Dipolar Bose–Einstein condensates in an array of double-well potentials realize an effective transverse Ising model with peculiar inter-layer interactions, that may result under proper conditions in an anomalous first-order ferromagnetic–antiferromagnetic phase transition, and non-trivial phases due to frustration. The considered setup allows as well for the study of Kibble–Zurek defect formation, whose kink statistics follows that expected from the universality class of the mean-field one-dimensional transverse Ising model. Furthermore, random occupation of each layer of the stack leads to random effective Ising interactions and local transverse fields, that may lead to the Anderson-like localization of imbalance perturbations.
Dipolar Bose–Einstein condensates in an array of double-well potentials realize an effective transverse Ising model with peculiar inter-layer interactions, that may result under proper conditions in an anomalous first-order ferromagnetic–antiferromagnetic phase transition, and non-trivial phases due to frustration. The considered setup allows as well for the study of Kibble–Zurek defect formation, whose kink statistics follows that expected from the universality class of the mean-field one-dimensional transverse Ising model. Furthermore, random occupation of each layer of the stack leads to random effective Ising interactions and local transverse fields, that may lead to the Anderson-like localization of imbalance perturbations.
Dipolar Bose-Einstein condensates in an array of double-well potentials realize an effective transverse Ising model with peculiar inter-layer interactions, that may result under proper conditions in an anomalous first-order ferromagnetic-antiferromagnetic phase transition, and non-trivial phases due to frustration. The considered setup allows as well for the study of Kibble-Zurek defect formation, whose kink statistics follows that expected from the universality class of the mean-field one-dimensional transverse Ising model. Furthermore, random occupation of each layer of the stack leads to random effective Ising interactions and local transverse fields, that may lead to the Anderson-like localization of imbalance perturbations.
We analyze the quantum phase transitions taking place in a one-dimensional transverse-field Ising model with long-range couplings that decay algebraically with distance. We are interested in the Kibble-Zurek universal scaling laws emerging in nonequilibrium dynamics and in the potential for the unambiguous observation of such behavior in a realistic experimental setup based on trapped ions. To this end, we determine the phase diagram of the model and the critical exponents characterizing its quantum phase transitions by means of density-matrix renormalization group calculations and finite-size scaling theory, which allows us to obtain good estimates for different range of ferromagnetic and antiferromagnetic interactions. Beyond critical equilibrium properties, we tackle a nonequilibrium scenario in which quantum Kibble-Zurek scaling laws may be retrieved. Here, it is found that the predicted nonequilibrium universal behavior, i.e., the scaling laws as a function of the quench rate and critical exponents, can be observed in systems comprising an experimentally feasible number of spins. Finally, a scheme is introduced to simulate the algebraically decaying couplings accurately by means of a digital quantum simulation with trapped ions. Our results suggest that quantum Kibble-Zurek physics can be explored and observed in state-of-the-art experiments with trapped ions realizing long-range Ising models.
The trapped-ion quantum simulator has demonstrated qualitative properties of different physical models for up to tens of ions.In particular, a linear ion chain naturally hosts long-range Ising interactions under the laser driving, which has been used for various phenomena, such as quantum phase transition, localization, thermalization, and information propagation.For near-term practical usage, a central task is to find more quantitative applications of the noisy quantum simulators that are robust to small errors in the parameters.Here we report the quantum simulation of a long-range transverse-field Ising model using up to 61 ions and probe the critical behavior of its quantum phase transition through the Kibble-Zurek mechanism.By calibrating and verifying the coupling coefficients, we realize the same model for increasing ion numbers, so as to extract a critical exponent free of the finite-size effect.For ferromagnetic interaction, our experimental result agrees with the previous numerical prediction.As for the antiferromagnetic case, signals are too weak to fit a critical exponent due to the frustration in the interaction, but still consistent with the theory.
Phase transitions are ubiquitous in our three-dimensional world. By contrast, most conventional transitions do not occur in infinite uniform low-dimensional systems because of the increased role of thermal fluctuations. The crossover between these situations constitutes an important issue, dramatically illustrated by Bose-Einstein condensation: a gas strongly confined along one direction of space may condense along this direction without exhibiting true long-range order in the perpendicular plane. Here we explore transverse condensation for an atomic gas confined in a novel trapping geometry, with a flat in-plane bottom, and we relate it to the onset of an extended (yet of finite-range) in-plane coherence. By quench crossing the transition, we observe topological defects with a mean number satisfying the universal scaling law predicted by Kibble-Zurek mechanism. The approach described can be extended to investigate the topological phase transitions that take place in planar quantum fluids. Interacting quantum many-body systems in low dimensions is an active research area in ultra-cold gases. Here, Chomaz et al.study the dimensional crossover of Bose–Einstein condensation and observe the emergence of phase coherence in an ultra-cold quasi-2D Bose gas confined to a flat-bottom trapping potential.
We investigate the phase diagram of a bilayer Kitaev honeycomb model with Ising interlayer interactions, deriving effective models via perturbation theory and performing Majorana mean-field theory calculations. We show that a diverse array of magnetic and topological phase transitions occur, depending on the direction of the interlayer Ising interaction and the relative sign of Kitaev interactions. When two layers have the same sign of the Kitaev interaction, a first-order transition from a Kitaev spin liquid to a magnetically ordered state takes place. The magnetic order points along the Ising axis and it is (anti)ferromagnetic for (anti)ferromagnetic Kitaev interactions. However, when two layers have opposite signs of the Kitaev interaction, we observe a notable weakening of magnetic ordering tendencies and the Kitaev spin liquid survives up to a remarkably larger interlayer exchange. Our mean-field analysis suggests the emergence of an intermediate gapped Z2 spin-liquid state, which eventually becomes unstable upon vison condensation. The confined phase is described by a highly frustrated ${120}^{\ensuremath{\circ}}$ compass model. We furthermore use perturbation theory to study the model with the Ising axis pointing along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ axis or lying in the xy plane. In both cases, our analysis reveals the formation of one-dimensional Ising chains, which remain decoupled in perturbation theory, resulting in a subextensive ground-state degeneracy. Our results highlight the interplay between topological order and magnetic ordering tendencies in bilayer quantum spin liquids.
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.