Abstract
We revisit the problem of the calculation of zero-temperature properties for the dilute two-dimensional Bose gas. By using Popov’s hydrodynamic approach and perturbation theory on the two-loop level, we recover not only the known expansion for the ground-state energy but also calculate for the first time the condensate density and Tan’s contact.
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Acknowledgements
We thank Prof. A. Rovenchak for stimulating discussions. This work was partly supported by Project FF-30F (No. 0116U001539) from the Ministry of Education and Science of Ukraine.
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Appendices
Appendices
1.1 A
The explicit analytic expressions for the diagrams determining corrections to the partition function logarithm \(-\beta \Delta E\) and presented in Fig. (1) are the following
(recall that \(|\mathbf{k}|,|\mathbf{q}|, |\mathbf{s}|<\varLambda \) and \(\alpha _k=E_k/\varepsilon _k\)), where the frequency sums in the diagrams b and c
and in e and d
were evaluated with the help of residue theorem in the zero-temperature limit.
1.2 B
In this section, we present some details of calculation of integrals determining constants in the ground-state energy (17), condensate depletion (18) and contact (20). We will not stop on the evaluation of variational derivatives \(\left( \frac{\delta E}{\delta \varepsilon _k}\right) _{n,\mathcal {T}}\), \(\left( \frac{\delta E}{\delta \mathcal {T}_k}\right) _{n}\) with the following integration in \(\mathbf{k}\)-space and only give the dimensionless expressions written in terms of triple integrals. After elimination of the explicit dependence on cutoff parameter \(\varLambda \) for \(\text {const}_E\), we obtained
here and below
The second-order correction to condensate density is determined by the following constant
Finally, \(\text {const}_{\mathcal {C}}\) after some rearrangements is given by
The results of numerical calculations of these integrals are presented in main text.
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Pastukhov, V. Ground-State Properties of a Dilute Two-Dimensional Bose Gas. J Low Temp Phys 194, 197–208 (2019). https://doi.org/10.1007/s10909-018-2082-1
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DOI: https://doi.org/10.1007/s10909-018-2082-1