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.     

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..     

Thermodynamics of SU(3) lattice gauge theory

The pressure and the energy density of the SU(3) gauge theory are calculated on lattices with temporal extent N_τ = 4, 6 and 8 and spatial extent N_σ = 16 and 32. The results are then extrapolated to the continuum limit. In the investigated temperature range up to five times T_c we observe a 15% deviation from the ideal gas limit. We also present new results for the critical temperature on lattices with temporal extent N_τ = 8 and 12. At the corresponding critical couplings the string tension is calculated on 32⁴ lattices to fix the temperature scale. An extrapolation to the continuum limit yields T_c/√σ = 0.629(3). We furthermore present results on the electric and magnetic condensates as well as the temperature dependence of the spatial string tension. These observables suggest that the temperature dependent running coupling remains large even at T ≅ 5T_c. For the spatial string tension we find √σ_s/T=0.566(13)g²2(T) with g² (5T_c) ≅ 1.5.     

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.     

Alternative scaling hypothesis forSU(2) andSU(3) lattice gauge theories

High precision data from a variety of sources forSU(2) andSU(3) Wilson action lattice gauge theory are analyzed with respect to the hypothesis of the possible existence of a zero temperature deconfining phase transition, in analogy with theU(1) theory. The internal energy, specific heat, string tension, and Wilson line, fit well to correlation length scaling laws associated with a finite order transition occurring at the weak coupling end of the crossover region for both theories. TheSU(2) theory is consistent with a correlation length exponent ν=2/3 and critical pointβ _c ≈2.47. ForSU(3) the data fit well to ν=1 andβ _c ≈6.69. Additional indirect evidence for the existence of such phase transitions is discussed, as is the possible crucial role of light dynamical fermions in the confinement mechanism.     

Vortex free energy and string tension at strong and intermediate coupling

We present results of high temperature expansions up to order g^?24 for the vortex free energy respectively string tension in pure lattice gauge theories with gauge groups SU(2) and Z₂ in 3 and 4 dimensions. For SU(2) in 4 dimensions the result is compared with Monte Carlo calculations of Creutz and is in good agreement. An intermediate coupling region is seen, where the string tension smoothly interpolates between strong coupling and weak coupling behaviour.     

Comments on |V ub /V cb | and |V ub | from non-leptonic B decays within the perturbative QCD approach

The Color String Percolation Model (CSPM) is used to determine the equation of state (EOS) of the Quark–Gluon Plasma (QGP) produced in central Au–Au collisions at $\sqrt{s_{\mathit{NN}}} = 200$  A GeV using STAR data at RHIC. When the initial density of interacting colored strings exceeds the 2D percolation threshold a cluster is formed, which defines the onset of color deconfinement. These interactions also produce fluctuations in the string tension which transforms the Schwinger particle (gluon) production mechanism into a maximum entropy thermal distribution analogous to QCD Hawking–Unruh radiation. The single string tension is determined by identifying the known value of the universal hadron limiting temperature T _c =167.7±2.6 MeV with the CSPM temperature at the critical percolation threshold parameter ξ _c =1.2. At midrapidity the initial Bjorken energy density and the initial temperature determine the number of degrees of freedom consistent with the formation of a ～2+1 flavor QGP. An analytic expression for the equation of state, the sound velocity $C_{s}^{2}(\xi)$ is obtained in CSPM. The CSPM $C_{s}^{2}(\xi)$ and the bulk thermodynamic values energy density ε/T ⁴ and entropy density s/T ³ are in excellent agreement in the phase transition region with recent lattice QCD simulations (LQCD) by the HotQCD Collaboration.     

Z(2) vortices and the SU(2) string tension

Topologically determined Z(2) variables in pure SU(2) lattice gauge theory are discussed. They count the number of “vortex souls”. The high temperature expansion for the corresponding Z(2) loops is examined. They obey an area law. The coefficient of the area is shown to be equal to the string tension to all orders of the high temperature expansion. This shows that the string tension is determined by the probability distribution of the vortex souls, at least in the high temperature region. The dependence of the string tension α(β, h) on an external field h that is coupled to the Z(2) field strength is calculated to lowest order of the high temperature expansion. In this approximation, α(β, h) is determined by the free energy of a 2-dimensional Ising model in an external magnetic field $\text{1}\text{2}\text{log}\text{(β/4}\text{tanh}\text{h)}$ at an inverse temperature $\text{1}\text{2}\text{log}\text{3}\text{4}\text{π = 0.429}$.     

SU(2) Lattice gauge theory in (2+1)D

Cluster expansion methods are applied to theSU(2) lattice gauge model in (2+1) dimensions. Strong-coupling series are calculated for the vacuum energy per site, the axial string tension, and the scalar mass gap; while ELCE approximants are used to estimate the string tension beyond its roughening transition. The simple scaling behaviour expected of this super-renormalizable theory is clearly seen, and we estimate that in the continuum limit the string tension σ～(0.14±0.01)g ⁴, while the mass gapM _s ～(2.2±0.25)g ². More accurate Monte Carlo simulations are needed to check the universality between the Hamiltonian and Euclidean versions of this model.     

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 static potential at large N

In a partial reduced Eguchi-Kawai model with temporal gauge fixing the large N static potential V(R) is determined for R ⩽ 5 and the string tension σ is extracted. Assuming asymptotic scaling, for the continuum value of the deconfinement temperature 0.60±0.13 is found in units of T_c/√σ. This is the first quantitative confirmation that large N is a goodapproximation to N = 3 (where T_c/√σ = 0.61).     

