LiHoF _{4} 4 is a magnetic material known for its Ising-type anisotropy, making it a model system for studying quantum magnetism. However, the theoretical description of LiHoF _{4} 4 using the quantum Ising model has shown discrepancies in its phase diagram, particularly in the regime dominated by thermal fluctuations. In this study, we investigate the role of off-diagonal dipolar terms in LiHoF _{4} 4 , previously neglected, in determining its properties. We analytically derive the low-energy effective Hamiltonian of LiHoF _{4} 4 , including the off-diagonal dipolar terms perturbatively, both in the absence and presence of a transverse field. Our results encompass the full phase diagram, confirming the significance of the off-diagonal dipolar terms in reducing the zero-field critical temperature and determining the critical temperature’s dependence on the transverse field. We also highlight the sensitivity of this mechanism to the crystal structure by comparing our calculations with the Fe _{8} 8 system.
LiHoF _{4} 4 is a magnetic material known for its Ising-type anisotropy, making it a model system for studying quantum magnetism. However, the theoretical description of LiHoF _{4} 4 using the quantum Ising model has shown discrepancies in its phase diagram, particularly in the regime dominated by thermal fluctuations. In this study, we investigate the role of off-diagonal dipolar terms in LiHoF _{4} 4 , previously neglected, in determining its properties. We analytically derive the low-energy effective Hamiltonian of LiHoF _{4} 4 , including the off-diagonal dipolar terms perturbatively, both in the absence and presence of a transverse field. Our results encompass the full phase diagram, confirming the significance of the off-diagonal dipolar terms in reducing the zero-field critical temperature and determining the critical temperature’s dependence on the transverse field. We also highlight the sensitivity of this mechanism to the crystal structure by comparing our calculations with the Fe _{8} 8 system.
The rare-earth material LiHoF4 is believed to be an experimental realization of the celebrated (dipolar) Ising model and, upon the inclusion of a transverse field Bx, an archetypal quantum Ising model. Moreover, by substituting the magnetic Ho ions by nonmagnetic Y ions, disorder can be introduced into the system, giving rise to a dipolar disordered magnet and at high disorders to a spin glass. Indeed, this material has been scrutinized experimentally, numerically, and theoretically over many decades with the aim of understanding various collective magnetic phenomena. One of the to-date open questions is the discrepancy between the experimental and theoretical ${B}_{x}\ensuremath{-}T$ phase diagram at low fields and high temperatures. Here we propose a mechanism, backed by numerical results, that highlights the importance of quantum fluctuations induced by the off-diagonal dipolar terms, in determining the critical temperature of anisotropic dipolar magnets in the presence and in the absence of a transverse field. We thus show that the description as a simple Ising system is insufficient to quantitatively describe the full phase diagram of LiHoF4, for the pure as well as for the dilute system.
The quantum Ising model is perhaps the simplest possible model of a quantum magnetic material. Despite its simplicity, its versatility and wide range of applications, from quantum computation, to combinatorial optimization, to biophysics, make it one of the most important models of modern physics. In this thesis, we develop a general framework for studying quantum Ising systems with an arbitrary single ion Hamiltonian, with emphasis on the effects of quantum fluctuations, and the quantum phase transition between paramagnetic and ferromagnetic states that occurs when a magnetic field is applied transverse to the easy axis of the system. The magnetic insulating crystal LiHoF₄ is a physical realization of the quantum Ising model, with the additional features that the dominant coupling between spins is the long range dipolar interaction, and each electronic spin is strongly coupled to a nuclear degree of freedom. These nuclear degrees of freedom constitute a spin bath environment acting on the system. In this thesis, we present an effective low temperature Hamiltonian for LiHoF₄ that incorporates both these features, and we analyze the effects of the nuclear spin bath on the system. We find the lowest energy crystal field excitation in the system is gapped at the quantum critical point by the presence of the nuclear spins, with spectral weight being transferred down to a lower energy electronuclear mode that fully softens to zero at the quantum critical point. Furthermore, we present a toy model, the spin half spin half model, that illustrates the effects of an anisotropic hyperfine interaction on a quantum Ising system. We find the critical transverse field is increased when the longitudinal hyperfine coupling is dominant, as well as an enhancement of both the longitudinal electronic susceptibility and an applied longitudinal field. In addition, we present a field theoretic formalism for incorporating the effects of fluctuations beyond the random phase approximation in general quantum Ising systems. We find that any regular on site interaction, such as a nuclear spin bath, does not fundamentally alter the critical properties of a quantum Ising system. This formalism is used to calculate corrections to the magnetization of LiHoF₄.
