A Monte Carlo simulation study of the critical and off-critical behavior of the Baxter-Wu model, which belongs to the universality class of the 4-state Potts model, was performed. We estimate the critical temperature window using known analytical results for the specific heat and magnetization. This helps us to extract reliable values of universal combinations of critical amplitudes with reasonable accuracy. Comparisons with approximate analytical predictions and other numerical results are discussed.
Monte Carlo (MC) simulations and series expansions (SE) data for the energy, specific heat, magnetization, and susceptibility of the three-state and four-state Potts model and Baxter-Wu model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states $q\le …
The operator content of the Baxter-Wu model with general toroidal boundary conditions is calculated analytically and numerically. These calculations were done by relating the partition function of the model with the generating function of a site-colouring problem in a hexagonal lattice. Extending the original Bethe-ansatz solution of the related colouring problem we are able to calculate the eigenspectra of both models by solving the associated Bethe-ansatz equations. We have also calculated, by exploring the conformal invariance at the critical point, the mass ratios of the underlying massive…
We propose that Baxter's Z-invariant six-vertex model at the rational gl(2) point on a planar but in general not rectangular lattice provides a way to study Yangian invariants. These are identified with eigenfunctions of certain monodromies of an auxiliary inhomogeneous spin chain. As a consequence they are special solutions to the eigenvalue problem of the associated transfer matrix. Excitingly, this allows to construct them using Bethe ansatz techniques. Conceptually, our construction generalizes to general (super) Lie algebras and general representations. Here we present the explicit form o…
We investigate the short-time universal behavior of the two dimensional Ashkin-Teller model at the Baxter line by performing time-dependent Monte Carlo Simulations. First, as preparatory results, we obtain the critical parameters by searching the optimal power law decay of the magnetization. Thus, the dynamic critical exponents $θ_{m}$ and $θ_{p}$, related to the magnetic and electric order parameters, as well as the persistence exponent $θ_{g}$, are estimated using heat-bath Monte Carlo simulations. In addition, we estimate the dynamic exponent $z$ and the static critical exponents $β$ and $ν…
Monte Carlo (MC) and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the two-dimensional 4-state Potts model in the vicinity of the critical point are analysed. The role of logarithmic corrections is discussed and an approach is proposed in order to account numerically for these corrections in the determination of critical amplitudes. Accurate estimates of universal amplitude ratios $A_+/A_-$, $Γ_+/Γ_-$, $Γ_T/Γ_-$ and $R_C^\pm$ are given, which arouse new questions with respect to previous works.
The scaling form of the free-energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic non-universal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the Tricritical Ising Model in two dimensions by using several theoretical methods from Perturbed Conformal F…
Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x^\star = \lim_{L\to\infty} ξ(L)/L and the first four magnetization moment ratios V_{2n} = <{\cal M}^{2n}>/<{\cal M}^2>^n. As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G_{2n}^*. We confirm these predictions by a high-precision Monte Carlo simulation.
We determine the specific heat amplitude ratio near a $m$-axial Lifshitz point and show its universal character. Using a recent renormalization group picture along with new field-theoretical $ε_{L}$-expansion techniques, we established this amplitude ratio at one-loop order. We estimate the numerical value of this amplitude ratio for $m=1$ and $d=3$. The result is in very good agreement with its experimental measurement on the magnetic material $MnP$. It is shown that in the limit $m \to 0$ it trivially reduces to the Ising-like amplitude ratio.
We study the behaviour of the 2d Ising model in the symmetric high temperature phase in presence of a small magnetic perturbation. We successfully compare the quantum field theory predictions for the shift in the mass spectrum of the theory with a set of high precision transfer matrix results. Our results rule out a prediction for the same quantity obtained some years ago with strong coupling methods.
The universal amplitude ratio $\tilde{R}_ξ$ for percolation in two dimensions is determined exactly using results for the dilute A model in regime 1, by way of a relationship with the q-state Potts model for q<4.
We complete the determination of the universal amplitude ratios of two-dimensional percolation within the two-kink approximation of the form factor approach. For the cluster size ratio, which has for a long time been elusive both theoretically and numerically, we obtain the value 160.2, in good agreement with the lattice estimate 162.5 +/- 2 of Jensen and Ziff.