High statistics study of the chiral condensate in quenched lattice QCD

We present the results of a high statistics study of the chiral condensate in quenched lattice QCD on an 8⁴ lattice at β = 5.4, 5.5, 5.6, 5.7, 5.8 and 6.0. We see clear evidence for deviation from asymptotic scaling in the range of β considered. Our results are in agreement with the behaviour anticipated from recent Monte Carlo renormalisation group studies of the β-function. We find indications of a common scaling behaviour for the condensate, the string tension and the deconfinement temperature.     

The estimates of correlations in two-dimensional Ising model

We investigate the correlation inequalities and the decay of correlations of stochastic Ising model in a rectangle with side length 2L×K(LlnL)^1/2, where K is some positive constant. With different boundary conditions, at inverse temperature β>β_c or β<β_c and zero external field, we show some estimates of the correlation functions for the two-dimensional Ising model.     

Linked cluster expansions for U(1) lattice gauge theory in 2 + 1 and 3 + 1 dimensions

Compact U(1) lattice gauge theory is studied in 2 + 1 and 3 + 1 dimensions using strong coupling series expansions and the recently proposed exact linled cluster expansion alborithm Results for the vacuum energy, specific heat and axial string tension in 2 + 1 dimensions are in agreement with previous finite lattice estimates. In 3 + 1 dimensions, we present new strong coupling series results (order g^?40) which together with the ELCE estimates show evidence of a continuous phase transition at x = 1/g⁴ = 0.72 ± 0.08. The associated critical index for the vanishing string tension is μ = 0.65 ± 0.12. The axial string tension in D = 3 + 1 appears to undergo a non-deconfining roughening transition at smaller x (0.56 ± 0.07).     

Crystallographic study of the phase transition in (Me4P)2ZnBr4

The compound (Me₄P)₂ZnBr₄, a member of the β-K₂SO₄ structure class, undergoes a phase transition at 84°C from the room temperature space group P12₁/c1 to the parent Pmcn structure. The room temperature structure corresponds to a ferrodistortive transition of B_1g symmetry at the zone center. At room temperature, the compound has lattice constants a=9.501(1), b=16.055(2), c=13.127(2) Å and β=90.43(1)°. For the high temperature phase, the orthorhombic cell has dimensions a=9.466(2), b=16.351(3) and c=13.284(2) Å. The structures consist of two crystallographically independent Me₄P⁺ cations and the ZnBr₄²⁻ anions. In the room temperature phase, all three ionic species show substantial displacement from the mirror plane perpendicular to the a-axis that exists in the high temperature phase, as well as rotations out of that plane. The thermal parameters of the cations are indicative of substantial librational motion. Measurements of lattice parameters have been made at 2-5°C intervals over the temperature range 40-140°C. The changes in the lattice constants appear continuous at T_c (within experimental limits) indicating that the phase transition is likely second-order. The a lattice constant shows an anomalous shortening as T_c is approached. Thermal expansion coefficients are calculated from this data. An application of Landau theory is used to derive the temperature dependencies of spontaneous shear strain and corresponding elastic stiffness constants associated with the primary order parameter.     

Physical mechanism of the (tri)critical point generation

We discuss some ideas resulting from a phenomenological relation recently declared between the tension of string connecting the static quark-antiquark pair and surface tension of corresponding cylindrical bag. This relation analysis leads to the temperature of vanishing surface tension coefficient of the QGP bags at zero baryonic charge density as T _?? = 152.9 ± 4.5 MeV. We develop the view point that this temperature value is not a fortuitous coincidence with the temperature of (partial) chiral symmetry restoration as seen in the lattice QCD simulations. Besides, we argue that T _?? defines the QCD (tri)critical endpoint temperature and claim that a negative value of surface tension coefficient recently discovered is not a sole result but is quite familiar for ordinary liquids at the supercritical temperatures.     

An investigation of the phase structure of the two-dimensional O(2) and O(4) symmetric nonlinear sigma-models

We investigate the phase structure of the two-dimensional O(2) and O(4) lattice σ-models by means of a Monte Carlo simulation using Binder's phenomenological renormalization group. For the O(2) model the transition temperature β_c⁻¹ is estimated and the expected line of critical points is found. For the O(4) model no signal of a phase transition is found in the range of couplings considered. This is in contradiction to a recent claim that the O(4) model has a critical point at finite β.     

Structural phase transitions at high-temperature in double perovskite Sr2GdRuO6

The crystal structure evolution of the Sr₂GdRuO₆ complex perovskite at high-temperature has been investigated over a wide temperature range between 298 K≤T≤1273 K. Powder X-ray diffraction measurements at room temperature and Rietveld analysis show that this compounds crystallizes in a monoclinic perovskite-type structure with P2₁/n (#14) space group and the 1:1 ordered arrangement of Ru⁵⁺ and Gd³⁺ cations over the six-coordinate M sites, with lattice parameters a=5.81032(8) Å, b=5.82341(4) Å, c=8.21939(7) Å, V=278.11(6) Å³ and angle β=90.311(2)^o. The high-temperature analysis shows that this material suffers two-phase transitions. At 373 K it adopts a monoclinic perovskite structure with I2/m space group, and lattice parameters a=5.81383(2) Å, b=5.82526(4) Å, c=8.22486(1) Å, V=278.56(2) Å³ and angle β=90.28(2)^o. Above of 773 K, it suffers a phase transition from monoclinic I2/m to tetragonal I4/m, with lattice parameters a=5.84779(1) Å, c=8.27261(1) Å, V=282.89(5) Å³ and angle β=90.02(9)^o. The high-temperature phase transition from monoclinic I2/m to tetragonal I4/m is characterized by strongly anisotropic displacements of the anions.