The wrong kind of gravity

The KPZ formula [V.G. Knizhnik, A.M. Polyakov, and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819] shows that coupling central charge c≤1 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 c. At the discrete level the coupling to 2D gravity is effected by putting the spin models on annealed ensembles of Φ³ planar random graphs or their dual triangulations, where the connectivity fluctuates on the same time-scale as the spins.Since the sole determining factor in the dressing is the central charge, one could contemplate putting a spin model on a quenched ensemble of 2D gravity graphs with the “wrong” c value. We might then expect to see the critical exponents appropriate to the c value used in generating the graphs. In such cases the KPZ formula could be interpreted as giving a continuous line of critical exponents which depend on this central charge. We note that rational exponents other than the KPZ values can be generated using this procedure for the Ising, tricritical Ising and 3-state Potts models.     

Nonequilibrium model on Archimedean lattices

On (4, 6, 12) and (4, 8²) Archimedean lattices, the critical properties of the majority-vote model are considered and studied using the Glauber transition rate proposed by Kwak et al. [Kwak et al., Phys. Rev. E, 75, 061110 (2007)] rather than the traditional majority-vote with noise [Oliveira, J. Stat. Phys. 66, 273 (1992)]. We obtain T _c and the critical exponents for this Glauber rate from extensive Monte Carlo studies and finite size scaling. The calculated values of the critical temperatures and Binder cumulant are T _c = 0.651(3) and U ₄ ^* = 0.612(5), and T _c = 0.667(2) and U ₄ ^* = 0.613(5), for (4, 6, 12) and (4, 8²) lattices respectively, while the exponent (ratios) β/ν, γ/ν and 1/ν are respectively: 0.105(8), 1.48(11) and 1.16(5) for (4, 6, 12); and 0.113(2), 1.60(4) and 0.84(6) for (4, 8²) lattices. The usual Ising model and the majority-vote model on previously studied regular lattices or complex networks differ from our new results.     

Thermodynamic properties of the φ4 lattice field theory near the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins with nearest-neighbor interactions, the partition function is transformed for large bare coupling constant λ into an Ising-like system with additional neighbor interactions. For d = 2 a mean field approximation is then used to estimate the difference in critical temperature between the lattice φ⁴ field theory and its Ising limit (λ = ∞). Expansions are obtained for the susceptibility and specific heat. The critical exponents are shown to be identical to the Ising exponents.     

Finite size scaling for O(N) φ4-theory at the upper critical dimension

A finite size scaling theory for the partition function zeroes and thermodynamic functions of O(N) φ⁴-theory in four dimensions is derived from renormalization group methods. The leading scaling behaviour is mean-field like with multiplicative logarithmic corrections which are linked to the triviality of the theory. These logarithmic corrections are independent of N for odd thermodynamic quantities and associated zeroes and are N dependent for the even ones. Thus a numerical study of finite size scaling in the Ising model serves as a non-perturbative test of triviality of φ⁴₄-theories for all N.     

Variational calculations on the energy levels of graphene quantum antidots

The critical properties of the two-dimensional Ising and Blume-Capel model on directedsmall-world lattices with quenched connectivity disorder are investigated. The disordered system is simulated by applying the Monte Carlo method with heat bath update algorithm and histogram re-weighting techniques. The critical temperature, as well as the critical exponents are obtained. For both models the critical parameters have been obtained for several values of the rewiring probability p. It is found that these disorder systems do not belong to the same universality class as two-dimensional ferromagnetic model on regular lattices. In particular, the Blume-Capel model, with zero crystal field interaction, on a directedsmall-world lattice presents a second-order phase transition for p < p _c , and a first-order phase transition for p > p _c , where p _c  ≈ 0.25. The critical exponents for p < p _c are different from those of the same model on a regular lattice, but are identical to the exponents of the Ising model on directedsmall-world lattice.     

A non-perturbative analysis of symmetry breaking in two-dimensional φ4 theory using periodic field methods

We describe the generalization of spherical field theory to other modal expansion methods. The main approach remains the same, to reduce a d-dimensional field theory into a set of coupled one-dimensional systems. The method we discuss here uses an expansion with respect to periodic-box modes. We apply the method to φ⁴ theory in two dimensions and compute the critical coupling and critical exponents. We compare with lattice results and predictions via universality and the two-dimensional Ising model.     

