Physics of Jordan cells
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1 Physics of Jordan cells Daniel Grumiller Institute for Theoretical Physics Vienna University of Technology IHP Workshop Advanced conformal field theory and application, September 2011
2 Outline Jordan cells in non-hermitian quantum mechanics Jordan cells in logarithmic conformal field theories Jordan cells in the holographic AdS 3 /LCFT 2 correspondence Jordan cells in condensed matter applications D. Grumiller Physics of Jordan cells 2/32
3 Outline Jordan cells in non-hermitian quantum mechanics Jordan cells in logarithmic conformal field theories Jordan cells in the holographic AdS 3 /LCFT 2 correspondence Jordan cells in condensed matter applications D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 3/32
4 Quantum mechanics One of the postulates of quantum mechanics: Time-evolution of closed system described by i t Ψ = H Ψ with hermitian Hamiltonian H Consequence of hermiticity: Eigenvalues are real D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 4/32
5 Quantum mechanics One of the postulates of quantum mechanics: Time-evolution of closed system described by i t Ψ = H Ψ with hermitian Hamiltonian H Consequence of hermiticity: Eigenvalues are real Very useful concept with many applications in physics! D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 4/32
6 Non-hermitian quantum mechanics Let us step back a bit Is assumption of hermiticity always justified? D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 5/32
7 Non-hermitian quantum mechanics Let us step back a bit Is assumption of hermiticity always justified? Not for open systems D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 5/32
8 Non-hermitian quantum mechanics Let us step back a bit Is assumption of hermiticity always justified? Not for open systems Open systems arise not just in quantum cosmology, but also in down-to-earth systems D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 5/32
9 Non-hermitian quantum mechanics Let us step back a bit Is assumption of hermiticity always justified? Not for open systems Open systems arise not just in quantum cosmology, but also in down-to-earth systems Perturbatively: use Breit Wigner method e iht e iht Γt Decay rate Γ = non-hermitian contribution to Hamiltonian D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 5/32
10 Non-hermitian quantum mechanics Let us step back a bit Is assumption of hermiticity always justified? Not for open systems Open systems arise not just in quantum cosmology, but also in down-to-earth systems Perturbatively: use Breit Wigner method e iht e iht Γt Decay rate Γ = non-hermitian contribution to Hamiltonian Non-hermiticity beyond perturbation theory useful for physics? D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 5/32
11 Non-hermitian quantum mechanics Let us step back a bit Is assumption of hermiticity always justified? Not for open systems Open systems arise not just in quantum cosmology, but also in down-to-earth systems Perturbatively: use Breit Wigner method e iht e iht Γt Decay rate Γ = non-hermitian contribution to Hamiltonian Non-hermiticity beyond perturbation theory useful for physics? Surpisingly, the answer is yes D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 5/32
12 Experimental example Example by Stefan Rotter et al. 04 Experimental setup: D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 6/32
13 Experimental example Example by Stefan Rotter et al. 04 Experimental setup: Total transmission probability: Strong effect of nonhermiticity! total transmission T(n) w/d=37% w/d=56% w/d=100% experiment experiment mode number n=kd/π D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 6/32 theory experiment theory theory (a) (b) (c)
14 Critical points and Jordan cells See Non-Hermitian quantum mechanics by Nimrod Moiseyev Consider the Hamiltonian H = ( 1 λ λ 1 ) with Eigenvalues E ± = ± 1 + λ 2 D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 7/32
15 Critical points and Jordan cells See Non-Hermitian quantum mechanics by Nimrod Moiseyev Consider the Hamiltonian H = ( 1 λ λ 1 ) with Eigenvalues E ± = ± 1 + λ 2 Non-hermitian critical points: λ ±i Eigenvector c ± = (±i, 1) self-orthogonal: c 2 ± = 0 D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 7/32
16 Critical points and Jordan cells See Non-Hermitian quantum mechanics by Nimrod Moiseyev Consider the Hamiltonian H = ( 1 λ λ 1 ) with Eigenvalues E ± = ± 1 + λ 2 Non-hermitian critical points: λ ±i Eigenvector c ± = (±i, 1) self-orthogonal: c 2 ± = 0 Similarity trafo J = A 1 HA: J = ( Simplest example of Jordan cell in nonhermitian critical quantum mechanics! ) D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 7/32
17 Berry phase Physical significance of critical points? Critical points and Jordan cells require infinite fine-tuning D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 8/32
18 Berry phase Physical significance of critical points? Critical points and Jordan cells require infinite fine-tuning Experimentally: can never infinitely fine-tune parameters D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 8/32
19 Berry phase Physical significance of critical points? Critical points and Jordan cells require infinite fine-tuning Experimentally: can never infinitely fine-tune parameters Instead: vary parameters adiabatically in vicinity of critical points Eigenvalue: λ = λ c + Re iφ D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 8/32
