  We report our combined experimental and theoretical study of magnetoelectric
properties of an antiferromagnet Sr(TiO)Cu$_4$(PO$_4$)$_4$, in comparison with
the isostructurals Ba(TiO)Cu$_4$(PO$_4$)$_4$ and Pb(TiO)Cu$_4$(PO$_4$)$_4$. The
family of compounds commonly possesses a low-symmetric magnetic unit called the
square cupola, which is a source of magnetoelectric responses associated with
the magnetic multipoles activated under simultaneous breaking of spatial
inversion and time reversal symmetries. Measuring the full magnetization curves
and the magnetic-field profiles of dielectric constant for
Sr(TiO)Cu$_4$(PO$_4$)$_4$ and comparing them with the theoretical analyses by
the cluster mean-field theory, we find that the effective $S=1/2$ spin model,
which was used for the previous studies for Ba(TiO)Cu$_4$(PO$_4$)$_4$ and
Pb(TiO)Cu$_4$(PO$_4$)$_4$, well explains the experimental results by tuning the
model parameters. Furthermore, elaborating the phase diagram of the model, we
find that the square cupolas could host a variety of magnetic multipoles, i.e.,
monopole, toroidal moment, and quadrupole tensor, depending on the parameters
that could be modulated by deformations of the magnetic square cupolas. Our
results not only provide a microscopic understanding of the series of the
square cupola compounds, but also stimulate further exploration of the
magnetoelectric behavior arising from cluster multipoles harboring in
low-symmetric magnetic units.

Strongly Correlated Electrons