On the universality class of the 3d Ising model with long-range-correlated disorder

We analyze a controversial topic about the universality class of the three-dimensional Ising model with long-range-correlated disorder. Whereas both theoretical and numerical studies agree on the validity of the extended Harris criterion [A. Weinrib, B.I. Halperin, Phys. Rev. B 27 (1983) 413] and indicate the existence of a new universality class, numerical values of the critical exponents found so far differ considerably. To resolve this discrepancy we perform extensive Monte Carlo simulations of a 3d Ising model with non-magnetic impurities being arranged in a form of lines along randomly chosen axes of a lattice. The Swendsen-Wang algorithm is used alongside with a histogram reweighting technique and finite-size scaling analysis to evaluate the values of critical exponents governing magnetic phase transition. Our estimates for these exponents differ from both previous numerical simulations and are in favor of a non-trivial dependency of the critical exponents on the peculiarities of long-range correlation decay.     

Ferromagnetic phase transition for the spanning-forest model (q-->0 limit of the Potts model) in three or more dimensions

We present Monte Carlo simulations of the spanning-forest model (q-->0 limit of the ferromagnetic Potts model) in spatial dimensions d=3, 4, 5. We show that, in contrast to the two-dimensional case, the model has a ferromagnetic second-order phase transition at a finite positive value w(c). We present numerical estimates of w(c) and of the thermal and magnetic critical exponents. We conjecture that the upper critical dimension is 6.     

Universal critical properties of the Eulerian bond-cubic model

We investigate the Eulerian bond-cubic model on the square lattice by means of Monte Carlo simulations,using an efficient cluster algorithm and a finite-size scaling analysis.The critical points and four critical exponents of the model are determined for several values of n.Two of the exponents are fractal dimensions,which are obtained numerically for the first time.Our results are consistent with the Coulomb gas predictions for the critical O(n) branch for n < 2 and the results obtained by previous transfer matrix calculations.For n=2,we find that the thermal exponent,the magnetic exponent and the fractal dimension of the largest critical Eulerian bond component are different from those of the critical O(2) loop model.These results confirm that the cubic anisotropy is marginal at n=2 but irrelevant for n < 2.     

Random-bond Ising chain in a transverse magnetic field: A finite-size scaling analysis

We investigate the zero-temperature quantum phase transition of the randombond Ising chain in a transverse magnetic field. Its critical properties are identical to those of the McCoy-Wu model, which is a classical Ising model in two dimensions with layered disorder. The latter is studied via Monte Carlo simulations and transfer matrix calculations and the critical exponents are determined with a finite-size scaling analysis. The magnetization and susceptibility obey conventional rather than activated scaling. We observe that the order parameter and correlation function probability distribution show a nontrivial scaling near the critical point, which implies a hierarchy of critical exponents associated with the critical behavior of the generalized correlation lengths.     

Nonequilibrium relaxation and critical aging for driven Ising lattice gases

We employ Monte?Carlo simulations to study the nonequilibrium relaxation of driven Ising lattice gases in two dimensions. Whereas the temporal scaling of the density autocorrelation function in the nonequilibrium steady state does not allow a precise measurement of the critical exponents, these can be accurately determined from the aging scaling of the two-time autocorrelations and the order parameter evolution following a quench to the critical point. We obtain excellent agreement with renormalization group predictions based on the standard Langevin representation of driven Ising lattice gases.     

Chern–Simons theory, 2d Yang–Mills, and Lie algebra wanderers

The dynamically triangulated random surface (DTRS) approach to Euclidean quantum gravity in two dimensions is considered for the case of the elemental building blocks being quadrangles instead of the usually used triangles. The well-known algorithmic tools for treating dynamical triangulations in a Monte Carlo simulation are adapted to the problem of these dynamical quadrangulations. The thus defined ensemble of 4-valent graphs is appropriate for coupling to it the 6- and 8-vertex models of statistical mechanics. Using a series of extensive Monte Carlo simulations and accompanying finite-size scaling analyses, we investigate the critical behaviour of the 6-vertex F model coupled to the ensemble of dynamical quadrangulations and determine the matter related as well as the graph related critical exponents of the model.     

Critical exponents for nuclear multifragmentation: dynamical lattice model

We present a dynamical and dissipative lattice model, designed to mimic nuclear multifragmentation. Monte Carlo simulations with this model show a clear signature of critical behaviour and reproduce experimentally observed correlations. In particular, using techniques devised for finite systems, we could obtain two of its critical exponents, whose values are in agreement with those of the universality class to which nuclear multifragmentation is supposed to belong.     