Symmetric vertex models on planar random graphs

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 lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models).The relations between the vertex weights and Ising model parameters in the 4-vertex model on Φ³ graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model.Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.     

Correlation functions in the multiple Ising model coupled to gravity

The model of p Ising spins coupled to 2d gravity, in the form of a sum over planar φ³ graphs, is studied and in particular the two-point and spin-spin correlation functions are considered. We first solve a toy model in which only a partial summation over spin configurations is performed and, using a modified geodesic distance, various correlation functions are determined. The two-point function has a diverging length scale associated with it. The critical exponents are calculated and it is shown that all the standard scaling relations apply. Next the full model is studied, in which all spin configurations are included. Many of the considerations for the toy model apply for the full model, which also has a diverging geometric correlation length associated with the transition to a branched polymer phase. Using a transfer function we show that the two-point and spin-spin correlation functions decay exponentially with distance. Finally, by assuming various scaling relations, we make a prediction for the critical exponents at the transition between the magnetized and branched polymer phases in the full model.     

Critical behavior of the 3D Ising model on a poissonian random lattice

The single-cluster Monte Carlo algorithm and the reweighting technique are used to simulate the 3D ferromagnetic Ising model on 3D Voronoi-Delauney lattices. It is assumed that the coupling factor J varies with the distance r between the first neighbors as J(r)∝e^−ar, with a≥0. The critical exponents γ/ν, β/ν, and ν are calculated, and according to the present estimates for the critical exponents, we argue that this random system belongs to the same universality class of the pure 3D ferromagnetic Ising model.     

A Monte Carlo study of the entropy,the pressure,and the critical behavior of the hard-square lattice gas

An approximate technique for estimating the entropyS with computer simulation methods, suggested recently by Meirovitch, is applied here to the Metropolis Monte Carlo (MC) simulation of the hard-square lattice gas in both the grand canonical and the canonical ensembles. The chemical potentialμ, calculated by Widom's method, andS enable one to obtain also the pressureP. The MC results are compared with results obtained with Padé approximants (PA) and are found to be very accurate; for example, at the critical activityz _c the MC and the PA estimates forS deviate by 0.5%. Beyondz _c this deviation decreases to 0.01% and comparable accuracy is detected forP. We argue that close toz _c our results forS, μ, andP are more accurate than the PA estimates. Independent of the entropy study, we also calculate the critical exponents by applying Fisher's finite-size scaling theory to the results for the long-range order, the compressibility and the staggered compressibility, obtained for several lattices of different size at z_c. The data are consistent with the critical exponents of the plane Ising latticeβ=1/8,ν=1,γ=7/4, andα=0. Our values forβ and ν agree with series expansion and renormalization group results, respectively,α=0 has been obtained also by matrix method studies; it differs, however, from the estimate of Baxteret al. α=0.09 ± 0.05. As far as we knowγ has not been calculated yet.     

Critical Behavior of the Annealed Ising Model on Random Regular Graphs

In Giardinà et al. (ALEA Lat Am J Probab Math Stat 13(1):121–161, 2016), the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in Can (Annealed limit theorems for the Ising model on random regular graphs, arXiv:1701.08639, 2017), we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem stating that the magnetization scaled by \(n^{3/4}\) converges to a specific random variable, with n the number of vertices of random regular graphs.     

The φ4 lattice field theory as an asymptotic expansion about the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins, an asymptotic expansion of expectations about the Ising limit is established in inverse powers of the bare coupling constant λ. In the thermodynamic limit (N → ∞), the expansion is expected to be valid in the noncritical region of the Ising system.     

The estimates of correlations in two-dimensional Ising model

We investigate the correlation inequalities and the decay of correlations of stochastic Ising model in a rectangle with side length 2L×K(LlnL)^1/2, where K is some positive constant. With different boundary conditions, at inverse temperature β>β_c or β<β_c and zero external field, we show some estimates of the correlation functions for the two-dimensional Ising model.     