20 Berry phase Physical significance of critical points? Critical points and Jordan cells require infinite fine-tuning Experimentally: can never infinitely fine-tune parameters Instead: vary parameters adiabatically in vicinity of critical points Energies on two branches: Eigenvalue: λ = λ c + Re iφ E ± (φ) = E c ± α Re iφ/2 D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 8/32
21 Berry phase Physical significance of critical points? Critical points and Jordan cells require infinite fine-tuning Experimentally: can never infinitely fine-tune parameters Instead: vary parameters adiabatically in vicinity of critical points Energies on two branches: Wavefunctions on two branches: Eigenvalue: λ = λ c + Re iφ E ± (φ) = E c ± α Re iφ/2 Ψ ± = A ± e iφ/4( Ψ c ± Re iφ/2 χ ) D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 8/32
22 Berry phase Physical significance of critical points? Critical points and Jordan cells require infinite fine-tuning Experimentally: can never infinitely fine-tune parameters Instead: vary parameters adiabatically in vicinity of critical points Energies on two branches: Wavefunctions on two branches: Eigenvalue: λ = λ c + Re iφ E ± (φ) = E c ± α Re iφ/2 Ψ ± = A ± e iφ/4( Ψ c ± Re iφ/2 χ ) Berry phase: rotate φ = 2πn, with n = 1, 2, 3, 4,... D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 8/32
23 Berry phase Energies on two branches: E ± (φ) = E c ± α Re iφ/2 Wavefunctions on two branches (A = ia + ): Ψ ± = A ± e iφ/4( Ψ c ± Re iφ/2 χ ) Berry phase: rotate φ = 2πn, with n = 1, 2, 3, 4,... n = 1: E ± (2π) = E (0) Ψ ± (2π) = Ψ (0) D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 8/32
24 Berry phase Energies on two branches: E ± (φ) = E c ± α Re iφ/2 Wavefunctions on two branches (A = ia + ): Ψ ± = A ± e iφ/4( Ψ c ± Re iφ/2 χ ) Berry phase: rotate φ = 2πn, with n = 1, 2, 3, 4,... n = 1: E ± (2π) = E (0) Ψ ± (2π) = Ψ (0) n = 2: E ± (4π) = E ± (0) Ψ ± (4π) = Ψ ± (0) D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 8/32
25 Berry phase Energies on two branches: E ± (φ) = E c ± α Re iφ/2 Wavefunctions on two branches (A = ia + ): Ψ ± = A ± e iφ/4( Ψ c ± Re iφ/2 χ ) Berry phase: rotate φ = 2πn, with n = 1, 2, 3, 4,... n = 1: E ± (2π) = E (0) Ψ ± (2π) = Ψ (0) n = 2: n = 4: E ± (4π) = E ± (0) Ψ ± (4π) = Ψ ± (0) E ± (8π) = E ± (0) Ψ ± (8π) = + Ψ ± (0) D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 8/32
26 Berry phase Energies on two branches: E ± (φ) = E c ± α Re iφ/2 Wavefunctions on two branches (A = ia + ): Ψ ± = A ± e iφ/4( Ψ c ± Re iφ/2 χ ) Berry phase: rotate φ = 2πn, with n = 1, 2, 3, 4,... n = 1: E ± (2π) = E (0) Ψ ± (2π) = Ψ (0) n = 2: n = 4: E ± (4π) = E ± (0) Ψ ± (4π) = Ψ ± (0) E ± (8π) = E ± (0) Ψ ± (8π) = + Ψ ± (0) Physical significance of critical points: geometrical (Berry) phases! D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 8/32
27 Quantum conclusions Hermitian quantum mechanics is useful D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 9/32
28 Quantum conclusions Hermitian quantum mechanics is useful Non-hermitian quantum mechanics is useful D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 9/32
29 Quantum conclusions Hermitian quantum mechanics is useful Non-hermitian quantum mechanics is useful Jordan cells at critical points in non-hermitian quantum systems D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 9/32
30 Quantum conclusions Hermitian quantum mechanics is useful Non-hermitian quantum mechanics is useful Jordan cells at critical points in non-hermitian quantum systems Critical points experimentally accessible through geometric phases D. Grumiller Physics of Jordan cells Jordan cells in non-hermitian quantum mechanics 9/32
31 Outline Jordan cells in non-hermitian quantum mechanics Jordan cells in logarithmic conformal field theories Jordan cells in the holographic AdS 3 /LCFT 2 correspondence Jordan cells in condensed matter applications D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 10/32
32 Conformal field theories CFT = Quantum field theory with invariance under translations, rotations + boosts, dilatations and special conformal transformations extends Poincare SO(p, q) to SO(p + 1, q + 1) if p = 1 and q = d 1: call this Lorentzian CFT d if p = 0 and q = d: call this Euclidean CFT d D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 11/32
33 Conformal field theories CFT = Quantum field theory with invariance under translations, rotations + boosts, dilatations and special conformal transformations extends Poincare SO(p, q) to SO(p + 1, q + 1) if p = 1 and q = d 1: call this Lorentzian CFT d if p = 0 and q = d: call this Euclidean CFT d In two dimensions: infinite dimensional symmetry algebra two copies of the Virasoro algebra with central charges c, c [L n, L m ] = (n m)l n+m + c 12 (n3 n) δ n+m,0 D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 11/32
34 Conformal field theories CFT = Quantum field theory with invariance under translations, rotations + boosts, dilatations and special conformal transformations extends Poincare SO(p, q) to SO(p + 1, q + 1) if p = 1 and q = d 1: call this Lorentzian CFT d if p = 0 and q = d: call this Euclidean CFT d In two dimensions: infinite dimensional symmetry algebra two copies of the Virasoro algebra with central charges c, c [L n, L m ] = (n m)l n+m + c 12 (n3 n) δ n+m,0 Each copy has an SL(2, R) subalgebra generated by L 0, L 1, L 1 where no central term appears D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 11/32