The susceptibility amplitude ratio in the neighborhood of a uniaxial Lifshitz point is calculated at one-loop level using field-theoretic and $ε_{L}$-expansion methods. We use the Schwinger parametrization of the propagator in order to split the quadratic and quartic part of the momenta, as well as a new special symmetry point suitable for renormalization purposes. For a cubic lattice (d = 3), we find the result $\frac{C_{+}}{C_{-}} = 3.85$.
The universal amplitude ratio $R_ξ$ for the ($q\leqslant 4$)-state Potts model in two dimensions is determined by using results for the dilute A model in regime 1. The nature of the relationship between the Potts model and the dilute A model, both related to $ϕ_{2,1}$ perturbed conformal field theory, is discussed.
Monte Carlo simulations and series expansion data for the energy, specific heat, magnetization and susceptibility of the 3-state Potts model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes.
We calculate the susceptibility amplitude ratio near a generic higher character Lifshitz point up to one-loop order. We employ a renormalization group treatment with $L$ independent scaling transformations associated to the various inequivalent subspaces in the anisotropic case in order to compute the ratio above and below the critical temperature and demonstrate its universality. Furthermore, the isotropic results with only one type of competition axes have also been shown to be universal. We describe how the simpler situations of $m$-axial Lifshitz points as well as ordinary (noncompeting) s…
Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window. The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios …
We analyze Monte Carlo simulation and series-expansion data for the susceptibility of the three-state Potts model in the critical region. The amplitudes of the susceptibility on the high- and the low-temperature sides of the critical point as extracted from the Monte Carlo data are in good agreement with those obtained from the series expansions and their (universal) ratio compares quite well with a recent quantum field theory prediction by Delfino and Cardy.
We investigate the three-dimensional O(2) model near the critical point by Monte Carlo simulations and calculate the major universal amplitude ratios of the model. The ratio U_0=A+/A- is determined directly from the specific heat data at zero magnetic field. The data do not, however, allow to extract an accurate estimate for alpha. Instead, we establish a strong correlation of U_0 with the value of alpha used in the fit. This numerical alpha-dependence is given by A+/A- = 1 -4.20(5) alpha + O(alpha^2). For the special alpha-values used in other calculations we find full agreement with the corr…
Monte Carlo and series expansion data for the energy, specific heat, magnetisation and susceptibility of the 4-state Potts model in the vicinity of the critical point are analysed. The role of logarithmic corrections is discussed. Estimates of universal ratios A_+/A_-, $Γ_+/Γ_L$, $Γ_T/Γ_L$ and R_c^+ are given.
We propose a new class of universal amplitude ratios which involve the first terms of the short distance expansion of the correlators of a statistical model in the vicinity of a critical point. We will describe the critical system with a conformal field theory (UV fixed point) perturbed by an appropriate relevant operator. In two dimensions the exact knowledge of the UV fixed point allows for accurate predictions of the ratios and in many nontrivial integrable perturbations they can even be evaluated exactly. In three dimensional O(N) scalar systems feasible extensions of some existing results…
In the last few years the derivative expansion of the Non-Perturbative Renormalization Group has proven to be a very efficient tool for the precise computation of critical quantities. In particular, recent progress in the understanding of its convergence properties allowed for an estimate of the error bars as well as the precise computation of many critical quantities. In this work we extend previous studies to the computation of several universal amplitude ratios for the critical regime of $O(N)$ models using the derivative expansion of the Non-Perturbative Renormalization Group at order $\ma…
We discuss how to calculate non-equilibrium universal amplitude ratios in the functional renormalization group approach, extending its applicability. In particular, we focus on the critical relaxation of the Ising model with non-conserved dynamics (model A) and calculate the universal amplitude ratio associated with the fluctuation-dissipation ratio of the order parameter, considering a critical quench from a high-temperature initial condition. Our predictions turn out to be in good agreement with previous perturbative renormalization-group calculations and Monte Carlo simulations.
