Logarithmic Corrections to Scaling in the XY2XY_2--Model

We study the distribution of partition function zeroes for the $XY$--model in two dimensions. In particular we find the scaling behaviour of the end of the distribution of zeroes in the complex external magnetic field plane in the thermodynamic limit (the Yang--Lee edge) and the form for the density of these zeroes. Assuming that finite--size scaling holds, we show that there have to exist logarithmic corrections to the leading scaling behaviour of thermodynamic quantities in this model. These logarithmic corrections are also manifest in the finite--size scaling formulae and we identify them numerically. The method presented here can be used to check the compatibility of scaling behaviour of odd and even thermodynamic functions in other models too.Comment: 3 pages, latex, 2 figure

Universal Amplitude Ratios for Constrained Critical Systems

The critical properties of systems under constraint differ from their ideal counterparts through Fisher renormalization. The mathematical properties of Fisher renormalization applied to critical exponents are well known: the renormalized indices obey the same scaling relations as the ideal ones and the transformations are involutions in the sense that re-renormalizing the critical exponents of the constrained system delivers their original, ideal counterparts. Here we examine Fisher renormalization of critical amplitudes and show that, unlike for critical exponents, the associated transformations are not involutions. However, for ratios and combinations of amplitudes which are universal, Fisher renormalization is involutory.Comment: JSTAT published versio

Critical Behaviour of the Two Dimensional Step Model

We use finite--size scaling of Lee--Yang partition function zeroes to study the critical behaviour of the two dimensional step or sgn $O(2)$ model. We present evidence that, like the closely related $XY$--model, this has a phase transition from a disordered high temperature phase to a low temperature massless phase where the model remains critical. The critical parameters (including logarithmic corrections) are compatible with those of the $XY$--model indicating that both models belong to the same universality class.Comment: 6 pages latex, 3 postscript figures, compressed and uuencoded (revised remarks on Lee_Yang theorem, version to appear in Phys Rev B

Phase Transition Strength through Densities of General Distributions of Zeroes

A recently developed technique for the determination of the density of partition function zeroes using data coming from finite-size systems is extended to deal with cases where the zeroes are not restricted to a curve in the complex plane and/or come in degenerate sets. The efficacy of the approach is demonstrated by application to a number of models for which these features are manifest and the zeroes are readily calculable.Comment: 16 pages, 12 figure

Logarithmic Corrections to Scaling in the Two Dimensional XYXY--Model

By expressing thermodynamic functions in terms of the edge and density of Lee--Yang zeroes, we relate the scaling behaviour of the specific heat to that of the zero field magnetic susceptibility in the thermodynamic limit of the $XY$--model in two dimensions. Assuming that finite--size scaling holds, we show that the conventional Kosterlitz--Thouless scaling predictions for these thermodynamic functions are not mutually compatable unless they are modified by multiplicative logarithmic corrections. We identify these logarithmic corrections analytically in the case of the specific heat and numerically in the case of the susceptibility. The techniques presented here are general and can be used to check the compatibility of scaling behaviour of odd and even thermodynamic functions in other models too.Comment: 11 pages, latex, 4 figure

Role of topological defects in the phase transition of modified XY model : A Monte Carlo study

Monte Carlo simulation has been performed on a classical two dimensional XY- model with a modified form of interaction potential to investigate the role of topological defects on the phase transition exhibited by the model. In simulations in a restricted ensemble without defects, the system appears to remain ordered at all temperatures. Suppression of topological defects on the square plaquettes in the modified XY- model leads to complete elimination of the phase transition observed in this model.Comment: 19 pages, 12 figures, Accepted for publication in Phys. Rev.

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.