Application of conformal field theory in the study of critical - - PowerPoint PPT Presentation

Aims & Motivations The Ising model Conformal transformations 2d CFT

Application of conformal field theory in the study of critical phenomena in 2d statistical systems

Artur Miroszewski

Institute of Physics, Jagiellonian University, Poland

Supervisor: dr Marcin Piątek

BLTP JINR and University of Szczecin, Poland

Dubna, July 25th, 2014

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Aims & Motivations

Aims & Motivations

Aims

Calculation of 1–, 2–, 3–point correlation functions for models of two-dimensional Conformal Field Theory (2d CFT) on the Riemann sphere C ∪ {∞}.

Motivations

Applications in the study of critical phenomena (second order phase transitions) in 2d statistical systems. Example: the Ising model

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Phase transitions Overview Critical exponents

Phase transitions

First order transitions — a finite jump in the latent heat (microscopic variable) at the transition temperature T = Tc. Continuous transitions — the derivatives of microscopic variables are discontinuous.

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Phase transitions Overview Critical exponents

Ising model overview

A square lattice of spins σi ∈ {+1, −1} Energy of a given configuration [σ]: H[σ] = −J

• ij

σiσj − h

• i

σi. Thermodynamic quantities of interest:

Pair correlation function: Γ(i − j) = σiσj Connected correlation function: (a measure of the mutual statistical dependence of the spins σi and σj): Γc(i − j) = σiσj − σiσj ∼ exp

• −|i − j|

ξ(T)

• .

ξ(T) – correlation length (typical distance over which spins are statistically correlated).

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Phase transitions Overview Critical exponents

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 → Tc), ξ(T) → ∞ and on the length scales ℓ: a ≪ ℓ ≪ ξ scale invariance emerges !!! (self-similar droplet structure of the system).

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Transformations in D dimensions Transformations in 2 dimensions

Conformal transformations in D dimensions

A conformal transformation of the coordinates is an invertible mapping x → x′ which leaves the metric gµν(x) invariant up to scale: g′

µν(x′) = Λ(x)gµν(x).

Let us consider the infinitesimal transformation xµ → x′µ = xµ + ǫµ(x). The requirement that the transformation be conformal implies ∂νǫµ + ∂µǫν = 2 D (∂ · ǫ)gµν. The solution is ǫµ(x) = αµ + βµ

ν xν + γµ νρxνxρ.

In particular, we have translations: xµ → xµ + αµ.

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Transformations in D dimensions Transformations in 2 dimensions

Conformal transformations in D dimensions

A conformal transformation of the coordinates is an invertible mapping x → x′ which leaves the metric gµν(x) invariant up to scale: g′

µν(x′) = Λ(x)gµν(x).

Let us consider the infinitesimal transformation xµ → x′µ = xµ + ǫµ(x). The requirement that the transformation be conformal implies ∂νǫµ + ∂µǫν = 2 D (∂ · ǫ)gµν. The solution is ǫµ(x) = αµ + βµ

ν xν + γµ νρxνxρ.

Lorentz rotations: xµ → xµ + ωµ

ν xν.

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Transformations in D dimensions Transformations in 2 dimensions

Conformal transformations in D dimensions

A conformal transformation of the coordinates is an invertible mapping x → x′ which leaves the metric gµν(x) invariant up to scale: g′

µν(x′) = Λ(x)gµν(x).

Let us consider the infinitesimal transformation xµ → x′µ = xµ + ǫµ(x). The requirement that the transformation be conformal implies ∂νǫµ + ∂µǫν = 2 D (∂ · ǫ)gµν. The solution is ǫµ(x) = αµ + βµ

ν xν + γµ νρxνxρ.

scale transformations: xµ → xµ + σxµ.

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Transformations in D dimensions Transformations in 2 dimensions

Conformal transformations in D dimensions

A conformal transformation of the coordinates is an invertible mapping x → x′ which leaves the metric gµν(x) invariant up to scale: g′

µν(x′) = Λ(x)gµν(x).

Let us consider the infinitesimal transformation xµ → x′µ = xµ + ǫµ(x). The requirement that the transformation be conformal implies ∂νǫµ + ∂µǫν = 2 D (∂ · ǫ)gµν. The solution is ǫµ(x) = αµ + βµ

ν xν + γµ νρxνxρ.

special conformal transformations: xµ→xµ + bµx2−2xµb · x

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Transformations in D dimensions Transformations in 2 dimensions

Conformal transformations in D = 2 dimensions

In 2d Euclidean space (gµν = δµν): ∂x1ǫ1 = ∂x2ǫ2, ∂x1ǫ2 = −∂x2ǫ1. Transition to complex variables: z = x1 − ix2, ¯ z = x1 + ix2 yields ∂z¯ ǫ(z, ¯ z) = 0, ∂¯

zǫ(z, ¯

z) = 0, where ǫ = ǫ1 − iǫ2, ¯ ǫ = ǫ1 + iǫ2. 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 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Transformations in D dimensions Transformations in 2 dimensions Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Definition of 2d CFT CWI Correlation functions Minimal Models

Conformal field theory in 2 dimensions

Two-dimensional conformal field theory is a conformally invariant euclidean 2d quantum field theory. 2d 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 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Definition of 2d CFT CWI Correlation functions Minimal Models

Conformal field theory in 2 dimensions

In particular, The Hilbert space of states H is built out with irreducible representations

• f the Virasoro algebra (c – central charge):

[Ln, Lm] = (n − m)Ln+m + c 12(n3 − n)δn+m,0. To any state |ξ ∈ H corresponds local operator (field) Φξ(z, ¯ z), z ∈ C which creates the state |ξ from the vacuum |0: |ξ = lim

z,¯ z→0 Φξ(z, ¯

z)|0. The simpliest of such operators are called primary fields: Φ∆, ¯

∆(z, ¯

z) → Φ′

∆, ¯ ∆(z′, ¯

z′) =

• ∂f (z)

∂z

∆

∂¯ f (¯ z) ∂¯ z

¯

∆

Φ∆, ¯

∆(f (z), ¯

f (¯ z)), where z → z′ = f (z) and (∆, ¯ ∆) – conformal weights (real numbers).

