The Phase Diagrams of the Schwinger and Gross-Neveu Models with Wilson Fermions

A new method to analytically determine the partition function zeroes of weakly coupled theories on finite-size lattices is developed. Applied to the lattice Schwinger model, this reveals the possible absence of a phase transition at fixed weak coupling. We show how finite-size scaling techniques on small or moderate lattice sizes may mimic the presence of a spurious phase transition. Application of our method to the Gross-Neveu model yields a phase diagram consistent with that coming from a saddle point analysis.Comment: Talk at LATTICE99, 3 pages, 2 figure

On the Correct Convergence of Complex Langevin Simulations for Polynomial Actions

There are problems in physics and particularly in field theory which are defined by complex valued weight functions $e^{-S}$ where $S$ is a polynomial action $S: R^n \rightarrow C$. The conditions under which a convergent complex Langevin calculation correctly simulates such integrals are discussed. All conditions on the process which are used to prove proper convergence are defined in the stationary limit.Comment: 8 pages, LaTeX file, preprint UNIGRAZ-UTP 29-09-9

Normalization of the chiral condensate in the massive Schwinger model

Within mass perturbation theory, already the first order contribution to the chiral condensate of the massive Schwinger model is UV divergent. We discuss the problem of choosing a proper normalization and, by making use of some bosonization results, we are able to choose a normalization so that the resulting chiral condensate may be compared, e.g., with lattice data.Comment: Latex file, 8 pages, 1 figure, needed macro: psbox.te

Critical Behavior of the Schwinger Model with Wilson Fermions

We present a detailed analysis, in the framework of the MFA approach, of the critical behaviour of the lattice Schwinger model with Wilson fermions on lattices up to $24^2$, through the study of the Lee-Yang zeros and the specific heat. We find compelling evidence for a critical line ending at $\kappa = 0.25$ at large $\beta$. Finite size scaling analysis on lattices $8^2,12^2,16^2, 20^2$ and $24^2$ indicates a continuous transition. The hyperscaling relation is verified in the explored $\beta$ region.Comment: 12 pages LaTeX file, 10 figures in one uuencoded compressed postscript file. Report LNF-95/049(P

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

Some approximate analytical methods in the study of the self-avoiding loop model with variable bending rigidity and the critical behaviour of the strong coupling lattice Schwinger model with Wilson fermions

Some time ago Salmhofer demonstrated the equivalence of the strong coupling lattice Schwinger model with Wilson fermions to a certain 8-vertex model which can be understood as a self-avoiding loop model on the square lattice with bending rigidity $\eta = 1/2$ and monomer weight $z = (2\kappa)^{-2}$. The present paper applies two approximate analytical methods to the investigation of critical properties of the self-avoiding loop model with variable bending rigidity, discusses their validity and makes comparison with known MC results. One method is based on the independent loop approximation used in the literature for studying phase transitions in polymers, liquid helium and cosmic strings. The second method relies on the known exact solution of the self-avoiding loop model with bending rigidity $\eta = 1/\sqrt{2}$. The present investigation confirms recent findings that the strong coupling lattice Schwinger model becomes critical for $\kappa_{cr} \simeq 0.38-0.39$. The phase transition is of second order and lies in the Ising model universality class. Finally, the central charge of the strong coupling Schwinger model at criticality is discussed and predicted to be $c = 1/2$.Comment: 22 pages LaTeX, 6 Postscript figure

Chiral Symmetry in Two-Color QCD at Finite Temperature

We study the chiral symmetry in two-color QCD with N massless flavors at finite temperature, using an effective theory. For the gauge group SU(2), the chiral symmetry is enlarged to SU(2N), which is then spontaneously broken to Sp(2N) at zero temperature. At finite temperature, and when the axial anomaly can be neglected, we find a first order phase transition occurring for two or more flavors. In the presence of instantons, the symmetry restoration unambiguously remains first order for three or more massless flavors. These results could be relevant for lattice studies of chiral symmetry at finite temperature and density.Comment: 10 pages, Revte

Critical Constraints on Chiral Hierarchies

We consider the constraints that critical dynamics places on models with a top quark condensate or strong extended technicolor (ETC). These models require that chiral-symmetry-breaking dynamics at a high energy scale plays a significant role in electroweak symmetry breaking. In order for there to be a large hierarchy between the scale of the high energy dynamics and the weak scale, the high energy theory must have a second order chiral phase transition. If the transition is second order, then close to the transition the theory may be described in terms of a low-energy effective Lagrangian with composite ``Higgs'' scalars. However, scalar theories in which there are more than one $\Phi^4$ coupling can have a {\it first order} phase transition instead, due to the Coleman-Weinberg instability. Therefore, top-condensate or strong ETC theories in which the composite scalars have more than one $\Phi^4$ coupling cannot always support a large hierarchy. In particular, if the Nambu--Jona-Lasinio model solved in the large-$N_c$ limit is a good approximation to the high-energy dynamics, then these models will not produce acceptable electroweak symmetry breaking.Comment: 10 pages, 1 postscript figure (appended), BUHEP-92-35, HUTP-92/A05

Non-Gaussian fixed point in four-dimensional pure compact U(1) gauge theory on the lattice

The line of phase transitions, separating the confinement and the Coulomb phases in the four-dimensional pure compact U(1) gauge theory with extended Wilson action, is reconsidered. We present new numerical evidence that a part of this line, including the original Wilson action, is of second order. By means of a high precision simulation on homogeneous lattices on a sphere we find that along this line the scaling behavior is determined by one fixed point with distinctly non-Gaussian critical exponent nu = 0.365(8). This makes the existence of a nontrivial and nonasymptotically free four-dimensional pure U(1) gauge theory in the continuum very probable. The universality and duality arguments suggest that this conclusion holds also for the monopole loop gas, for the noncompact abelian Higgs model at large negative squared bare mass, and for the corresponding effective string theory.Comment: 11 pages, LaTeX, 2 figure

Universality of the Ising Model on Sphere-like Lattices

We study the 2D Ising model on three different types of lattices that are topologically equivalent to spheres. The geometrical shapes are reminiscent of the surface of a pillow, a 3D cube and a sphere, respectively. Systems of volumes ranging up to O($10^5$) sites are simulated and finite size scaling is analyzed. The partition function zeros and the values of various cumulants at their respective peak positions are determined and they agree with the scaling behavior expected from universality with the Onsager solution on the torus ($\nu=1$). For the pseudocritical values of the coupling we find significant anomalies indicating a shift exponent $\neq 1$ for sphere-like lattice topology.Comment: 24 pages, LaTeX, 8 figure