This is a question from Altland and Simons book "Condensed Matter Field Theory".(adsbygoogle = window.adsbygoogle || []).push({});

In the second exercise on page 64, the book claims that if we define [itex] \hat P_s, \hat P_d [/itex] to be the operators that project onto the singly and doubly occupied subspaces respectively, then at half-filling the following equality holds

[itex] \hat P_s \hat H_t \hat P_s = 0 [/itex] and [itex] \hat P_d \hat H_t \hat P_d =0 [/itex],

where [itex]\hat H_t[/itex] is the hopping term in the Hubbard model. I am having trouble to see why these two conditions are true. Do I have to write out the projection operator in terms of creation and annihilation operators and directly calculate?

Thanks in advance.

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# Projection in Hubbard Model

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