From logarithmic scale invariance to logarithmic Conformal Field Theory Philippe Ruelle Renormalization Group & Related Topics Dubna, 1–5 September 2008 RG2008 – Dubna – Sept 08 1 Plan 1. Logarithmic scaling another (generalized) way to realize scale invariance 2. Logarithmic CFT field theoretic realization of this new form of scaling Bottom line: scaling becomes non-diagonalizable ! RG2008 – Dubna – Sept 08 2 History Logarithmic scaling is not really new: (about) first appearance in polymers and percolation (Saleur ’92), but first systematic study of logCFT by Gurarie in ’93. Since then, they have played a prominent role in many topics: • percolation • spanning trees • polymers • quantum Hall effect • WZW models • string theory • 2d turbulence • dimer models • disordered systems • logarithmic minimal models • sandpile models • W-algebras ... RG2008 – Dubna – Sept 08 3 RG transformations Depends on rescaling parameter b, ′ ∗ ∗ K = Rb(K) with fixed point K = Rb(K ). Linearization around K∗ yields ′ ∗ ∗ Kα − Kα = Lαβ (Kβ − Kβ). Xβ If L is diagonalizable, we form scaling variables (eigenvectors) α ∗ ′ ui = ci (Kα − Kα) =⇒ ui = λiui λi = λi(b) Xα ′ ′ yi Semi-group property L(b)L(b )= L(bb ) implies λi = b . The exponents are directly related to the critical exponents. RG2008 – Dubna – Sept 08 4 Scaling operators Scaling operators couple to scaling variables H = ui φi = ui φi(~r) ∼ ui d~rφi(~r) Z Xi Xi X~r Xi Invariance of H requires that under r → r′ = r/b, they transform as RG ′ ′ d−yi φi(r) −→ φi(r )= b φi(r/b) Set xi = d − yi the scaling dimension of φi. Scaling (and translation) invariance implies that correlators obey −xi−xj aij hφi(r1) φj(r2)i = b hφi(r1/b) φj(r2/b)i = . xi+xj |r1 − r2| Power laws, (and more) well accounted for by ordinary CFTs. RG2008 – Dubna – Sept 08 5 Conformal symmetry Note that previous correlator is also invariant under special conformal r + ar2 r′ = , 1+2a · r + a2 r2 provided ′ xi/d Dr ′ 2 2 −xi ′ φi(r) −→ φi(r )=(1+2a · r + a r ) φi(r ) . Dr With rotations, these transformations form the global conformal group SO(d +1, 1) (Euclidean). In d =2, this global invariance can be supplemented with local conformal covariance, leading to CFTs. Then φ above is primary field. RG2008 – Dubna – Sept 08 6 So far Scale transformations r → r/b are ‘diagonalized’ : Scaling variables transform multiplicatively ′ yi ui = b ui Conjugate operators transform homogeneously ′ ′ xi φi(r )= b φi(r/b) Assuming local conformal symmetry, higher correlators can be computed, and have algebraic singularities only. RG2008 – Dubna – Sept 08 7 Question is ... What happens if linearized RG transformations are no longer diagonalizable ?? RG2008 – Dubna – Sept 08 8 Jordan ... Natural to think of Jordan blocks for degenerate eigenvalues: λ 1 canonical form is for rank 2. 0 λ Assume two scaling variables have same eigenvalue and transform non-diagonally (in the Jordan way). RG2008 – Dubna – Sept 08 9 Jordan ... Natural to think of Jordan blocks for degenerate eigenvalues: λ 1 canonical form is for rank 2. 0 λ Assume two scaling variables have same eigenvalue and transform non-diagonally (in the Jordan way). Under linear RG transformations, we write v′ by ∗ v ′ = u 0 by u RG2008 – Dubna – Sept 08 9 Jordan ... Natural to think of Jordan blocks for degenerate eigenvalues: λ 1 canonical form is for rank 2. 0 λ Assume two scaling variables have same eigenvalue and transform non-diagonally (in the Jordan way). Under linear RG transformations, we write ′ y v b ∗ v y 1 f(b) v ′ = = b u 0 by u 0 1 u RG2008 – Dubna – Sept 08 9 Jordan ... Natural to think of Jordan blocks for degenerate eigenvalues: λ 1 canonical form is for rank 2. 0 λ Assume two scaling variables have same eigenvalue and transform non-diagonally (in the Jordan way). Under linear RG transformations, we write ′ y v b ∗ v y 1 f(b) v ′ = = b u 0 by u 0 1 u RG composition law requires f(b)+ f(b′)= f(bb′), namely f(b)= A log b RG2008 – Dubna – Sept 08 9 Logarithmic scaling Non-diagonal scaling for two degenerate scaling variables takes the logarithmic form ′ v y 1 A log b v ′ = b u 0 1 u with a rank 2 Jordan cell. RG2008 – Dubna – Sept 08 10 Logarithmic scaling Non-diagonal scaling for two degenerate scaling variables takes the logarithmic form ′ v y 1 A log b v ′ = b u 0 1 u with a rank 2 Jordan cell. Hamiltonian containing operators coupling to u, v H = u dr φ(r)+ v drψ(r)+ . Z Z is invariant provided ′ ψ x 1 −A log b ψ ′ = b φ 0 1 φ RG2008 – Dubna – Sept 