35 Conformal field theories CFT = Quantum field theory with invariance under translations, rotations + boosts, dilatations and special conformal transformations extends Poincare SO(p, q) to SO(p + 1, q + 1) if p = 1 and q = d 1: call this Lorentzian CFT d if p = 0 and q = d: call this Euclidean CFT d In two dimensions: infinite dimensional symmetry algebra two copies of the Virasoro algebra with central charges c, c [L n, L m ] = (n m)l n+m + c 12 (n3 n) δ n+m,0 Each copy has an SL(2, R) subalgebra generated by L 0, L 1, L 1 where no central term appears CFTs arise in physics in statistical mechanics, condensed matter physics, quantum field theory, string theory, gauge/gravity duality etc. D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 11/32
36 Conformal field theories CFT = Quantum field theory with invariance under translations, rotations + boosts, dilatations and special conformal transformations extends Poincare SO(p, q) to SO(p + 1, q + 1) if p = 1 and q = d 1: call this Lorentzian CFT d if p = 0 and q = d: call this Euclidean CFT d In two dimensions: infinite dimensional symmetry algebra two copies of the Virasoro algebra with central charges c, c [L n, L m ] = (n m)l n+m + c 12 (n3 n) δ n+m,0 Each copy has an SL(2, R) subalgebra generated by L 0, L 1, L 1 where no central term appears CFTs arise in physics in statistical mechanics, condensed matter physics, quantum field theory, string theory, gauge/gravity duality etc. Isometry group of (Lorentzian) AdS d : SO(2, d 1) same as (Lorentzian) CFT d 1 Relevant for AdS d /CFT d 1 correspondence! D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 11/32
37 Conformal field theories in two dimensions Any CFT has a conserved traceless energy momentum tensor. T µ µ = 0 µ T µν = 0 D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 12/32
38 Conformal field theories in two dimensions Any CFT has a conserved traceless energy momentum tensor. T µ µ = 0 µ T µν = 0 Useful: light-cone gauge for metric ds 2 = 2 dz d z T z z = 0 T zz = O L (z) T z z = O R ( z) D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 12/32
39 Conformal field theories in two dimensions Any CFT has a conserved traceless energy momentum tensor. Useful: light-cone gauge for metric T µ µ = 0 µ T µν = 0 ds 2 = 2 dz d z T z z = 0 T zz = O L (z) T z z = O R ( z) The 2- and 3-point correlators are fixed by conformal Ward identities. O R ( z) O R (0) = c R 2 z 4 O L (z) O L (0) = c L 2z 4 c R O R ( z) O R ( z ) O R (0) = z 2 z 2 ( z z ) 2 O L (z) O L (z ) O L c L (0) = z 2 z 2 (z z ) 2 Central charges c L/R determine key properties of CFT. D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 12/32
40 Correlators The c = 0 catastrophe Primary field O M with conformal weights (h, h): O M (z, z)o M (0, 0) = A z2h z 2 h OPE: O M (z, z)o M (0, 0) A ( 1 + 2h ) z2h z 2 h c z2 O L (0) +... Problem: divergence for c 0 D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 13/32
41 Correlators The c = 0 catastrophe Primary field O M with conformal weights (h, h): O M (z, z)o M (0, 0) = A z2h z 2 h OPE: O M (z, z)o M (0, 0) A ( 1 + 2h ) z2h z 2 h c z2 O L (0) +... Problem: divergence for c 0 Possible resolutions in limit c 0: weights vanish (h, h) (0, 0) normalization vanishes A 0 other operator(s) arise with h 2, which cancel divergence Focus on last possibility D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 13/32
42 Correlators in logarithmic conformal field theories Aghamohammadi, Khorrami & Rahimi Tabar 97; Kogan & Nichol 01; Rasmussen 04 Suppose now that primary has conformal weights (2 + ε, ε): O M (z, z) O M (0, 0) = z Suppose that limits exist: b L := lim c L c L 0 ε 0 Define log operator: Obtain 2-point correlators: O log = b L O L c L O L (z)o L (0, 0) = 0 O L (z)o log (0, 0) = b L 2z 4 ˆB 4+2ε z 2ε ( 2 ) B := lim ˆB + c L 0 c L + b L 2 OM O log (z, z)o log (0, 0) = b Lln(m 2 L z 2 ) z 4 D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 14/32
43 Critical points and Jordan cells If EMT acquires log partner Hamiltonian cannot be diagonalized ( ) O log H = O L ( ) ( O log O L Consider only situations where J is diagonalizable: ( ) ( ) ( O log 2 0 O log J = 0 2 O L O L ) ) Appearance of Jordan cell = defining feature of log CFTs Note: Jordan cell can be higher rank than 2, but consider only rank 2 case here LCFTs: Gurarie 93 Reviews on LCFTs: Flohr 01; Gaberdiel 01 D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 15/32
44 Logarithmic conclusions c 0 limit in CFT intriguing ( c = 0 catastrophe ) D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 16/32
45 Logarithmic conclusions c 0 limit in CFT intriguing ( c = 0 catastrophe ) EMT can acquire log partner D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 16/32
46 Logarithmic conclusions c 0 limit in CFT intriguing ( c = 0 catastrophe ) EMT can acquire log partner correlators and OPEs acquire logarithms D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 16/32
47 Logarithmic conclusions c 0 limit in CFT intriguing ( c = 0 catastrophe ) EMT can acquire log partner correlators and OPEs acquire logarithms Hamiltonian acquires Jordan cell structure D. Grumiller Physics of Jordan cells Jordan cells in logarithmic conformal field theories 16/32
