Potts Models on Feynman Diagrams

We investigate numerically and analytically Potts models on ``thin'' random graphs -- generic Feynman diagrams, using the idea that such models may be expressed as the N --> 1 limit of a matrix model. The thin random graphs in this limit are locally tree-like, in distinction to the ``fat'' random graphs that appear in the planar Feynman diagram limit, more familiar from discretized models of two dimensional gravity. The interest of the thin graphs is that they give mean field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree-like structures such as the Bethe lattice. q-state Potts models display a first order transition in the mean field for q>2, so the thin graph Potts models provide a useful test case for exploring discontinuous transitions in mean field theories in which many quantities can be calculated explicitly in the saddle point approximation.Comment: 10 pages, latex, + 6 postscript figure

Phase diagram of the mean field model of simplicial gravity

We discuss the phase diagram of the balls in boxes model, with a varying number of boxes. The model can be regarded as a mean-field model of simplicial gravity. We analyse in detail the case of weights of the form $p(q) = q^{-\beta}$, which correspond to the measure term introduced in the simplicial quantum gravity simulations. The system has two phases~: {\em elongated} ({\em fluid}) and {\em crumpled}. For $\beta\in (2,\infty)$ the transition between these two phases is first order, while for $\beta \in (1,2]$ it is continuous. The transition becomes softer when $\beta$ approaches unity and eventually disappears at $\beta=1$. We then generalise the discussion to an arbitrary set of weights. Finally, we show that if one introduces an additional kinematic bound on the average density of balls per box then a new {\em condensed} phase appears in the phase diagram. It bears some similarity to the {\em crinkled} phase of simplicial gravity discussed recently in models of gravity interacting with matter fields.Comment: 15 pages, 5 figure

Decorating Random Quadrangulations

On various regular lattices (simple cubic, body centred cubic..) decorating an edge with an Ising spin coupled by bonds of strength L to the original vertex spins and competing with a direct anti-ferromagnetic bond of strength alpha L can give rise to three transition temperatures for suitable alpha. The system passes through ferromagnetic, paramagnetic, anti-ferromagnetic and paramagnetic phases respectively as the temperature is increased. For the square lattice on the other hand multiple decoration is required to see this effect. We note here that a single decoration suffices for the Ising model on planar random quadrangulations (coupled to 2D quantum gravity). Other random bipartite lattices such as the Penrose tiling are more akin to the regular square lattice and require multiple decoration to have any affect.Comment: 6 pages + 5 figure

The effects of arbuscular mycorrhizal fungi (AMF) and Rhizophagus irregularis on soil microorganisms assessed by metatranscriptomics and metaproteomics

Arbuscular mycorrhizal fungi (AMF) form symbioses with approximately 80% of plant species and potentially benefit their hosts (e.g. nutrient acquisition) and the soil environment (e.g. soil aggregation). AMF also affect soil microbiota and soil multifunctionality. We manipulated AMF presence (via inoculation of non-sterile soil with Rhizophagus irregularis and using a hyphal compartment design) and used RNA-seq and metaproteomics to assess AMF roles in soil. The results indicated that AMF drove an active soil microbial community expressing transcripts and proteins related to nine metabolic functions, including the metabolism of C and N. We suggest two possible mechanisms: 1) the AMF hyphae produce exudates that select a beneficial community, or, 2) the hyphae compete with other soil microbes for available nutrients and consequently induce the community to mineralize nutrients from soil organic matter. We also identified candidate proteins that are potentially related to soil aggregation, such as Lpt and HSP60. Our results bridge microbial ecology and ecosystem functioning. We show that the AMF hyphosphere contains an active community related to soil respiration and nutrient cycling, thus potentially improving nutrient mineralization from soil organic matter and nutrient supply to the plants

Potts Models with (17) Invisible States on Thin Graphs

The order of a phase transition is usually determined by the nature of the symmetry breaking at the phase transition point and the dimension of the model under consideration. For instance, q-state Potts models in two dimensions display a second order, continuous transition for q = 2,3,4 and first order for higher q. Tamura et al recently introduced Potts models with "invisible" states which contribute to the entropy but not the internal energy and noted that adding such invisible states could transmute continuous transitions into first order transitions. This was observed both in a Bragg-Williams type mean-field calculation and 2D Monte-Carlo simulations. It was suggested that the invisible state mechanism for transmuting the order of a transition might play a role where transition orders inconsistent with the usual scheme had been observed. In this paper we note that an alternative mean-field approach employing 3-regular random ("thin") graphs also displays this change in the order of the transition as the number of invisible states is varied, although the number of states required to effect the transmutation, 17 invisible states when there are 2 visible states, is much higher than in the Bragg-Williams case. The calculation proceeds by using the equivalence of the Potts model with 2 visible and r invisible states to the Blume-Emery-Griffiths (BEG) model, so a by-product is the solution of the BEG model on thin random graphs.Comment: (2) Minor typos corrected, references update

The Gonihedric Ising Model and Glassiness

The Gonihedric 3D Ising model is a lattice spin model in which planar Peierls boundaries between + and - spins can be created at zero energy cost. Instead of weighting the area of Peierls boundaries as the case for the usual 3D Ising model with nearest neighbour interactions, the edges, or "bends" in an interface are weighted, a concept which is related to the intrinsic curvature of the boundaries in the continuum. In these notes we follow a roughly chronological order by first reviewing the background to the formulation of the model, before moving on to the elucidation of the equilibrium phase diagram by various means and then to investigation of the non-equilibrium, glassy behaviour of the model.Comment: To appear as Chapter 7 in Rugged Free-Energy Landscapes - An Introduction, Springer Lecture Notes in Physics, 736, ed. W. Janke, (2008

Magnetic properties of strongly disordered electronic systems

We present a unified, global perspective on the magnetic properties of strongly disordered electronic systems, with special emphasis on the case where the ground state is metallic. We review the arguments for the instability of the disordered Fermi liquid state towards the formation of local magnetic moments, and argue that their singular low temperature thermodynamics are the ``quantum Griffiths'' precursors of the quantum phase transition to a metallic spin glass; the local moment formation is therefore not directly related to the metal-insulator transition. We also review the the mean-field theory of the disordered Fermi liquid to metallic spin glass transition and describe the separate regime of ``non-Fermi liquid'' behavior at higher temperatures near the quantum critical point. The relationship to experimental results on doped semiconductors and heavy-fermion compounds is noted.Comment: 25 pages; Contribution to the Royal Society Discussion Meeting on "The Metal-Non Metal Transition in Macroscopic and Microscopic Systems", March 5-6, 199