Desmond A. Johnston

Heriot-Watt University · School of Mathematical and Computer Sciences

We investigate whether the dynamical lattice supersymmetry discussed for various Hamiltonians, including one-dimensional quantum spin chains, by Fendley et.al. and Hagendorf et.al. might also exist for the Markov matrices of any one-dimensional exclusion processes, since these can be related by conjugation to quantum spin chain Hamiltonians. We fin...

We review some recent investigations of the 3d plaquette Ising model. This displays a strong first-order phase transition with unusual scaling properties due to the size-dependent degeneracy of the low-temperature phase. In particular, the leading scaling correction is modified from the usual inverse volume behaviour 1/L^3 to 1/L^2. The degeneracy...

An anisotropic limit of the 3d plaquette Ising model, in which the plaquette couplings in one direction were set to zero, was solved for free boundary conditions by Suzuki (1972) [1], who later dubbed it the fuki-nuke, or “no-ceiling”, model. Defining new spin variables as the product of nearest-neighbour spins transforms the Hamiltonian into that...

An anisotropic limit of the 3d plaquette Ising model, in which the plaquette couplings in one direction were set to zero, was solved for free boundary conditions by Suzuki (Phys.Rev.Lett. 28 (1972) 507), who later dubbed it the fuki-nuke, or "no-ceiling", model. Defining new spin variables as the product of nearest-neighbour spins transforms the Ha...

A well-known feature of first-order phase transitions is that fixed boundary conditions can strongly influence finite-size corrections, modifying the leading corrections for an L 3 lattice in 3d from order 1/L 3 under periodic boundary conditions to 1/L. A rather similar effect, albeit of completely different origin, occurs when the system possesse...

The purely plaquette 3d Ising Hamiltonian with the spins living at the vertices of a cubic lattice displays several interesting features. The symmetries of the model lead to a macroscopic degeneracy of the low-temperature phase and prevent the definition of a standard magnetic order parameter. Consideration of the strongly anisotropic limit of the...

In this paper we conduct a careful multicanonical simulation of the isotropic $3d$ plaquette ("gonihedric") Ising model and confirm that a planar, fuki-nuke type order characterises the low-temperature phase of the model. From consideration of the anisotropic limit of the model we define a class of order parameters which can distinguish the low- an...

The three-dimensional purely plaquette gonihedric Ising model and its dual are investigated to resolve inconsistencies in the literature for the values of the inverse transition temperature of the very strong temperature-driven first-order phase transition that is apparent in the system. Multicanonical simulations of this model allow us to measure...

In a recent paper Lipstein and Reid-Edwards discussed a discretized formulation of 2-form abelian and non-abelian gauge fields on d-dimensional hypercubic lattices. In this note we recall that the Hamiltonian of a $\mathbb{Z}_2$ variant of such theories is one of the family of generalized Ising models originally considered by Wegner. For such "$\ma...

We note that the standard inverse system volume scaling for finite-size corrections at a first-order phase transition (i.e., 1/L^3 for an L×L×L lattice in 3D) is transmuted to 1/L^2 scaling if there is an exponential low-temperature phase degeneracy. The gonihedric Ising model which has a four-spin interaction, plaquette Hamiltonian provides an exe...

It is known that fixed boundary conditions modify the leading finite-size corrections for an L^3 lattice in 3d at a first-order phase transition from 1/L^3 to 1/L. We note that an exponential low-temperature phase degeneracy of the form 23L will lead to a different leading correction of order 1/L^2. A 3d gonihedric Ising model with a four-spin inte...

The number of so-called invisible states which need to be added to the q-state Potts model to transmute its phase transition from continuous to first order has attracted recent attention. In the q=2 case, a Bragg-Williams, mean-field approach necessitates four such invisible states while a 3-regular, random-graph formalism requires seventeen. In bo...

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...

A 3D Ising model with a purely plaquette, 4-spin interaction displays a planar flip symmetry intermediate between a global and a gauge symmetry and as a consequence has a highly degenerate low temperature phase and no standard magnetic order parameter. This plaquette Hamiltonian is a particular case of a family of 3D Gonihedric Ising models defined...

It is clear from both the non-perturbative and perturbative approaches to two-dimensional quantum gravity that a new strong coupling regime is setting in at d = 1, independent of the genus of the worldsheet being considered. It has been suggested that a Kosterlitz–Thouless (KT) phase transition in the Liouville theory is the cause of this behavior....