The LiHo_xY_{1-x}F_4 magnetic material in a transverse magnetic \nfield B_x perpendicular to the Ising spin direction has long been used \nto study tunable quantum phase transitions in pure and random disordered systems. \nWe first present analytical and numerical evidences for the validity of an effective spin-1/2 \napproach to the description of a general dipolar spin \nglass model with strong uniaxial Ising anisotropy and subject to weak \nB_x. \n We relate this toy model to the LiHo_xY_{1-x}F_4 transverse field Ising material. \n We show that an effective spin-1/2 \n model is able to capture both the qualitative and quantitative aspects of the \nphysics at small B_x. \nAfter confirming the validity of the effective spin-1/2 approach, we show that the field-induced magnetization along the \nx direction, \ncombined with the local random dilution-induced \ndestruction of crystalline mirror symmetries \ngenerates, via the predominant dipolar interactions between Ho^{3+} ions, \n random fields along the Ising z direction. \nThis identifies LiHo_xY_{1-x}F_4 in B_x as a new \nrandom field Ising system. \nWe show that the random fields explain the smearing \nof the nonlinear susceptibility at the spin glass transition \nwith increasing B_x. \nIn this thesis, we also investigate the phase diagram of non-diluted LiHoF_4 in the presence of B_x, by performing \nMonte-Carlo simulations. A previous quantum Monte Carlo (QMC) simulation found that even for small B_x where quantum fluctuations are small, close to the classical critical \npoint, there is a discrepancy between experiment and the QMC results. We revisit this problem, focusing on \nweak B_x close to the classical T_c, using an alternative approach. For small B_x, by applying a so-called cumulant \nexpansion, the quantum fluctuations around the classical T_c are taken into account perturbatively. We derived an effective \nperturbative classical Hamiltonian, on which MC simulations are performed. With this method we \ninvestigate different proposed sources of uncertainty which can affect the numerical results. \n We fully reproduce the previous QMC results at small B_x. Unfortunately, we find that \n none of the modifications to the microscopic Hamiltonian that we explore are able to provide a B_x-T phase diagram compatible with the experiments in the small \nsemi-classical B_x regime.
Several magnetic materials, in the first approximation, can be described by idealised theoretical models, such as classical Ising or Heisenberg spin systems, and, to some extent, such models are able to qualitatively expose many experimentally observed phenomena. But often, to account for complex behavior of magnetic matter, such models have to be refined by including more terms in Hamiltonian. The compound LiHo_xY_{1-x}F_4, by increasing concentration of nonmagnetic yttrium can be tuned from a diluted ferromagnet to a spin glass. LiHoF_4 is a good realisation of the transverse field Ising model, the simplest model exhibiting a quantum phase transition. In the pure case the magnetic behaviour of this material is well described by mean-field theory. It was believed that when diluted, LiHo_xY_{1-x}F_4 would also manifest itself as a diluted transverse field Ising model which continue to be well described by mean-field theory, and, at sufficient dilution, at zero temperature, exhibit a quantum spin-glass transition. The experimental data did not support such a scenario, and it was pointed out that, to explain physics of LiHo_xY_{1-x}F_4 in transverse magnetic field, the effect of a transverse-field-generated longitudinal random field has to be considered. We explore this idea further in local mean-field studies in which all three parameters: temperature, transverse field and concentration can be consistently surveyed, and where the transverse-field-generated longitudinal random field is explicitly present in the effective spin-1/2 Hamiltonian. We suggest other materials that are possible candidates for studying quantum criticality in the transverse field Ising model, and in the diluted case, for studying the effects of transverse and longitudinal random fields. The compounds we consider are RE(OH)_3, where RE are the rare earth ions Tb^{3+}, Dy^{3+} and Ho^{3+}. Using mean-field theory, we estimate the values of the transverse magnetic field that, at zero temperature, destroy ferromagnetic order to be B_x^c=4.35 T, B_x^c=5.03 T and B_x^c=54.81 T for Ho(OH)_3, Dy(OH)_3 and Tb(OH)_3, respectively. We confirm that Ho(OH)_3 and Tb(OH)_3, similarly to LiHoF_4, can be described by an effective spin-1/2 Hamiltonian. In the case of Dy(OH)_3 there is a possibility of a first order phase transition