Related papers: Discrete Holomorphicity at Two-Dimensional Critica…
Over the past few years it has been discovered that an "observable" can be set up on the lattice which obeys the discrete Cauchy-Riemann equations. The ensuing condition of discrete holomorphicity leads to a system of linear equations which…
We consider critical curves -- conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field…
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…
We introduce a new version of discrete holomorphic observables for the critical planar Ising model. These observables are holomorphic spinors defined on double covers of the original multiply connected domain. We compute their scaling…
Schramm-Loewner evolution appears as the scaling limit of interfaces in lattice models at critical point. Critical behavior of these models can be described by minimal models of conformal field theory. Certain CFT correlation functions are…
Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin, Polyakov, and Zamolodchikov [BPZ84a]. Both exhibit exactly solvable structures in two dimensions. A…
Critical points of classical and quantum lattice models are often described by scale-invariant Lifshitz theories which are anisotropic in the continuum limit, as characterized by a dynamical critical exponent $z\neq1$. This type of critical…
It is believed that the large-scale geometric properties of two-dimensional critical percolation are described by a logarithmic conformal field theory, but it has been challenging to exhibit concrete examples of logarithmic singularities…
Scharmm-Loewner evolution (SLE) and conformal field theory (CFT) are popular and widely used instruments to study critical behavior of two-dimensional models, but they use different objects. While SLE has natural connection with lattice…
We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward…
Logarithmic Conformal Field Theories (LCFT) play a key role, for instance, in the description of critical geometrical problems (percolation, self avoiding walks, etc.), or of critical points in several classes of disordered systems…
These lecture notes provide a (almost) self-contained account on conformal invariance of the planar critical Ising and FK-Ising models. They present the theory of discrete holomorphic functions and its applications to planar statistical…
Conformally-invariant curves that appear at critical points in two-dimensional statistical mechanics systems, and their fractal geometry have received a lot of attention in recent years. On the one hand, Schramm has invented a new rigorous…
The Schramm-Loewner evolution (SLE) is a powerful tool to describe fractal interfaces in 2D critical statistical systems. Yet the application of SLE is well established for statistical systems described by quantum field theories satisfying…
Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, ... This has led to numerous exact (but non-rigorous) predictions of their scaling…
We explore the connection between the transfer matrix formalism and discrete complex analysis approach to the two dimensional Ising model. We construct a discrete analytic continuation matrix, analyze its spectrum and establish a direct…
We construct lattice parafermions - local products of order and disorder operators - in nearest-neighbor Z(N) models on regular isotropic planar lattices, and show that they are discretely holomorphic, that is they satisfy discrete…
The large-scale behavior of two-dimensional critical percolation is expected to be described by a conformal field theory (CFT). Moreover, this putative CFT is believed to be of the logarithmic type, exhibiting logarithmic corrections to the…
One of the most natural and challenging issues in discrete complex analysis is to prove the convergence of discrete holomorphic functions to their continuous counterparts. This article is to solve the open problem in the general setting. To…
The laws of quantum-critical scaling theory of quantum fidelity, dependent on the underlying system dimensionality $D$, have so far been verified in exactly solvable $1D$ models, belonging to or equivalent to interacting, quadratic…