We perform Monte-Carlo simulations using the Wolff cluster algorithm of the q=2 (Ising), 3, 4 and q=10 Potts models on dynamical phi-cubed graphs of spherical topology with up to 5000 nodes. We find that the measured critical exponents are in reasonable agreement with those from the exact solution of the Ising model and with those calculated from K...

Earlier simulations of dynamically triangulated random surfaces with a pure Gaussian (Polyakov) action have suggested that the incorporation of a term which is equivalent to the square of the scalar curvature, R2, in the continuum can affect the properties of the surfaces, despite the fact that such a term appears to be irrelevant on dimensional gr...

We note that two formulations of dual gonihedric Ising models in 3d, one based on using Wegner's general framework for duality to construct a dual Hamiltonian for codimension one surfaces, the other on constructing a dual Hamiltonian for two-dimensional surfaces, are related by a variant of the standard decoration/iteration transformation. The dual...

We investigate the dual of the κ = 0 gonihedric Ising model on a 3D cubic lattice, which may be written as an anisotropically coupled Ashkin–Teller model. The original κ = 0 gonihedric model has a purely plaquette interaction, displays a first order transition and possesses a highly degenerate ground state. We find that the dual model admits a simi...

The gonihedric Ising Hamiltonians defined in three and higher dimensions by Savvidy and Wegner provide an extensive, and little explored, catalogue of spin models on (hyper)cubic lattices with many interesting features. In three dimensions the kappa=0 gonihedric Ising model on a cubic lattice has been shown to possess a degenerate low-temperature p...

The application of information geometric ideas to statistical mechanics using a metric on the space of states, pioneered by Ruppeiner and Weinhold, has proved to be a useful alternative approach to characterizing phase transitions. Some puzzling anomalies become apparent, however, when these methods are applied to the study of black hole thermodyna...

We note that a tridiagonal matrix representation of the algebra of the partially asymmetric exclusion process (PASEP) lends itself to interpretation as the transfer matrix for weighted Motzkin lattice paths. A continued-fraction (‘J fraction’) representation of the lattice-path-generating function is particularly well suited to discussing the PASEP...

We review a class of 3D lattice spin models in which planar Peierls boundaries between + and - spins can be created at zero energy cost. These so-called Gonihedric Ising models have (in general) specially tuned nearest neighbour, next-to-nearest neighbour and plaquette interactions, which endow the models with some novel properties both in and out...

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 re...

We consider the effect of geometric frustration induced by the random distribution of loop lengths in the "fat" graphs of the dynamical triangulations model on coupled antiferromagnets. While the influence of such connectivity disorder is rather mild for ferromagnets in that an ordered phase persists and only the properties of the phase transition...

Multiplicative logarithmic corrections frequently characterize critical behavior in statistical physics. Here, a recently proposed theory relating the exponents of such terms is extended to account for circumstances which often occur when the leading specific-heat critical exponent vanishes. Also, the theory is widened to encompass the correlation...

DOI:https://doi.org/10.1103/PhysRevLett.97.169901

Multiplicative logarithmic corrections to scaling are frequently encountered in the critical behavior of certain statistical-mechanical systems. Here, a Lee-Yang zero approach is used to systematically analyze the exponents of such logarithms and to propose scaling relations between them. These proposed relations are then confronted with a variety...

All of the thermodynamic information on a statistical mechanical system is encoded in the locus and density of its partition function zeroes. Recently, a new technique was developed which enables the extraction of the latter using finite-size data of the type typically garnered from a computational approach. Here that method is extended to deal wit...

There is only limited experimental evidence for the existence in nature of phase transitions of Ehrenfest order greater than two. However, there is no physical reason for their non-existence, and such transitions certainly exist in a number of theoretical models in statistical physics and lattice field theory. Here, higher-order transitions are ana...

This paper gives a brief introduction to using two-dimensional discrete and Euclidean quantum gravity approaches as a laboratory for studying the properties of fluctuating and frozen random graphs in interaction with "matter fields" represented by simple spin or vertex models. Due to the existence of numerous exact analytical results and prediction...

Experimental evidence for the existence of strictly higher-order phase transitions (of order three or above in the Ehrenfest sense) is tenuous at best. However, there is no known physical reason why such transitions should not exist in nature. Here, higher-order transitions characterized by both discontinuities and divergences are analysed through...

It has recently been observed that the normalization of a one-dimensional out-of-equilibrium model, the asymmetric exclusion process (ASEP) with random sequential dynamics, is exactly equivalent to the partition function of a two-dimensional lattice path model of one-transit walks, or equivalently Dyck paths. This explains the applicability of the...

