Integrals of Motion for Critical Dense Polymers and Symplectic Fermions

We consider critical dense polymers ${\cal L}(1,2)$. We obtain for this model the eigenvalues of the local integrals of motion of the underlying Conformal Field Theory by means of Thermodynamic Bethe Ansatz. We give a detailed description of the relation between this model and Symplectic Fermions including the indecomposable structure of the transfer matrix. Integrals of motion are defined directly on the lattice in terms of the Temperley Lieb Algebra and their eigenvalues are obtained and expressed as an infinite sum of the eigenvalues of the continuum integrals of motion. An elegant decomposition of the transfer matrix in terms of a finite number of lattice integrals of motion is obtained thus providing a reason for their introduction.Comment: 53 pages, version accepted for publishing on JSTA

A participatory methodology for large scale field trials in the UK

Farmer participation was essential in developing a uniquely useful set of wheat variety trials data on a wide range of organic farms over two years. Although the trials were successful, it became clear that some of the participating farmers felt there were some limitations in the process. These included a lack of ownership in the project and a concern for more researcher help. It was clear that a greater time in-vestment was needed at the start of the project to help with farmer understanding and ownership. De-spite the negative comments, farmers appreciated their involvement, particularly in contrasting their own views and information with that from the wider scene. Farmer participation is essential for systems-level research and this project helped to develop a small core of trained farmers and researchers

Solvable Critical Dense Polymers

A lattice model of critical dense polymers is solved exactly for finite strips. The model is the first member of the principal series of the recently introduced logarithmic minimal models. The key to the solution is a functional equation in the form of an inversion identity satisfied by the commuting double-row transfer matrices. This is established directly in the planar Temperley-Lieb algebra and holds independently of the space of link states on which the transfer matrices act. Different sectors are obtained by acting on link states with s-1 defects where s=1,2,3,... is an extended Kac label. The bulk and boundary free energies and finite-size corrections are obtained from the Euler-Maclaurin formula. The eigenvalues of the transfer matrix are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields a selection rule for the physically relevant solutions to the inversion identity and explicit finitized characters for the associated quasi-rational representations. In particular, in the scaling limit, we confirm the central charge c=-2 and conformal weights Delta_s=((2-s)^2-1)/8 for s=1,2,3,.... We also discuss a diagrammatic implementation of fusion and show with examples how indecomposable representations arise. We examine the structure of these representations and present a conjecture for the general fusion rules within our framework.Comment: 35 pages, v2: comments and references adde

Using legume-based mixtures to enhance the nitrogen use efficiency and economic viability of cropping systems - Final report (LK09106/HGCA3447)

As costs for mineral fertilisers rise, legume-based leys are recognised as a potential alternative nitrogen source for crops. Here we demonstrate that including species-rich legume-based leys in rotations helps to maximise synergies between agricultural productivity and other ecosystem services. By using functionally diverse plant species mixtures, these services can be optimised and fine-tuned to regional and farm-specific needs. Replicated field experiments were conducted over three years at multiple locations, testing the performance of 12 legume species and 4 grass species sown in monocultures, as well as in a mixture of 10 of the legumes and all 4 grasses (called the All Species Mix, ASM). In addition, we compared this complex mixture to farmer-chosen ley mixtures on 34 sites across the UK. The trials showed that there is a large degree of functional complementarity among the legume species. No single species scored high on all evaluation criteria. In particular, the currently most frequently used species, white clover, is outscored by other legume species on a number of parameters such as early development and resistance to decomposition. Further complementarity emerged from the different responses of legume species to environmental variables, with soil pH and grazing or cutting regime being among the more important factors. For example, while large birdsfoot trefoil showed better performance on more acidic soils, the opposite was true for sainfoin, lucerne and black medic. In comparison with the monocultures, the ASM showed increased ground cover, increased above-ground biomass and reduced weed biomass. Benefits of mixing species with regard to productivity increased over time. In addition, the stability of biomass production across sites was greater in the ASM than in the legume monocultures. Within the on-farm trials, we further found that on soils low in organic matter the biomass advantage of the ASM over the Control ley was more marked than on the soils with higher organic matter content. Ecological modelling revealed that the three best multifunctional mixtures all contained black medic, lucerne and red clover. Within the long term New Farming Systems (NFS) rotational study, the use of a clover bi-crop showed improvement to soil characteristics compared to current practice (e.g. bulk density and water infiltration rate). Improvements in wheat yield were also noted with respect to the inclusion of a clover bi-crop in 2010, but there was evidence of a decline in response as the N dose was increased. Cumulatively, over both the wheat crop and the spring oilseed rape crop, the clover bi-crop improved margin over N. The highest average yield response (~9%) was associated with the ASM legume species mix cover cropping approach

Excited Boundary TBA in the Tricritical Ising Model

By considering the continuum scaling limit of the $A_{4}$ RSOS lattice model of Andrews-Baxter-Forrester with integrable boundaries, we derive excited state TBA equations describing the boundary flows of the tricritical Ising model. Fixing the bulk weights to their critical values, the integrable boundary weights admit a parameter $\xi$ which plays the role of the perturbing boundary field $\phi_{1,3}$ and induces the renormalization group flow between boundary fixed points. The boundary TBA equations determining the RG flows are derived in the $\mathcal{B}_{(1,2)}\to \mathcal{B}_{(2,1)}$ example. The induced map between distinct Virasoro characters of the theory are specified in terms of distribution of zeros of the double row transfer matrix.Comment: Latex, 14 pages - Talk given at the Landau meeting "CFT and Integrable Models", Sept. 2002 - v2: some statements about $\phi_{1,2}$ perturbations correcte

W-Extended Fusion Algebra of Critical Percolation

Two-dimensional critical percolation is the member LM(2,3) of the infinite series of Yang-Baxter integrable logarithmic minimal models LM(p,p'). We consider the continuum scaling limit of this lattice model as a `rational' logarithmic conformal field theory with extended W=W_{2,3} symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture. We find that the representation content of the ensuing closed fusion algebra contains 26 W-indecomposable representations with 8 rank-1 representations, 14 rank-2 representations and 4 rank-3 representations. We identify these representations with suitable limits of Yang-Baxter integrable boundary conditions on the lattice and obtain their associated W-extended characters. The latter decompose as finite non-negative sums of W-irreducible characters of which 13 are required. Implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit Cayley table. The closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry.Comment: 30 page