Critical behavior of 3D SU(2) gauge theory at finite temperature: exact results from universality

We show that universality arguments, namely the Svetitsky-Yaffe conjecture, allow one to obtain exact results on the critical behavior of 3D SU(2) gauge theory at the finite temperature deconfinement transition,through a mapping into the 2D Ising model. In particular, we consider the finite-size scaling behavior of the plaquette operator, which can be mapped into the energy operator of the 2D Ising model. We obtain exact predictions for the dependence of the plaquette expectation value on the size and shape of the lattice and we compare them to Monte Carlo results, finding complete agreement. We discuss the application of this method to the computation of more general correlators of the plaquette operator at criticality, and its relevance to the study of the color flux tube structure.Comment: 10 pages, LaTeX file + 3 eps figure

Block Spin Effective Action for 4d SU(2) Finite Temperature Lattice Gauge Theory

The Svetitsky-Yaffe conjecture for finite temperature 4d SU(2) lattice gauge theory is confirmed by observing matching of block spin effective actions of the gauge model with those of the 3d Ising model. The effective action for the gauge model is defined by blocking the signs of the Polyakov loops with the majority rule. To compute it numerically, we apply a variant of the IMCRG method of Gupta and Cordery.Comment: LaTeX2e, 22 pages, 8 Figure

Theoretical Analysis of Acceptance Rates in Multigrid Monte Carlo

We analyze the kinematics of multigrid Monte Carlo algorithms by investigating acceptance rates for nonlocal Metropolis updates. With the help of a simple criterion we can decide whether or not a multigrid algorithm will have a chance to overcome critial slowing down for a given model. Our method is introduced in the context of spin models. A multigrid Monte Carlo procedure for nonabelian lattice gauge theory is described, and its kinematics is analyzed in detail.Comment: 7 pages, no figures, (talk at LATTICE 92 in Amsterdam

Effective Field Theories

Effective field theories encode the predictions of a quantum field theory at low energy. The effective theory has a fairly low ultraviolet cutoff. As a result, loop corrections are small, at least if the effective action contains a term which is quadratic in the fields, and physical predictions can be read straight from the effective Lagrangean. Methods will be discussed how to compute an effective low energy action from a given fundamental action, either analytically or numerically, or by a combination of both methods. Basically,the idea is to integrate out the high frequency components of fields. This requires the choice of a "blockspin",i.e. the specification of a low frequency field as a function of the fundamental fields. These blockspins will be the fields of the effective field theory. The blockspin need not be a field of the same type as one of the fundamental fields, and it may be composite. Special features of blockspins in nonabelian gauge theories will be discussed in some detail. In analytical work and in multigrid updating schemes one needs interpolation kernels \A from coarse to fine grid in addition to the averaging kernels $C$ which determines the blockspin. A neural net strategy for finding optimal kernels is presented. Numerical methods are applicable to obtain actions of effective theories on lattices of finite volume. The constraint effective potential) is of particular interest. In a Higgs model it yields the free energy, considered as a function of a gauge covariant magnetization. Its shape determines the phase structure of the theory. Its loop expansion with and without gauge fields can be used to determine finite size corrections to numerical data.Comment: 45 pages, 9 figs., preprint DESY 92-070 (figs. 3-9 added in ps format

Multigrid Monte Carlo Algorithms for SU(2) Lattice Gauge Theory: Two versus Four Dimensions

We study a multigrid method for nonabelian lattice gauge theory, the time slice blocking, in two and four dimensions. For SU(2) gauge fields in two dimensions, critical slowing down is almost completely eliminated by this method. This result is in accordance with theoretical arguments based on the analysis of the scale dependence of acceptance rates for nonlocal Metropolis updates. The generalization of the time slice blocking to SU(2) in four dimensions is investigated analytically and by numerical simulations. Compared to two dimensions, the local disorder in the four dimensional gauge field leads to kinematical problems.Comment: 24 pages, PostScript file (compressed and uuencoded), preprint MS-TPI-94-

Kinematics of Multigrid Monte Carlo

We study the kinematics of multigrid Monte Carlo algorithms by means of acceptance rates for nonlocal Metropolis update proposals. An approximation formula for acceptance rates is derived. We present a comparison of different coarse-to-fine interpolation schemes in free field theory, where the formula is exact. The predictions of the approximation formula for several interacting models are well confirmed by Monte Carlo simulations. The following rule is found: For a critical model with fundamental Hamiltonian H(phi), absence of critical slowing down can only be expected if the expansion of in terms of the shift psi contains no relevant (mass) term. We also introduce a multigrid update procedure for nonabelian lattice gauge theory and study the acceptance rates for gauge group SU(2) in four dimensions.Comment: 28 pages, 8 ps-figures, DESY 92-09

Critical Exponents of the 3D Ising Universality Class From Finite Size Scaling With Standard and Improved Actions

We propose a method to obtain an improved Hamiltonian (action) for the Ising universality class in three dimensions. The improved Hamiltonian has suppressed leading corrections to scaling. It is obtained by tuning models with two coupling constants. We studied three different models: the +1,-1 Ising model with nearest neighbour and body diagonal interaction, the spin-1 model with states 0,+1,-1, and nearest neighbour interaction, and phi**4-theory on the lattice (Landau-Ginzburg Hamiltonian). The remarkable finite size scaling properties of the suitably tuned spin-1 model are compared in detail with those of the standard Ising model. Great care is taken to estimate the systematic errors from residual corrections to scaling. Our best estimates for the critical exponents are nu= 0.6298(5) and eta= 0.0366(8), where the given error estimates take into account the statistical and systematic uncertainties.Comment: 55 pages, 12 figure

Computing Masses from Effective Transfer Matrices

We study the use of effective transfer matrices for the numerical computation of masses (or correlation lengths) in lattice spin models. The effective transfer matrix has a strongly reduced number of components. Its definition is motivated by a renormalization group transformation of the full model onto a 1-dimensional spin model. The matrix elements of the effective transfer matrix can be determined by Monte Carlo simulation. We show that the mass gap can be recovered exactly from the spectrum of the effective transfer matrix. As a first step towards application we performed a Monte Carlo study for the 2-dimensional Ising model. For the simulations in the broken phase we employed a multimagnetical demon algorithm. The results for the tunnelling correlation length are particularly encouraging.Comment: (revised version: a few references added) LaTeX file, 25 pages, 6 PostScript figures, (revised version: a few references added

Algebraic Computation of the Hierarchical Renormalization Group Fixed Points and their ϵ\epsilon-Expansions

Nontrivial fixed points of the hierarchical renormalization group are computed by numerically solving a system of quadratic equations for the coupling constants. This approach avoids a fine tuning of relevant parameters. We study the eigenvalues of the renormalization group transformation, linearized around the non-trivial fixed points. The numerical results are compared with $\epsilon$-expansion.Comment: LaTex file, 24 pages, 5 figures appended as 1 PostScript file, preprint MS-TPI-94-

Quantum broadening of k-strings in gauge theories

We study the thickness of the confining flux tube generated by a pair of sources in higher representations of the gauge group. Using a simple geometric picture we argue that the area of the cross-section of the flux tube, as measured by a Wilson loop probe, grows logarithmically with source separation, as a consequence of the quantum fluctuations of the underlying k-string. The slope of the logarithm turns out to be universal, i.e. it is the same for all the representations and all the gauge theories. We check these predictions in a 3D Z_4 lattice gauge model by comparing the broadening of the 1-string and the 2-string.Comment: 15 pages, 5 figure