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Definition of 2d CFT CWI Correlation functions Minimal Models

Conformal Ward Identities

We are interested in calculation of correlation functions of primary fields : 0|Φn(wn, ¯ wn)...Φ1(w1, ¯ w1)|0, wi ∈ C. Conformal symmetry implies the so-called conformal Ward identities:

n

• i=1
• 2wi∆i + w2

i ∂wi

• 0|Φn(wn, ¯

wn)...Φ1(w1, ¯ w1)|0 = 0,

n

• i=1

(∆i + wi∂wi) 0|Φn(wn, ¯ wn)...Φ1(w1, ¯ w1)|0 = 0,

n

• i=1

∂wi0|Φn(wn, ¯ wn)...Φ1(w1, ¯ w1)|0 = 0.

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Definition of 2d CFT CWI Correlation functions Minimal Models

1-, 2-, 3-point correlation functions of primary fields

Due to orthogonality the 1-point correlation function vanishes 0|Φ∆, ¯

∆(z, ¯

z)|0 = 0. From CWI we get a system of differential equations for the 2-point correlation function 0|Φ∆2, ¯

∆2(z2, ¯

z2)Φ∆1, ¯

∆1(z1, ¯

z1)|0 ≡ G(z1, z2) (here holomorphic part only):

2

• i

∂zi G = 0 ⇒ G = G(z1 − z2);

2

• i

(∆i + zi∂zi )G = 0 ⇒ G = const. × (z1 − z2)−∆1−∆2,

2

• i

(2zi∆i + z2

i ∂zi )G = 0

⇒ G = δ∆1,∆2(z1 − z2)−∆1−∆2 ,

• const. = 1.

Using the same method as above we can calculate the 3-point correlation function (zij = zi − zj): 0|

3

• i=1

Φ∆i , ¯

∆i (zi, ¯

zi)|0 = C123 × z∆3−∆2−∆1

12

z∆1−∆2−∆3

23

z∆2−∆3−∆1

31

× c.c. .

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Definition of 2d CFT CWI Correlation functions Minimal Models

Unitary Minimal Models

A family M(p, p′) of 2d CFT’s with central charge c and conformal weights ∆rs: c = 1 − 6(p − p′)2 pp′ ∆rs = (pr − p′s)2 − (p − p′)2 4pp′ , 1 r < p′, 1 s < p. Minimal model M(4, 3): c = 1

2 and

fields conformal weights 1 (∆11, ¯ ∆11) = (0, 0) Φ12 (∆12, ¯ ∆12) = ( 1

16, 1 16)

Φ21 (∆21, ¯ ∆21) = ( 1

2, 1 2)

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Definition of 2d CFT CWI Correlation functions Minimal Models

Minimal model M(4, 3) vs. critical Ising model

A general form of the 2-point correlation function: 0|Φ∆2, ¯

∆2(z2, ¯

z2)Φ∆1, ¯

∆1(z1, ¯

z1)|0 = = δ∆2,∆1(z2 − z1)−∆2−∆1 × c.c. = |z2 − z1|−4∆. 2-point correlation functions in the minimal model M(4, 3): 0|Φ12(z, ¯ z)Φ12(0, 0)|0 = |z|−1/4, 0|Φ21(z, ¯ z)Φ21(0, 0)|0 = |z|−2. Spin-spin σnσ0 and energy-energy ǫnǫ0 correlation functions in 2d critical Ising model (D = 2; η = −1/4, ν = 1 – critical Ising exponents): σnσ0 ∝ |n|−D+2−η = |n|−1/4, ǫnǫ0 ∝ |n|−2D−2/ν = |n|−2. Hence, one can identify σ = Φ12 and ǫ = Φ21.

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Definition of 2d CFT CWI Correlation functions Minimal Models

Conclusions

Correspondence between 2-point functions in the M(4, 3) minimal model and the critical Ising model provides a convincing evidence that the dynamics of the latter is described by 2d CFT. There are more examples of such correspondence, cf.: tricritical Ising model, 3-state Potts model, . . .. For further studies: applications of 2d CFT to string theory, geometric correspondences in supersymmetric gauge theories, integrable systems.

Artur Miroszewski 2d CFT & Ising model

Aims & Motivations The Ising model Conformal transformations 2d CFT Definition of 2d CFT CWI Correlation functions Minimal Models

Conclusions

Correspondence between 2-point functions in the M(4, 3) minimal model and the critical Ising model provides a convincing evidence that the dynamics of the latter is described by 2d CFT. There are more examples of such correspondence, cf.: tricritical Ising model, 3-state Potts model, . . .. For further studies: applications of 2d CFT to string theory, geometric correspondences in supersymmetric gauge theories, integrable systems. Thanks for your attention!

Artur Miroszewski 2d CFT & Ising model