The gonihedric Ising model is a particular case of the class of models defined by Savvidy and Wegner intended as discrete versions of string theories on cubic lattices. In this Letter we perform a high statistics analysis of the phase transition exhibited by the 3d gonihedric Ising model with k=0 in the light of a set of recently stated scaling law...

The one-dimensional asymmetric exclusion process (ASEP) is a paradigm for nonequilibrium dynamics, in particular driven diffusive processes. It is usually considered in a canonical ensemble in which the number of sites is fixed. We observe that the grand-canonical partition function for the ASEP is remarkably simple. It allows a simple direct deriv...

The introduction of a metric onto the space of parameters in models in statistical mechanics and beyond gives an alternative perspective on their phase structure. In such a geometrisation, the scalar curvature, , plays a central role. A non-interacting model has a flat geometry , while diverges at the critical point of an interacting one. Here, the...

A recently developed technique for the determination of the density of partition function zeroes using data coming from finite-size systems is extended to deal with cases where the zeroes are not restricted to a curve in the complex plane and/or come in degenerate sets. The efficacy of the approach is demonstrated by application to a number of mode...

Although the notion of entropy lies at the core of statistical mechanics, it is not often used in statistical mechanical models to characterize phase transitions, a role more usually played by quantities such as various order parameters, specific heats or suscept ibilities. The relative entropy induces a metric, the so-called information or Fisher-...

Motivated by the observation that geometrizing statistical mechanics offers an interesting alternative to more standard approaches, we calculate the scaling behavior of the curvature R of the information geometry metric for the spherical model. We find that R approximately epsilon(-2), where epsilon=beta(c)-beta is the distance from criticality. Th...

It has been suggested that an information geometric view of statistical mechanics in which a metric is introduced onto the space of parameters provides an interesting alternative characterization of the phase structure, particularly in the case where there are two such parameters, such as the Ising model with inverse temperature beta and external f...

The identification of phase transition points, beta(c), with the percolation thresholds of suitably defined clusters of spins has proved immensely fruitful in many areas of statistical mechanics. Some time ago, Kertesz suggested that such percolation thresholds for models defined in field might also have measurable physical consequences for regions...

A recently developed technique to determine the order and strength of phase transitions by extracting the density of partition function zeroes (a continuous function) from finite-size systems (a discrete data set) is generalized to systems for which (i) some or all of the zeroes occur in degenerate sets and/or (ii) they are not confined to a singul...

In various statistical-mechanical models the introduction of a metric onto the space of parameters (e.g. the temperature variable, $\beta$, and the external field variable, $h$, in the case of spin models) gives an alternative perspective on the phase structure. For the one-dimensional Ising model the scalar curvature, ${\cal R}$, of this metric ca...

It is known that the exact renormalization transformations for the one-dimensional Ising model in a field can be cast in the form of the logistic map f(x)=4x(1-x) with x a function of the Ising couplings K and h. The locus of the Lee-Yang zeros for the one-dimensional Ising model in the K,h plane is given by the Julia set of the logistic map. In th...

We study damage spreading in the ferromagnetic Ising model on small world networks using Monte Carlo simulation with Glauber dynamics. The damage spreading temperature T(d) is determined as a function of rewiring probability p for small world networks obtained by rewiring the two-dimensional square and three dimensional cubic lattices. We find that...

We discuss a Pareto macroeconomy (a) in a closed system with fixed total wealth and (b) in an open system with average mean wealth, and compare our results to a similar analysis in a super-open system (c) with unbounded wealth [J.-P. Bouchaud and M. Mézard, Physica A 282, 536 (2000)]. Wealth condensation takes place in the social phase for closed a...

On various regular lattices (simple cubic, body centred cubic, etc) 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 alphaL can give rise to three transition temperatures for suitable alpha. The system passes through ferromagnetic, param...

We study a class of homogeneous finite-dimensional Ising models which were recently shown to exhibit glassy properties. Monte Carlo simulations of a particular three-dimensional model in this class show that the glassy phase obtained under slow cooling is dominated by large-scale excitations whose energy E(l) scales with their size l as E(l) approx...

Four dimensional simplicial gravity has been studied by means of Monte Carlo simulations for some time 1 and an extensive numerical documentation of the properties of the model has been gathered. The main outcome of the studies is that the model undergoes a discontinuous phase transition 2 between the elongated and the crumpled phase when one chang...

We show that it is possible to determine the locus of Fisher zeroes in the thermodynamic limit for the Ising model on planar (``fat'') phi4 random graphs and their dual quadrangulations by matching up the real part of the high- and low-temperature branches of the expression for the free energy. Similar methods work for the mean-field model on gener...

