Four-dimensional Lattice Gauge Theory Research Articles

Tensor renormalization group study of (3+1)-dimensional ℤ2 gauge-Higgs model at finite density

We investigate the critical endpoints of the (3+1)-dimensional ℤ2 gauge-Higgs model at finite density together with the (2+1)-dimensional one at zero density as a benchmark using the tensor renormalization group method. We focus on the phase transition between the Higgs phase and the confinement phase at finite chemical potential along the critical end line. In the (2+1)-dimensional model, the resulting endpoint is consistent with a recent numerical estimate by the Monte Carlo simulation. In the (3+1)-dimensional case, however, the location of the critical endpoint shows disagreement with the known estimates by the mean-field approximation and the Monte Carlo studies. This is the first application of the tensor renormalization group method to a four-dimensional lattice gauge theory and a key stepping stone toward the future investigation of the phase structure of the finite density QCD.

Symmetric Mass Generation in Lattice Gauge Theory

We construct a four-dimensional lattice gauge theory in which fermions acquire mass without breaking symmetries as a result of gauge interactions. Our model consists of reduced staggered fermions transforming in the bifundamental representation of an SU(2)×SU(2) gauge symmetry. This fermion representation ensures that single-site bilinear mass terms vanish identically. A symmetric four-fermion operator is however allowed, and we give numerical results that show that a condensate of this operator develops in the vacuum.

The U(1) lattice gauge theory universally connects all classical models with continuous variables,including background gravity

We show that the partition function of many classical models with continuous degrees of freedom,e.g. Abelian lattice gauge theories and statistical mechanical models, can be written as thepartition function of an (enlarged) four-dimensional lattice gauge theory (LGT) with gauge groupU(1). This result is very general in that it includes models in differentdimensions with different symmetries. In particular, we show that aU(1) LGT defined in a curved spacetime can be mapped to aU(1) LGT with a flat background metric. The result is achieved by expressing theU(1) LGT partition function as an inner product between two quantum states.

CPT anomaly: a rigorous result in four dimensions

The existence of a CPT anomaly is established for a particular four-dimensional Abelian lattice gauge theory with Ginsparg–Wilson fermions.

Four-dimensional lattice gauge theory with ribbon categories and the Crane–Yetter state sum

Lattice Gauge Theory in four-dimensional Euclidean space–time is generalized to ribbon categories which replace the category of representations of the gauge group. This provides a framework in which the gauge group becomes a quantum group while space–time is still given by the “classical” lattice. At the technical level, this construction generalizes the spin foam model dual to lattice gauge theory and defines the partition function for a given triangulation of a closed and oriented piecewise-linear four-manifold. This definition encompasses both the standard formulation of d=4 pure Yang–Mills theory on a lattice and the Crane–Yetter invariant of four-manifolds. The construction also implies that certain classes of spin foam models formulated using ribbon categories are well-defined even if they do not correspond to a topological quantum field theory.

Adjoint “quark” color fields in four-dimensional lattice gauge theory: Vacuum screening and penetration

The fields generated by “quarks” in the adjoint representation of SU(2) color are analyzed in the scaling region of the four-dimensional lattice theory. Evidence of vacuum screening of adjoint quarks is obtained from a comparison of quark-antiquark ( Q Q ) flux-tubes for quarks in the adjoint (“isospin” j = 1) and fundamental ( j = 1 2 ) representations. The component ε j of the color-electric field strength in the direction parallel to the Q Q axis is calculated. Near the quarks the ratio of fields ε j=1 ε j= 1 2 approaches the value 8 3 , which is equal to the ratio of SU(2) Casimirs. In between the quarks, the ratio falls well below 8 3 at large R. ε j also falls off rapidly as a function of distance x ⊥ perpendicular to the Q Q axis. However, the ratio ε j=1 ε j= 1 2 depends very weakly on x ⊥. The flux-tubes in the two representations thus appear to have very similar cross-sections. This result could imply that the QCD vacuum is dual to a type I superconductor.

Self-dual models with theta terms

There exists a large class of models which are self-dual with respect to inversion of coupling constants. When a theta term is added, this duality may be extended to an invariance under an action of an infinite discrete modular group on the coupling parameter space. In particular, four-dimensional abelian lattice gauge theories possess a Sp(2k, Z) modular symmetry, where k is the number of simple factors in the gauge group. We generalize this result to models of arbitrary dimension, and show that the partition functions of two-dimensional Z N spin models with 2 k flavors and string compactifications on k-dimensional tori are invariant under O(k, k; Z) . We also draw some suggestive parallels to the fractional quantum Hall effect.

Monte Carlo analysis of the 2D chiral SU(2) × SU(2) lattice theory with a spin- [formula omitted]action

Using Monte Carlo simulations, the phase structure of the two-dimensional SU(2)×SU(2) lattice theory with an action defined through the spin- 3 2 representation of the SU(2) matrices has been determined. A second-order phase transitions has been found, like the SO(3)×SO(3) chiral model, in contrast to the equivalent four-dimensional lattice gauge theory with the 3 2 representation action that exhibits a clear first-order transition.

Mean-field approximation for Abelian and non-Abelian lattice gauge theories at finite temperature.

