Frustration-induced degenerate spin state with up-up-down-down ordering in corner-connected Heisenberg square-plaquettes

Abstract

We investigate frustrated magnetism of a corner-connected square plaquette Heisenberg model with exchange interactions along the edges $(J_1)$, along inter-plaquette links $(J_2)$, and along square diagonals $(J_3)$. Using Luttinger--Tisza (LT) minimization of the Fourier interaction matrix $\mathcal J(\mathbf q)$ together with large-scale Monte Carlo (MC) simulations, we obtain a classical low-temperature magnetic phase diagram in the normalized plane $(J_1/|J_2|, J_3/|J_2|)$. The two methods play complementary roles: the LT analysis provides the zero-temperature candidate ordering wave vectors, while the MC simulations elucidate the resulting ordering tendencies under the hard-spin constraint at low but finite temperatures. Three regimes emerge at low temperatures, an antiferromagnetic phase (AF), a ferromagnetic phase (FM) and a degenerate spin state with \textit{up-up-down-down (uudd)} ordering (DS). For the frustrated degenerate spinstate, LT exhibits line-like minima along $q_x=\pm q_y$ in the $hk$-plane, revealing a highly degenerate spin configuration which violates the hard-spin constraint. The MC results uncover a state with quasi-two-dimensional ordering. The DS regime is intrinsically multi-$\mathbf{q}$: the ordered texture assembles itself from symmetry-related modes on the lines, producing a ``\textit{uudd}'' spin arrangement comprising of distinctive strong antiferromagnetic correlations on diagonals, ferromagnetic correlations on corner links, and highly suppressed correlations on the edges. The field-temperature phase diagram for a representative parameter point in the DS regime, determined by MC simulations, yields field induced distinct regions of negatively and positively correlated layers. These two regions are separated by a curve corresponding to negligible inter-layer correlations.