Quantum spin liquid: The Heisenberg antiferromagnet on the three-dimensional pyrochlore lattice

We calculate the correlation functions of the $S=\frac{1}{2}$ Heisenberg antiferromagnet on the three-dimensional pyrochlore lattice. We use a perturbative expansion in terms of spin blocks. This method is self-controlled by the extracted correlation length. We find that the ground state is a spin-liquid state: the spin-spin correlation functions decay exponentially with distance and the correlation length never exceeds the interatomic distance, even at $T=0.\mathrm{}$ The static structure factor becomes dispersive, indicating a selection of modes by thermal and quantum fluctuations. We also calculate the magnetic neutron diffraction cross section in two planes of the reciprocal space and we find a very good agreement with neutron experiments performed on ${\mathrm{Y}\left(\mathrm{Sc}\right)\mathrm{Mn}}_2$ and ${\mathrm{CsNiCrF}}_6.$