Intergranular diffusion rates from the analysis of garnet surfaces: implications for metamorphic equilibration

Novel approaches to garnet analysis have been used to assess rates of intergranular diffusion between different matrix phases and garnet porphyroblasts in a regionally metamorphosed staurolite-mica-schist from the Barrovian-type area in Scotland. X-ray maps and chemical traverses of planar porphyroblast surfaces reveal chemical heterogeneity of the garnet grain boundary linked to the nature of the adjacent matrix phase. The garnet preserves evidence of low temperature retrograde exchange with matrix minerals and diffusion profiles documenting cation movement along the garnet boundaries. Garnet–quartz and garnet–plagioclase boundaries preserve evidence of sluggish Mg, Mn and Fe diffusion at comparable rates to volume diffusion in garnet, whereas diffusion along garnet–biotite interfaces is much more effective. Evidence of particularly slow Al transport, probably coupled to Fe3+ exchange, is locally preserved on garnet surfaces adjacent to Fe-oxide phases. The Ca distribution on the garnet surface shows the most complex behaviour, with long-wavelength heterogeneities apparently unrelated to the matrix grain boundaries. This implies that the Ca content of garnet is controlled by local availability and is thought likely to reflect disequilibrium established during garnet growth. Geochemical anomalies on the garnet surfaces are also linked to the location of triple junctions between the porphyroblasts and the matrix phases, and imply enhanced transport along these channels. The slow rates of intergranular diffusion and the characteristics of different boundary types may explain many features associated with the prograde growth of garnet porphyroblasts. Thus, minerals such as quartz, Fe-oxides and plagioclase whose boundaries with garnet are characterized by slow intergranular diffusion rates appear to be preferentially trapped as inclusions within porphyroblasts. As such grain boundary diffusion rates may be a significant kinetic impediment to metamorphic equilibrium and garnet may struggle to maintain chemical and textural equilibrium during growth in pelites

The degree of commutativity and lamplighter groups

The degree of commutativity of a group $G$ measures the probability of choosing two elements in $G$ which commute. There are many results studying this for finite groups. In [AMV17], this was generalised to infinite groups. In this note, we compute the degree of commutativity for wreath products of the form $\mathbb{Z}\wr \mathbb{Z}$ and $F\wr \mathbb{Z}$ where $F$ is any finite group.Comment: 9 pages, accepted, International Journal of Algebra and Computation (IJAC

Supersymmetry and the Hartmann Potential of Theoretical Chemistry

The ring-shaped Hartmann potential $V = \eta \sigma^{2} \epsilon_{0} \left( \frac{2 a_{0}}{r} - \frac{\eta a_{0}^{2}}{r^{2} sin^{2} \theta} \right)$ was introduced in quantum chemistry to describe ring-shaped molecules like benzene. In this article, the supersymmetric features of the Hartmann potential are discussed. We first review the results of a previous paper in which we rederived the eigenvalues and radial eigenfunctions of the Hartmann potential using a formulation of one-dimensional supersymmetric quantum mechanics (SUSYQM) on the half-line $\left[ 0, \infty \right)$. A reformulation of SUSYQM in the full line $\left( -\infty, \infty \right)$ is subsequently developed. It is found that the second formulation makes a connection between states having the same quantum number $L$ but different values of $\eta \sigma^{2}$ and quantum number $N$. This is in contrast to the first formulation, which relates states with identical values of the quantum number $N$ and $\eta \sigma^{2}$ but different values of the quantum number $L$.Comment: 24 pages; uses LaTex; to be published at the Theoretica Chimica Acta; hard copy available from the author upon request (use address: [email protected]

Density Matrix Renormalization Group Lagrangians

We introduce a Lagrangian formulation of the Density Matrix Renormalization Group (DMRG). We present Lagrangians which when minimised yield the optimal DMRG wavefunction in a variational sense, both within the general matrix product ansatz, as well as within the canonical form of the matrix product that is constructed within the DMRG sweep algorithm. Some of the results obtained are similar to elementary expressions in Hartree-Fock theory, and we draw attention to such analogies. The Lagrangians introduced here will be useful in developing theories of analytic response and derivatives in the DMRG.Comment: 6 page