48 Outline Jordan cells in non-hermitian quantum mechanics Jordan cells in logarithmic conformal field theories Jordan cells in the holographic AdS 3 /LCFT 2 correspondence Jordan cells in condensed matter applications D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 17/32
49 Motivations for studying gravity in 3 dimensions Quantum gravity Address conceptual issues of quantum gravity Black hole evaporation, information loss, black hole microstate counting, virtual black hole production,... Technically much simpler than 4D or higher D gravity Integrable models: powerful tools in physics (Coulomb problem, Hydrogen atom, harmonic oscillator,...) Models should be as simple as possible, but not simpler Gauge/gravity duality Deeper understanding of black hole holography AdS 3 /CFT 2 correspondence best understood Quantum gravity via AdS/CFT? (Witten 07, Li, Song, Strominger 08) Applications to 2D condensed matter systems? Gauge gravity duality beyond standard AdS/CFT: warped AdS, asymptotic Lifshitz, non-relativistic CFTs, logarithmic CFTs,... Physics Cosmic strings (Deser, Jackiw, t Hooft 84, 92) Black hole analog systems in condensed matter physics (graphene, BEC, fluids,...) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 18/32
50 Gravity in three dimensions (Witten 07) Gravity in three dimensions is not power-counting renormalizable D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 19/32
51 Gravity in three dimensions (Witten 07) Gravity in three dimensions is not power-counting renormalizable Einstein gravity in three dimensions is locally trivial D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 19/32
52 Gravity in three dimensions (Witten 07) Gravity in three dimensions is not power-counting renormalizable Einstein gravity in three dimensions is locally trivial Globally and in the presence of boundaries Einstein gravity becomes non-trivial, particularly for negative cosmological constant D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 19/32
53 Gravity in three dimensions (Witten 07) Gravity in three dimensions is not power-counting renormalizable Einstein gravity in three dimensions is locally trivial Globally and in the presence of boundaries Einstein gravity becomes non-trivial, particularly for negative cosmological constant Higher derivative theories of gravity can have massive graviton excitations and thus are locally non-trivial Example: Topologically massive gravity (Deser, Jackiw & Templeton 82) I TMG = 1 d 3 x [ g R π G l µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) ] D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 19/32
54 Gravity in three dimensions (Witten 07) Gravity in three dimensions is not power-counting renormalizable Einstein gravity in three dimensions is locally trivial Globally and in the presence of boundaries Einstein gravity becomes non-trivial, particularly for negative cosmological constant Higher derivative theories of gravity can have massive graviton excitations and thus are locally non-trivial Example: Topologically massive gravity (Deser, Jackiw & Templeton 82) I TMG = 1 d 3 x [ g R π G l µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) ] Equations of motion: with the Cotton tensor defined as R µν 1 2 g µνr 1 l 2 g µν + 1 µ C µν = 0 C µν = 1 2 ε µ αβ α R βν + (µ ν) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 19/32
55 Asymptotically AdS Advantages of a negative cosmological constant in 3D gravity: Black holes exist Do not have to discuss S-matrix Can use AdS 3 /CFT 2 correspondence Black holes have positive specific heat D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 20/32
56 Asymptotically AdS Advantages of a negative cosmological constant in 3D gravity: Black holes exist Do not have to discuss S-matrix Can use AdS 3 /CFT 2 correspondence Black holes have positive specific heat Working definiton of asymptotically locally AdS 3 : g = ḡ + h ḡ µν dx µ dx ν = dx+ dx + dy 2 y 2 with the state-dependent part h near the boundary y = 0: h ++ = o(1/y 2 ) h + = O(1) h +y = O(1) h = o(1/y 2 ) h y = O(1) h yy = O(1) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 20/32
57 Asymptotic symmetry group Kinematics of asymptotically AdS 3 ASG = boundary preserving trafos / trivial gauge trafos D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 21/32
58 Asymptotic symmetry group Kinematics of asymptotically AdS 3 ASG = boundary preserving trafos / trivial gauge trafos Boundary preserving trafos L ξ (ḡ + h) = O(h) generated by vector fields ξ with ξ + = ε + (x + ) + y 2... ξ = ε (x ) + y 2... ξ y = y 2 ε + O(y3 ) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 21/32
59 Asymptotic symmetry group Kinematics of asymptotically AdS 3 ASG = boundary preserving trafos / trivial gauge trafos Boundary preserving trafos L ξ (ḡ + h) = O(h) generated by vector fields ξ with ξ + = ε + (x + ) + y 2... ξ = ε (x ) + y 2... ξ y = y 2 ε + O(y3 ) asymptotic symmetry algebra generated by ε + (x + ) + = n L n e inx+ ε (x ) = n L n e inx D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 21/32
60 Canonical realization of ASG Dynamics of asymptotically AdS 3 Asymptotic symmetry algebra: two copies of Witt algebra (theory-independent!) [L n, L m ] = (n m)l n+m [ L n, L m ] = (n m) L n+m D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 22/32
61 Canonical realization of ASG Dynamics of asymptotically AdS 3 Asymptotic symmetry algebra: two copies of Witt algebra (theory-independent!) [L n, L m ] = (n m)l n+m [ L n, L m ] = (n m) L n+m Brown, Henneaux 86: canonical realization of ASG can lead to central extension (theory-dependent!) In Einstein gravity: [L n, L m ] = (n m)l n+m + c 12 (n3 n) δ n+m,0 c = c = 3l 2G D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 22/32
