Random Transverse Field Spin-Glass Model on the Cayley tree : phase transition between the two Many-Body-Localized Phases

The quantum Ising model with random couplings and random transverse fields on the Cayley tree is studied by Real-Space-Renormalization in order to construct the whole set of eigenstates. The renormalization rules are analyzed via large deviations. The phase transition between the paramagnetic and the spin-glass Many-Body-Localized phases involves the activated exponent $\psi =1$ and the correlation length exponent $\nu =1$. The spin-glass-ordered cluster containing $N_{SG}$ spins is found to be extremely sparse with respect to the total number $N$ of spins : its size grows only logarithmically at the critical point $N_{SG}^{criti}\propto \ln N$, and it is sub-extensive $N_{SG} \propto N^{\theta}$ in the finite region of the spin-glass phase where the continuously varying exponent $\theta$ remains in the interval $0<\theta <1$.