Finite-size scaling analysis of the critical behavior of a general epidemic process in 2D

We investigate the critical behavior of a stochastic lattice model describing a General Epidemic Process. By means of a Monte Carlo procedure, we simulate the model on a regular square lattice and follow the spreading of an epidemic process with immunization. A finite size scaling analysis is employed to determine the critical point as well as some critical exponents. We show that the usual scaling analysis of the order parameter moment ratio does not provide an accurate estimate of the critical point. Precise estimates of the critical quantities are obtained from data of the order parameter variation rate and its fluctuations. Our numerical results corroborate that this model belongs to the dynamic isotropic percolation universality class. We also check the validity of the hyperscaling relation and present data collapse curves which reinforce the accuracy of the estimated critical parameters.     

Monte Carlo study of self-avoiding surfaces

Two models of self-avoiding surfaces on the cubic lattice are studied by Monte Carlo simulations. Both the first model with fluctuating boundary and the second one with a fixed boundary are found to belong to the universality class of branched polymers. The algorithms as well as the methods used to extract the critical exponents are described in detail. The results are compared to other recent estimates in the literature.     

Monte Carlo simulation of quantum statistical lattice models

In this article we review recent developments in computational methods for quantum statistical lattice problems. We begin by giving the necessary mathematical basis, the generalized Trotter formula, and discuss the computational tools, exact summations and Monte Carlo simulation, that will be used to examine explicit examples. To illustrate the general strategy, the method is applied to an analytically solvable, non-trivial, model: the one-dimensional Ising model in a transverse field. Next it is shown how to generalized Trotter formula most naturally leads to different path-integral representations of the partition function by considering one-dimensional fermion lattice models. We show how to analyze the different representations and discuss Monte Carlo simulation results for one-dimensional fermions. Then Monte Carlo work on one- and two-dimensional spin-$\text{1}\text{2}$ models based upon the Trotter formula approach is reviewed and the more dedicated Handscomb Monte Carlo method is discussed. We consider electron-phonon models and discuss Monte Carlo simulation data on the Molecular Crystal Model in one, two and three dimensions and related one-dimensional polaron models. Exact numerical results are presented for free fermions and free bosons in the canonical ensemble. We address the main problem of Monte Carlo simulations of fermions in more than one dimension: the cancellation of large contributions. Free bosons on a lattice are compared with bosons in a box and the effects of finite size on Bose-Einstein condensation are discussed.     

Coulomb and liquid dimer models in three dimensions

We study classical hard-core dimer models on three-dimensional lattices using analytical approaches and Monte Carlo simulations. On the bipartite cubic lattice, a local gauge field generalization of the height representation used on the square lattice predicts that the dimers are in a critical Coulomb phase with algebraic, dipolar correlations, in excellent agreement with our large-scale Monte Carlo simulations. The nonbipartite fcc and Fisher lattices lack such a representation, and we find that these models have both confined and exponentially deconfined but no critical phases. We conjecture that extended critical phases are realized only on bipartite lattices, even in higher dimensions.     

Evidence for an unconventional universality class from a two-dimensional dimerized quantum heisenberg model

The two-dimensional J-J' dimerized quantum Heisenberg model is studied on the square lattice by means of (stochastic series expansion) quantum Monte Carlo simulations as a function of the coupling ratio alpha=J'/J. The critical point of the order-disorder quantum phase transition in the J-J' model is determined as alpha_c=2.5196(2) by finite-size scaling for up to approximately 10 000 quantum spins. By comparing six dimerized models we show, contrary to the current belief, that the critical exponents of the J-J' model are not in agreement with the three-dimensional classical Heisenberg universality class. This lends support to the notion of nontrivial critical excitations at the quantum critical point.     

Deconfinement transition and dimensional cross-over in the 3D gauge Ising model

We present a high-precision Monte Carlo study of the finite-temperature gauge theory in 2 + 1 dimensions. The duality with the 3D Ising spin model allows us to use powerful cluster algorithms for the simulations. For temporal extensions of up to N_t = 16 we obtain the inverse critical temperature with a statistical accuracy comparable with the most accurate results for the bulk phase transition of the 3D Ising model. We discuss the predictions of T.W. Capehart and M.E. Fisher for the dimensional cross-over from 2 to 3 dimensions. Our precise data for the critical exponents and critical amplitudes confirm the Svetitsky-Yaffe conjecture. We find deviations from Olesen's prediction for the critical temperature of about 20%.     