The phase diagrams of the Z(2), Z(3), and SU(2) four-dimensional lattice gauge theories at finite temperature are analyzed by the mean-field technique including Gaussian corrections. The effect of the finite size of the lattice in the time Euclidean direction is considered at lowest order by the Bethe-Peierls approximation. We discuss the gauge dependence of the results. The predictions for Abelian models are excellent but for SU(2) we show that in order to reproduce the Monte Carlo--simulations results, the Gaussian corrections are not enough and higher-loop calculations may be necessary.

Effective transfer matrix for low-lying glueball states in lattice gauge theory

In four-dimensional lattice gauge theory without quarks the lowest particle states in the strong coupling region correspond to a spin-zero and a spin-two glueball. They are degenerate in the lowest order of the strong coupling expansion. There exists an effective transfer matrix, acting in the lowest order eigenspace, whose eigenvalues are identical with the corresponding ones of the full transfer matrix to all orders in the strong coupling expansion. It is constructed here up to the eighth nontrivial order for different gauge groups. Diagonalization yields the energy-momentum dispersion relations for glueball states.

Topology of 2D lattice gauge fields

A local topological charge density for two-dimensional (2d) U(1) lattice gauge field is defined. A Monte Carlo (MC) simulation of pure 2d U(1) lattice gauge theory is carried out and results are obtained for the topological susceptibility and the mean value of the topological charge density in units of the string tension in a θ vacuum. The dilute gas approximation appears to be very successful. Definitions of the topological charge not based on a local charge density, analogous to the ones used in MC studies of four-dimensional lattice gauge theories, are also considered and detailed comparisons are made. These non-local charges are found to have several undesirable features.

√2 renormalization group transformation for lattice gauge theories in four dimensions

A real space renormalization group transformation with scale factor √2 for four-dimensional lattice gauge theories is presented. No such transformation on a hypercube whose block lattice points are also site points can have a smaller scale factor. The transformation is utilized here to study the critical behaviour of SU(2) lattice gauge theory. Because of the small scale factor, lattices as small as 2 4 sites can be considered. Results are in excellent agreement with standard conclusions.

Analytic study of θ vacua on the lattice

A new definition of the topological charge density for four-dimensional lattice gauge theory is given. Using a systematic expansion we find a cusp in the vacuum energy at θ = π signaling the spontaneous breaking of CP there. Unlike its two-dimensional analogue (QED 2), QCD confines at θ = π. As a by-product an expression for the topological mass term for (2+1)-dimensional lattice gauge theory is obtained.

High-statistics study of the phase transition in U(2) four-dimensional simplicial lattice gauge theory

Monte-Carlo simulations are used to study pure U(2) gauge fields on a four-dimensional simplicial space-time lattice. Using Wilson's form of the action, a first-order phase transition is clearly seen at beta c approximately=3.00+or-0.05.

U(2) four-dimensional simplicial lattice gauge theory

Monte Carlo simulations for pureU (2) gauge theory on a four-dimensional simplicial lattice with six sites in each direction are reported. Wilson loops and the string tensions for squares and triangles are presented. A first-order phase transition, similar to that found for the hypercubical lattice, is observed and found to confineSU (2) colour and deconfineU (1) charge.

Improved variational treatment of large-Nlattice gauge theories

An improved variational ansatz is used to study the phase structure of four-dimensional lattice gauge theories in the large-$N$ limit. Although the large-$N$ limit is characterized by a stationary condition for an effective action, fluctuations peculiar to this limit still remain, which can best be probed by our improved treatment. To be specific, we treat the mixed $\mathrm{SU}(N)\ensuremath{-}\frac{\mathrm{SU}(N)}{{Z}_{2}}$ system for even $N$ large, by utilizing a two-parameter trial action.

Wilson loop calculations in four-dimensional lattice gauge theory on the CDC cyber 205

Pure SU(4) gauge theory is simulated by Monte Carlo methods on an 8 4 lattice. The method of Metropolis et al. is used to equilibriate the space-time lattice. All Wilson loops up to size 4×4 are calculated. Because of memory requirements we work on the 2 Mword CDC CYBER 205 at Colorado State University and take full advantage of the parallel processing capabilities of this vector machine.

Comparison of Monte Carlo data with strong-coupling expansions forU(N)lattice gauge theories

The average action per plaquette is calculated using strong-coupling expansions up to 15th order for the pure $\mathrm{U}(N)$, $N=2, 3, 4, 5, \mathrm{and} 6$, four-dimensional Euclidean lattice gauge theory. We compare these expansions with Monte Carlo-generated data and find agreement to be better than 1% over the whole strong-coupling region.

Variational analysis of lattice gauge theories

A variational approach to the study of four-dimensional lattice gauge theories is initiated. We illustrate the technique by first analyzing the simpler SU (2) chiral model. A systematic treatment of the SU (2) lattice gauge theory with a mixed fundamental-adjoint action is presented next. This leads to an improvement over the standard mean-field (MF) results; in particular, a "second-order deconfining transition line" in the negative SO (3) coupling region previously identified by a MF technique now disappears. All transitions found are first order. The procedure is then generalized to the large-$N$ limit. In particular, a first-order transition line corresponding to the breaking of a (local) U (1) symmetry (at vanishing fundamental coupling) is again identified.

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

Monte Carlo results for the pure U(3) lattice gauge theory on a 6 4 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..