Using Monte Carlo simulations we study crystallization in the three-dimensional Ising model with four-spin interaction. We monitor the morphology of crystals which grow after placing crystallization seeds in a supercooled liquid. Defects in such crystals constitute an intricate and very stable network that separates various domains by tensionless d...

Various authors have suggested that the loci of partition function zeros can profitably be regarded as phase boundaries in the complex temperature or field planes. We obtain the Fisher zeros for Ising and Potts models on non-planar (`thin') regular random graphs using this approach, and note that the locus of Fisher zeros on a Bethe lattice is iden...

We show that it is possible to determine the locus of Fisher zeroes in the thermodynamic limit for the Ising model on planar (“fat”) φ4 random graphs and their dual quadrangulations by matching up the real part of the high and low temperature branches of the expression for the free energy. The form of this expression for the free energy also means...

We study disordered spin models using the recently introduced concept of small world graphs. Small world graphs are constructed by considering in turn all the bonds of a regular lattice and with a probability p replacing them with a randomly chosen long-range bond. By varying p, it is possible to study the crossover from a regular lattice to a rand...

We discuss a series of Monte Carlo studies of systems with quenched connectivity disorder - spin models defined on quenched, random lattices generated in simulations of 2D quantum gravity, paying particular attention to non-self-averaging properties.

We discuss a Pareto macro-economy (a) in a closed system with fixed total wealth and (b) in an open system with average mean wealth and compare our results to a similar analysis in a super-open system (c) with unbounded wealth. Wealth condensation takes place in the social phase for closed and open economies, while it occurs in the liberal phase fo...

It is known that the (exact) renormalization transformations for the one-dimensional Ising model in field can be cast in the form of a logistic map f(x) = 4 x (1 - x) with x a function of the Ising couplings. Remarkably, the line bounding the region of chaotic behaviour in x is precisely that defining the Yang-Lee edge singularity in the Ising mode...

Using Monte Carlo simulations we study the dynamics of three-dimensional Ising models with nearest-, next-nearest-, and four-spin (plaquette) interactions. During coarsening, such models develop growing energy barriers, which leads to very slow dynamics at low temperature. As already reported, the model with only the plaquette interaction exhibits...

Using Monte Carlo simulations we study cooling-rate effects in a three-dimensional Ising model with four-spin interactions. During coarsening, this model develops growing energy barriers, which at low temperature lead to very slow dynamics. We show that the characteristic zero-temperature length increases very slowly with the inverse cooling rate,...

Slow dynamics and metastability are often seen in models with quenched disorder, but rather harder to find in situations where no such disorder is present and energy or entropy barriers must be generated dynamically. Using Monte Carlo simulations we show that the 3D four-spin interaction Ising model, which possesses no quenched disorder, exhibits r...

Under certain conditions phase transitions in systems with quenched disorder are expected to exhibit a different behaviour than in the corresponding pure system. Here we discuss a series of Monte Carlo studies of a special type of such disordered systems, namely spin models defined on quenched, random lattices exhibiting geometrical disorder in the...

We investigate the non-self-averaging properties of the dynamics of Ising, 4-state Potts and 10-state Potts models in single-cluster Monte Carlo simulations on quenched ensembles of planar, trivalent (3 ) random graphs, which we use as an example of relevant quenched connectivity disorder. We employ a novel application of scaling techniques to th...

We discuss the finite size behaviour in the canonical ensemble of the balls in boxes model. We compare theoretical predictions and numerical results for the finite size scaling of cumulants of the energy distribution in the canonical ensemble and perform a detailed analysis of the first and third order phase transitions which appear for different p...

We report on single-cluster Monte Carlo simulations of the Ising, 4-state Potts and 10-state Potts models on quenched ensembles of planar, tri-valent random graphs. We confirm that the first-order phase transition of the 10-state Potts model on regular 2D lattices is softened by the quenched connectivity disorder represented by the random graphs an...

The KPZ formula shows that coupling central charge less than one spin models to 2D quantum gravity dresses the conformal weights to get new critical exponents, where the relation between the original and dressed weights depends only on the central charge. At the discrete level the coupling to 2D gravity is effected by putting the spin models on ann...

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−β, which correspond to the measure term introduced in the simplicial quantum gravity simulations. The system has two phases: el...

We note that it is possible to construct a bond vertex model that displays q-state Potts criticality on an ensemble of 3 random graphs of arbitrary topology, which we denote as `thin' random graphs in contrast to the fat graphs of the planar diagram expansion. Since the four-vertex model in question also serves to describe the critical behaviour o...

