Aims & Motivations The Ising model Conformal transformations 2d CFT Application of conformal field theory in the study of critical phenomena in 2 d statistical systems Artur Miroszewski Institute of Physics, Jagiellonian University, Poland Supervisor: dr Marcin Piątek BLTP JINR and University of Szczecin, Poland Dubna , July 25 th , 2014 Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations The Ising model Aims & Motivations Conformal transformations 2d CFT Aims & Motivations Aims Calculation of 1–, 2–, 3–point correlation functions for models of two-dimensional Conformal Field Theory (2 d CFT) on the Riemann sphere C ∪ {∞} . Motivations Applications in the study of critical phenomena (second order phase transitions) in 2 d statistical systems. Example: the Ising model Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations Phase transitions The Ising model Overview Conformal transformations Critical exponents 2d CFT Phase transitions First order transitions — a finite jump in the latent heat (microscopic variable) at the transition temperature T = T c . Continuous transitions — the derivatives of microscopic variables are discontinuous. Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations Phase transitions The Ising model Overview Conformal transformations Critical exponents 2d CFT Ising model overview A square lattice of spins σ i ∈ { + 1 , − 1 } Energy of a given configuration [ σ ] : � � H [ σ ] = − J σ i σ j − h σ i . � ij � i Thermodynamic quantities of interest: Pair correlation function: Γ( i − j ) = � σ i σ j � Connected correlation function: (a measure of the mutual statistical dependence of the spins σ i and σ j ): � −| i − j | � Γ c ( i − j ) = � σ i σ j � − � σ i �� σ j � ∼ exp . ξ ( T ) ξ ( T ) – correlation length (typical distance over which spins are statistically correlated). Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations Phase transitions The Ising model Overview Conformal transformations Critical exponents 2d CFT Critical behavior and scale invariance For generic values of temperature T there is no symmetry of the Ising model under scale transformations, because of the two length scales: lattice spacing a, correlation length ξ ( T ) . For special values of external parameters ≡ critical point ( T → T c ), ξ ( T ) → ∞ and on the length scales ℓ : a ≪ ℓ ≪ ξ scale invariance emerges !!! (self-similar droplet structure of the system). Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations The Ising model Transformations in D dimensions Conformal transformations Transformations in 2 dimensions 2d CFT Conformal transformations in D dimensions A conformal transformation of the coordinates is an invertible mapping x → x ′ which leaves the metric g µν ( x ) invariant up to scale: g ′ µν ( x ′ ) = Λ( x ) g µν ( x ) . Let us consider the infinitesimal transformation x µ → x ′ µ = x µ + ǫ µ ( x ) . The requirement that the transformation be conformal implies ∂ ν ǫ µ + ∂ µ ǫ ν = 2 D ( ∂ · ǫ ) g µν . The solution is ǫ µ ( x ) = α µ + β µ ν x ν + γ µ νρ x ν x ρ . In particular, we have translations : x µ → x µ + α µ . Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations The Ising model Transformations in D dimensions Conformal transformations Transformations in 2 dimensions 2d CFT Conformal transformations in D dimensions A conformal transformation of the coordinates is an invertible mapping x → x ′ which leaves the metric g µν ( x ) invariant up to scale: g ′ µν ( x ′ ) = Λ( x ) g µν ( x ) . Let us consider the infinitesimal transformation x µ → x ′ µ = x µ + ǫ µ ( x ) . The requirement that the transformation be conformal implies ∂ ν ǫ µ + ∂ µ ǫ ν = 2 D ( ∂ · ǫ ) g µν . The solution is ǫ µ ( x ) = α µ + β µ ν x ν + γ µ νρ x ν x ρ . Lorentz rotations : x µ → x µ + ω µ ν x ν . Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations The Ising model Transformations in D dimensions Conformal transformations Transformations in 2 dimensions 2d CFT Conformal transformations in D dimensions A conformal transformation of the coordinates is an invertible mapping x → x ′ which leaves the metric g µν ( x ) invariant up to scale: g ′ µν ( x ′ ) = Λ( x ) g µν ( x ) . Let us consider the infinitesimal transformation x µ → x ′ µ = x µ + ǫ µ ( x ) . The requirement that the transformation be conformal implies ∂ ν ǫ µ + ∂ µ ǫ ν = 2 D ( ∂ · ǫ ) g µν . The solution is ǫ µ ( x ) = α µ + β µ ν x ν + γ µ νρ x ν x ρ . scale transformations : x µ → x µ + σ x µ . Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations The Ising model Transformations in D dimensions Conformal transformations Transformations in 2 dimensions 2d CFT Conformal transformations in D dimensions A conformal transformation of the coordinates is an invertible mapping x → x ′ which leaves the metric g µν ( x ) invariant up to scale: g ′ µν ( x ′ ) = Λ( x ) g µν ( x ) . Let us consider the infinitesimal transformation x µ → x ′ µ = x µ + ǫ µ ( x ) . The requirement that the transformation be conformal implies ∂ ν ǫ µ + ∂ µ ǫ ν = 2 D ( ∂ · ǫ ) g µν . The solution is ǫ µ ( x ) = α µ + β µ ν x ν + γ µ νρ x ν x ρ . special conformal transformations : x µ → x µ + b µ x 2 − 2 x µ b · x Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations The Ising model Transformations in D dimensions Conformal transformations Transformations in 2 dimensions 2d CFT Conformal transformations in D = 2 dimensions In 2 d Euclidean space ( g µν = δ µν ): ∂ x 1 ǫ 1 = ∂ x 2 ǫ 2 , ∂ x 1 ǫ 2 = − ∂ x 2 ǫ 1 . Transition to complex variables: z = x 1 − ix 2 , ¯ z = x 1 + ix 2 yields ∂ z ¯ ǫ ( z , ¯ z ) = 0 , ∂ ¯ z ǫ ( z , ¯ z ) = 0 , where ǫ = ǫ 1 − i ǫ 2 , ¯ ǫ = ǫ 1 + i ǫ 2 . Hence, the solutions ǫ = ǫ ( z ) , ¯ ǫ = ¯ ǫ (¯ z ) are arbitrary infinitesimal holomorphic functions of z , ¯ z respectively. Corresponding finite (local) conformal transformations are of the form: z → z ′ = f ( z ) , z ′ = ¯ ¯ z → ¯ f (¯ z ) , where f ( z ) is an arbitrary holomorphic function of z ∈ C . Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations The Ising model Transformations in D dimensions Conformal transformations Transformations in 2 dimensions 2d CFT Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations Definition of 2d CFT The Ising model CWI Conformal transformations Correlation functions 2d CFT Minimal Models Conformal field theory in 2 dimensions Two-dimensional conformal field theory is a conformally invariant euclidean 2 d quantum field theory. 2 d CFT can be defined axiomatically, see: A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite Conformal Symmetry In Two-Dimensional Quantum Field Theory , Nucl. Phys. B 241, (1984). Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations Definition of 2d CFT The Ising model CWI Conformal transformations Correlation functions 2d CFT Minimal Models Conformal field theory in 2 dimensions In particular, The Hilbert space of states H is built out with irreducible representations of the Virasoro algebra ( c – central charge): [ L n , L m ] = ( n − m ) L n + m + c 12 ( n 3 − n ) δ n + m , 0 . To any state | ξ � ∈ H corresponds local operator (field) Φ ξ ( z , ¯ z ) , z ∈ C which creates the state | ξ � from the vacuum | 0 � : | ξ � = lim z → 0 Φ ξ ( z , ¯ z ) | 0 � . z , ¯ The simpliest of such operators are called primary fields: � ¯ � ∆ � ∆ ∂ ¯ � ∂ f ( z ) f (¯ z ) ∆ ( f ( z ) , ¯ z ) → Φ ′ ∆ ( z ′ , ¯ z ′ ) = Φ ∆ , ¯ ∆ ( z , ¯ Φ ∆ , ¯ f (¯ z )) , ∆ , ¯ ∂ z ∂ ¯ z where z → z ′ = f ( z ) and (∆ , ¯ ∆) – conformal weights (real numbers). Artur Miroszewski 2 d CFT & Ising model
Aims & Motivations Definition of 2d CFT The Ising model CWI Conformal transformations Correlation functions 2d CFT Minimal Models Conformal Ward Identities We are interested in calculation of correlation functions of primary fields : � 0 | Φ n ( w n , ¯ w n ) ... Φ 1 ( w 1 , ¯ w 1 ) | 0 � , w i ∈ C . Conformal symmetry implies the so-called conformal Ward identities : n � � � 2 w i ∆ i + w 2 � 0 | Φ n ( w n , ¯ w n ) ... Φ 1 ( w 1 , ¯ w 1 ) | 0 � = 0 , i ∂ w i i = 1 n � (∆ i + w i ∂ w i ) � 0 | Φ n ( w n , ¯ w n ) ... Φ 1 ( w 1 , ¯ w 1 ) | 0 � = 0 , i = 1 n � ∂ w i � 0 | Φ n ( w n , ¯ w n ) ... Φ 1 ( w 1 , ¯ w 1 ) | 0 � = 0 . i = 1 Artur Miroszewski 2 d CFT & Ising model
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