and HFB approximation is the way that the averages are performed in the Hartree—Fock term.

The many-body t-matrix approximation It is worthwhile to compare the present B—P approximation with the many-body t-matrix approximation discussed in Section 4. The similarities and differences between these two approaches are schematically shown in Fig. To first order in the vertex function C, the self-energy diagrams included in these two approaches are exactly the same. However, the higher order self-energy diagrams are different. For R , the many-body t-matrix approximation includes only 11 a , a and a , but not a , a , a , and a ; and for R , the many-body t-matrix approximation include 0 1 2 3 4 5 6 12 only b and b , but not b , b ,2, b.

On the other hand, the many-body t-matrix approximation 0 1 2 3 6 also includes the set of diagrams shown in Fig. Among these extra diagrams in Fig. Overlap and difference between the self-energies kept in the many-body t-matrix and the B—P approximations. Extra self-energy diagrams included in the many-body t-matrix approximation. The frequency sum in Eq. This is why a is not important in 7 the B—P approximation. A similar analysis applies to b for R.

This can be seen from the extra diagrams the type-II diagrams, which are labeled a — for R and b — for R which are kept in the B—P approxima26 11 26 12 tion. The collisions between condensate—condensate and condensate—excited atoms are treated in an improved manner, as compared to the HFB approximation. This leads to a gapless approximation, as proved [23] by the fact that the chemical potential satisfies the Hugenholtz—Pines theorem.

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Compared to all the other approximations mentioned above, there is a key difference in the self-energy diagrams of the B—P theory. This means that in 5 6 the B—P approximation, one takes into account the contributions of three-operator averages such as SaL saL aL T, etc. These contributions are crucial and thus the B—P theory represents a significant improvement over the other approximations mentioned above.

It is always difficult to justify why one includes a certain set of higher order diagrams but not the other, and especially so in Bose-condensed systems. We have concentrated here on what is done in the B—P approximation, without going into a thorough analysis as whether it is justifiable or not.

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With a clear understanding of what is included and the consequences, it is then possible to examine such questions. Evaluation of the B—P self-energies In Section 5. We now use these results to obtain more explicit expressions for the self-energies. On the right-hand side of Eq. Taking advantage of this, we can use Eqs. This distinction between these two kinds of terms is less obvious in the expression 5. E E n 1 2 A further simplification [compare with Eq. We next calculate the contribution of the type-I diagrams.

We note that the term containing J p in Eq.

Clearly, the remaining integral in Eq. However, this does not cause any trouble because, as we shall see shortly, this term is canceled by another divergent contribution to R p II in Eq.

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This completes our evaluation of type-I and type-II diagrams. Now, we combine the contributions of the two types to find R and R. By combining Eqs. E E n 1 2 where the function g is defined in Eq. Similarly, combining Eqs. E E n 1 2 q where the function h is defined in Eq. In Eqs. Energy of excitations in the B—P approximation Once we have obtained a specific approximation for R and R , we can substitute these into 11 12 Eqs. Due to the lengthy expressions in Eqs. However, 11 12 there is a way [34] to simplify such results so that the poles and their residues of G and G are 11 12 more clearly exhibited, and the second-order corrections to the energy of excitations are shown in 60 H.

To do so, it is convenient to separate out contributions which are explicitly linear and quadratic in fI , 0 R p " R 1 p R 2 p , 5.

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