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Triplet character of 2D-fermion dimers arising from $s$-wave attraction via spin-orbit coupling and Zeeman splitting

by Ulrich Ebling, Ulrich Zülicke, Joachim Brand

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Submission summary

Authors (as registered SciPost users): Joachim Brand · Ulrich Ebling
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-12-20 23:01
Submitted by: Brand, Joachim
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Theory
Approach: Theoretical


We theoretically study spin-$\frac{1}{2}$ fermions confined to two spatial dimensions and experiencing isotropic short-range attraction in the presence of both spin-orbit coupling and Zeeman spin splitting -- a prototypical system for developing topological superfluidity in the many-body sector. Exact solutions for two-particle bound states are found to have a triplet character that can become dominant in the regime of weak attractive $s$-wave interaction when the energy scale of spin-orbit coupling is comparable to a combination of the Zeeman and center-of-mass kinetic energies. The center-of-mass momentum is one of the parameters determining the existence of bound states, which we map out for both two- and one-dimensional types of spin-orbit coupling. Distinctive features emerging in the orbital part of the bound-state wave function, including but not limited to its $p$-wave character, provide observable signatures of unconventional pairing.

Current status:
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Reports on this Submission

Anonymous Report 2 on 2022-3-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2112.09336v1, delivered 2022-03-22, doi: 10.21468/SciPost.Report.4743


1. This study is well-motivated and timely.

2. Interesting new results on two-body bound states of spin-orbit coupled fermions.

3. The results will be useful as a starting point for developing many-body theories of SO-interacting fermions.


Presentation lack sufficient detail (see requested changes below).


This manuscript presents a theoretical study of interacting spin-1/2 fermions in two dimensions. The interactions are composed of isotropic short-range attraction and the spin-orbit coupling, and the Zeeman interactions. The authors find triplet bound states which could be observed in the regime of sufficiently strong spin-orbit (SO) coupling, which leads to interesting unconventional pairing mechanisms. These mechanisms could give rise to topological superfluidity with Majorana-fermion excitations. This study is therefore well-motivated and timely.

The major new aspect of this work is the inclusion of the Zeeman splitting on the same footing as center-of-mass momentum, which has apparently not been done before. In addition, the authors examine the spin properties of the bound state in detail and consider a usefully wide range of SO couplings, such as Dirac, Rashba, and Dresselhaus in 2D and the p_x \sigma_x coupling in 1D. They first solve the two-body problem with a general SO coupling to obtain the wavefunction |\psi_b(p)> and then project the solution onto the total spin states of the two-particle system (Eqs. 25). They finally consider the different types of SO couplings and analyze the properties of the bound states obtained with each type of coupling.

Some of the interesting results obtained by the authors include: (i) the COM momentum acts as an effective magnetic field and (ii) the two-body bound states disappear when the COM momentum exceeds a certain threshold.

Perhaps even more importantly, the results obtained in this manuscript will be useful as a starting point for developing many-body theories of SO-interacting fermions. Thus, in my opinion, the Expectation “Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work” is satisfied.

It should also be noted that the manuscript is clearly and concisely written. The general acceptance criteria will also be satisfied after the authors address my comments below.
As such, I recommend this manuscript for publication in SciPost Physics.

Requested changes

1. The parameter b defined below Eq. (5) gives the relative strength of the Zeeman and SO interaction seems to have the dimension of momentum. Does this parameter then adequately reflects the relative strength of these Zeeman and SO interactions. Would it not be better to use a dimensionless ratio of interaction strengths?

2. Equations (9) establish that the relative motion of two particles in the COM frame depends on the COM momentum P. This is an unusual situation because the reason one introduces the COM and relative coordinates in the first place is to decouple the Hamiltonian into two commuting parts (the COM and internal Hamiltonians).

Hence, two points should be clarified. First, is there really any practical advantage to using the coordinates in Eq. (6)? Second, it would be helpful to mention exactly which interactions are responsible for the coupling between the external and internal degrees of freedom. Are these the SO coupling terms of the kind \sigma_x p_y?

3. In Section 2.2 the authors use the Green function approach to obtain the bound states of s-wave interacting particles. Is this the only approach that can be used?
It would also be helpful for the general reader to understand how this approach works using a simple example. A reference to the approach being applied to, e.g., two interacting particles in the absence of the SO interaction, would be helpful.

  • validity: high
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

Report 1 by Peter Schmelcher on 2022-2-7 (Invited Report)

  • Cite as: Peter Schmelcher, Report on arXiv:2112.09336v1, delivered 2022-02-07, doi: 10.21468/SciPost.Report.4334


See my referee report


See my referee report


See attached file.

Requested changes

See attached file.


  • validity: high
  • significance: good
  • originality: high
  • clarity: good
  • formatting: excellent
  • grammar: excellent

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