The authors show that an interpolated loop expansion produces a convex effective potential for Higgs fields in the vector representations of SU(N) and SO(N) and the adjoint representation of any simple Lie group, provided one considers the Higgs fields as a sector of a gauge theory and use the gauge fixing freedom to choose a 't Hooft-type gauge fi...

We solve a 4-(bond)-vertex model on an ensemble of 3-regular (Φ3) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent – a symmetric 8-vertex model on the honeycomb lat...

Using Monte Carlo simulations we show that the three-dimensional Ising model with four-spin (plaquette) interactions has some characteristic glassy features. The model dynamically generates diverging energy barriers, which give rise to slow dynamics at low temperature. Moreover, in a certain temperature range the model possesses a metastable (super...

We discuss the statistical mechanics of vertex models on both generic (“thin”) and planar (“fat”) random graphs. Such models can be formulated as the N → 1 and N → ∞ limits of N × Ncomplex matrix models, respectively. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directe...

This paper has been withdrawn

Lattice animals provide a discretized model for the -transition displayed by branched polymers in solvent. Exact graph enumeration studies have given some indications that the phase diagram of such lattice animals may contain two collapsed phases as well as an extended phase. This has not been confirmed by studies using other means. We use the exac...

INTRODUCTION The analytical investigation of spin glasses on random graphs of various sorts has a long and honourable history [1,2], though there has been little in the way of numerical simulations. Random graphs with a fixed or fixed average connectivity have a locally tree like structure, which means that loops in the graph are predominantly larg...

We investigate the Yang-Lee edge singularity on non-planar random graphs, which we consider as the Feynman diagrams of various d = 0 field theories, in order to determine the value of the edge exponent . We consider the hard dimer model on and random graphs to test the universality of the exponent with respect to coordination number, and the Ising...

Recently we have found identical behaviour for various spin models on `thin' random graphs - Feynman diagrams - and the corresponding Bethe lattices. In this paper we observe that the ratios of the saddle-point equations in the random graph approach are identical to the fixed point(s) of the recursion relations which are used to solve the models on...

For the three-dimensional gonihedric Ising models defined by Savvidy and Wegner the bare string tension is zero and the energy of a spin interface depends only on the number of bends and self-intersections, in antithesis to the standard nearest-neighbour three-dimensional Ising action. When the parameter , weighting the self-intersections, is small...

Four dimensional simplicial gravity has been studied by means of Monte Carlo simulations for some time, the main outcome of the studies being that the model undergoes a discontinuous phase transition between an elongated and a crumpled phase when one changes the curvature (Newton) coupling. In the crumpled phase there are singular vertices growing...

In a series of papers we have found identical behaviour for various spin models on thin random graphs - Feynman diagrams - and the corresponding Bethe lattices. In this note we observe that in all cases the ratios of various saddle point equations in the random graph approach are identical in form to the fixed point(s) of the recursion relations wh...

In a recent paper hep-lat/9704020 we investigated 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 models displayed first order transitions for all q greater than 2, giving identical behaviour to the corresponding Bethe lattice. We use he...

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 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 fa...

For the 3D gonihedric Ising models defined by Savvidy and Wegner the bare string tension is zero and the energy of a spin interface depends only on the number of bends and self-intersections, in antithesis to the standard nearest-neighbour 3D Ising action. When the parameter kappa weighting the self-intersections is small the model has a first orde...

We use the cluster variation method (CVM) to investigate the phase structure of the 3d gonihedric Ising actions defined by Savvidy and Wegner. The geometrical spin cluster boundaries in these systems serve as models for the string worldsheets of the gonihedric string embedded in . The models are interesting from the statistical mechanical point of...

We investigate a three-dimensional Ising action which corresponds to a class of models defined by Savvidy and Wegner, originally intended as discrete versions of string theories on cubic lattices. These models have vanishing bare surface tension and the couplings are tuned in such a way that the action depends only on the angles of the discrete sur...

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 82C26 Dynamic and nonequilibrium phase transitions (general) 82D30 Random media, disordered materials (including liquid crystals and spin glasses)

We perform extensive Monte Carlo simulations of the 10-state Potts model on quenched two-dimensional Φ3 gravity graphs to study the effect of quenched connectivity disorder on the phase transition, which is strongly first order on regular lattices. The numerical data provides strong evidence that, due to the quenched randomness, the discontinuous f...

We analyse the properties of a very simple “balls-in-boxes” model which can exhibit a phase transition between a fluid and a condensed phase, similar to the behaviour encountered in models of random geometries in one, two and four dimensions. This model can be viewed as a generalization of the backgammon model introduced by Ritort as an example of...