Gapped paramagnetic state in a frustrated spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the cross-striped square lattice

We implement the coupled cluster method to very high orders of approximation to study the spin-$\frac{1}{2}$ $J_1$–$J_2$ Heisenberg model on a cross-striped square lattice. Every nearest-neighbour pair of sites on the square lattice has an isotropic antiferromagnetic exchange bond of strength $J_1>0$, while the basic square plaquettes in alternate columns have either both or neither next-nearest-neighbour (diagonal) pairs of sites connected by an equivalent frustrating bond of strength $J_2\equiv \alpha J_1>0$. By studying the magnetic order parameter (i.e., the average local on-site magnetization) in the range $0\le \alpha \le 1$ of the frustration parameter we find that the quasiclassical antiferromagnetic N'{e}el and (so-called) double N'{e}el states form the stable ground-state phases in the respective regions $\alpha <{\alpha }_{1a}^c=0.46(1)$ and $\alpha > \alpha_{1b}^{c} = 0.615(5)$. The double N'{e}el state has N'{e}el ($\cdots \uparrow \downarrow \uparrow \downarrow \cdots$) ordering along the (column) direction parallel to the stripes of squares with both or no $J_2$ bonds, and spins alternating in a pairwise ($\cdots \uparrow \uparrow \downarrow \downarrow \uparrow \uparrow \downarrow \downarrow \cdots$) fashion along the perpendicular (row) direction, so that the parallel pairs occur on squares with both $J_2$ bonds present. Further explicit calculations of both the triplet spin gap and the zero-field uniform transverse magnetic susceptibility provide compelling evidence that the ground-state phase over all or most of the intermediate regime ${\alpha }_{1a}^c<\alpha <{\alpha }_{1b}^c$ is a gapped state with no discernible long-range magnetic order.