08 10 Consequences on correlators Scaling form for 2-pt correlators hφ φi, hφψi, hψψi ? RG2008 – Dubna – Sept 08 11 Consequences on correlators Scaling form for 2-pt correlators hφ φi, hφψi, hψψi ? Invariance under non-diagonal scale transformations determines completely the three correlators (A fixed to 1) a hφ(r1) φ(r2)i = 2x , |r1 − r2| RG2008 – Dubna – Sept 08 11 Consequences on correlators Scaling form for 2-pt correlators hφ φi, hφψi, hψψi ? Invariance under non-diagonal scale transformations determines completely the three correlators (A fixed to 1) a hφ(r1) φ(r2)i = 2x , |r1 − r2| −2x hφ(r1) ψ(r2)i = b hφ(r1/b) ψ(r2/b)i + hφ(r1/b) φ(r2/b)i log b n o −2x a = b hφ(r1/b) ψ(r2/b)i + 2x log b |r1 − r2| RG2008 – Dubna – Sept 08 11 Consequences on correlators Scaling form for 2-pt correlators hφ φi, hφψi, hψψi ? Invariance under non-diagonal scale transformations determines completely the three correlators (A fixed to 1) a hφ(r1) φ(r2)i = 2x , |r1 − r2| ′ a − a log |r1 − r2| hφ(r1) ψ(r2)i = 2x , |r1 − r2| RG2008 – Dubna – Sept 08 11 Consequences on correlators Scaling form for 2-pt correlators hφ φi, hφψi, hψψi ? Invariance under non-diagonal scale transformations determines completely the three correlators (A fixed to 1) a hφ(r1) φ(r2)i = 2x , |r1 − r2| ′ a − a log |r1 − r2| hφ(r1) ψ(r2)i = 2x , |r1 − r2| ′′ ′ 2 a − 2a log |r1 − r2| + a log |r1 − r2| hψ(r1) ψ(r2)i = 2x , |r1 − r2| Now contain logarithmic singularities ! Forms dictated by translation (L−1) and scale invariance (L0) only. Not conformally invariant yet ... RG2008 – Dubna – Sept 08 11 Under conformal transformations If we assume invariance under special conformal transformations (L1), it implies a =0 in the previous formulas, which simplify to hφ(r1) φ(r2)i = 0 ←− true for n-pt !! a′ hφ(r1) ψ(r2)i = 2x |r1 − r2| ′′ ′ a − 2a log |r1 − r2| hψ(r1) ψ(r2)i = 2x |r1 − r2| The log2 r term disappears. Generic 2-pt functions in LogCFT for pair of fields transforming in the Jordan way: the fields (φ, ψ) make up a logarithmic pair; φ is the primary field, ψ is the logarithmic partner of φ. RG2008 – Dubna – Sept 08 12 Higher rank Easily generalized to higher rank cells. F.i. the rank 3 case ′ A1A2 2 w 1 A1 log b 2 log b + A3 log b w ′ y v = b 0 1 A2 log b v u′ 00 1 u involves log b and log2 b terms. In general, rank r Jordan cells lead to • log b terms to maximal power r − 1 in RG transformations, • log |r1 − r2| terms to maximal power r − 1 in 2-pt functions, • n-pt correlator of primary partner hφ(r1) ...φ(rn)i =0. RG2008 – Dubna – Sept 08 13 Summary Usually (no log) Diagonal scaling ⇔ RG transformations diagonalizable yi Homogeneous transfos for scaling parameters ui (and b ) Tensorial transfos for scaling operators −→ power laws Realized by CFTs in the continuum (local scale inv/cov in d =2) Jordan cells Logarithmic scaling ⇔ RG no longer diagonalizable Inhomogeneous transfos for scaling parameters ui with log b factors Inhomogeneous transfos for scaling operators → power laws + logs Principles of local scale inv −→ LogCFTs (more complicated) RG2008 – Dubna – Sept 08 14 Example Simplest and most studied LogCFT. 1 S = ∂θ∂¯θ˜ (symplectic fermions) π Z • θ and θ˜ are scalar, anticomm. fields, with canonical dimension 0 −→ four fields I, θ, θ,˜ ω =:θθ˜ : of dimension 0, two are bosonic • Wick contraction θ(z, z¯) θ˜(w, w¯)= − log |z − w| • stress-energy tensor T (z)= −2 :∂θ∂θ˜: • Virasoro algebra has central charge c = −2 ′ 6(p−p′)2 • may be thought of as minimal model (p,p )=(1, 2), c =1 − pp′ RG2008 – Dubna – Sept 08 15 Jordan cell The identity I and ω = :θθ˜: form a logarithmic pair with x =0. From OPE T (z)ω(w), one finds, under infinitesimal dilation, L0I =0 , L0ω = I Likewise, φ = ∂∂¯(θθ˜ ) and ψ = θθ˜ ∂∂¯(θθ˜ ) form another logarithmic pair with x =2 L0φ = φ, L0ψ = ψ + φ Each of these pairs (+ many more) generates an indecomposable representation of the Virasoro algebra (because of Jordan cell). RG2008 – Dubna – Sept 08 16 Correlators ? Can we understand the structure of 2-pt functions ? hφ(r1) φ(r2)i = 0, ←− true for n-pt !! a′ a′′ − 2a′ log |r − r | φ r ψ r , ψ r ψ r 1 2 h ( 1) ( 2)i = 2x h ( 1) ( 2)i = 2x |r1 − r2| |r1 − r2| RG2008 – Dubna – Sept 08 17 Correlators ? Can we understand the structure of 2-pt functions ? hφ(r1) φ(r2)i = 0, ←− true for n-pt !! a′ a′′ − 2a′ log |r − r | φ r ψ r , ψ r ψ r 1 2 h ( 1) ( 2)i = 2x h ( 1) ( 2)i = 2x |r1 − r2| |r1 − r2| Because of zero modes of θ, θ˜ (remember dθ0 =0) R h I i =0 [and also hφ(1)φ(2) .