Quantum critical dynamics simulation of dirty boson systems

Recently, the scaling result z=d for the dynamic critical exponent at the Bose glass to superfluid quantum phase transition has been questioned both on theoretical and numerical grounds. This motivates a careful evaluation of the critical exponents in order to determine the actual value of z. We study a model of quantum bosons at T=0 with disorder in 2D using highly effective worm Monte?Carlo simulations. Our data analysis is based on a finite-size scaling approach to determine the scaling of the quantum correlation time from simulation data for boson world lines. The resulting critical exponents are z=1.8±0.05, ν=1.15±0.03, and η=-0.3±0.1, hence suggesting that z=2 is not satisfied.     

Nonequilibrium critical poisoning in a single-species model

A simple model is introduced, which exhibits a poisining transition similar to that observed in studies of catalyc surface reactions. Steady-state behavior and critical exponents are determined via series expansion and Monte Carlo simulations. The relation between catalytic surface models and reggeon field theory is examined.     

Spin-glass transition of the three-dimensional Heisenberg spin glass

It is shown, by means of Monte Carlo simulation and finite size scaling analysis, that the Heisenberg spin glass undergoes a finite-temperature phase transition in three dimensions. There is a single critical temperature, at which both a spin glass and a chiral glass ordering develop. The Monte Carlo algorithm, adapted from lattice gauge theory simulations, makes it possible to thermalize lattices of size L = 32, larger than in any previous spin-glass simulation in three dimensions. High accuracy is reached thanks to the use of the Marenostrum supercomputer. The large range of system sizes studied allows us to consider scaling corrections.     

Critical Dynamics of Transverse-field Quantum Ising Model using Finite-Size Scaling and Matrix Product States

The study of phase transition is usually done by numerical simulation of finite system. Conventional methods such as Monte Carlo simulations and phenomenological renormalization group methods obtain the critical exponents without obtaining the quantum wavefunction of the system. The Matrix Product States formalism allows one to obtain accurate numerical wavefunctions of short ranged interacting quantum many-body systems. In this study we combine the Finite Size Scaling theory and Matrix Product States formalism to study the critical dynamics of one-dimensional quantum Ising model. Finite size simulations of 20, 40, 60, 80, 100 and 120 spins are done using the Density Matrix Renormalization Group to obtain the ground state wavefunction of the system. The thermodynamic quantities such as the magnetization, susceptibility and correlation function are calculated. The critical exponents independently calculated are respectively β/ν = 0.1235(1), γ/ν = 1.7351(2), and η = 0.249(1). They conform with the theoretical values from analytical solution and fulfil the hyperscaling relation. We showed that both methods combined can reliably study the critical dynamics of one-dimensional Ising-like quantum lattice systems. Application of the study on water-ice phase transition of single-file water in nanopores is proposed.

     

Height Representation,Critical Exponents,and Ergodicity in the Four-State Triangular Potts Antiferromagnet

We study the four-state antiferromagnetic Potts model on the triangular lattice. We show that the model has six types of defects which diffuse and annihilate according to certain conservation laws consistent with their having a vector-valued topological charge. Using the properties of these defects, we deduce a (2+2)-dimensional height representation for the model and hence show that the model is equivalent to the three-state Potts antiferromagnet on the Kagomé lattice and to bond-coloring models on the triangular and honeycomb lattices. We also calculate critical exponents for the ground-state ensemble of the model. We find that the exponents governing the spin–spin correlation function and spin fluctuations violate the Fisher scaling law because of constraints on path length which increase the effective wavelength of the spin operator on the height lattice. We confirm our predictions by extensive Monte Carlo simulations of the model using the Wang–Swendsen–Kotecký cluster algorithm. Although this algorithm is not ergodic on lattices with toroidal boundary conditions, we prove that it is ergodic on lattices whose topology has no noncontractible loops of infinite order, such as the projective plane. To guard against biases introduced by lack of ergodicity, we perform our simulations on both the torus and the projective plane.     

Phase transitions of bilayer spin-1/2 Baxter-Wu model with repulsive interlayer interactions

Using a Monte Carlo simulation and the single histogram reweighting technique, we study the critical behaviors and phase transitions of the Baxter-Wu model on a two-layer triangular lattice with repulsive interlayer interactions. Via the finite-size analysis, we obtain the transition temperatures and critical exponents at different interlayer coupling strengths. The reduced energy cumulants and the histograms reveal pseudo-first-order phase transitions, and the critical exponents derivate from the standard Baxter-Wu model if the coupling is weak. As the increase of the coupling, the behavior of the pseudo-first order transition disappears and the critical exponents are in accordance with the ones known for the 2D 4-state Potts model.