Vladimir S. Dotsenko

Renormalisation-group calculation of correlation functions for the 2D random bond Ising and Potts models

Abstract We find the crossover behaviour of the disorder averaged spin-spin correlation function for the 2D Ising and 3-state Potts model with random bonds at the critical point. The procedure employed is the renormalisation approach of the perturbation series around the conformal field theories representing the pure models. We obtain a crossover in the amplitude for the correlation function for the Ising model, which does not change the critical exponent, and a shift in the critical exponent produced by randomness in the case of the Potts model. A comparison with numerical data is discussed briefly.

Parafermionic theory with the symmetry Z5

Abstract A parafermionic conformal theory with the symmetry Z5 is constructed, based on the second solution of Fateev–Zamolodchikov for the corresponding parafermionic chiral algebra. The primary operators of the theory, which are the singlet, doublet 1, doublet 2, and disorder operators, are found to be accommodated by the weight lattice of the classical Lie algebra B2. The finite Kac tables for unitary theories are defined and the formula for the conformal dimensions of primary operators is given.

Parafermionic theory with the symmetry Z(N), for N even

Abstract We construct a parafermionic conformal theory with the symmetry ZN, for N odd, based on the second solution of Fateev–Zamolodchikov for the corresponding parafermionic chiral algebra. Primary operators are classified according to their transformation properties under the dihedral group DN, as singlet, doublet 1,2,…, N−1 2 , and disorder operators. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the weight lattice of the Lie algebra B(N−1)/2. The unitary theories are representations of the coset SOn(N)×SO2(N)/SOn+2(N), with n=1,2,… . Physically, they realise the series of multicritical points in statistical theories having a DN symmetry.

Renormalization Group Solution for the Two-Dimensional Random Bond Potts Model with Broken Replica Symmetry

We find a new solution of the renormalization group for the Potts model with ferromagnetic random valued coupling constants. The solution exhibits universality and broken replica symmetry. It is argued that the model reaches this universality class if the replica symmetry is broken initially. Otherwise the model stays with the replica-symmetric renormalization group flow and reaches the fixed point which has been considered before.

The third parafermionic chiral algebra with the symmetry Z3

Abstract We have constructed the parafermionic chiral algebra with the principal parafermionic fields Ψ , Ψ + having the conformal dimension Δ Ψ = 8 / 3 and realizing the symmetry Z 3 .

RANDOM BOND POTTS MODEL : THE TEST OF THE REPLICA SYMMETRY BREAKING

Abstract The averaged spin-spin correlation function squared is calculated for the ferro-magnetic random bond Potts model in two dimensions. The technique being used is the renormalization group plus conformal field theory. The results are of the E-expansion type fixed point calculation, E being the deviation of the central charge (or the number of components) of the Potts model from the Ising model value. Calculations are done both for the replica symmetric and the replica symmetry broken fixed points. The results obtained allow for numerical simulation tests to decide between the two different criticalities of the random bond Potts model.

Coupled Potts models: Self-duality and fixed point structure

Abstract We consider q -state Potts models coupled by their energy operators. Restricting our study to self-dual couplings, numerical simulations demonstrate the existence of non-trivial fixed points for 2 ⩽ q ⩽ 4. These fixed points were first predicted by perturbative renormalisation group calculations. Accurate values for the central charge and the multiscaling exponents of the spin and energy operators are calculated using a series of novel transfer matrix algorithms employing clusters and loops. These results compare well with those of the perturbative expansion, in the range of parameter values where the latter is valid. The criticality of the fixed-point models is independently verified by examining higher eigenvalues in the even sector, and by demonstrating the existence of scaling laws from Monte Carlo simulations. This might be a first step towards the identification of the conformal field theories describing the critical behaviour of this class of models.

Classification of conformal field theories based on Coulomb gases. Application to loop models

Abstract We present a method for classifying conformal field theories based on Coulomb gases (bosonic free-field construction). Given a particular geometric configuration of the screening charges, we give necessary conditions for the existence of degenerate representations and for the closure of the vertex-operator algebra. The resulting classification contains, but is more general than, the standard one based on classical Lie algebras. We then apply the method to the Coulomb gas theory for the two-flavoured loop model of Jacobsen and Kondev. The purpose of the study is to clarify the relation between Coulomb gas models and conformal field theories with extended symmetries.

Quantum Field Theory and String Theory

Fractal Structure in 4d Gravity -- A One Dimensional Ideal Gas of Spinons, or Some Exact Results on the XXX Spin Chain with Long Range Interaction -- Kodaira-Spencer Theory of Gravity -- 3d Gravity and Gauge Theories -- On the W-Gravity Spectrum and its G-Structure -- Light-Cone Quantization of Matrix Models at c>1 -- Multicritical Points of 2-Matrix Models -- The Super Self-Dual Matreoshka -- The Phenomenology of Strings and Clusters in the 3-d Ising Model -- Conformai Field Theory Techniques in Large N Yang-Mills Theory -- to Differential W-Geometry -- Topological Strings and QCD in Two Dimensions -- Continuum QCD2 in Terms of Discrete Random Surfaces with Local Weights -- Strings and Causality -- Loop Equation and Area Law in Turbulence -- The Two-Dimensional String as a Topological Field Theory -- Linear Systems for 2d Poincare Supergravities -- Quantization of Mirror Symmetry -- Integrable Qft2 Encoded on Products of Dynkin Diagrams -- Remarks on Topological String Theories -- Hamiltonian Reduction of the BRST Complex and N=2 SUSY -- Lattice Models and N=2 Supersymmetry -- Canonical Construction of Liouville Field Operators with Arbitrary Spin -- Bethe Ansatz for the Bloch Particle in Magnetic Field.

Correlation function of four spins in the percolation model

Abstract By using the Coulomb gas technics we calculate the four-spin correlation function in the percolation q → 1 limit of the Potts model. It is known that the four-point functions define the actual fusion rules of a particular model. In this respect, we find that fusion of two spins, of dimension Δ σ = 5 96 , produce a new channel, in the 4-point function, which is due to the operator with dimension Δ = 5 / 8 .