at transverse field close to B_x^c, and Dy(OH)_3 cannot be described by a spin-1/2 effective Hamiltonian. While diluted dipolar Ising spin glass has been studied experimentally in LiHo_xY_{1-x}F_4 and in numerical simulations, there are no studies of the Heisenberg case. Example materials that are likely candidates to be realisations of the diluted dipolar Heisenberg spin glass are (Gd_xY_{1-x})_2Ti_2O_7, (Gd_xY_{1-x})_2Sn_2O_7 and (Gd_xY_{1-x})_3Ga_5O_{12}. To stimulate interest in experimental studies of these systems we present results of Monte of Carlo simulations of the diluted dipolar Heisenberg spin glass. By performing finite-size scaling analysis of the spin-glass correlation length and the spin-glass susceptibility, we provide a compelling evidence of a thermodynamical spin-glass transition in the model. Frustrated pyrochlore magnets, depending on the character of single ion anisotropy and interplay of different types of interaction over a broad range of energy scales, exhibit a large spectrum of exotic phases and novel phenomena. The pyrochlore antiferromagnet Er_2Ti_2O_7 is characterised by a strong planar anisotropy. Experimental studies reveal that Er_2Ti_2O_7 undergoes a continuous phase transition to a long-range ordered phase with a spin configuration that, in this thesis, is referred to as the Champion-Holdsworth state. Such results are not in agreement with the theoretical prediction that the ground state of the pyrochlore easy-plane antiferromagnet with dipolar interactions complementing the nearest neighbour exchange interactions, is not the Champion-Holdsworth state but the so-called Palmer-Chalker state. On the other hand, Monte Carlo simulations of the easy-plane pyrochlore antiferromagnet indicate a thermal order-by-disorder selection of the Champion-Holdsworth state. To answer the question of whether order-by-disorder selection can be the mechanism at play in Er_2Ti_2O_7, we performed Monte Carlo simulations of the easy-plane pyrochlore antiferromagnet with weak dipolar interactions. We estimate the range strengths of the dipolar interaction such that order-by-disorder selection of the Champion-Holdsworth state is not suppressed. The estimated value of the allowed strength of the dipolar interactions indicates that the model studied is likely insufficient to explain the physics of Er_2Ti_2O_7 and other types of interactions or quantum effects should be considered.
The spin-glass phase in the LiHo x Y 1− x F 4 compound is considered. At zero transverse field this system is well described by the classical Ising model. At finite transverse field deviations from the transverse field quantum Ising model are significant, and one must take properly into account the hyperfine interactions, the off-diagonal terms in the dipolar interactions, and details of the full J = 8 spin Hamiltonian to obtain the correct physical picture. In particular, the system is not a spin glass at finite transverse fields and does not show quantum criticality.
In this work, we investigate the thermodynamic properties of the quantum Blume-Capel model with spin S=5/2 in the presence of transverse and random crystalline fields. The system is described by a Hamiltonian that includes ferromagnetic exchange interactions between nearest neighbors, a longitudinal single-ion anisotropy, and a transverse single-ion anisotropy. Using a mean-field approach based on Bogoliubov's inequality for the Gibbs free energy, we derive the fundamental thermodynamic potential and the equation of state for the magnetization. The influence of the longitudinal and transverse anisotropy parameters on the magnetic ordering and phase transitions is analyzed in detail. We present magnetization versus temperature diagrams for various combinations of the anisotropies, exploring both positive and negative values. Our results reveal that the system exhibits standard second-order phase transitions for most parameter ranges, with no evidence of tricritical behavior. However, for certain positive values of the anisotropies, the model displays a first-order phase transition within the ordered phase, characterized by a jump from a higher-spin ordered state to a lower-spin ordered state. The critical temperatures are shown to be sensitive to the magnitude and sign of the anisotropy parameters. In particular, negative transverse anisotropies favor magnetic order, raising the critical temperature, while positive anisotropies promote disorder, lowering the critical temperature. This study provides a comprehensive analysis of the phase diagram of the S=5/2 quantum Blume-Capel model and highlights the role of transverse fields in modifying the critical behavior.