N\'eel ordered versus quantum disordered behavior in doped spin-Peierls and Haldane gap systems

I consider a theoretical description of recent experiments on doping the spin-Peierls compound CuGeO$_3$ and the Haldane gap compounds PbNi$_2$V$_2$O$_8$ and Y$_2$BaNiO$_5$. The effective theory is the one of randomly distributed spin-$1/2$ moments interacting with an exchange decaying exponentially with distance. The model has two phases in the (doping, interchain coupling) plane: (i) A N\'eel ordered phase at small doping; (ii) A quantum disordered phase at larger doping and small interchain interactions. The spin-Peierls compound CuGeO$_3$ and the Haldane gap Nickel oxides PbNi$_2$V$_2$O$_8$ and Y$_2$BaNiO$_5$ fit well into this phase diagram. At small temperature, the N\'eel phase is found to be reentrant into the quantum disordered region. The N\'eel transition relevant for CuGeO$_3$ and PbNi$_2$V$_2$O$_8$ can be described in terms of a classical disordered model. A simplified version of this model is introduced, and is solved on a hierarchical lattice structure, which allows to discuss the renormalization group flow of the model. It is found that the system looks non disordered at large scale, which is not against available susceptibility experiments. In the quantum disordered regime relevant for Y$_2$BaNiO$_5$, the two spin model and the cluster RG in the 1D regime show a power law susceptibility, in agreement with recent experiments on Y$_2$BaNiO$_5$. It is found that there is a succession of two distinct quantum disordered phases as the temperature is decreased. The classical disordered model of the doped spin-$1$ chain contains already a physics relevant to the quantum disordered phase.