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.     

The thermodynamics of SU(3) lattice gauge theory with a light isodoublet of quarks

Using hybrid differential equations with stochastic pseudo-fermion fields, we simulated SU(3) lattice gauge theory with a light isodoublet of quarks on a 4 × 8³ lattice. The bare quark mass is 0.0125 in lattice units which corresponds to 7.75 MeV if the finite temperature chiral transition occurs at T_c=155 MeV as suggested by phenomenology. The transition is first order and metastable states are found over a narrow range in temperature. Results for 4⁴ lattices are also presented for 2 and 3 light flavors.     

The large-N deconfinement phase transition in a partially reduced Eguchi-Kawal model

We construct generalized twisted Eguchi-Kawai models which for large-N reduce space-time to a lattice of arbitrary size. Large-N lattice gauge theory at finite temperature is investigated in a model on a lattice with L₀ time slices and two lattice points in very time slice. We observe the large-N deconfinement phase transition in the weak coupling region. Assuming asymptotic scaling we find a transition temperature T_c = (101±4)Λ_L.     

Plaquette expansion study of compact U(1) lattice gauge theory in (2 + 1) dimensions

The plaquette expansion is applied to compact U(1) lattice gauge theory in (2 + 1) dimensions to high order: 1/N ⁸ for the ground state energy density and 1/N ⁷ for the anti-symmetric or photon mass gap, where N is defined as the number of plaquettes. Evidence of scaling in the photon mass gap is observed for low order (≤ 1/N ⁴) for inverse coupling values β = 0. 7 to 1. 25 with scaling behaviour given by the weak-coupling formula: M ²/β = exp(-5. 01β + 5. 82) in good agreement with other studies. Higher order results appear to diverge from the scaling slope past the transition point portenting the prospect that the strong coupling trial state in this region gives vise to an asymtotic series in 1/N for the photon mass gap.     

Phase transition in four-dimensional compact QED

The energy and the specific heat of the four-dimensional U(1) lattice gauge model is evaluated by Monte Carlo simulations on lattices of size L⁴, where L = 4, 5 and 6, evidence is presented for the occurence of a second-order phase transition. A finite size scaling analysis of our results gives the critical value of the coupling constant e²_c = 0.995 and a correlation length exponent $\text{v ≈}\text{1}\text{3}$.     

The calculation of critical amplitudes in SU(2) lattice gauge theory

We calculate the critical amplitudes of the Polyakov loop and its susceptibility at the deconfinement transition of (3 + 1)-dimensional SU(2) gauge theory. To this end we study the corrections due to irrelevant exponents in the scaling functions. As a guiding line for determining the critical amplitudes we use envelope equations which we derive from the finite size scaling formulae of the observables. We have produced new high precision data on N_{σ³ × 4} lattices for N_χ = 12, 18, 26 and 36. With these data we find different corrections to the asymptotic scaling behaviour above and below the transition. Our result for the universal ratio of the susceptibility amplitudes is C₊/C₋ = 4.72(11) and thus in excellent agreement with a recent measurement for the 3d Ising model.     

On the phase structure of five-dimensional SU(2) gauge theories with anisotropic couplings

The phase diagram of five-dimensional SU(2) gauge theories is explored using Monte Carlo simulations of the theory discretized on a Euclidean lattice using the Wilson plaquette action and periodic boundary conditions. We simulate anisotropic gauge couplings which correspond to different lattice spacings a₄ in four dimensions and a₅ along the extra dimension. In particular we study the case where a₅>a₄. We identify a line of first order phase transitions which separate the confined from the deconfined phase. We perform simulations in large volume at the bulk phase transition staying in the confined vacuum. The static potential measured in the hyperplanes orthogonal to the extra dimension hints at dimensional reduction. We also locate and analyze second order phase transitions related to breaking of the center along one direction.     

