Phase structure of QCD at high temperature with massive quarks and finite quark density: A Z(3) paradigm

The confinement/deconfinement phase transition in SU(3) lattice gauge theories at high temperatures is analogous to that of the Z(3) gauge theories. We study various Z(3) gauge-matter theories that result from replacing the gauge group SU(3) with its center Z(3). We include large-mass fermions in the Wilson formulation and allow a chemical potential. We show that in the limit of strong coupling and high temperature the (3 + 1)-dimensional theory becomes a three state, three-dimensional Potts model with uniform external fields of real and imaginary strengths related to the fermion mass and chemical potential. By studying the phase structure of the q = 3, d = 3 Potts model with external fields we argue that the confinement/deconfinement phase transition is first order, but highly sensitive to external fields, and that it does not occur at “strong coupling” in a Z(3) gauge theory if there is a light enough fermion present. We discuss the consequences of this result for QCD.     

Determination of the phase structure of the four-dimensional coupled gauge-Higgs Potts model

Using Monte Carlo simulations the phase structure of the four-dimensional N-state gauge Potts model coupled to Higgs fields is determined. A three-phase diagram is established. In the Z(2) case, a first and a second-order transition lines are present. For Z(5) and Z(10) only first-order transition lines appear. The results are consistent with previous mean field predictions.     

Migdal renormalization group approach to deconfinement phase transition

The Migdal renormalization group approach is applied to a finite temperature lattice gauge theory. Imposing the periodic boundary condition in the timelike orientation, the phase structure of the finite temperature lattice gauge system with a gauge groupG in (d+1)-dimensional space is determined by two kinds of recursion equations, describing spacelike and timelike correlations, respectively. One is the recursion equation for ad-dimensional gauge system with the gauge groupG, and the other corresponds to ad-dimensional spin system for which the effective theory is described by the nearest neighbor interaction of the Wilson lines. Detailed phase structure is investigated for theSU(2) gauge theory in (3+1)-dimensional space. Deconfinement phase transition is obtained. Using the recursion equation for the three dimensional spin system of the Wilson lines, it is shown that the flow of the renormalization group trajectories leads to a phase transition of the three dimensional Ising model.     

The Kadanoff lowerbound renormalization transformation for the q-state Potts model

The Kadanoff lowerbound renormalization transformation when applied to the 2-dimensional q-state Potts model, is found to show a bifurcation phenomenon at q = 4, that might be considered as signalling the onset to the first-order transition. At the value of the free parameter where the bifurcation is found, the specific heat exponent takes almost the value predicted by weak universality $\text{α(4) =}\text{2}\text{3}$, while the cross-over exponent in the Potts-lattice gas direction becomes marginal. The cross-over exponent in the cubic direction is found already to be irrelevant for q > 3.3. Further a duality relation for a class of models obtained by a partial breaking of the Potts symmetry in the hamiltonian, including the cubic model is derived.     

Histogram data analysis for a three-dimensional diluted ferromagnetic 3- and 4-state potts models

On the basis of a histogram data analysis, phase transitions (PTs) in a three-dimensional diluted ferromagnetic 3- and 4-state Potts models are investigated. Systems with linear dimensions of L = 20–60 and spin concentrations of p = 1.00, 0.95, and 0.65 are studied. It is shown that the introduction of weak disorder (p ～ 0.95) into the system in the three-dimensional Potts model with q = 3 is sufficient to change a first-order phase transition to a second-order one, whereas, in the three-dimensional Potts model with q = 4, the change of a first-order phase transition to a second-order one occurs only in the presence of strong disorder (p ～ 0.65).     

Effective scalar field theory for the electroweak phase transition

We investigate an effective model for the finite-temperature symmetry-restoration phase transition of the electroweak theory. It is obtained by dimensional reduction of the (3 + 1)-dimensional full theory and by subsequent integration over all static gauge degrees of freedom. The resulting theory corresponds to a 3-dimensional O(4) ferromagnet containing cubic and quartic terms of the field in its potential function. Possible nonperturbative effects of a magnetic screening mass are parametrically included in the potential. We analyse the theory using mean-field and numerical Monte Carlo (MC) simulation methods. At the value of the physical Higgs mass, m_H = 37 GeV, considered in the present investigation, we find a discontinuous symmetry-restoring phase transition. We determine the critical temperature, order parameter jump, interface tension and latent heat characteristics of the transition. The Monte Carlo results indicate a somewhat weaker first-order phase transition as compared to the mean-field treatment, demonstrating that non-perturbative fluctuations of the Higgs field are relevant. This effect is especially important for the interface tension. Any observation of hard first-order transition could result only from non-perturbative effects related to the gauge degrees of freedom.     

Discontinuity of the Wilson string tension in the 4-dimensional lattice pure gauge Potts model

We consider the 4-dimensionalq-state pure gauge Potts model. Forq large enough, we give a new proof of the existence of a unique coupling constant β _t , where a first order phase transition occurs. Moreover we prove the following new results: The string tension is discontinuous at β _t , the Wilson parameter exhibits at β _t a direct transition from an area law decay (quark confinement) to a perimeter law decay (quark deconfinement).     

Phase transitions in the two-dimensional ferro- and antiferromagnetic potts models on a triangular lattice

The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.     

Variational approach for the N-state spin and gauge Potts model

A hamiltonian variational treatment is applied both to the spin Potts model and to its gauge version for any number of states N and spatial dimensions d?2. Regarding the former we reproduce the correct critical coupling and latent heat for not too low N and d. For the latter, our approach turns the gauge theory into an equivalent d-dimensional classical spin model, which evaluated for d + 1 = 4 gives results in agreement with 1/N expansions.     