We have derived analytic expressions for the amplitudes of correction terms in the critical free energy and inverse correlation length expansions for square, honeycomb, and plane-triangular lattices in strip geometry. We have found that ratio of amplitudes for all of these lattices is universal. We have also obtained similar expansions for the ground state energy and the first energy gap of a critical quantum spin chain and find that the amplitude ratios have the same values.
In this Letter we compute analytically the effect of conformal symmetry on the radiative corrections to the amplitude ratios for O($N$) $λϕ^{4}$ massless scalar field theories in curved spacetime for probing the two-scale-factor universality hypothesis. For that we employ three distinct and independent field-theoretic renormalization group methods. The amplitude ratios values obtained are identical when computed through the three distinct methods, thus showing their universal character. Furthermore, they are the same as that obtained in flat spacetime, then satisfying the two-scale-factor univ…
We present analytical and numerical results for the specific heat and susceptibility amplitude ratios in parallel plate geometries. The results are derived using field-theoretic techniques suitable to describe the system in the bulk limit. Advantages and drawbacks of our method are discussed on the light of other approaches previously reported in the literature.
We compute the radiative quantum corrections to the critical exponents and amplitude ratios for O($N$) $λϕ^{4}$ scalar high energy nonextensive $q$-field theories. We employ the field theoretic renormalization group approach through six methods for evaluating the high energy nonextensive critical exponents up to next-to-leading order while the high energy nonextensive amplitude ratios are computed up to leading level by applying three methods. Later we generalize these high energy nonextensive finite loop order results for any loop level. We find that the high energy nonextensive critical expo…
We present a new way of probing the universality class of the site-diluted two-dimensional Ising model. We analyse Monte Carlo data for the magnetic susceptibility, introducing a new fitting procedure in the critical region applicable even for a single sample with quenched disorder. This gives us the possibility to fit simultaneously the critical exponent, the critical amplitude and the sample dependent pseudo-critical temperature. The critical amplitude ratio of the magnetic susceptibility is seen to be independent of the concentration $q$ of the empty sites for all investigated values of $q\…
Motivated by the results of two-dimensional conformal field theory (CFT) we investigate the finite-size scaling of the mass spectrum of an Ising model on three-dimensional lattices with a spherical cross section. Using a cluster-update Monte Carlo technique we find a linear relation between the masses and the corresponding scaling dimensions, in complete analogy to the situation in two dimensions. Amplitude ratios as well as the amplitudes themselves appear to be universal in this case.
The critical properties of systems under constraint differ from their ideal counterparts through Fisher renormalization. The mathematical properties of Fisher renormalization applied to critical exponents are well known: the renormalized indices obey the same scaling relations as the ideal ones and the transformations are involutions in the sense that re-renormalizing the critical exponents of the constrained system delivers their original, ideal counterparts. Here we examine Fisher renormalization of critical amplitudes and show that, unlike for critical exponents, the associated transformati…
We compute a number of universal amplitude ratios in the three-dimensional Ising universality class. To this end, we perform Monte Carlo simulations of the improved Blume-Capel model on the simple cubic lattice. For example, we obtain A_+/A_-=0.536(2) and C_+/C_-=4.713(7), where A_+- and C_+- are the amplitudes of the specific heat and the magnetic susceptibility, respectively. The subscripts + and - indicate the high and the low temperature phase, respectively. We compare our results with those obtained from previous Monte Carlo simulations, high and low temperature series expansions, field t…
The phase transition for the $(2+1)$-dimensional spin-$S=1$ $XY$ model was investigated numerically. Because of the boson-vortex duality, the spin stiffness $ρ_s$ in the ordered phase and the vortex-condensate stiffness $ρ_v$ in the disordered phase should have a close relationship. We employed the exact diagonalization method, which yields the excitation gap directly. As a result, we estimate the amplitude ratios $ρ_{s,v}/Δ$ ($Δ$: Mott insulator gap) by means of the scaling analyses for the finite-size cluster with $N \le 22$ spins. The ratio $ρ_s/ρ_v$ admits a quantitative measure of deviati…
We study the influence of boundary conditions on the finite-size corrections of a one-dimensional (1D) quantum spin model by exact and perturbative theoretic calculations. We obtain two new infinite sets of universal amplitude ratios for the finite-size correction terms of the 1D quantum spin model of $N$ sites with free and antiperiodic boundary conditions. The results for the lowest two orders are in perfect agreement with a perturbative conformal field theory scenario proposed by Cardy [Nucl. Phys. B {\bf 270}, 186 (1986)].
