Derivation of the low- T phase diagram of LiHo_xY_1−xF₄: A dipolar quantum Ising magnet

The LiHo_xY_1−xF₄ compound is widely considered to be the archetypal dipolar quantum Ising system, with longitudinal dipolar interactions V_ij^zz between Ho spins {i,j} competing with transverse field–induced tunneling, to give a T=0 quantum phase transition. By varying the Ho concentration x, the typical strength V₀ of V_ij^zz can be varied over many orders of magnitude, and so can the transverse field H_⊥. A new effective Hamiltonian is derived, starting from the electronuclear degrees of freedom, which is valid at low and intermediate temperatures. For any such dipolar quantum Ising system, the hyperfine interaction will dominate the physics at low temperatures, even if its strength A₀<V₀: One must therefore go beyond an electronic transverse field quantum Ising model. We derive the full phase diagram of this system, including all nuclear levels, as a function of transverse field H_⊥, temperature T, and dipole concentration x. For LiHo_xY_1−xF₄ we predict a re-entrant critical field as a function of x. We also predict the phase diagram for x=0.045 and the behavior of the system in magnetic-resonance and muon-spin-relaxation experiments.