Corner transfer matrices in statistical mechanics

Corner transfer matrices are a useful tool in the statistical mechanics of simple two-dimensinal models. They can be very effective way of obtaining series expansions of unsolved models, and of calculating the order parameters of solved ones. Here we review these features and discuss the reason why the method fails to give the order parameter of the chiral Potts model.Comment: 18 pages, 4 figures, for Proceedings of Conference on Symmetries and Integrability of Difference Equations. (SIDE VII), Melbourne, July 200

Mathematics, statistics and archaeometry: the past 50 years or so

This review of developments in the use of mathematics and statistics in archaeometry over the past 50 years is partial, personal and 'broad-brush'. The view is expressed that it is in the past 30 years or so that the major developments have taken place. The view is also expressed that, with the exception of methods for analysing radiocarbon dates and increased computational power, mathematical and statistical methods that are currently used, and found to be useful in widespread areas of application such as provenance studies, don't differ fundamentally from what was being done 30 years ago

Two-dimensional Rydberg gases and the quantum hard squares model

We study a two-dimensional lattice gas of atoms that are photo-excited to high-lying Rydberg states in which they interact via the van-der-Waals interaction. We explore the regime of dominant nearest neighbor interaction where this system is intimately connected to a quantum version of Baxter's hard squares model. We show that the strongly correlated ground state of the Rydberg gas can be analytically described by a projected entangled pair state that constitutes the ground state of the quantum hard squares model. This correspondence allows us to identify a first order phase boundary where the Rydberg gas undergoes a transition from a disordered (liquid) phase to an ordered (solid) phase

Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices

We present an efficient algorithm for computing the partition function of the q-colouring problem (chromatic polynomial) on regular two-dimensional lattice strips. Our construction involves writing the transfer matrix as a product of sparse matrices, each of dimension ~ 3^m, where m is the number of lattice spacings across the strip. As a specific application, we obtain the large-q series of the bulk, surface and corner free energies of the chromatic polynomial. This extends the existing series for the square lattice by 32 terms, to order q^{-79}. On the triangular lattice, we verify Baxter's analytical expression for the bulk free energy (to order q^{-40}), and we are able to conjecture exact product formulae for the surface and corner free energies.Comment: 17 pages. Version 2: added 4 further term to the serie

Neogene paleoceanography of the eastern equatorial Pacific based on the radiolarian record of IODP drill sites off Costa Rica

The Integrated Ocean Drilling Program (IODP) Expedition 344 drilled cores following a transect across the convergent margin off Costa Rica. Two of the five sites (U1381 and U1414) are the subject of the present study. Major radiolarian faunal breaks and characteristic species groups were defined with the aid of cluster analysis, nodal analysis, and discriminant analysis of principal components. A middle-late Miocene to Pleistocene age (radiolarian zones RN5 to RN16) was determined for the sites, which agrees with the nannofossil zonations and 40Ar/39Ar and tephra layers. Considering the northward movement of the Cocos plate (∼7.3 cm/yr), and a paleolatitude calculator, it is assumed that during the Miocene the two sites were located ∼1000 km to the southwest of their current position, slightly south of the equator. The radiolarian faunas retrieved were thus seemingly formed under the influence of different oceanic currents and sources of nutrients. Changes in the radiolarian assemblages at Site U1414 point at dissimilar environmental settings associated with the colder South Equatorial Current and the warmer Equatorial Countercurrent, as well as to coastal upwelling. These differences are best reflected by changes in the abundance of the morphotype Spongurus spp., with noticeably higher values during the Miocene, than in the Pliocene and the Pleistocene. Because Spongurus spp. is generally associated with cooler waters, these abundance variations (as well as those of several other species) suggest that during the Miocene the area had a stronger influence of colder waters than during younger periods. During the Pliocene and the lowermost Pleistocene, biogenic remains are scarce, presumably due to the terrigenous input, which could have diluted and affected the preservation of pelagic fossils, as well as to the displacement of the site to warmer waters. A typically tropical fauna characterized the Pleistocene, yet with widespread presence of colder water species, most probably indicative of the influence of coastal upwelling processes.Fil: Sandoval, María I.. Universidad de Costa Rica; Costa Rica. Universite de Lausanne; SuizaFil: Boltovskoy, Demetrio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Ecología, Genética y Evolución de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Ecología, Genética y Evolución de Buenos Aires; ArgentinaFil: Baxter, Alan T.. University of New England; Australia. McGill University; CanadáFil: Baumgartner, Peter O.. Universite de Lausanne; Suiz