62 Canonical realization of ASG Dynamics of asymptotically AdS 3 Asymptotic symmetry algebra: two copies of Witt algebra (theory-independent!) [L n, L m ] = (n m)l n+m [ L n, L m ] = (n m) L n+m Brown, Henneaux 86: canonical realization of ASG can lead to central extension (theory-dependent!) [L n, L m ] = (n m)l n+m + c 12 (n3 n) δ n+m,0 In Einstein gravity: c = c = 3l 2G In topologically massive gravity (Kraus & Larsen 05): c = 3l 2G ( 1 1 µl ) c = 3l 2G ( µl ) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 22/32
63 Critical points in massive gravity TMG at the chiral point is TMG with the tuning µ l = 1 between the cosmological constant and the Chern Simons coupling. Why special? (Li, Song & Strominger 08) c = 0 c = 3l G D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 23/32
64 Critical points in massive gravity TMG at the chiral point is TMG with the tuning µ l = 1 between the cosmological constant and the Chern Simons coupling. Why special? (Li, Song & Strominger 08) c = 0 c = 3l G Interesting possibilities: Dual CFT could be chiral (Li, Song & Strominger 08) Dual CFT could be logarithmic (DG & Johansson 08) Note: similar critical points exist in generic higher derivative gravity (DG, Johansson & Zojer 10) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 23/32
65 Checks of LCFT conjecture Jordan cell structure Linearization around AdS background. Line-element ḡ µν of pure AdS: g µν = ḡ µν + h µν d s 2 AdS = ḡ µν dx µ dx ν = l 2( cosh 2 ρ dτ 2 + sinh 2 ρ dφ 2 + dρ 2) Isometry group: SL(2, R) L SL(2, R) R Useful to introduce light-cone coordinates u = τ + φ, v = τ φ. The SL(2, R) L generators L 0 = i u [ L ±1 = ie ±iu cosh 2ρ sinh 2ρ 1 u sinh 2ρ v i ] 2 ρ obey the algebra [L 0, L ±1 ] = L ±1, [L 1, L 1 ] = 2L 0. The SL(2, R) R generators L 0, L ±1 obey same algebra, but with u v, L L D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 24/32
66 Checks of LCFT conjecture Jordan cell structure Linearization around AdS background. g µν = ḡ µν + h µν leads to linearized EOM that are third order PDE G (1) µν + 1 µ C(1) µν = (D R D L D M h) µν = 0 (1) with three mutually commuting first order operators (D L/R ) µ ν = δ ν µ ± l ε µ αν α (D M ) µ ν = δ ν µ + 1 µ ε µ αν α D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 24/32
67 Checks of LCFT conjecture Jordan cell structure Linearization around AdS background. g µν = ḡ µν + h µν leads to linearized EOM that are third order PDE G (1) µν + 1 µ C(1) µν = (D R D L D M h) µν = 0 (1) with three mutually commuting first order operators (D L/R ) µ ν = δ ν µ ± l ε µ αν α (D M ) µ ν = δ ν µ + 1 µ ε µ αν α Three linearly independent solutions to (1): ( D L h L) µν = 0 ( D R h R) µν = 0 ( D M h M) µν = 0 D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 24/32
68 Checks of LCFT conjecture Jordan cell structure Linearization around AdS background. g µν = ḡ µν + h µν leads to linearized EOM that are third order PDE G (1) µν + 1 µ C(1) µν = (D R D L D M h) µν = 0 (1) with three mutually commuting first order operators (D L/R ) µ ν = δ ν µ ± l ε µ αν α (D M ) µ ν = δ ν µ + 1 µ ε µ αν α Three linearly independent solutions to (1): ( D L h L) µν = 0 ( D R h R) µν = 0 ( D M h M) µν = 0 At chiral point left (L) and massive (M) branches coincide! D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 24/32
69 Checks of LCFT conjecture Jordan cell structure Linearization around AdS background. g µν = ḡ µν + h µν leads to linearized EOM that are third order PDE G (1) µν + 1 µ C(1) µν = (D R D L D M h) µν = 0 (1) with three mutually commuting first order operators (D L/R ) µ ν = δ ν µ ± l ε µ αν α (D M ) µ ν = δ ν µ + 1 µ ε µ αν α At chiral point: get log solution (DG & Johansson 08) h log h M µν(µl) h L µν µν = lim µl 1 µl 1 with property ( D L h log) µν = ( D M h log) µν 0, ( (D L ) 2 h log) µν = 0 D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 24/32
70 Checks of LCFT conjecture Jordan cell structure Linearization around AdS background. g µν = ḡ µν + h µν Log mode exhibits interesting property: ( ) ( ) ( h log 2 1 h log H h L = 0 2 h L ( ) ( ) ( h log 2 0 h log J = 0 2 h L h L ) ) Here H = L 0 + L 0 t is the Hamilton operator and J = L 0 L 0 φ the angular momentum operator. Such a Jordan form of H and J is defining property of a logarithmic CFT! D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 24/32
71 Checks of LCFT conjecture Finiteness Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 25/32
72 Checks of LCFT conjecture Finiteness Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge Energy of logarithmic mode is finite E log = G l 3 and negative instability! (DG & Johansson 08) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 25/32
73 Checks of LCFT conjecture Finiteness Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge Energy of logarithmic mode is finite E log = G l 3 and negative instability! (DG & Johansson 08) Logarithmic mode is asymptotically AdS ds 2 = dρ 2 + ( γ (0) ij e2ρ/l + γ (1) ij ρ + γ(0) ij + γ (2) ij e 2ρ/l +... ) dx i dx j but violates Brown Henneaux boundary conditions! (γ (1) ij BH = 0) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 25/32
