Random Bonds Effects in the Spin-$\frac{1}{2}$ Heisenberg Antifferomagnet on the Square Lattice

In one dimension, it is well know that the spin-$\frac{1}{2}$ Heisenberg antiferromagnetic (AF) chain, governed by ${\mathcal{H}}_{1d}=\sum_{i} J_i {\vec{S}}_i \cdot {\vec{S}}_{i+1},$ is unstable against any strength of bond randomness [1]. The {\it{quasi}}-long-range-ordered phase is indeed destroyed $\forall \langle J_{i}^{2} \rangle \ne 0$ and then replaced by the so-called {\it{Random Singlet Phase}} [1]. Here, we adress the question of the two-dimensionnal case on the square lattice. When non-frustrating randomness in the AF exchanges is introduced, we show that the situation for the following Hamiltonian ${\mathcal{H}}_{2d}=\sum_{\langle i,j \rangle} J_{i,j} {\vec{S}}_i \cdot {\vec{S}}_{j}$, is completely different from the one dimensionnal case. In fact, the extreme robustness of the $T=0$ AF order parameter as well as the appearance of localized excitations with increasing disorder has been studied with the help of several theoretical tools: Exact Diagonalizations, modified Spin-Waves calculations and Quantum Monte Carlo simulations performed at extremely low temperature over thousands of disordered samples and for systems up to $32\times 32$. Our results also lead to more general consideration about Griffiths singularities in random quantum magnets.\\ $[1]$ D. S. Fisher, Phys. Rev. B {\bf 50}, 3799 (1994).