keep $t_a=t_b=t$ for simplicity (see however, the discussion of particle-hole symmetry in Sec. \[sec:discussion\] surrounding Eqs. \[ph1\]-\[ph2\]). In general, there are also hopping processes connecting $a$ and $b$ orbitals. Due to the symmetry of the orbitals in Fig. 1, these occur only for next-nearest neighbors, $$H^{\prime\prime} = \sum_{\langle\langle ij\rangle\rangle} t_{ab}
  {\rm sign}[(x_i-x_j)(y_i-y_j)] \left(
    a_i^\dagger b_j^{\vphantom\dagger} + {\rm h.c.}\right),
  \label{nnn}$$ where the double angular brackets denote a sum over next nearest neighbors $i$ and $j$. Note that the hopping matrix elements in Eq. \[nnn\] are real and vary in sign. The sign variations reflect the symmetry differences (under rotations) between the $s$ and $d$ orbitals. The reality of the coefficients is a matter of convention, which we fix by choosing the orbital wavefunctions to be real. We will assume, as appropriate in this example, that $t_{ab} \ll t$, so that $H^{\prime\prime}$ is small, and can therefore be treated perturbatively.

It is sometimes an important perturbation, because it reduces the symmetry of the Hamiltonian. In particular, all of the terms in $H_0+H_1+H'$ conserve the number of $a$ and $b$ particles separately. Neglecting the $a$–$b$ hopping, therefore, the model has $SU(2)\times
U(1)\times U(1)$ continuous symmetries, corresponding to conservation of spin, and $a$ and 