The random-field Ising model (RFIM) is central to the study of disordered systems. Yet, for a long time it eluded realization in ferromagnetic systems because of the difficulty to produce locally random magnetic fields. Recently, it was shown that in anisotropic dipolar magnetic insulators, the archetype of which is the ${\text{LiHo}}_{x}{\text{Y}}_{1\ensuremath{-}x}{\text{F}}_{4}$ system, the RFIM can be realized in both ferromagnetic and spin-glass phases. The interplay between an applied transverse field and the off-diagonal terms of the dipolar interaction produce effective longitudinal fields, which are random in sign and magnitude as a result of spatial dilution. In this paper, we use exact numerical diagonalization of the full Hamiltonian of Ho pairs in ${\text{LiHo}}_{x}{\text{Y}}_{1\ensuremath{-}x}{\text{F}}_{4}$ to calculate the effective longitudinal field beyond the perturbative regime. In particular, we find that nearby spins can experience an effective field larger than the intrinsic dipolar broadening (of quantum states in zero field) which can therefore be evidenced in experiments. We then calculate the magnetization and susceptibility under several experimental protocols, and show how these protocols can produce direct measurement of the effective longitudinal field.
The random-field Ising model (RFIM) is central to the study of disordered systems. Yet, for a long time it eluded realization in ferromagnetic systems because of the difficulty to produce locally random magnetic fields. Recently, it was shown that in anisotropic dipolar magnetic insulators, the archetype of which is the LiHoxY1\ensuremath−xF4 system, the RFIM can be realized in both ferromagnetic and spin-glass phases. The interplay between an applied transverse field and the off-diagonal terms of the dipolar interaction produce effective longitudinal fields, which are random in sign and magnitude as a result of spatial dilution. In this paper, we use exact numerical diagonalization of the full Hamiltonian of Ho pairs in LiHoxY1\ensuremath−xF4 to calculate the effective longitudinal field beyond the perturbative regime. In particular, we find that nearby spins can experience an effective field larger than the intrinsic dipolar broadening (of quantum states in zero field) which can therefore be evidenced in experiments. We then calculate the magnetization and susceptibility under several experimental protocols, and show how these protocols can produce direct measurement of the effective longitudinal field.
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 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.
Anisotropic dipolar systems are considered. Such systems in an external magnetic field are expected to be a good experimental realization of the transverse field Ising model. With random interactions, this model yields a spin glass to paramagnet phase transition as a function of the transverse field. We show that the off-diagonal dipolar interaction, although effectively reduced, induces a finite correlation length and thus destroys the spin-glass order at any finite transverse field. We thus explain the behavior of the nonlinear susceptibility in the experiments on LiHo(x)Y(1-x)F(4), and argue that a crossover to the paramagnetic phase, and not quantum criticality, is observed.
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.
Microscopics of Quantum Annealing in the Disordered Dipolar Ising Ferromagnet LiHoxY1−xF4
Quantum annealing (QA) refers to an optimization process that uses quantum fluctuations to find the global minimum of a rugged energy landscape with many local minima. Conceptually, QA is often framed in the context of the disordered transverse field Ising model, in which a magnetic field applied perpendicular to the Ising axis tunes the quantum fluctuations and enables the system to tunnel through energy barriers, and hence reach the ground state more quickly. A solid state material closely related to this model, LiHo0.45Y0.55F4, was shown to exhibit faster dynamics after a QA protocol, compared to thermal annealing (TA), but little is known about the actual process of optimization involved or the nature of the spin correlations in the state that is reached. Here, we report on the microscopics of QA in this material using diffuse magnetic neutron scattering. Comparing a QA to a TA protocol that reach the same end-point, we find very similar final diffuse scattering which consists of pinch-point scattering, largely consistent with critical scattering near a phase boundary in the dipolar Ising ferromagnetic model. However, comparing the time evolution at the end of the protocols, we find that the spin correlations evolve more significantly after TA, suggesting that QA produces a state closer to equilibrium. We also observe experimental evidence that the transverse field produces random fields, which had been previously predicted for this material and studied in other contexts. Thus, while the material does exhibit a quantum speedup under quantum annealing conditions, it is not a simple annealing problem; the energy landscape being optimized is changing as the optimization proceeds.