74 Checks of LCFT conjecture Finiteness Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge Energy of logarithmic mode is finite E log = G l 3 and negative instability! (DG & Johansson 08) Logarithmic mode is asymptotically AdS ds 2 = dρ 2 + ( γ (0) ij e2ρ/l + γ (1) ij ρ + γ(0) ij + γ (2) ij e 2ρ/l +... ) dx i dx j but violates Brown Henneaux boundary conditions! (γ (1) ij BH = 0) Consistent log boundary conditions replacing Brown Henneaux (DG & Johansson 08, Martinez, Henneaux & Troncoso 09) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 25/32
75 Checks of LCFT conjecture Finiteness Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge Energy of logarithmic mode is finite E log = G l 3 and negative instability! (DG & Johansson 08) Logarithmic mode is asymptotically AdS ds 2 = dρ 2 + ( γ (0) ij e2ρ/l + γ (1) ij ρ + γ(0) ij + γ (2) ij e 2ρ/l +... ) dx i dx j but violates Brown Henneaux boundary conditions! (γ (1) ij BH = 0) Consistent log boundary conditions replacing Brown Henneaux (DG & Johansson 08, Martinez, Henneaux & Troncoso 09) Brown York stress tensor is finite and traceless, but not chiral D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 25/32
76 Checks of LCFT conjecture Finiteness Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge Energy of logarithmic mode is finite E log = G l 3 and negative instability! (DG & Johansson 08) Logarithmic mode is asymptotically AdS ds 2 = dρ 2 + ( γ (0) ij e2ρ/l + γ (1) ij ρ + γ(0) ij + γ (2) ij e 2ρ/l +... ) dx i dx j but violates Brown Henneaux boundary conditions! (γ (1) ij BH = 0) Consistent log boundary conditions replacing Brown Henneaux (DG & Johansson 08, Martinez, Henneaux & Troncoso 09) Brown York stress tensor is finite and traceless, but not chiral Log mode persists non-perturbatively, as shown by Hamilton analysis (DG, Jackiw & Johansson 08, Carlip 08) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 25/32
77 Checks of LCFT conjecture Correlators Reminder: any CFT has conserved traceless EMT T z z = 0 T zz = O L (z) T z z = O R ( z) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 26/32
78 Checks of LCFT conjecture Correlators Reminder: any CFT has conserved traceless EMT T z z = 0 T zz = O L (z) T z z = O R ( z) 2- and 3-point correlators fixed by conformal Ward identities O R ( z) O R (0) = c R 2 z 4 O L (z) O L (0) = c L 2z 4 O L (z) O R (0) = 0 c R O R ( z) O R ( z ) O R (0) = z 2 z 2 ( z z ) 2 O L (z) O L (z ) O L c L (0) = z 2 z 2 (z z ) 2 O L (z) O R ( z ) O R (0) = 0 O L (z) O L (z ) O R (0) = 0 Central charges c L/R determine key properties of CFT. D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 26/32
79 Checks of LCFT conjecture Correlators Reminder: any CFT has conserved traceless EMT T z z = 0 T zz = O L (z) T z z = O R ( z) 2- and 3-point correlators fixed by conformal Ward identities Central charges c L/R determine key properties of CFT. Suppose there is an additional operator O M with conformal weights h = 2 + ε, h = ε ˆB O M (z, z) O M (0, 0) = z4+2ε z 2ε which degenerates with O L in limit c L ε 0 D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 26/32
80 Checks of LCFT conjecture Correlators Reminder: any CFT has conserved traceless EMT T z z = 0 T zz = O L (z) T z z = O R ( z) 2- and 3-point correlators fixed by conformal Ward identities Central charges c L/R determine key properties of CFT. Suppose there is an additional operator O M with conformal weights h = 2 + ε, h = ε ˆB O M (z, z) O M (0, 0) = z4+2ε z 2ε which degenerates with O L in limit c L ε 0 Then energy momentum tensor acquires logarithmic partner O log where O log = b L O L c L + b L 2 OM b L := lim c L 0 c L ε 0 D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 26/32
81 Checks of LCFT conjecture Correlators Reminder: any CFT has conserved traceless EMT T z z = 0 T zz = O L (z) T z z = O R ( z) 2- and 3-point correlators fixed by conformal Ward identities Central charges c L/R determine key properties of CFT. Suppose there is an additional operator O M with conformal weights h = 2 + ε, h = ε which degenerates with O L in limit c L ε 0 Then energy momentum tensor acquires logarithmic partner O log Some 2-point correlators: O L (z)o L (0, 0) = 0 O L (z)o log (0, 0) = b L 2z 4 O log (z, z)o log (0, 0) = b L ln (m 2 L z 2 ) z 4 New anomaly b L determines key properties of logarithmic CFT. D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 26/32
82 Checks of LCFT conjecture Correlators If LCFT conjecture is correct then following procedure must work: Calculate non-normalizable modes for left, right and logarithmic branches by solving linearized EOM on gravity side According to AdS 3 /LCFT 2 dictionary these non-normalizable modes are sources for corresponding operators in the dual CFT Calculate 2- and 3-point correlators on the gravity side, e.g. by plugging non-normalizable modes into second and third variation of the on-shell action These correlators must coinicde with the ones of a logarithmic CFT D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 26/32
83 Checks of LCFT conjecture Correlators If LCFT conjecture is correct then following procedure must work: Calculate non-normalizable modes for left, right and logarithmic branches by solving linearized EOM on gravity side According to AdS 3 /LCFT 2 dictionary these non-normalizable modes are sources for corresponding operators in the dual CFT Calculate 2- and 3-point correlators on the gravity side, e.g. by plugging non-normalizable modes into second and third variation of the on-shell action These correlators must coinicde with the ones of a logarithmic CFT Except for value of new anomaly b L no freedom in this procedure. Either it works or it does not work. D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 26/32
