Monte Carlo study of chiral structure: The Gross-Neveu model

The two-dimensional Gross-Neveu model is studied by Monte Carlo integration of Fermi fields. A clear transition is seen from a strong-coupling, massive, chiral symmetric phase to the weak-coupling chiral broken phase found in the continuum model. The weak-coupling behaviour of the dynamically generated scale agrees well with asymptotic freedom results.     

A Monte Carlo study of the asymmetric clock or chiral Potts model in two dimensions

The phase diagram of the two-dimensional, three-state chiral Potts or asymmetric clock model is studied using Monte Carlo techniques. The phase boundaries are compared to those obtained using the finite-size renormalization group and the free fermion approximation. The incommensurate phase is described in detail and crossover effects near the Lifshitz point are discussed.     

A Monte Carlo study of the Gross-Neveu model

Several lattice versions of the Gross-Neveu model are constructed and studied using Monte Carlo methods. The expected shiral structures are confirmed by the numerical results. The correct asymptotic freedom behaviour is recovered with the appropriate number of species taken into account. The models differ in their number of soft modes and their strong coupling behaviour. In some of them, chiral symmetry is restored at a finite coupling. The large-N, finite-temperature transition is also found in good agreement with the theoretical value.     

A Monte Carlo study of the XY model

Spin systems which have a phase transition without an order parameter, like the XY model, have great importance in physics. Here we show how it is possible to study the behaviour of such a system by computer simulation without studying the spin-spin correlation functions.     

Monte Carlo study of U(1) and SU(2) gauge fields on four-dimensional random lattices

The asymptotic behavior of the random lattice gauge model is examined. Numerical results of the specific heat of U(1) and SU(2) on the random lattices of a different number of sites in four dimensions are reported and the phase structure is discussed.     

Monte Carlo study of long-range chiral structure in QCD

Results of a Monte Carlo study of long-range chiral structure is massless 4-dimensional QCD are presented. The algorithm employed integrates fermion degrees of freedom exactly. The behavior of chiral correlations in both abelian and SU(2) color theories is compared. We find strong evidence for chiral breakdown, with a solution of the U(1) problem, in lattice QCD at large β.     

Monte Carlo studies of wilson loops in four-dimensional U(2) gauge theory

Wilson loops are calculated using Monte Carlo simulations for pure U(2) gauge theory on a 6⁴ lattice. The loops appear to contain an area law piece in both the high and low temperature regions. The string tension is discontinuous at β = β_c, where β_c is the critical inverse temperature. This suggests that the first-order phase transition in U(2) gauge theory is not a deconfining phase transition. The determinant of the Wilson loop, however, extracts the U(1) part of the theory and appears to lose the area law at low temperature.     

A Monte Carlo analysis of an SU(5) scalar model

Spontaneous symmetry breaking is studied in an SU(5) model of adjoint scalars with quartic couplings λ₁ (tr φ²)² + λ₂ trφ⁴ and with a negative mass term. The breaking for λ₂ > 0 is to SU(3)?SU(2)?U(1), and for λ₂ > 0 to SU(3)?SU(2)?U(1), in agreement with the standard tree-level analysis of the scalar potential.     

Monte Carlo study of the two-dimensional spin-1/2 XY model

The generalized Trotter formula is used to map the two-dimensional spin-1/2 XY model onto a three-dimensional Ising model with complicated many-spin interactions. This relationship is used to construct an efficient Monte Carlo algorithm. Simulation data for the specific heat and vortex correlation functions give strong evidence for the existence of a phase transition.     

How well do Monte Carlo simulations distinguish between U(1) and SU(2)?: A study of the string tension in U(1) lattice gauge theory

To investigate how a system with a known deconfining phase transition behaves when studied on finite lattices via Monte Carlo simulations, we have made such studies of compact U(1) lattice gauge theory for 8⁴, 10⁴, and 12⁴ lattices. We have concentrated on the mean plaquette energy and the string tension. The string tension does not vanish on a finite lattice, but using finite size scaling arguments the indications are that it does vanish on an infinite lattice, where we predict the critical coupling β_c = 1.008 and the correlation length exponent $\text{ν =}\text{1}\text{3}$. We compare our results to those for SU(2) and find that although there are differences, they are not yet definitive.     

Worm Monte Carlo study of the honeycomb-lattice loop model

We present a Markov-chain Monte Carlo algorithm of worm   type that correctly simulates the O(n)$O(n)$ loop model on any (finite and connected) bipartite cubic graph, for any real n>0$n>0$, and any edge weight, including the fully-packed limit of infinite edge weight. Furthermore, we prove rigorously that the algorithm is ergodic and has the correct stationary distribution. We emphasize that by using known exact mappings when n=2$n=2$, this algorithm can be used to simulate a number of zero-temperature Potts antiferromagnets for which the Wang–Swendsen–Kotecký cluster algorithm is non-ergodic, including the 3-state model on the kagome lattice and the 4-state model on the triangular lattice. We then use this worm algorithm to perform a systematic study of the honeycomb-lattice loop model as a function of n?2$n?2$, on the critical line and in the densely-packed and fully-packed phases. By comparing our numerical results with Coulomb gas theory, we identify a set of exact expressions for scaling exponents governing some fundamental geometric and dynamic observables. In particular, we show that for all n?2$n?2$, the scaling of a certain return time in the worm dynamics is governed by the magnetic dimension of the loop model, thus providing a concrete dynamical interpretation of this exponent. The case n>2$n>2$ is also considered, and we confirm the existence of a phase transition in the 3-state Potts universality class that was recently observed via numerical transfer matrix calculations.     

Monte Carlo renormalization group study of the XY model

The Δβ method has been used to investigate the phase structure of the XY model. A line of fixed points was found in the low-temperature phase, while a renormalization group trajectory (one relevant and no marginal coupling parameters) in the high-temperature phase. A study of higher (3rd and 4th) order derivatives of the free energy indicate that the phase transition is of infinite order and that hyperscaling is valid for the system.     

Critical dynamics of the three-dimensional Ising model: A Monte Carlo study

Extensive Monte Carlo simulations have been performed to analyze the dynamical behavior of the three-dimensional Ising model with local dynamics. We have studied the equilibrium correlation functions and the power spectral densities of odd and even observables. The exponential relaxation times have been calculated in the asymptotic one-exponential time region. We find that the critical exponentz=2.09 ±0.02 characterizes the algebraic divergence with lattice size for all observables. The influence of scaling corrections has been analyzed. We have determined integrated relaxation times as well. Their dynamical exponentz _int agrees withz for correlations of the magnetization and its absolute value, but it is different for energy correlations. We have applied a scaling method to analyze the behavior of the correlation functions. This method verifies excellent scaling behavior and yields a dynamical exponentz _scal which perfectly agrees withz.     

U(3) four-dimensional lattice gauge theory and Monte Carlo calculations

Monte Carlo results for the pure U(3) lattice gauge theory on a 6⁴ lattice are reported. Wilson loops and the string tension are presented. The first-order phase transition in U(3) is reflected quite clearly in a discontinuity in the string tension at β = β_c. The U(1) factor of U(3) is extracted using the determinant of the Wilson loops. As expected, the U(1) component appears to deconfine at the phase transition..