Please correct me if I make any mistakes along the way.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose we have a simple tight-binding Hamiltonian

[itex]H=\sum_i \epsilon _i c_i^\dagger c_i - t\sum_{\langle i j\rangle} c_i^\dagger c_j + h.c.,[/itex]

In half-filling systems, we tend to impose a constraint such that each site has only one electron on average, i.e.,

[itex]\langle c^\dagger_i c_i\rangle = 1[/itex]

Suppose I were to ensure that this constraint is in the Hamiltonian, I add a Lagrange multiplier (which ends up to be a chemical potential?)

[itex]H=\sum_i \epsilon _i c_i^\dagger c_i - t\sum_{\langle i j\rangle} c_i^\dagger c_j + h.c. - \mu_i (c^\dagger_i c_i-1),[/itex]

Now my confusion comes in, how does the constraint work here, since the 1 is a constant, and can be effectively ignored to obtain a Hamiltonian like

[itex]H=\sum_i (\epsilon_i - \mu_i) c_i^\dagger c_i - t\sum_{\langle i j\rangle} c_i^\dagger c_j + h.c.,[/itex]

so how can I ensure that my constraint is really doing anything in mean-field (MF) since I could have the constraint be any other number?

I'm working only in the mean-field approximation.

I received a suggestion to square the constraint, and hence end up with a quartic term, that can be solved perturbatively. However, it seems at zeroth (first?) order in mean-field, the entire term disappears

[itex](c^\dagger_i c_i-1)^2 = c_i^\dagger c_i c_i^\dagger c_i -2 c_i^\dagger c_i +1[/itex]

Under MF, with the equation [itex]AB \approx \langle A\rangle B + A\langle B\rangle - \langle A\rangle \langle B \rangle,[/itex]

[itex] 2\langle c_i^\dagger c_i \rangle c_i^\dagger c_i - \langle c_i^\dagger c_i \rangle \langle c_i^\dagger c_i \rangle -2 c_i^\dagger c_i +1 = 2 c_i^\dagger c_i - 1\cdot 1 - 2c_i^\dagger c_i +1=0[/itex]

I hope someone can relieve some of my confusion.

Specifically, I am working with slave fermions and want to impose [itex]\langle f_i^\dagger f_i \rangle=2[/itex] for a spin, [itex]S=1[/itex] system, and with itinerant electrons with [itex]\langle c_i^\dagger c_i \rangle=1[/itex]

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# Constraints in Hamiltonian

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