The cavity method for quantum disordered systems: from transverse random field ferromagnets to directed polymers in random media

After reviewing the basics of the cavity method in classical systems, we show how its quantum version, with some appropriate approximation scheme, can be used to study a system of spins with random ferromagnetic interactions and a random transverse field. The quantum cavity equations describing the ferromagnetic-paramagnetic phase transition can be transformed into the well-known problem of a classical directed polymer in a random medium. The glass transition of this polymer problem translates into the existence of a 'Griffiths phase' close to the quantum phase transition of the quantum spin problem, where the physics is dominated by rare events. The physical behaviour of random transverse-field ferromagnets on the Bethe lattice is found to be very similar to that found in finite-dimensional systems, and the quantum cavity method gets back the known exact results of the one-dimensional problem.