Theory of the magnetic phase diagram of $\mathrm{Li}\mathrm{Ho}{\mathrm{F}}_4$

The properties of $\mathrm{Li}\mathrm{Ho}{\mathrm{F}}_4$ are believed to be well described by a long-range dipolar Ising model. We go beyond mean-field theory and calculate the phase diagram of the Ising model in a transverse field using a quantum Monte Carlo method. The relevant Ising degrees of freedom are obtained using a nonperturbative projection onto the low-lying crystal-field eigenstates. We explicitly take the domain structure into account, and the strength of the near-neighbor exchange interaction is obtained as a fitting parameter. The on-site hyperfine interaction is approximately taken into account through a renormalization of the transverse applied magnetic field. Finally, we propose a spectroscopy experiment to precisely measure the most important parameter controlling the location of the phase boundary.

Figure 1

Lifting of ground-state degeneracy by the transverse field.

Figure 2

The effective magnetic field.

Figure 3

The phase diagram as a function of effective temperature and effective magnetic field obtained from quantum Monte Carlo simulations on different lattice sizes. The upper curve is the mean-field solution.

Figure 4

Exchange energy needed to tune the critical temperature.

Figure 5

Critical temperature as a function of effective transverse field, with an exchange interaction.

Figure 6

Comparison of the single-ion longitudinal susceptibilies with and without the hyperfine interaction. We make the approximation of assuming that the increase in longitudinal susceptibility caused by the nuclei can be modeled by simply renormalizing the transverse field downwards by the appropriate amount.

Figure 7

The complete phase diagram of $\mathrm{Li}\mathrm{Ho}{\mathrm{F}}_4$. Experimental data is from Ref. 2.

Figure 8

Ion energy spectrum (for a spin-1/2 nucleus) with the hyperfine interaction treated in first order perturbation theory.