Existence theorem for solitary waves on lattices

In this article we give an existence theorem for localized travelling wave solutions on one-dimensional lattices with Hamiltonian $$H = \sum\limits_{n \in \mathbb{Z}} {\left( {\tfrac{1}{2}p_n^2 + V(q_{n + 1} - q_n )} \right)} ,$$ whereV(·) is the potential energy due to nearest-neighbour interactions. Until now, apart from rare integrable lattices like the Toda latticeV(φ)=ab ^?1(e ^?bφ+bφ?1), the only evidence for existence of such solutions has been numerical. Our result in particular recovers existence of solitary waves in the Toda lattice, establishes for the first time existence of solitary waves in the (nonintegrable) cubic and quartic latticesV(φ)= 1/2φ ² + 1/3aφ ³,V(φ) = 1/2φ ² + 1/4bφ ⁴, thereby confirming the numerical findings in [1] and shedding new light on the recurrence phenomena in these systems observed first by Fermi, Pasta and Ulam [2], and shows that contrary to widespread belief, the presence of exact solitary waves is not a peculiarity of integrable systems, but “generic” in this class of nonlinear lattices. The approach presented here is new and quite general, and should also be applicable to other forms of lattice equations: the travelling waves are sought as minimisers of a naturally associated variational problem (obtained via Hamilton's principle), and existence of minimisers is then established using modern methods in the calculus of variations (the concentration-compactness principle of P.-L. Lions [3]).     

Complex zeroes of the d = 3 Ising model: Finite-size scaling and critical amplitudes

By means of an MC simulation of lattices of size 4³ to 8³, we study the distribution of the complex zeroes closest to the real β axis of the d = 3 Ising model. We observe they do scale and we measure in this way ν and A₊/A_-. We obtain the prediction A₊/A_- = 0.45±0.07.     

Nontrivial critical behaviour in a lattice model of random surfaces

A model of “planar random surfaces without spikes” on hypercubical lattices was introduced some years ago as a discretization of quantum string theory. We review some general properties of this model and present results from a Monte Carlo study of its critical behaviour in d = 4, 8 and 10 dimensions. In d = 4 dimensions we find a Hausdorff dimension d_H ≈ 4 and an anomalous dimensions η ≈ 1. These critical exponents imply a deviation from mean field theory in contrast to other lattice random surface models. Furthermore, we find evidence for mean field behaviour in 8 and 10 dimensions, indicating an upper critical dimension d_c^u ⩽ 8.     

Ising ferromagnet with biquadratic exchange

A spin one Ising system with biquadratic exchange, is investigated, using Green's function technique in random phase approximation (RPA). Transition temperature T_c and <(S^z)²> at T_c, are found to increase with biquadratic exchange parameter α for sc, bcc and fcc lattices. The variation of <(S^z)²> at T_c with α is found to be the same for the above lattices.     

Thermodynamics of the quantum and classical Ising models with skew magnetic field

The thermodynamics of the unitary (normalized spin) quantum and classical Ising models with skew magnetic field, for |J|β?0.9, is derived for the ferromagnetic and antiferromagnetic cases. The high-temperature expansion (β-expansion) of the Helmholtz free energy is calculated up to order β⁷ for the quantum version (spin S≥1/2) and up to order β¹⁹ for the classical version. In contrast to the S=1/2 case, the thermodynamics of the transverse Ising and that of the XY model for S>1/2 are not equivalent. Moreover, the critical line of the T=0 classical antiferromagnetic Ising model with skew magnetic field is absent from this classical model, at least in the temperature range of |J|β?0.9.     

Strong-Disorder Paramagnetic-Ferromagnetic Fixed Point in the Square-Lattice ±J Ising Model

We consider the random-bond ±J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T ^*=0.9527(1), p ^*=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T≈0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T ^*, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents ν=1.50(4), η=0.128(8), and β=0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T ^*. Our results for the critical exponents are consistent with the hyperscaling relation 2β/ν?η=d?2=0.     

Monte Carlo simulation of Ising model on decagonal covering structure

We study the Ising model on a two-dimensional quasilattice developed from the decagonal covering structure. The periodic boundary conditions are applied to a patch of rhombus-like covering pattern. By means of the Monte Carlo simulation and the finite-size scaling analysis the critical temperature is estimated as 2.317±0.002. Two critical exponents are obtained being 1/v=0.992±0.003 and η=0.247±0.002, which are close to the values of the two-dimensional regular lattices as well as the Penrose tilings.