84 Checks of LCFT conjecture Correlators If LCFT conjecture is correct then following procedure must work: Calculate non-normalizable modes for left, right and logarithmic branches by solving linearized EOM on gravity side According to AdS 3 /LCFT 2 dictionary these non-normalizable modes are sources for corresponding operators in the dual CFT Calculate 2- and 3-point correlators on the gravity side, e.g. by plugging non-normalizable modes into second and third variation of the on-shell action These correlators must coinicde with the ones of a logarithmic CFT Except for value of new anomaly b L no freedom in this procedure. Either it works or it does not work. Works at level of 2-point correlators (Skenderis, Taylor & van Rees 09, DG & Sachs 09) Works at level of 3-point correlators (DG & Sachs 09) Value of new anomaly: b L = c R = 3l/G D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 26/32
85 Checks of LCFT conjecture 1-loop partition function (Gaberdiel, DG & Vassilevich 10) Structure of low-lying states in LCFT: T (L 0 2), L 0 t L 2 L 2 Ω Total partition function of Virasoro descendants ZLCFT 0 = Z Ω + Z t = 1 ( 1 q n q2 ) 1 q 2 n=2 D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 27/32
86 Checks of LCFT conjecture 1-loop partition function (Gaberdiel, DG & Vassilevich 10) Structure of low-lying states in LCFT: Total partition function of Virasoro descendants T (L 0 2), L 0 t L 2 L 2 Ω Z TMG = torus b.c. ZLCFT 0 = Z Ω + Z t = 1 ( 1 q n q2 ) 1 q 2 n=2 Comparison with 1-loop calculation in Euclidean path integral approach to quantum gravity: Dh µν ghost exp ( δ 2 S(h) ) = Z Ein det(d L ) 1/2 Calculating 1-loop determinant yields Einstein gravity result times another determinant. D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 27/32
87 Checks of LCFT conjecture 1-loop partition function (Gaberdiel, DG & Vassilevich 10) Structure of low-lying states in LCFT: Total partition function of Virasoro descendants T (L 0 2), L 0 t L 2 L 2 Ω ln det(d L ) 1/2 = n=1 ZLCFT 0 = Z Ω + Z t = 1 ( 1 q n q2 ) 1 q 2 n=2 Comparison with 1-loop calculation in Euclidean path integral approach to quantum gravity: 1 n q 2n (1 q n )(1 q n ) Heat kernel methods allow to determine the new 1-loop determinant. D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 27/32
88 Checks of LCFT conjecture 1-loop partition function (Gaberdiel, DG & Vassilevich 10) Structure of low-lying states in LCFT: Total partition function of Virasoro descendants T (L 0 2), L 0 t L 2 L 2 Ω Z TMG = n=2 1 1 q n 2 ZLCFT 0 = Z Ω + Z t = 1 ( 1 q n q2 ) 1 q 2 n=2 Comparison with 1-loop calculation in Euclidean path integral approach to quantum gravity: m=2 m=0 1 1 qm q m The final result consists of two factors, an Einstein piece and a new contribution from the log modes. D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 27/32
89 Checks of LCFT conjecture 1-loop partition function (Gaberdiel, DG & Vassilevich 10) Structure of low-lying states in LCFT: Total partition function of Virasoro descendants T (L 0 2), L 0 t L 2 L 2 Ω Z TMG = Z 0 LCFT + h, h ZLCFT 0 = Z Ω + Z t = 1 ( 1 q n q2 ) 1 q 2 n=2 Comparison with 1-loop calculation in Euclidean path integral approach to quantum gravity: N h, h q h q h n=1 1 1 q n 2 All multiplicity coefficients N h, h can be shown to be non-negative. Fairly non-trivial test of the LCFT conjecture! D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 27/32
90 Generalizations Log CFTs have taught us a great deal about critical TMG! D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 28/32
91 Generalizations Log CFTs have taught us a great deal about critical TMG! Generalizations beyond topologically massive gravity at the critical point? D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 28/32
92 Generalizations Log CFTs have taught us a great deal about critical TMG! Generalizations beyond topologically massive gravity at the critical point? 3D: generic massive gravity theories (DG, Johansson, Zojer 10) New massive gravity, generalized massive gravity, higher curvature gravity,... higher rank Jordan cells possible in some of these models qualitatively new log CFTs arise: no log partner of energy-momentum tensor beyond the SUGRA approximation? D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 28/32
93 Generalizations Log CFTs have taught us a great deal about critical TMG! Generalizations beyond topologically massive gravity at the critical point? 3D: generic massive gravity theories (DG, Johansson, Zojer 10) New massive gravity, generalized massive gravity, higher curvature gravity,... higher rank Jordan cells possible in some of these models qualitatively new log CFTs arise: no log partner of energy-momentum tensor beyond the SUGRA approximation? 4D log gravity? (Lü, Pope 11) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 28/32
94 Generalizations Log CFTs have taught us a great deal about critical TMG! Generalizations beyond topologically massive gravity at the critical point? 3D: generic massive gravity theories (DG, Johansson, Zojer 10) New massive gravity, generalized massive gravity, higher curvature gravity,... higher rank Jordan cells possible in some of these models qualitatively new log CFTs arise: no log partner of energy-momentum tensor beyond the SUGRA approximation? 4D log gravity? (Lü, Pope 11) Higher spin log gravity? (Chen, Long, Wu; Bagchi, Lal, Saha, Sahoo 11) D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 28/32
