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Recipe to determine symmetries of quadratic fermionic Hamiltonian in second quantisation

Physics Asked on July 31, 2021

Consider an arbitrary 1D chain (of length $N$) of fermions with an arbitrary quadratic Hamiltonian of the form

$$mathcal{H}=hat{Psi}^dagger H hat{Psi}$$

with

$$hat{Psi}=left(a_1, a_2, …,a_N,a_1^dagger, a_2^dagger, …,a_N^dagger right)^T$$

a vector of fermionic operators where $a_n^dagger$ creates a fermion at site $n$.

Are there some straight forward recipes for determining whether the Hamiltonian has any symmetries, specifically chiral, time-reversal, and particle-hole symmetry etc.?

One Answer

Just use the eyeball technique: the form of $hat{Psi}$ suggests that you express the single particle hamiltonian $H$ as a $2 times 2$-block operator and look for relations between the blocks. Hence, start with complex conjugation, the three Pauli matrices and products of Pauli matrices and complex conjugation.

Answered by Max Lein on July 31, 2021

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