Random transverse field Ising model in dimension d > 1: scaling analysis in the disordered phase from the directed polymer model

For the quantum Ising model with ferromagnetic random couplings J(i, j) > 0 and random transverse fields h(i) > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in (J(i, j)/h(i)) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as lnC(r) similar to -r/xi(typ) + r(omega)u, where xi(typ) is the typical correlation length, u is a random variable and omega coincides with the droplet exponent omega(DP)(D = d - 1) of the directed polymer with D = (d - 1) transverse directions. Our main conclusions are as follows. (i) Whenever omega > 0, the quantum model is governed by an infinite-disorder fixed point: there are two distinct correlation length exponents related by nu(typ) = (1 - omega )nu(av); the distribution of the local susceptibility chi(loc) presents the power-law tail P(chi(loc)) similar to 1/chi(1+mu)(loc) , where mu vanishes as xi(-omega)(av) so that the averaged local susceptibility diverges in a finite neighborhood 0 < mu < 1 before criticality (Griffiths phase); the dynamical exponent z diverges near criticality as z = d/mu similar to xi(omega)(av). (ii) In dimensions d <= 3, any infinitesimal disorder flows toward this infinite-disorder fixed point with omega(d) > 0 (for instance omega(d = 2) = 1/3 and omega(d = 3) similar to 0.24). (iii) In finite dimensions d > 3, a finite disorder strength is necessary to flow toward the infinite-disorder fixed point with omega(d) > 0 (for instance omega(d = 4) similar or equal to 0.19), whereas a finite-disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension d = infinity, where omega = 0, we discuss the similarities and differences with the case of finite dimensions.