Calculation of the energetics of water incorporation in majorite garnet

Interpretation of lateral variations in upper mantle seismic wave speeds requires constraints on the relationship between elasticity and water concentration at high pressure for all major mantle minerals, including the garnet component. We have calculated the structure and energetics of charge-balanced hydrogen substitution into tetragonal MgSiO3 majorite up to P = 25 GPa using both classical atomistic simulations and complementary first-principles calculations. At the pressure conditions of Earth’s transition zone, hydroxyl groups are predicted to be bound to Si vacancies (o) as the hydrogarnet defect, [oSi+4OHO]X, at the Si2 tetrahedral site or as the [oMg+2OHO]X defect at the octahedral Mg3 site. The hydrogarnet defect is more favorable than the [oMg+2OHO]X defect by 0.8–1.4 eV/H at 20 GPa. The presence of 0.4 wt% Al2O3 substituted into the octahedral sites further increases the likelihood of the hydrogarnet defect by 2.2–2.4 eV/H relative to the [oMg+2OHO]X defect at the Mg3 site. OH defects affect the seismic ratio, R = dlnvs/dlnvp, in MgSiO3 majorite (?R = 0.9–1.2 at 20 GPa for 1400 ppm wt H2O) differently than ringwoodite at high pressure, yet may be indistinguishable from the thermal dlnvs/dlnvp for ringwoodite. The incorporation of 3.2 wt% Al2O3 also decreases R(H2O) by ~0.2–0.4. Therefore, to accurately estimate transition zone compositional and thermal anomalies, hydrous majorite needs to be considered when interpreting seismic body wave anomalies in the transition zone

Topological stability of broken symmetry on fuzzy spheres

We study the spontaneous symmetry breaking of O(3) scalar field on a fuzzy sphere $S_F^2$. We find that the fluctuations in the background of topological configurations are finite. This is in contrast to the fluctuations around a uniform configuration which diverge, due to Mermin-Wagner-Hohenberg-Coleman theorem, leading to the decay of the condensate. Interesting implications of enhanced topological stability of the configurations are pointed out.Comment: Version to appear in MPLA, 9 pages, 6 figure

Monte Carlo approach to nonperturbative strings -- demonstration in noncritical string theory

We show how Monte Carlo approach can be used to study the double scaling limit in matrix models. As an example, we study a solvable hermitian one-matrix model with the double-well potential, which has been identified recently as a dual description of noncritical string theory with worldsheet supersymmetry. This identification utilizes the nonperturbatively stable vacuum unlike its bosonic counterparts, and therefore it provides a complete constructive formulation of string theory. Our data with the matrix size ranging from 8 to 512 show a clear scaling behavior, which enables us to extract the double scaling limit accurately. The ``specific heat'' obtained in this way agrees nicely with the known result obtained by solving the Painleve-II equation with appropriate boundary conditions.Comment: 15 pages, 10 figures, LaTeX, JHEP3.cls; references added, typos correcte

Spontaneous symmetry breakdown in fuzzy spheres

We study and analyse the questions regarding breakdown of global symmetry on noncommutative sphere. We demonstrate this by considering a complex scalar field on a fuzzy sphere and isolating Goldstone modes. We discuss the role of nonlocal interactions present in these through geometrical considerations.Comment: 8 pages, 7 figure

Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere

We address a detailed non-perturbative numerical study of the scalar theory on the fuzzy sphere. We use a novel algorithm which strongly reduces the correlation problems in the matrix update process, and allows the investigation of different regimes of the model in a precise and reliable way. We study the modes associated to different momenta and the role they play in the ``striped phase'', pointing out a consistent interpretation which is corroborated by our data, and which sheds further light on the results obtained in some previous works. Next, we test a quantitative, non-trivial theoretical prediction for this model, which has been formulated in the literature: The existence of an eigenvalue sector characterised by a precise probability density, and the emergence of the phase transition associated with the opening of a gap around the origin in the eigenvalue distribution. The theoretical predictions are confirmed by our numerical results. Finally, we propose a possible method to detect numerically the non-commutative anomaly predicted in a one-loop perturbative analysis of the model, which is expected to induce a distortion of the dispersion relation on the fuzzy sphere.Comment: 1+36 pages, 18 figures; v2: 1+55 pages, 38 figures: added the study of the eigenvalue distribution, added figures, tables and references, typos corrected; v3: 1+20 pages, 10 eps figures, new results, plots and references added, technical details about the tests at small matrix size skipped, version published in JHE

Twisted Covariant Noncommutative Self-dual Gravity

A twisted covariant formulation of noncommutative self-dual gravity is presented. The formulation for constructing twisted noncommutative Yang-Mills theories is used. It is shown that the noncommutative torsion is solved at any order of the $\theta$-expansion in terms of the tetrad and some extra fields of the theory. In the process the first order expansion in $\theta$ for the Pleba\'nski action is explicitly obtained.Comment: 23+1 pages, no figures, corrected typos, references and Appendix B is adde

Probing the fuzzy sphere regularisation in simulations of the 3d \lambda \phi^4 model

We regularise the 3d \lambda \phi^4 model by discretising the Euclidean time and representing the spatial part on a fuzzy sphere. The latter involves a truncated expansion of the field in spherical harmonics. This yields a numerically tractable formulation, which constitutes an unconventional alternative to the lattice. In contrast to the 2d version, the radius R plays an independent r\^{o}le. We explore the phase diagram in terms of R and the cutoff, as well as the parameters m^2 and \lambda. Thus we identify the phases of disorder, uniform order and non-uniform order. We compare the result to the phase diagrams of the 3d model on a non-commutative torus, and of the 2d model on a fuzzy sphere. Our data at strong coupling reproduce accurately the behaviour of a matrix chain, which corresponds to the c=1-model in string theory. This observation enables a conjecture about the thermodynamic limit.Comment: 31 pages, 15 figure

Noncommutative vector bundles over fuzzy CP^N and their covariant derivatives

We generalise the construction of fuzzy CP^N in a manner that allows us to access all noncommutative equivariant complex vector bundles over this space. We give a simplified construction of polarization tensors on S^2 that generalizes to complex projective space, identify Laplacians and natural noncommutative covariant derivative operators that map between the modules that describe noncommuative sections. In the process we find a natural generalization of the Schwinger-Jordan construction to su(n) and identify composite oscillators that obey a Heisenberg algebra on an appropriate Fock space.Comment: 34 pages, v2 contains minor corrections to the published versio

Simulation of a scalar field on a fuzzy sphere

The phi^4 real scalar field theory on a fuzzy sphere is studied numerically. We refine the phase diagram for this model where three distinct phases are known to exist: a uniformly ordered phase, a disordered phase, and a non-uniform ordered phase where the spatial SO(3) symmetry of the round sphere is spontaneously broken and which has no classical equivalent. The three coexistence lines between these phases, which meet at a triple point, are carefully located with particular attention paid to the one between the two ordered phases and the triple point itself. In the neighbourhood of the triple point all phase boundaries are well approximated by straight lines which, surprisingly, have the same scaling. We argue that unless an additional term is added to enhance the effect of the kinetic term the infinite matrix limit of this model will not correspond to a real scalar field on the commutative sphere or plane.Comment: 30 pages, 10 figures, accepted in International Journal of Modern Physics