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EDIT: Nevermind, I figured it out. When you apply the expression on a state, you can clearly see what it does: The new state created by applying the Majorana operators must have the same energy as the original one. So when applied on a zero-energy state, this creates a degeneracy and the so-called Majorana zero mode.


I am currently trying to grasp this lecture by Frank Wilczek about Majorana zero modes on an 1-dimensional wire. In equation (31), he states that a general condition for such a zero mode is $$ \left[ H, \sum_{j=1}^{2L} c_j \gamma_j \right] = 0, $$ where the $c_j$ are coefficients and the $\gamma_j$ are the Majorana operators for the sites of the chain. I'm not sure I understand why this is a general condition for a zero mode if the two operators commute. Can anyone shed some light on this?

Thanks in advance!

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Post the edit as an answer and accept it--- this is a fine question, even though you figured out the answer yourself. –  Ron Maimon Jul 29 '12 at 17:41
    
Ok. This was my first post here, so thanks for the advice. Apparently, I cannot answer my own question now, but will be allowed to after 2 more hours. –  sds Jul 29 '12 at 18:13

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When you apply the expression on a state, you can clearly see what it does: The new state created by applying the Majorana operators must have the same energy as the original one. So when applied on a zero-energy state, this creates a degeneracy and the so-called Majorana zero mode.

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