  Using large-scale determinant quantum Monte Carlo simulations in combination
with the stochastic analytical continuation, we study two-particle dynamical
correlation functions in the anisotropic square lattice of weakly coupled
one-dimensional (1D) Hubbard chains at half-filling and in the presence of weak
frustration. The evolution of the static spin structure factor upon increasing
the interchain coupling is suggestive of the transition from the power-law
decay of spin-spin correlations in the 1D limit to long-range antiferromagnetic
order in the quasi-1D regime and at $T=0$. In the numerically accessible regime
of interchain couplings, the charge sector remains gapped. The low-energy
momentum dependence of the spin excitations is well described by the linear
spin-wave theory with the largest intensity located around the
antiferromagnetic wave vector. This magnon mode corresponds to a bound state of
two spinons. At higher energies the spinons deconfine and we observe signatures
of the two-spinon continuum which progressively fade away as a function of
interchain hopping.