95 Holographic conclusions 3-dimensional gravity duals for 2-dimensional log CFTs seem to exist D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 29/32
96 Holographic conclusions 3-dimensional gravity duals for 2-dimensional log CFTs seem to exist learned a lot about gravity applying log CFT technology D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 29/32
97 Holographic conclusions 3-dimensional gravity duals for 2-dimensional log CFTs seem to exist learned a lot about gravity applying log CFT technology trend will probably continue in higher spin gravity D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 29/32
98 Holographic conclusions 3-dimensional gravity duals for 2-dimensional log CFTs seem to exist learned a lot about gravity applying log CFT technology trend will probably continue in higher spin gravity possibly can learn also something about strongly coupled log CFTs D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 29/32
99 Holographic conclusions 3-dimensional gravity duals for 2-dimensional log CFTs seem to exist learned a lot about gravity applying log CFT technology trend will probably continue in higher spin gravity possibly can learn also something about strongly coupled log CFTs intriguing perspective: higher dimensional log CFTs? D. Grumiller Physics of Jordan cells Jordan cells in the holographic AdS 3 /LCFT 2 correspondence 29/32
100 Outline Jordan cells in non-hermitian quantum mechanics Jordan cells in logarithmic conformal field theories Jordan cells in the holographic AdS 3 /LCFT 2 correspondence Jordan cells in condensed matter applications D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 30/32
101 Systems with quenched disorder LCFTs arise in systems with quenched disorder. D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 31/32
102 Systems with quenched disorder LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 31/32
103 Systems with quenched disorder LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 31/32
104 Systems with quenched disorder LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 31/32
105 Systems with quenched disorder LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point Infamous denominator in correlators: ( Dφ exp I[φ] d 2 z V (z )O(z ) ) O(z) O(0) O(z) O(0) = DV P [V ] ( Dφ exp I[φ] d 2 z V (z )O(z ) ) D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 31/32
106 Systems with quenched disorder LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point Infamous denominator in correlators: ( Dφ exp I[φ] d 2 z V (z )O(z ) ) O(z) O(0) O(z) O(0) = DV P [V ] ( Dφ exp I[φ] d 2 z V (z )O(z ) ) Different ways to deal with denominator (replica trick, SUSY) D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 31/32
107 Systems with quenched disorder LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point Infamous denominator in correlators: ( Dφ exp I[φ] d 2 z V (z )O(z ) ) O(z) O(0) O(z) O(0) = DV P [V ] ( Dφ exp I[φ] d 2 z V (z )O(z ) ) Different ways to deal with denominator (replica trick, SUSY) Result: operators degenerate and correlators acquire logarithmic behavior, exactly as in LCFT (Cardy 99) D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 31/32
108 Systems with quenched disorder LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point Infamous denominator in correlators: ( Dφ exp I[φ] d 2 z V (z )O(z ) ) O(z) O(0) O(z) O(0) = DV P [V ] ( Dφ exp I[φ] d 2 z V (z )O(z ) ) Different ways to deal with denominator (replica trick, SUSY) Result: operators degenerate and correlators acquire logarithmic behavior, exactly as in LCFT (Cardy 99) Exploit LCFTs to compute correlators of quenched random systems D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 31/32
109 Systems with quenched disorder LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point Infamous denominator in correlators: ( Dφ exp I[φ] d 2 z V (z )O(z ) ) O(z) O(0) O(z) O(0) = DV P [V ] ( Dφ exp I[φ] d 2 z V (z )O(z ) ) Different ways to deal with denominator (replica trick, SUSY) Result: operators degenerate and correlators acquire logarithmic behavior, exactly as in LCFT (Cardy 99) Exploit LCFTs to compute correlators of quenched random systems Idea: Apply AdS 3 /LCFT 2 to describe strongly coupled LCFTs! D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 31/32
110 Some literature on condensed matter applications of LCFTs Cardy 99 Logarithmic correlations in Quenched Random Magnets and Polymers Gurarie & Ludwig 99 Conformal algebras of 2D disordered systems Rahimi Tabar 00 Quenched Averaged Correlation Functions of the Random Magnets Reviews: Flohr 01 and Gaberdiel 01 D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 32/32
111 Some literature on condensed matter applications of LCFTs Cardy 99 Logarithmic correlations in Quenched Random Magnets and Polymers Gurarie & Ludwig 99 Conformal algebras of 2D disordered systems Rahimi Tabar 00 Quenched Averaged Correlation Functions of the Random Magnets Reviews: Flohr 01 and Gaberdiel 01 More applications awaiting to be discovered! D. Grumiller Physics of Jordan cells Jordan cells in condensed matter applications 32/32
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