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Recently, Majorana zero mode becomes very hot in condensed matter physics. I remember there was a lot of study of fermion zero mode in quantum field theory, where advanced math, such as index theorem, was used. I would like to know if there were previous studies of fermion zero mode where fermions have no $U(1)$ particle number conservation? (Such fermionic zero mode without fermion number conservation corresponds to Majorana zero mode.)

Edit: Here, a Majorana zero mode is a zero-energy mode localized at the center of some defect (like vortex or monopole).

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If one stays at the level of free fermions, Majorana zero modes are nothing but the zero modes of BdG equations. One can apply the rich mathematical results like Atiyah-Singer index theorem to the BdG equations if they have the right structure. The U(1) symmetry does not seem to play a key role here. A particular example is to consider the topological insulator surface in proximity to a s-wave superconductor. The BdG equation is exactly a Dirac equation with mass term. However, to apply the Atiyah-Singer index theorem one needs to have chiral symmetry in the Dirac equation which is equivalent to no doping(or chemical potential set to zero). Then one can directly infer that the number of zero modes is equal to the winding number of the defect in the mass term.

When there is doping the chiral symmetry of the Dirac equation is broken and the index theorem ceases to work. I do not know there are other mathematical results that could apply here. But physically we know that in this case the stable zero modes is given by the winding number mod 2. I have been wondering for a nice "index theorem" for this general case for a while but unfortunately has not found anything. Teo and Kane has a general classification for zero modes in defects, utilizing the K-theory classification(see http://arxiv.org/abs/1006.0690).

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Regarding to "One can apply the rich mathematical results like Atiyah-Singer index theorem to the BdG equations if they have the right structure", do you have any references, in particular that related to Majorana zero mode? I have an impression that this was well known in field theory community in 1980's. –  Xiao-Gang Wen Jun 19 '12 at 13:01
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At the experimental level, those condensed matter Majorana degrees of freedom are the pioneering examples (assuming that the claims are true). The only other Majorana fields we know in the world around us are the neutrino fields but even though there are strong theoretical reasons to think that the neutrino species we know are Majorana and not Dirac, we can't really experimentally show it is the case.

Theoretical physics is full of Majorana fields, however. The world sheet fermions in string theory are Majorana fermions – well, in 2 dimensions, much like in 10 dimensions and any $8k+2$ dimensions, one may impose the Majorana and Weyl conditions simultaneously so we're working with Majorana-Weyl fermions.

Similarly, there are lots of hypothetical Majorana (but not Weyl at the same moment) fields in $d=4$ according to supersymmetry (and some other models of new physics). The superpartners to any bosonic field of the Standard Model – the Higgs and the gauge bosons – are Majorana fermions. Neutralinos may be the lightest example: they may be the lightest superpartners (LSPs) and in many models, they account for most of the dark matter. This dark matter would annihilate in pairs, something we expect for Majorana excitations that naturally carry a conserved ${\mathbb Z}_2$ quantum number.

I would slightly disagree that the absence of a $U(1)$ charge is equivalent to the Majorana condition. This identification of the two conditions holds in one direction and it is "economic" in the other but it doesn't have to be the case. The neutrinos could be Majorana but they could still refuse to carry any conserved $U(1)$ charges. Majorana degrees of freedom mean that the fields transform as Majorana spinors (spinor representations that obey a reality condition); more generally, these fermions are the canonical momenta to themselves so that one may write $\{\theta_a,\theta_b\}=\delta_{ab}$ anticommutators without any daggers while $\theta^\dagger=\theta$ for these components.

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Thank you for the nice summary for the Majorana fields. Do you know any field theory refs about Majorana zero mode, which are the zero-energy mode localized at the center of some defect (like vortex or monopole). –  Xiao-Gang Wen Jun 1 '12 at 4:16
Thanks for your interest. Maybe you may start with arxiv.org/abs/cond-mat/0609556 and some literature mentioned in it. –  Luboš Motl Jun 1 '12 at 6:09
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