Hey all! Thanks for reading. I'm currently following along in some reading and had some trouble with re-writing a Hamiltonian in Bogluibov-de Gennes form using Nambu notation (Nambu spinors). Here is the low down:(adsbygoogle = window.adsbygoogle || []).push({});

Say we have a Hamiltonian:

[tex] \frac{1}{2} \sum_{i=1}^{N} c_{i}^{\dagger} D c_{i} + \frac{1}{2}\sum_{i=1}^{N-1}c^{\dagger}_{i+1}Tc_{i} + c^{\dagger}_{i}T^{\dagger}c_{i+1}

[/tex]

where we have

[tex] D = (\frac{\hbar^2}{ma^2} - \mu)\tau_{z} [/tex]

and

[tex] T = (-\frac{\hbar^2}{2ma^2}\tau_{z} - \frac{i\Delta}{2a}\tau_{x}) [/tex]

The tau are Pauli matrices. We are to be able to write the Hamiltonian as a 2Nx2N matrix (N being the number of particles), by defining Nambu spinors

[tex] \tilde{c} = (c_{1}, c_{2},...,c_{N})^T [/tex]

which is of length 2N since each c_{i} is a 2 spinor. Finally, we can simplify the Hamiltonian to

[tex] H = \frac{1}{2} \tilde{c^{\dagger}}H\tilde{c} [/tex]

Where H is a tridiagonal matrix consisting of T's and D's. I understand how the two are equivalent, but how could I just look at a Hamiltonian and tell if I could simplify it using Nambu spinors? When using the Bogluibov quasiparticles we see the same type of thing to get a BdG Hamiltonian. I can check the equivalence by working backwards, but how could I start with the original H in my statement and rearrange it to ultimately get the Nambu form? Hopefully I'm being clear enough :P

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# A Nambu Spinor Notation in Kitaev spinless p-wave model

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