The low-$T$ phase diagram of ${\rm LiHo_xY_{1-x}F_4}$

TL;DR

This paper derives a comprehensive phase diagram for the dipolar Quantum Ising system ${\rm LiHo_xY_{1-x}F_4}$, accounting for hyperfine interactions and nuclear levels, predicting re-entrant critical fields and experimental behaviors.

Contribution

It introduces a new effective Hamiltonian including nuclear degrees of freedom, extending the understanding of the quantum phase transition in dipolar systems.

Findings

Predicts a re-entrant critical field as a function of x.

Provides the phase diagram for x=0.045.

Describes behavior in magnetic resonance and $\mu$SR$ experiments.

Abstract

The ${\rm LiHo_xY_{1-x}F_4}$ compound is widely considered to be the archetypal dipolar Quantum Ising system, with longitudinal dipolar interactions $V_{ij}^{zz}$ between ${\rm Ho}$ spins $\{i,j \}$ competing with transverse field-induced tunneling, to give a T=0 quantum phase transition. By varying the ${\rm Ho}$ concentration x, the typical strength $V_0$ of $V_{ij}^{zz}$ can be varied over many orders of magnitude; and so can the transverse field $H_{\perp}$. A new effective Hamiltonian is derived, starting from the electronuclear degrees of freedom, and 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 $A_0 < V_0$: 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 $H_{\perp}$, temperature $T$, and dipole concentration x. For ${\rm LiHo_xY_{1-x}F_4}$ 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 $\mu$SR experiments.