Chemical and textural equilibration of garnet during amphibolite-facies metamorphism: The influence of coupled dissolution-reprecipitation

Metamorphic equilibration requires chemical communication between minerals and may be inhibited through sluggish volume diffusion and or slow rates of dissolution in a fluid phase. Relatively slow diffusion and the perceived robust nature of chemical growth zoning may preclude garnet porphyroblasts from readily participating in low temperature amphibolite-facies metamorphic reactions. Garnet is widely assumed to be a reactant in staurolite-isograd reactions, and the evidence for this has been assessed in the Late Proterozoic Dalradian pelitic schists of the Scottish Highlands. Three-D imaging of garnet porphyroblasts in staurolite-bearing schists reveal a good crystal shape and little evidence of marginal dissolution, however there is also lack of evidence for the involvement of either chlorite or chloritoid in the reaction. Staurolite forms directly adjacent to the garnet, and its nucleation is strongly associated with deformation of the muscovite-rich fabrics around the porphyroblasts. “Cloudy” fluid inclusion-rich garnet forms in both marginal and internal parts of the garnet porphyroblast and is linked both to the production of staurolite and to the introduction of abundant quartz inclusions within the garnet. Such cloudy garnet typically has a Mg-rich, Mn-poor composition and is interpreted to have formed during a coupled dissolution-reprecipitation process, triggered by a local influx of fluid. All garnet in the muscovite-bearing schists present in this area is potentially reactive, irrespective of the garnet composition, but very few of the schists contain staurolite. The staurolite-producing reaction appears to be substantially overstepped during the relatively high pressure Barrovian regional metamorphism reflecting the limited permeability of the schists in peak metamorphic conditions. Fluid influx and hence reaction progress appear to be strongly controlled by subtle differences in deformation history. The remaining garnet fails to achieve chemical equilibrium during the reaction creating distinctive patchy compositional zoning. Such zoning in metamorphic garnet created during coupled dissolution-reprecipitation reactions may be difficult to recognize in higher grade pelites due to subsequent diffusive re-equilibration. Fundamental assumptions about metamorphic processes are questioned by the lack of chemical equilibrium during this reaction and the restricted permeability of the regional metamorphic pelitic schists. In addition the partial loss of prograde chemical and textural information from the garnet porphyroblasts cautions against their routine use as a reliable monitor of metamorphic history. However the partial re-equilibration of the porphyroblasts during coupled dissolution-reprecipitation opens possibilities of mapping reaction progress in garnet as a means of assessing fluid access during peak metamorphic conditions

Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm

We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general molecules. We describe an optimal tree tensor network state algorithm for quantum chemistry. We introduce the concept of half-renormalization which greatly improves the efficiency of the calculations. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states versus matrix product states. We carry out benchmark calculations both on tree systems (hydrogen trees and \pi-conjugated dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree molecules, whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling associated with tree algorithms. In tree like molecules, tree network states are easily superior to matrix product states. As an ilustration, our largest dendrimer calculation with tree tensor network states correlates 110 electrons in 110 active orbitals.Comment: 15 pages, 19 figure

First principles coupled cluster theory of the electronic spectrum of the transition metal dichalcogenides

The electronic properties of two-dimensional transition metal dichalcogenides (2D TMDs) have attracted much attention during the last decade. We show how a diagrammatic ab initio coupled cluster singles and doubles (CCSD) treatment paired with a careful thermodynamic limit extrapolation in two dimensions can be used to obtain converged band gaps for monolayer materials in the MoS₂ family. We find CCSD gaps to lie in the upper range of the spread of GW approximation based on density functional theory (DFT) simulations, and also find slightly higher effective hole masses compared to previous reports. We also investigate the ability of CCSD to describe trion states, finding a reasonable qualitative structure, but poor excitation energies due to the lack of screening of three-particle excitations in the effective Hamiltonian. Our study provides an independent high-level benchmark of the role of many-body effects in 2D TMDs and showcases the potential strengths and weaknesses of diagrammatic coupled cluster approaches for realistic materials