The Phase Structure of the Weakly Coupled Lattice Schwinger Model

The weak coupling expansion is applied to the single flavour Schwinger model with Wilson fermions on a symmetric toroidal lattice of finite extent. We develop a new analytic method which permits the expression of the partition function as a product of pure gauge expectation values whose zeroes are the Lee-Yang zeroes of the model. Application of standard finite-size scaling techniques to these zeroes recovers previous numerical results for the small and moderate lattice sizes to which those studies were restricted. Our techniques, employable for arbitrarily large lattices, reveal the absence of accumulation of these zeroes on the real hopping parameter axis at constant weak gauge coupling. The consequence of this previously unobserved behaviour is the absence of a zero fermion mass phase transition in the Schwinger model with single flavour Wilson fermions at constant weak gauge coupling.Comment: 8 pages, 2 figures, insert to figure 2 include

The Structure of the Aoki Phase at Weak Coupling

A new method to determine the phase diagram of certain lattice fermionic field theories in the weakly coupled regime is presented. This method involves a new type of weak coupling expansion which is multiplicative rather than additive in nature and allows perturbative calculation of partition function zeroes. Application of the method to the single flavour Gross-Neveu model gives a phase diagram consistent with the parity symmetry breaking scenario of Aoki and provides new quantitative information on the width of the Aoki phase in the weakly coupled sector.Comment: 9 pages, 1 figure (minor changes) To be published in Phys. Lett.

Scaling and Density of Lee-Yang Zeroes in the Four Dimensional Ising Model

The scaling behaviour of the edge of the Lee--Yang zeroes in the four dimensional Ising model is analyzed. This model is believed to belong to the same universality class as the $\phi^4_4$ model which plays a central role in relativistic quantum field theory. While in the thermodynamic limit the scaling of the Yang--Lee edge is not modified by multiplicative logarithmic corrections, such corrections are manifest in the corresponding finite--size formulae. The asymptotic form for the density of zeroes which recovers the scaling behaviour of the susceptibility and the specific heat in the thermodynamic limit is found to exhibit logarithmic corrections too. The density of zeroes for a finite--size system is examined both analytically and numerically.Comment: 17 pages (4 figures), LaTeX + POSTSCRIPT-file, preprint UNIGRAZ-UTP 20-11-9

Universal finite-size scaling for percolation theory in high dimensions

We present a unifying, consistent, finite-size-scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions $d_c$. Behaviour at the critical point is non-universal in $d>d_c=6$ dimensions. Proliferation of the largest clusters, with fractal dimension $4$, is associated with the breakdown of hyperscaling there when free boundary conditions are used. But when the boundary conditions are periodic, the maximal clusters have dimension $D=2d/3$, and obey random-graph asymptotics. Universality is instead manifest at the pseudocritical point, where the failure of hyperscaling in its traditional form is universally associated with random-graph-type asymptotics for critical cluster sizes, independent of boundary conditions.Comment: Revised version, 26 pages, no 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

The Weakly Coupled Gross-Neveu Model with Wilson Fermions

The nature of the phase transition in the lattice Gross-Neveu model with Wilson fermions is investigated using a new analytical technique. This involves a new type of weak coupling expansion which focuses on the partition function zeroes of the model. Its application to the single flavour Gross-Neveu model yields a phase diagram whose structure is consistent with that predicted from a saddle point approach. The existence of an Aoki phase is confirmed and its width in the weakly coupled region is determined. Parity, rather than chiral symmetry breaking naturally emerges as the driving mechanism for the phase transition.Comment: 15 pages including 1 figur

Griffiths singularities in the two dimensional diluted Ising model

We study numerically the probability distribution of the Yang-Lee zeroes inside the Griffiths phase for the two dimensional site diluted Ising model and we check that the shape of this distribution is that predicted in previous analytical works. By studying the finite size scaling of the averaged smallest zero at the phase transition we extract, for two values of the dilution, the anomalous dimension, $\eta$, which agrees very well with the previous estimated values.Comment: 11 pages and 4 figures, some minor changes in Fig. 4, available at http://chimera.roma1.infn.it/index_papers_complex.htm

Ising spins coupled to a four-dimensional discrete Regge skeleton

Regge calculus is a powerful method to approximate a continuous manifold by a simplicial lattice, keeping the connectivities of the underlying lattice fixed and taking the edge lengths as degrees of freedom. The discrete Regge model employed in this work limits the choice of the link lengths to a finite number. To get more precise insight into the behavior of the four-dimensional discrete Regge model, we coupled spins to the fluctuating manifolds. We examined the phase transition of the spin system and the associated critical exponents. The results are obtained from finite-size scaling analyses of Monte Carlo simulations. We find consistency with the mean-field theory of the Ising model on a static four-dimensional lattice.Comment: 19 pages, 7 figure

Density of states, Potts zeros, and Fisher zeros of the Q-state Potts model for continuous Q

The Q-state Potts model can be extended to noninteger and even complex Q in the FK representation. In the FK representation the partition function,Z(Q,a), is a polynomial in Q and v=a-1(a=e^-T) and the coefficients of this polynomial,Phi(b,c), are the number of graphs on the lattice consisting of b bonds and c connected clusters. We introduce the random-cluster transfer matrix to compute Phi exactly on finite square lattices. Given the FK representation of the partition function we begin by studying the critical Potts model Z_{CP}=Z(Q,a_c), where a_c=1+sqrt{Q}. We find a set of zeros in the complex w=sqrt{Q} plane that map to the Beraha numbers for real positive Q. We also identify tilde{Q}_c(L), the value of Q for a lattice of width L above which the locus of zeros in the complex p=v/sqrt{Q} plane lies on the unit circle. We find that 1/tilde{Q}_c->0 as 1/L->0. We then study zeros of the AF Potts model in the complex Q plane and determine Q_c(a), the largest value of Q for a fixed value of a below which there is AF order. We find excellent agreement with Q_c=(1-a)(a+3). We also investigate the locus of zeros of the FM Potts model in the complex Q plane and confirm that Q_c=(a-1)^2. We show that the edge singularity in the complex Q plane approaches Q_c as Q_c(L)~Q_c+AL^-y_q, and determine the scaling exponent y_q. Finally, by finite size scaling of the Fisher zeros near the AF critical point we determine the thermal exponent y_t as a function of Q in the range 2<Q<3. We find that y_t is a smooth function of Q and is well fit by y_t=(1+Au+Bu^2)/(C+Du) where u=u(Q). For Q=3 we find y_t~0.6; however if we include lattices up to L=12 we find y_t~0.50.Comment: to appear in Physical Review

Critical phenomena on scale-free networks: logarithmic corrections and scaling functions

In this paper, we address the logarithmic corrections to the leading power laws that govern thermodynamic quantities as a second-order phase transition point is approached. For phase transitions of spin systems on d-dimensional lattices, such corrections appear at some marginal values of the order parameter or space dimension. We present new scaling relations for these exponents. We also consider a spin system on a scale-free network which exhibits logarithmic corrections due to the specific network properties. To this end, we analyze the phase behavior of a model with coupled order parameters on a scale-free network and extract leading and logarithmic correction-to-scaling exponents that determine its field- and temperature behavior. Although both non-trivial sets of exponents emerge from the correlations in the network structure rather than from the spin fluctuations they fulfil the respective thermodynamic scaling relations. For the scale-free networks the logarithmic corrections appear at marginal values of the node degree distribution exponent. In addition we calculate scaling functions, which also exhibit nontrivial dependence on intrinsic network properties.Comment: 15 pages, 4 figure