Perturbative Cumulant Monte Carlo Study of LiHoF$_4$ in a Weak Transverse Magnetic Field

Results from a recent quantum Monte Carlo (QMC) (P.B. Chakraborty {\it et al.}, Phys. Rev. B {\bf 70}, 144411 (2004)) study of the LiHoF$_4$ Ising magnetic material in an applied transverse magnetic field $B_x$ show a discrepancy with the experimental results, even for small $B_x$ where quantum fluctuations are small. This discrepancy persists asymptotically close to the classical ferromagnet to paramagnet phase transition. We numerically reinvestigate the phase diagram of LiHoF$_4$ in the regime of weak $B_x$. In this regime, we derive an effective temperature-dependent classical Hamiltonian that incorporates perturbatively the small quantum fluctuations in the vicinity of the classical phase transition at $B_x=0$. Via this effective classical Hamiltonian, we study the $B_x-T$ phase diagram via classical Monte Carlo simulations. In particular, we investigate the influence of various effects that may be at the source of the discrepancy. We also show how our method can be generalized to numerically study the diluted LiHo$_x$Y$_{1-x}$F$_4$ in the small $B_x$ regime.