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Suppose we have a bipartite lattice( A and B). By tuning some parameters in the Hamiltonian, we get a region of parameter space where sublattice magnetizations, $m_{A}$ and $m_{B}$ are both of same sign but not equal in magnitude. I mean to say , I have finite staggered magnetization, $m_{s}=(m_{A}-m_{B})/2$ as well as uniform magnetization, $m_{f}=(m_{A}+m_{B})/2$. Can I call it ferrimagnetic ?

Normally, if the sublattice magnetizations are oppositely oriented and there also happens to be some difference in their magnitudes, we call it ferri. But here I have $m_{A},m_{B}$ having same orientation but differing magnitudes.

According to my understanding, even if A and B have same direction of magnetizations but unequal magnitude, it means that A is more up, B is less up $\equiv$ B is more down. So, it can be called ferrimagnetic. Any clarification in this regard will be appreciated.

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The typical, textbook definitions of ferrimagnetic do mention sublattices with unequal and opposite moments, so you should probably avoid calling it that. I think what you describe is better called a ferromagnet, albeit with a two-site unit cell. People have studied ferromagnetic mixed-spin Hamiltonians (e.g. chains coupling sites with magnetic moment $S=3/2$ to $s=1$ to $S=3/2$ and so on) and those systems tend to be called mixed-spin ferromagnets even though they will have unequal magnetization in the fully polarized case.

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