2d ising cft

PDF

Analyzing the two dimensional Ising model with conformal eld

plication to the statistical mechanical Ising model of ferromagnetism is presented This approach allows to highlight the connections between phenomena at criticality and quantum eld theory in particular the model of a free fermion Starting from a classical picture the Ising model is quantized

PDF

2D CFT and the Ising Model

2D CFT and the Ising Model Alex Atanasov December 27 2017 Abstract In this lecture we review the results of Appendix E from the seminal paper by Belavin Polyakov and Zamolodchikov on the critical scaling exponents of the 2D Ising model by studying the conformal eld theory of a free massless fermion system 1 Review of 2D CFT Basics

PDF

Conformal Field Theory

Aim: to understand CFT’s in 2d Why? (A)Second order phase transitions in 2d systems (so-called critical phenomena) For example the 2d Ising model Take a 2d lattice of critical sites (in principal this is assumed to be in nite): a ˙spin1 2 These spins are equipped with a nearest neighbor interaction

Which coset model describes a 2D CFT with ZK Symme-try?
k + 2 k + 2 which is equal to (5.94). Indeed, this coset model describes the 2d CFT with Zk symme-try [FZ85, ISZ88-No.14]. This model contains a current with fractional spin (conformal dimension), which does obey neither bose nor fermi statistics, and is rather subject to para-statistics. Therefore, it is called parafermion. Zk
Which 2D CFT is the most important?
Since the Ising model is the most important 2d CFT, let us investigate it more in detail. For the Ising model on a lattice of (N M) sites, the statistical partition function is where we define K = J/(kBT). One can use the identity At high temperature, tanh . K is small and one can take expansion.
What is an example of a 2D Ising model?
For example, the 2d Ising model. Take a 2d lattice of These spins are equipped with a nearest neighbor interaction. The 2. j #i). is the inverse temperature. j. ! ! 0. If ! and the system reaches criticality. Ising model at criticality. meaning. It concerns itself with the time evolution of 1d objects reparametrization invariant and Weyl invariant.
Can radial quantization be used in a 2D CFT?
However, in a 2d CFT, it is most convenient to work in the complex coordinate with the Euclidean signature. In fact, there is a natural choice in the context of string theory by using radial quantization. We may first define our theory on an infinite space-time cylinder, with time t going from to along the “flat” direc-

1 Introduction

Conformal field theories have been at the center of much attention during the last few decades since they have been relevant for at least three different areas of modern theoretical physics: quantum field theories at an infra-red fixed point, two-dimensional critical phenomena, string theory. 2d CFTs are moreover placed on mathematically rig-orous

1.1 Renormalization group flow and Wilson-Fisher fixed points

In quantum field theory, a partition function is conceptually defined by using Feyn-man path integral. The basic integration variables are the Fourier modes fk of a field f. With a cutoff L, we schematically write arxiv.org

CFT2

By naive dimensional analysis, the possible parameter space becomes finite-dimensional. Although one can consider infinitely many interaction terms arxiv.org

Sint = Z ddx å giOi(f) ,

the terms with d < i are suppressed at low energy, which are called irrelevant. D Oi Therefore, we just need to consider finitely many terms Oi with d i at low energy. D To understand the behavior of a theory for these terms, we need to consider the beta function b(g) which describes how the coupling parameter g changes in terms of energy scale : arxiv.org

3.4 Example: Free boson

Now we will study the free boson field as an example. We first review some basic concepts in field theory. arxiv.org

Radial quantization

In the previous discussion, we find that operator formalism distinguishes a time di-rection from a space direction. However, in a 2d CFT, it is most convenient to work in the complex coordinate with the Euclidean signature. In fact, there is a natural choice in the context of string theory by using radial quantization. We may first define our theor

Iw dz a(z)b(w) = dz a(z)b(w) dz b(w)a(z) = [A, b(w)] , (3.92)

where the operator A is the contour integral of a(z) at a fixed time arxiv.org

4 Free boson and free fermion

In this section, we are going to demonstrate more simple examples in CFT. arxiv.org

4.1 Free boson

In the previous sections, we have learned the calculation of some basic OPEs in the massless free boson theory. We will further study this theory. arxiv.org

2 z w

When w z, this result coincides with the previous one. So, the boundary condition does not affect the two-point function locally. However, globally when z or w is taken around the origin, there arises a minus sign due to the boundary condition. We find that different boundary conditions lead to different two-point functions on the plane. arxiv.org

Z Z2.

fore, it is sufficient to check whether a theory is modular invariant by studying its behav-ior under T and S transformations. arxiv.org

Torus partition function

Now we will define a torus partition function Z in terms of Virasoro generators. The story is analogous to the partition function in statistical mechanics arxiv.org

24 cAA(t) .