On solving four-dimensional SU (2) gauge theory by numerically finding its partition function

We demonstrate a method to directly simulate the partition function of non-abelian lattice theories. We determine the partition function of the SU(2) lattice gauge theory in four dimensions both for the full SU(2) group and the 120 element icosahedral subgroup on a variety of lattice actions for lattices of size up to 4⁴. All the phenomena (transitions, crossovers, etc.) of these theories are readily observed in our simulation. In addition, even from small lattice simulations, we can distinguish potential critical behavior from rapid changes in order parameters. With the Wilson and adjoint actions we also see a clear line of zeros pointing to the zero temperature (g₀² = 0) fixed point of this theory. We discuss how a finite size scaling analysis of the position of such zeros would yield the beta function of the theory.     

Polyakov loop distributions near deconfinement inSU(2) lattice gauge theory

The distribution function of the Polyakov loop is investigated on a 16³×3 lattice in the neighbourhood of the deconfinement transition ofSU(2) gauge theory. We find, that well above the transition the distribution is a Gaussian; when the coupling approaches the critical point it is modified due to phase flip attempts of the system. Corresponding distributions for the plaquettes remain, however, Gaussian. For one coupling close to the transition we study the distributions on 8³, 12³ and 18³×4 lattices and show that strong finite size effects are present. Using the maximum values of the Gaussian parts of the distributions we construct a more physical (and therefore scaling) order parameter whose critical exponent is in excellent agreement with the universality hypothesis.     

Liquid–gas and other unusual thermal phase transitions in some large-N magnets

Much insight into the low temperature properties of quantum magnets has been gained by generalizing them to symmetry groups of order N, and then studying the large-N limit. In this paper we consider an unusual aspect of their finite temperature behavior—their exhibiting a phase transition between a perfectly paramagnetic state and a paramagnetic state with a finite correlation length at N=∞. We analyze this phenomenon in some detail in the large “spin” (classical) limit of the SU(N) ferromagnet which is also a lattice discretization of the CP^N−1 model. We show that at N=∞ the order of the transition is governed by lattice connectivity. At finite values of N, the transition goes away in one or less dimension but survives on many lattices in two dimensions and higher, for sufficiently large N. The latter conclusion contradicts a recent conjecture of Sokal and Starinets [Nucl. Phys. B 601 (2001) 425], yet is consistent with the known finite temperature behavior of the SU(2) case. We also report closely related first order paramagnet–ferromagnet transitions at large N and shed light on a violation of Elitzur's theorem at infinite N via the large-q limit of the q state Potts model, reformulated as an Ising gauge theory.     

On the thermodynamics and scaling behaviour of SU(2) gauge theory with fermion feedback

We study SU(2) lattice gauge theory with four species (N_f = 4) of light dynamical fermions by microcanonical simulation methods. Relatively large lattices, long runs and small quark masses are considered. On a 8³ × 16 lattice $\text{〈}\text{ψ}\text{ψ〉}$ is measured and good evidence for asymptotic freedom with fermion feedback is found. The scaling window begins at β = 4/g² ≈ 1.85. On a 12³ × 6 lattice SU(2) thermodynamics is studied systematically. The chiral symmetry restoration transition is found at β = 1.925 ± .025. The crossover from hadronic matter to the quark-gluon plasma is abrupt.     

First-order chiral phase transition in lattice QCD

The chiral phase transition in lattice QCD has been studied for light fermions of mass ma=0.025 on lattices of size 4⁴ and 8³×4 using the hybrid algorithm. We find evidence for a first-order chiral phase transition with a large latent heat. A comparison with 10³×6 data shows violations of asymptotic scaling for T_ch which are similar in magnitude to those observed in the pure gauge sector.     

A finite size scaling analysis of the O(4)-invariant lattice δ4 theory

The critical behaviour of the three- and four-dimensional N=4 vector model is investigated by means of a Monte Carlo simulation on lattices with size between 4³ and 16³, and between 4⁴ and 12⁴, respectively. For obtaining information about some critical properties of the model, we use a method due to Binder which is based on the theory of finite size scaling. For the three-dimensional model we get estimates of the critical exponents ν and η which are compatible with estimates obtained from the ϵ-expansion. In four dimensions we study for two different values of the bare self-coupling λ (λ=1 in our normalization, and λ=∞) the scaling behaviour of some Green function ratios at the phase boundary. In both cases we find compatibility with the “predicted” scaling behaviour at the gaussian fixed point. This is another independent numerical hint that the continuum limit of the four-dimensional O(4)-invariant lattice δ⁴-model is a free field theory.     

