# How to find the projected Hamiltonian for lowest flat-band in general?

In [1], starting with the bosonic Hamiltonian (Eqn. 1) for the dice lattice model with half flux density (with Ahronov-Bohm phases incorporated), $$$$H=-t\sum_{\langle j,\mu\rangle}(a^\dagger_\mu a_je^{iA_{\mu j}}+\text{h.c.})+\frac{U_\Delta}{2}\sum_{\mu \in \Delta,\nabla}a^\dagger_\mu a_\mu(a^\dagger_\mu a_\mu-1)+\frac{U_*}{2}\sum_{j\in *}a^\dagger_j a_j(a^\dagger_j a_j-1)$$$$ where the form of the Hamiltonian is well understood (which is what we care for, in this question). Let the constants involved be arbitrarily chosen with main focus on the functional form of the operators involved. Single particle spectrum can be easily deduced from the above Hamiltonian.

Further, the authors, in the special regime $$nU_{*/\Delta}/t\ll 1$$, make the lowest flat band approximation leading to the Hamiltonian (Eqn. 2), $$$$H_{\text{proj}}=\gamma_1\sum_i n_i(n_i-1)+\gamma_2\sum_{\langle i,j \rangle}[n_in_j+(c^\dagger)^2c_j^2+(c_j^\dagger)^2c_i^2]+\gamma_3\sum_{\Delta(i,j,k)}[\sigma^{ij}_{kk}c^\dagger_ic^\dagger_jc_k^2+\sigma^{ik}_{jk}c^\dagger_ic_jn_k+\text{h.c.}]$$$$ where $$\gamma_i$$ are some specially chosen numbers and $$n_{\lambda}=a^\dagger_\lambda a_\lambda$$ is the number operator for that mode. $$\sigma^{qr}_{st}=\exp [i(A_{q\mu}+A_{r\mu}+A_{\mu s}+A_{\mu t})]$$ are the phase factors. This projected Hamiltonan works perfectly fine for the lowest flat-band dynamics but it is not proved how to arrive at it by a projection action. However, by construction it works well enough. Again, we shall only focus on the functional form of the Hamiltonian. My questions are:

• How to take the lowest band, especially lowest flat-band projection, of a given Hamiltonian in general? I am looking for a matrix projector method.
• How to show the above with respect to the given research in [1]?

[1] G. Moller, N. R. Cooper. Correlated Phases of Bosons in the Flat Lowest Band of the Dice Lattice.Phys. Rev. Lett. 108 (4), 045306. Jan, 2012.