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Accurate projective two-band description of topological superfluidity in spin-orbit-coupled Fermi gases

by J. Brand, L. A. Toikka, U. Zuelicke

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Submission summary

Authors (as registered SciPost users): Joachim Brand · Lauri Toikka · Ulrich Zuelicke
Submission information
Preprint Link:  (pdf)
Date accepted: 2018-07-31
Date submitted: 2018-06-19 02:00
Submitted by: Zuelicke, Ulrich
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


The interplay of spin-orbit coupling and Zeeman splitting in ultracold Fermi gases gives rise to a topological superfluid phase in two spatial dimensions that can host exotic Majorana excitations. Theoretical models have so far been based on a four-band Bogoliubov-de Gennes formalism for the combined spin-1/2 and particle-hole degrees of freedom. Here we present a simpler, yet accurate, two-band description based on a well-controlled projection technique that provides a new platform for exploring analogies with chiral p-wave superfluidity and detailed future studies of spatially non-uniform situations.

Author comments upon resubmission

We would like to thank the referee for the supportive evaluation of our work and constructive feedback on the manuscript. In the resubmitted version, we have taken all suggestions for improvement onboard. Here we respond in detail to each point raised by the referee. A complete list of changes made in the resubmitted manuscript is provided separately.

To 1: The decoupling approximation made in Eq. (5) relies on neglecting terms of order |Delta|^2/(h_k epsilon_k) and becomes exact when Delta = 0. For finite Delta, the dispersion relations and eigenvectors have the correct asymptotic behavior for large k. Integrated quantities will indeed become correct when |Delta|/h << 1. We have modified the discussion of Eq. (5) to clarify the nature of the approximation. We have further added a paragraph with pertinent comments at the end of section 2 ("By construction ... |Delta|\ll h."). A caveat to the limit |Delta|/h << 1 is that it is not sufficient if the approximate self-consistent Delta obtained in the projected 2x2 approach is small, as this can occur by accident during the summation of Eq. (13), but it is required that the exact value of Delta is small. A relevant comment has been added to the discussion of Fig. 7 in Sec. 4.2 ("From the derivation ... self-consistent |Delta|.").

To 2: We are happy to follow the suggestion to show more extensive comparisons between the 2x2 and 4x4 theories. For this purpose, we have included the new Figure 2 and also revised Figure 3 (former Figure 2) to be more informative about the limits where the 2x2 theory breaks down.

To 3: The perturbative Schrieffer-Wolff (SW) transformation and our 2x2-projection technique are formally different, and the approximate dispersions obtained by these approaches become accurate in opposite limits. Both agree to lowest order in 1/h near the topological gap. The SW result is ill-defined at the s-wave gap, while our 2x2-projection result is well-behaved everywhere. SW approximates the exact dispersions better than our result for small k, but SW differs significantly from the exact dispersions at large k, while our 2x2-projection approach exactly captures the latter. Both the divergence at the s-wave gap and the incorrect large-k behavior make it impossible to use the SW-transformation results as input for solving the self-consistency conditions. We include a brief comment comparing our approach to the SW transformation in the revised manuscript at the end of section 2 ("Our Feshbach-projection ... as small quantities.").

To 4: The referee is correct; it is the ambiguity in the conventions for defining the 2D scattering length that has caused different expressions to be given in the literature for the prefactor in the relation between E_b and a_2D. We have clarified this point in the new footnote 4.

To 5: We are very grateful to the referee for raising this issue. Prompted by this comment, we have re-examined the properties of the contributions (17b) to densities and found that there is no logarithmic divergence as we had mistakenly stated. Rather, we have identified the term n_{k\uparrow}^\downarrow to be pathological due to an unphysical pole caused by the denominator of (17b) having a zero in this case. Otherwise this term contributes negligibly to n_{k\uparrow}, hence we continue to omit it. However, the perfectly well-behaved n_{k\downarrow}^\uparrow is now included when calculating particle densities within the 2x2 approach, significantly improving agreement with the exact 4x4 results. See the revised Figure 3.

To 6: We have revised this figure (current Figure 3, former Figure 2) to only include self-consistent results. Our new choice of fixed parameters in the calculation of densities shown in the figure panels is intended to also address the referee’s suggestion (made in point 2 above) to show more broadly how 2x2 and 4x4-theory results compare.

To 7: The previous discrepancy between values for the chemical potential obtained within the 2x2 theory and the analytical approximations of Eqs. (21) and (23) has largely been rectified by including the contribution n_{k\downarrow}^\uparrow to the total density. The chemical potential is now very well reproduced by the 2x2-projected approach, except at small Zeeman energy h where the Feshbach projection is expected to fail. See Figure 4.

To 8: Our main motivation for developing the projected BdG equations was to simplify the numerical procedures for solving the inhomogeneous and time-dependent BdG equations, which will be the subject of future work. We have added a new paragraph to the introduction (“One of the main ... Eq. (20)].”) in order to comment more explicitly on the benefits of our approach.

List of changes

Changes are listed in the order in which they appear in the manuscript:

- added paragraph "One of the main benefits ... [see Eq. (2)]." in Sec. 1

- reformulated discussion around Eq. (5)

- added new Figure 2

- added discussion of Figure 2 and two paragraphs at the end of Sec. 2 ("Further comparison between ... |Delta|\ll h."), including also new footnotes 2 and 3

- added new footnote 4

- revised Figure 3 (former Figure 2) and reformulated discussion pertaining to this figure in the paragraph below Eq. (17b)

- revised Figure 4 (former Figure 3) and reformulated discussion pertaining to this figure in the paragraph below Eq. (23)

- revised Table 1 to also show results for Delta^{2x2} and mu^{2x2}, rationalised color scheme, and updated discussion pertaining to the table in paragraphs below Eq. (26)

- moved part of the first paragraph of Sec. 4.2 into new footnote 6 and added new paragraph "From the derivation ... self-consistent Delta."

- reformulated parts of the 2nd and 3rd paragraphs in Sec. 5 to reflect revisions to Figures 3 and 4

- included new Refs. [11,39,49,51]

Published as SciPost Phys. 5, 016 (2018)

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