We study the critical behaviour of the $q$-state Potts model on an uncorrelated scale-free network having a power-law node degree distribution with a decay exponent $λ$. Previous data show that the phase diagram of the model in the $q,λ$ plane in the second order phase transition regime contains three regions, each being characterized by a different set of critical exponents. In this paper we complete these results by finding analytic expressions for the scaling functions and critical amplitude ratios in the above mentioned regions. Similar to the previously found critical exponents, the scali…
In the vicinity of boundaries the bulk universality class of critical phenomena splits into several boundary universality classes, depending upon whether the tendency to order in the boundary is smaller or larger than in the bulk. For Ising universality class there are five different boundary universality classes: periodic, antiperiodic, free, fixed and mixed (mixture of the last two). In this paper we present the new set of the universal amplitude ratios for the mixed boundary universality class. The results are in perfect agreement with a perturbated conformal field theory scenario proposed …
The amplitude ratio of the susceptibility (or second size-moment) for two-dimensional percolation is calculated by two series methods and also by Monte-Carlo simulation. The first series method is a new approach based upon integrating approximations to the scaling function. The second series method directly uses low- and high-density series expansions of the susceptibility, going to unprecedented orders for both bond and site percolation on the square lattice. Putting all methods together we find a consistent value $Γ^{-}/Γ^{+} = 162.5 \pm 2$, a significant improvement over previous results th…
We use a high-precision Monte Carlo simulation to determine the universal specific-heat amplitude ratio A+/A- in the three-dimensional Ising model via the impact angle ϕof complex temperature zeros. We also measure the correlation-length critical exponent νfrom finite-size scaling, and the specific-heat exponent αthrough hyperscaling. Extrapolations to the thermodynamic limit yield ϕ= 59.2(1.0) degrees, A+/A- = 0.56(3), ν= 0.63048(32) and α= 0.1086(10). These results are compatible with some previous estimates from a variety of sources and rule out recently conjectured exact values.
We study the (analytic) finite-size corrections in the Ising model on the strip with fixed ($+ -$) boundary conditions. We find that subdominant finite-size corrections to scaling should be to the form $a_k/N^{2k-1}$ for the free energy $f_N$ and $b_k^{(n)}/N^{2k-1}$ for inverse correlation length $ξ_n^{-1}$, with integer value of $k$. We investigate the set $\{a_k, b_k^{(n)}\}$ by exact evaluation and their changes upon varying anisotropy of coupling. We find that the amplitude ratios $b_k^{(n)}/a_k$ remain constant upon varying coupling anisotropy. Such universal behavior are correctly repro…
We simulate lattice models in the three-dimensional XY universality class in the low and the high temperature phase. This allows us to compute a number of universal amplitude ratios with unprecedented precision: R_Υ=0.411(2), R_B=2.83(1), R_ξ^+=0.3562(10) and R_ξ^-=0.850(5). These results can be compared with those obtained from other theoretical methods, such as field theoretic methods or the high temperature series expansion and also with experimental results for the lambda-transition of $^4$He. In addition to the XY model, we study the three-dimensional two-component $ϕ^4$ model on the simp…
The quantum $XY$ spin chain with the interactions decaying as a power law $1/r^{1+ σ}$ of the distance between spins $r$ was investigated with the exact diagonalization method. Here, the constituent spin is set to $S=1$, which enables us to incorporate the biquadratic interactions so as to realize the order-disorder transition with the O$(2)$ symmetry maintained. Thereby, in the ordered phase, we resolved the Higgs mass $m_H$ out of the Goldstone-excitation continuum by specifying Higgs-particle's quantum numbers to adequate indices. We then turn to the analysis of the critical amplitude ratio…