Strong-coupling planar diagrams,random surfaces,and the large-N phase transition in U(N) lattice gauge theories

The strong-coupling expansion of U(N) gauge theory on a D-dimensional lattice is reformulated in the limit N → ∞ through a set of diagrammatic rules directly for the free energy and Wilson loops. The strong-coupling planar diagrams are interpreted as surfaces embedded in the lattice. The large-N phase transition is related to the entropy of these surfaces. It is shown that the strong-coupling phase of the U(∞) gauge theory terminates with a phase transition of Gross-Witten type only in 2 and 3 dimensions. When D⩾4 the large-N singularity takes place in a metastable phase because of an earlier first-order transition to the weak-coupling phase of the theory.     

Topological aspects of gauge and spin potts models

For the d-dimensional lattice gauge Potts model the representation of the partition function is derived, where the field configuration summation is reduced to the summation over submanifolds (oriented and nonoriented), constructed from the plaquettes of the lattice. The topological invariants (Betti numbers) of these two-dimensional submanifolds are used in an essential way. Some possible applications of this representation for the partition function are discussed.     

One-loopN-point functions in the light-cone gauge

We consider, in the light-cone gauge, the possible structure of the counterterms arising in the solution to the renormalization equation. Using the structure of these counterterms and the noncovariant formalism of the integrals as guidelines, we also examine to one-loop order the divergence and nonlocality of theN-point gluon vertex functions forN≧5.     

Three-dimensional 3-state Potts model revisited with new techniques

We report a fairly detailed finite-size scaling analysis of the first-order phase transition in the three-dimensional 3-state Potts model on cubic lattices with emphasis on recently introduced quantities whose infinite-volume extrapolations are governed only by exponentially small terms. In these quantities no asymptotic power series in the inverse volume are involved which complicate the finite-size scaling behaviour of standard observables related to the specific-heat maxima or Binder-parameter minima. Introduced initially for strong first-order phase transitions in q-state Potts models with “large enough” q, the new techniques prove to be surprisingly accurate for a q value as small as 3. On the basis of the high-precision Monte Carlo data of Alves et al. [Phys. Rev. B 43 (1991) 5846], this leads to a refined estimate of β_t = 0.550 565(10) for the infinite-volume transition point.     

Investigation of the critical behavior of the three-dimensional weakly diluted Potts model

The static critical behavior of the three-dimensional weakly diluted Potts model with the state q = 3 on a simple cubic lattice has been investigated by the Monte Carlo method using the Wolff single-cluster algorithm. It is shown that at the spin concentrations p = 0.9 and 0.8 a second-order phase transition is observed in the three-dimensional weakly diluted Potts model with the state q = 3. On the basis of the finite-size scaling theory, we calculated the static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the correlation-length exponent v.     

On phase transitions in Schlögl's second model

We study Schlögl's second model, characterized by chemical reactions $$\begin{array}{*{20}c} {2X\underset{{k_2 }}{\overset{{k_1 }}{\longleftrightarrow}}3X,} & {X\underset{{k_4 }}{\overset{{k_3 }}{\longleftrightarrow}}0,} \\ \end{array} $$ ind-dimensional space. The reactions are assumed to be local; local fluctuations are fully taken into account, and particle transport occurs via diffusion. In contrast to previous investigations, we find no phase transition whenk ₄≠0 andd<4. Fork ₄=0,k ₃≠0, and 1≦d<4, we find a second-order phase transition which is in the same universality class as the transition in Schlögl's first model. Only ford≧4 we do find the first-order transition found also by previous authors. These claims are supported by extensive Monte Carlo calculations for various realizations of this process on discrete space-time lattices.     

Critical behaviour of random-bond Potts models: a transfer matrix study

We study the two-dimensional Potts model on the square lattice in the presence of quenched random-bond impurities. For q > 4 the first-order transitions of the pure model are softened due to the impurities, and we determine the resulting universality classes by combining transfer matrix data with conformal invariance. The magnetic exponent β/v varies continuously with q, assuming non-Ising values for q > 4, whereas the correlation length exponent ν is numerically consistent with unity. We present evidence for the correctness of a formerly proposed phase diagram, unifying pure, percolative and non-trivial random behaviour.     

Phase transition analysis in Z2 and U(1) lattice gauge theories

The transition region of Z₂ lattice gauge theory is investigated by inverting the strong coupling series of the average plaquette energy E_P(J). We find a clear evidence for a first-order transition and the existence of a metastable phase. In the U(1) case we confirm a second-order phase transition even if there is a little discrepancy on the critical point position as indicated by Monte Carlo simulations.     

The particle structure ofv-dimensional Ising models at low temperatures

In this work we study thev-dimensional Ising model at low temperatures and establish the existence of an upper gap in the energy-momentum spectrum of the two-point function forv3. Forv=2, it is known that this gap is absent.Supported in part by Conselho Nacional de Pesquisas (CNPq-Brazil), Universidade Federal de Minas Gerais (Brazil) and the National Science Foundation under Grant PHY76-17191     

Discontinuity of surface tensions in the q-state Potts model

For the ferromagnetic scalar q-state Potts model on a d-dimensional cubic lattice we prove the following results: (1) We derive a correlation inequality and then we prove that the surface tension between two ordered phases exists in dimension d ⩾ 2 whenever q ⩾ 2 and it is discontinuous at the transition point whenever q is large enough. (2) At the limit q↗ ∞ the surface tension between an ordered phase and the disordered one vanishes everywhere except at the transition point.