On the other hand, the modular S-transformation leads to arxiv.org

Kac Determinant

A representation of the Virasoro algebra is said to be unitary if it contains no negative-norm states. In the section of Verma module, we have mentioned that the negative of central charge and highest weight will make the states non-unitary. As we will see below, the unitarity imposes strong constraints on (c, h). In addition to unitarity, one has

3-state Potts model (p0, p) = (6, 5)

On a square lattice, the hamiltonian of the three-state Potts model is given by J arxiv.org

o , g(n)

i in this way. Then, ignoring an overall constant, we have the equivalence of the statistical partition functions å exp h arxiv.org

g = R (g )

Since the coarse-graining does not affect the theory, it is scale-invariant, implying that it is conformal. As briefly explained in the introduction, critical phenomena are described by these fixed points. arxiv.org

Affinization

Hence, one can straightforwardly generalize the current associated to a Lie algebra g as follows. First, the current taking its value on a Lie algebra g can be written arxiv.org

S G

where is generally a two-dimensional manifold, called a Riemann surface. Under the S variation of the action arxiv.org

(z w)2 i fabc . åc (z w)

Introducing the modes Ja n from the Laurent expansion arxiv.org

T(z) = åz n 2Ln . (7.42) n2Z

With the OPEs listed above, we can compute the complete affine Lie and Virasoro algebra c arxiv.org

z w

This can be generalized to define a WZNW primary field. Any WZNW primary field fl,m with l and m specified the representation in the holomorphic and antiholomorphic sector respectively should satisfy the following OPE arxiv.org

Tg/h := Tg Th . bgkg/bhkh

Then, we have Jb h(w) ¶w jb ̇ h(w) Tg(z)Jb h(w) = + + arxiv.org

Rational conformal field theories

The minimal models, the WZNW models and the coset models belong to a class of rational conformal field theories (RCFTs). A theory is endowed with a chiral algebra A that contains the Virasoro algebra as a subalgebra. The definition of an RCFT is (roughly) given as follows. It is unitary with a unique vacuum. There are finitely many primary fields f

Boundary

Figure 32: Holographic entanglement entropy As we will see at the end, when the region A is large enough, the minimal surface (submanifold) wraps the horizon of AdS black hole, and the RT formula can explain the Bekenstein-Hawking entropy as a special case. However, minimal surfaces (submani-folds) are more general so that the RT formula can be int

The 2D Ising Model Monte Carlo Simulation Using the Metropolis Algorithm

2d Ising model Animation : Transition of ferromagnetic to Paramagnetic phase Spin Flip Dynamics

PDF	2D CFT and the Ising Model 27 déc. 2017 Ising model by studying the conformal field theory of a free massless fermion system. 1 Review of 2D CFT Basics.

PDF	The Ising model in Conformal Field Theory 11 mar. 2013 Motivation: ? CFT “in action”: physical application! ? Part I: Overview of statistical phyisics and the 2D Ising model.

PDF	Analyzing the two dimensional Ising model with conformal field theory 10 mar. 2013 In this report focus will be put on the one and two dimensional spin Ising models and other dimensionalities will only be briefly mentioned. In ...

PDF	CFT and SLE and 2D statistical physics An example: 2D Ising model. Squares of two colors representing spins s=±1. Nearby spins want to be the same

PDF	CFT and 2d chiral topological phases: an introduction 20 juil. 2017 fermions and it is related to the simplest possible 2d CFT: the one of the critical 2d Ising model. For this reason

PDF	3D Ising Model and Conformal Bootstrap

PDF	Extracting boundary CFT data from lattice models with tensor 5 mar. 2021 Boundary CFT (BCFT) = CFT with open boundary ... Study CFT emergent on lattice models numerically ... Ex.) 2d critical Ising model.

PDF	Conformal Field theory for 2d Statistical Mechanics 1 avr. 2020 ables (like Sj ? ?Sj in Ising) then 0 is invariant under this symmetry. • All excited states of the CFT Hamiltonian H on a periodic ...

PDF	Topological Lines in 2D 25 jui. 2018 ubiquitous in 2D CFT (e.g. 2D Ising model). ... •In 2D if certain non-symmetry topological line is preserved along a RG flow

PDF	Bootstrapping mixed correlators in the 2D Ising model 22 avr. 2019 Recently developed tools in conformal field theory (CFT) have shed some light. Fixed points of the RG are often CFTs [3]. A CFT is so ...

Share on Facebook Share on Whatsapp

Choose PDF

More..

PDF	2D CFT and the Ising Model - Alex Atanasov 27 déc 2017 · Ising model by studying the conformal field theory of a free massless fermion system 1 Review of 2D CFT Basics Recall that in the complex

PDF	The two dimensional Ising model Contents 1 Introduction 4 2 Correspondence between a critical 2-D Ising model and a Free Fermion CFT 7 2 1 2-D Ising model vs 1-D Quantum Ising model

PDF	The Ising model in Conformal Field Theory 11 mar 2013 · Motivation: → CFT “in action”: physical application ▫ Part I: Overview of statistical phyisics and the 2D Ising model ▫ Part II: From the 2D Ising

PDF	Conformal Field Theory Chemistry for his solution of the 2d Ising model at criticality Page 4 4 (B) String theory provides other examples where CFT's can be given

PDF	Integrals of Motion in the 2 Dimensional Ising Model and Lattice 1 5 5 CFT on the Torus and Modular Invariance 3 2 The 2D Anisotropic Ising Model 3 4 1 Boundary Conditions and CFT Operator Content

PDF	CFT and SLE and 2D statistical physics An example: 2D Ising model Squares of two colors, representing spins s=±1 Nearby spins want to be the same, parameter x : Prob x#{+-neighbors}

PDF	CFT and 2d chiral topological phases: an - Indico de lIN2P3 20 juil 2017 · I am assuming that the reader is familiar with the 2d Ising model and 2 1 Preliminary: a refresher on conformal blocks in the Ising CFT 16