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This is a question from Altland and Simons book "Condensed Matter Field Theory".
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.
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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