Yang-Lee Edge Singularity Research Articles

Directed lattice animals and the Lee-Yang edge singularity

The exponents nu perpendicular to , gamma of directed animals in d dimensions are shown to equal those for the Lee-Yang problem in d-1 dimensions. For d=2 the author obtains the exact results nu perpendicular to = theta =1/2, and for d<or=7 the scaling relation theta =(d-1) nu perpendicular to .

Site and bond directed branched polymers for arbitrary dimensionality: evidence supporting a relation with the Lee-Yang edge singularity

Parisi and Sourlas (1981) have argued that the isotropic branched polymer problem in dimension d+1 is related to the Lee-Yang edge singularity problem in dimension d-1. To test if there is a relation with the directed branched polymer problem, the authors calculate the generating functions for both site and bond directed branched polymers for arbitrary dimension d to order smax=10 and bmax=8. Their analysis lends support to the proposal that theta (d+1)-1= theta D(d)= sigma LY(d-1)+1, where theta and theta D are the critical exponents for lattice animals and directed lattice animals, and sigma LY is the Lee-Yang edge singularity exponent. They also obtain expansions for the growth parameter in the variable sigma -1=(2d-1)-1 for bond and site directed lattice animals.

Critical behavior ofφ3field theories in three dimensions

The methods of renormalized perturbation theory are used to find the critical exponents of the percolation and Yang-Lee edge singularity problems. Both of the above critical phenomena are described by ${\ensuremath{\varphi}}^{3}$ field theories, and the Wilson functions are calculated explicitly in three dimensions to order three loops and summed using the Pad\'e-Borel method. The resulting critical exponent values are consistent with direct series expansion results.

Bond percolation and the Yang-Lee edge singularity problems in three dimensions

Field theories with a trilinear interaction are used to describe both the random bond percolation problem and the Yang-Lee edge singularity problems in three dimensions at criticality. Renormalisation group functions are calculated to order four for the exponents eta , omega and the beta function, and to order five for the remaining critical exponents. Estimations of the critical coupling constant and the exponents for percolation theory are found by summing conformal transformations of the Borel transformed series.

Critical exponents for the percolation problem and the Yang-Lee edge singularity

Gives details of a calculation of critical exponents for a class of field theory models that have an interaction cubic in the fields. The results have already been reported, give the exponents to third order in epsilon where epsilon =6-d and d is the dimensionality of space. The class of models includes the percolation problem and the Yang-Lee edge singularity, so the authors give explicit results for the exponents to order epsilon 3 in these cases. By resummation methods, based on the symptotic behaviour of the epsilon expansion, they obtain numerical estimates for these exponents for a number of interesting values of d.

The Yang-Lee edge singularity by the phenomenological renormalisation group

The phenomenological renormalisation group is applied to the 1D transverse Ising model with longitudinal imaginary field in order to investigate the Yang-Lee edge singularity (1952) of the equivalent 2D Ising model. The method is extended to calculate directly the exponent sigma of the magnetisation.

Critical Behavior of Branched Polymers and the Lee-Yang Edge Singularity

The exponents for large branched dilute polymers (which are also connected with the exponents of the lattice animals) are related in $D$ dimensions to the exponents of the Lee-Yang edge singularity of the Ising model in $D\ensuremath{-}2$ dimensions. From the exact solution of the Ising model in zero and one dimension, one gets the polymer exponents in two and three dimensions, $\ensuremath{\theta}(D=2)=1$ and $\ensuremath{\theta}(D=3)=\frac{3}{2}$.

Equation of state and dynamical properties near the Yang-Lee edge singularity

Statics and dynamics of the Yang-Lee edge singularity are investigated by field theoretic renormalization group techniques. Exploiting a continuous symmetry under a shift of the order parameter we calculate the static critical exponent η to order ɛ2=(6−d)2, in accordance with previous results; in addition, we derive the equation of state and its asymptotic behaviour. The dynamic scaling exponentz is calculated to order ɛ2 from a purely relaxational model with non-conserved order parameter; joining the e-expansion to an exact result ford=0 in a [2/1] Pade approximant we estimatez=1.81 ford=3.

Anderson localization, branched polymers and the Yang-Lee edge singularity

We show that a recently proposed field-theoretic model of Anderson localization has the same critical behaviour as that found in the problem of branched polymers. Thus this model in D dimensions is in the same universality class as that studied by Parisi and Sourlas, which includes the Yang-Lee edge singularity in D — 2 dimensions. We stress that critical properties can be studied either at constant order parameter or at constant field and that critical exponents in the two cases are related by a Fisher renormalization. We note that constant order parameter exponents diverge at a critical dimension Dc.

The Yang-Lee edge singularity studied by a four-level quantum renormalization-group blocking method

The Yang-Lee edge singularity studied by a four-level quantum renormalization-group blocking method

Critical exponents to order ϵ3for ϕ3models of critical phenomena in 6-ϵ dimensions

The authors study scalar field theories for which the interaction term of the Hamiltonian is cubic in the fields. They indicate the circumstances for which field theory models of this type represent continuous phase transitions. The renormalisation group functions for these models are presented up to, and including, three-loop contributions, giving critical exponents to order epsilon 3 in 6- epsilon dimensions. The exponent sigma which characterises the Yang-Lee edge singularity is given explicitly to this order.

1D transverse Ising model in a longitudinal real or complex field

A real space RG method is used to study the quantum Ising chain in a real and complex symmetry breaking field. The later case corresponds to the study the Yang-Lee edge singularity of the equivalent classical Ising model.

Comments on the field-theoretic formulation of the Yang-Lee edge singularity

Fisher has recently shown that the character of the Yang-Lee edge singularity may be obtained from a one-component field theory with pure imaginary phi 3 coupling. The authors point out that phi 4 interactions are not required to stabilise this theory and that, even if they are present, their contribution is likely to be insignificant compared with that from the phi 3 interactions, as the dimension of space is lowered through four.

The Yang-Lee edge singularity in spherical models

The density of Yang-Lee zeros in the thermodynamic limit is discussed for ferromagnetic spherical models of general dimensionalities and arbitrary range of interaction. In all cases the zeros lie on the imaginary axis in the complex magnetic field planeH=H′+iH″ with a densityℊ (H″) that exhibits a square root singularityℊ(H″) ∼(H″-H 0)σ, withσ=1/2, as the edge of the gap atH″=H 0(T) is approached forT>T c. WhenT→T c one hasH 0(T)∼(T∼T c )Δ with critical exponentΔ=β+γ.