Yacine Ikhlef

Bio: Yacine Ikhlef is an academic researcher from University of Geneva. The author has contributed to research in topics: Holomorphic function & Square lattice. The author has an hindex of 16, co-authored 39 publications receiving 665 citations. Previous affiliations of Yacine Ikhlef include Centre national de la recherche scientifique & University of Paris-Sud.

Classical dimers with aligning interactions on the square lattice

Abstract: We present a detailed study of a model of close-packed dimers on the square lattice with an interaction between nearest-neighbor dimers The interaction favors parallel alignment of dimers, resulting in a low-temperature crystalline phase With large-scale Monte Carlo and Transfer Matrix calculations, we show that the crystal melts through a Kosterlitz-Thouless phase transition to give rise to a high-temperature critical phase, with algebraic decays of correlations functions with exponents that vary continuously with the temperature We give a theoretical interpretation of these results by mapping the model to a Coulomb gas, whose coupling constant and associated exponents are calculated numerically with high precision Introducing monomers is a marginal perturbation at the Kosterlitz-Thouless transition and gives rise to another critical line We study this line numerically, showing that it is in the Ashkin-Teller universality class, and terminates in a tricritical point at finite temperature and monomer fugacity In the course of this work, we also derive analytic results relevant to the non-interacting case of dimer coverings, including a Bethe Ansatz (at the free fermion point) analysis, a detailed discussion of the effective height model and a free field analysis of height fluctuations

Integrable Spin Chain for the SL(2,R)/U(1) Black Hole Sigma Model

Abstract: We introduce a spin chain based on finite-dimensional spin-$1/2$ SU(2) representations but with a non-Hermitian ``Hamiltonian'' and show, using mostly analytical techniques, that it is described at low energies by the $\mathrm{SL}(2,R)/\mathrm{U}(1)$ Euclidian black hole conformal field theory. This identification goes beyond the appearance of a noncompact spectrum; we are also able to determine the density of states, and show that it agrees with the formulas in [J. Maldacena, H. Ooguri, and J. Son, J. Math. Phys. (N.Y.) 42, 2961 (2001)] and [A. Hanany, N. Prezas, and J. Troost, J. High Energy Phys. 04 (2002) 014], hence providing a direct ``physical measurement'' of the associated reflection amplitude.

Discretely holomorphic parafermions and integrable loop models

Abstract: We define parafermionic observables in various lattice loop models, including examples where no Kramers–Wannier duality holds. For a particular rhombic embedding of the lattice in the plane and a value of the parafermionic spin these variables are discretely holomorphic (they satisfy a lattice version of the Cauchy–Riemann equations) as long as the Boltzmann weights satisfy certain linear constraints. In the cases considered, the weights then also satisfy the critical Yang–Baxter equations, with the spectral parameter being related linearly to the angle of the elementary rhombus.

A staggered six-vertex model with non-compact continuum limit

Abstract: The antiferromagnetic critical point of the Potts model on the square lattice was identified by Baxter as a staggered integrable six-vertex model. In this work, we investigate the integrable structure of this model. It enables us to derive some new properties, such as the Hamiltonian limit of the model, an equivalent vertex model, and the structure resulting from the Z_2 symmetry. Using this material, we discuss the low-energy spectrum, and relate it to geometrical excitations. We also compute the critical exponents by solving the Bethe equations for a large lattice width N. The results confirm that the low-energy spectrum is a collection of continua with typical exponent gaps of order 1/(log N)^2.

Exactly Solved Models in Statistical Mechanics

Abstract: R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Quantum Inverse Scattering Method and Correlation Functions

Abstract: One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

A guide to quantum groups

Abstract: By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$

Abstract: We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to $\sqrt{2+\sqrt 2}$. This value has been derived non rigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to SLE(8/3).