Extended Scaling for the high dimension and square lattice Ising Ferromagnets

In the high dimension (mean field) limit the susceptibility and the second moment correlation length of the Ising ferromagnet depend on temperature as chi(T)=tau^{-1} and xi(T)=T^{-1/2}tau^{-1/2} exactly over the entire temperature range above the critical temperature T_c, with the scaling variable tau=(T-T_c)/T. For finite dimension ferromagnets temperature dependent effective exponents can be defined over all T using the same expressions. For the canonical two dimensional square lattice Ising ferromagnet it is shown that compact "extended scaling" expressions analogous to the high dimensional limit forms give accurate approximations to the true temperature dependencies, again over the entire temperature range from T_c to infinity. Within this approach there is no cross-over temperature in finite dimensions above which mean-field-like behavior sets in.Comment: 6 pages, 6 figure

Comment on `Series expansions from the corner transfer matrix renormalization group method: the hard-squares model'

Earlier this year Chan extended the low-density series for the hard-squares partition function $\kappa(z)$ to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity $z_d$ which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at $z_d$ with exponents $\theta=0.83333(2)$ and $\theta'= 1.6676(3)$. We thus confirm that the exponent $\theta$ has the exact value $\frac56$ as observed by Dhar.Comment: 5 pages, 1 figure, IoP macros. Expanded second and final versio

Scaling of Entanglement Entropy in the Random Singlet Phase

We present numerical evidences for the logarithmic scaling of the entanglement entropy in critical random spin chains. Very large scale exact diagonalizations performed at the critical XX point up to L=2000 spins 1/2 lead to a perfect agreement with recent real-space renormalization-group predictions of Refael and Moore [Phys. Rev. Lett. {\bf 93}, 260602 (2004)] for the logarithmic scaling of the entanglement entropy in the Random Singlet Phase with an effective central charge ${\tilde{c}}=c\times \ln 2$. Moreover we provide the first visual proof of the existence the Random Singlet Phase thanks to the quantum entanglement concept.Comment: 4 pages, 3 figure

Resource recovery and remediation of highly alkaline residues : a political-industrial ecology approach to building a circular economy

Highly alkaline industrial residues (e.g., steel slag, bauxite processing residue (red mud) and ash from coal combustion) have been identified as stocks of potentially valuable metals. Technological change has created demand for metals, such as vanadium and certain rare earth elements, in electronics associated with renewable energy generation and storage. Current raw material and circular economy policy initiatives in the EU and industrial ecology research all promote resource recovery from residues, with research so far primarily from an environmental science perspective. This paper begins to address the deficit of research into the governance of resource recovery from a novel situation where re-use involves extraction of a component from a bulk residue that itself represents a risk to the environment. Taking a political industrial ecology approach, we briefly present emerging techniques for recovery and consider their regulatory implications in the light of potential environmental impacts. The paper draws on EU and UK regulatory framework for these residues along with semi-structured interviews with industry and regulatory bodies. A complex picture emerges of entwined ownerships and responsibilities for residues, with past practice and policy having a lasting impact on current possibilities for resource recovery

Critical and Tricritical Hard Objects on Bicolorable Random Lattices: Exact Solutions

We address the general problem of hard objects on random lattices, and emphasize the crucial role played by the colorability of the lattices to ensure the existence of a crystallization transition. We first solve explicitly the naive (colorless) random-lattice version of the hard-square model and find that the only matter critical point is the non-unitary Lee-Yang edge singularity. We then show how to restore the crystallization transition of the hard-square model by considering the same model on bicolored random lattices. Solving this model exactly, we show moreover that the crystallization transition point lies in the universality class of the Ising model coupled to 2D quantum gravity. We finally extend our analysis to a new two-particle exclusion model, whose regular lattice version involves hard squares of two different sizes. The exact solution of this model on bicolorable random lattices displays a phase diagram with two (continuous and discontinuous) crystallization transition lines meeting at a higher order critical point, in the universality class of the tricritical Ising model coupled to 2D quantum gravity.Comment: 48 pages, 13 figures, tex, harvmac, eps