Mean field and Monte Carlo studies of SU(N) chiral models in three dimensions

SU(N) chiral models defined on three-dimensional cubic lattices arestudied using mean field and Monte Carlo techniques. Mean field theory predicts first-order transitions for all finite N greater than two. The mean field estimates of the transition temperature and discontinuity of the order parameter are in good agreement with computer simulations for N = 3 and 4. The N → ∞ limit of mean field theory has a first-order phase transition.     

Continuum QCD2 from a fixed-point lattice action

Gauge invariant expectation values for lattice gauge theory with a general local action in two dimensions may be expressed as functions of the single plaquette averages. The value of these averages at the fixed point of the renormalization group can be determined exactly, and the corresponding lattice theory is shown to reproduce the continuum results. The limit N_e = ∞ is investigated in detail, and fixed point values for all the averages are explicitly determined. Wilson's action results agree only to first order in weak coupling.     

The nature of the transition in d = 4 U(1) lattice gauge theory

The phase transition for the d = 4 compact U(1) lattice gauge theory has been studied using the Monte Carlo renormalization-group method. A single relevant eigenvalue is observed on the Wilson axis. The MCRG estimate for the exponent v changes with the coupling β; an extrapolation towards β^∞_c provides an estimate v ∼ 0.42 rather than ∼ 0.33 as obtained by finite-size scaling. It is confirmed that loops of monopole current are the mechanism driving the transition. It is shown that at the transition the largest loop of monopole current undergoes a discontinuity in size and begins to span the lattice. On the basis of these findings, it is conjectured that on the Wilson axis the discontinuity is a finite-size effect.     

Black hole/string transition for the small Schwarzschild black hole of AdS 5 × S5 and critical unitary matrix models

In this paper we discuss the black hole–string transition of the small Schwarzschild black hole of AdS ₅×S⁵ using the AdS/CFT correspondence at finite temperature. The finite temperature gauge theory effective action, at weak and strong coupling, can be expressed entirely in terms of constant Polyakov lines which are SU(N) matrices. In showing this we have taken into account that there are no Nambu–Goldstone modes associated with the fact that the 10-dimensional black hole solution sits at a point in S⁵. We show that the phase of the gauge theory in which the eigenvalue spectrum has a gap corresponds to supergravity saddle points in the bulk theory. We identify the third order N=∞ phase transition with the black hole–string transition. This singularity can be resolved using a double scaling limit in the transition region where the large N expansion is organized in terms of powers of N^-2/3. The N=∞ transition now becomes a smooth crossover in terms of a renormalized string coupling constant, reflecting the physics of large but finite N. Multiply wound Polyakov lines condense in the crossover region. We also discuss the implications of our results for the resolution of the singularity of the lorenztian section of the small Schwarzschild black hole.     

Scaling in lattice QCD: Glueball masses and string tension

We study the scaling behaviour of lattice quantum chromodynamics by comparing the β dependence of the string tension and the 0⁺⁺ glueball mass. We use a source method at β=5.7, β=5.9 and β=6.1, on lattices from 9³· 24 to 16³· 32. Assuming a string tension of about (420 MeV)², the lattice spacing ranges from 0.16 to 0.08 fm. In order to separate finite volume from scaling violation effects we have compared data from lattices having approximately the same overall physical size at the different values of β. We find deviations from scaling to be very small.     

The critical exponents of crystalline random surfaces

We report on a high statistics numerical study of the crystalline random surface model with extrinsic curvature on lattices of up to 64² points. The critical exponents at the crumpling transition are determined by a number of methods all of which are shown to agree within estimated errors. The correlation length exponent is found to be ν = 0.71(5) from the tangent-tangent correlation function whereas we find ν = 0.73(6) by assuming finite size scaling of the specific heat peak and hyperscaling. These results imply a specific heat exponent α = 0.58(10); this is a good fit to the specific heat on a 64² lattice with a χ² per degree of freedom of 1.7 although the best direct fit to the specific heat data yields a much lower value of a. We have measured the normal-normal correlation function in the crumpled phase and find that, within the accuracy of our simulations, the data can be described by a super-renormalizable field theory.