Fractals : Spectral properties Statistical physics

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1 Fractals : Spectral properties Statistical physics Course 1 Eric Akkermans I S R A T I O N E L S C I E N C E F O U N D A 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals, June 13-17, 2017

Gerald Dunne (UConn.) Alexander Teplyaev (UConn.

2 Benefitted from discussions and collaborations with: Technion: Evgeni Gurevich (KLA-Tencor) Dor Gittelman Eli Levy (+ Rafael) Ariane Soret (ENS Cachan) Or Raz (HUJI, Maths) Omrie Ovdat Yaroslav Don Rafael: Assaf Barak Amnon Fisher Elsewhere: Gerald Dunne (UConn.) Alexander Teplyaev (UConn.) Jacqueline Bloch (LPN, Marcoussis) Dimitri Tanese (LPN, Marcoussis) Florent Baboux (LPN, Marcoussis) Alberto Amo (LPN, Marcoussis) Eva Andrei (Rutgers) Jinhai Mao (Rutgers) Arkady Poliakovsky (Maths. BGU)

3 Benefitted from discussions and collaborations with: Technion: Evgeni Gurevich (KLA-Tencor) Dor Gittelman Eli Levy (+ Rafael) Ariane Soret (ENS Cachan) Or Raz (HUJI, Maths) Omrie Ovdat Yaroslav Don Rafael: Assaf Barak Amnon Fisher Elsewhere: Gerald Dunne (UConn.) Alexander Teplyaev (UConn.) Jacqueline Bloch (LPN, Marcoussis) Dimitri Tanese (LPN, Marcoussis) Florent Baboux (LPN, Marcoussis) Alberto Amo (LPN, Marcoussis) Eva Andrei (Rutgers) Jinhai Mao (Rutgers) Arkady Poliakovsky (Maths. BGU)

4 Plan of the 4 talks Course 1 : Spectral properties of fractals - Application in statistical physics Talk : quantum phase transition - scale anomaly and fractals Course 2 : topology and fractals - measuring topological numbers with waves. Elaboration : Renormalisation group and Efimov physics

5 Program for today Introduction : spectral properties of self similar fractals. Heat kernel - Asymptotic behaviour - Weyl expansion - Spectral volume. Thermodynamics of the fractal blackbody. Summary - Phase transitions.

$fractals.$

6 Introduction : spectral properties of self similar fractals. attractive objects - Bear exotic names Julia sets

7 Anatomy of the Hofstadter butterfly Hofstadter butterfly Sierpinski carpet Sierpinski gasket

$Diamond fractals Triadic Cantor set Convey the idea of highly symmetric$

8 Diamond fractals Triadic Cantor set Convey the idea of highly symmetric objects yet with an unusual type of symmetry and a notion of extreme subdivision

9 Fractal : Iterative graph structure n Sierpinski gasket Diamond fractals

10 Fractal : Iterative graph structure n Sierpinski gasket Diamond fractals

11 As opposed to Euclidean spaces characterised by translation symmetry, fractals possess a dilatation symmetry. Fractals are self-similar objects

12 Fractal Self-similar Discrete scaling symmetry

13 But not all fractals are obvious, good faith geometrical objects. Sometimes, the fractal structure is not geometrical but it is hidden at a more abstract level. Exemple : quasi-periodic stack of dielectric layers of 2 types A, B Fibonacci sequence : F 1 = B; F 2 = A; F j 3 = F j 2 F j 1 Defines a cavity whose mode spectrum is fractal.

$But generally, not all fractals are obvious, good faith geometrical objects. Sometimes, the fractal structure is not geometrical but it is hidden at a more abstract level.$

14 But generally, not all fractals are obvious, good faith geometrical objects. Sometimes, the fractal structure is not geometrical but it is hidden at a more abstract level. Exemple : Quasi-periodic chain of layers of 2 types A, B Fibonacci sequence : F 1 = B; F 2 = A; F j 3 = F j 2 F j 1 Defines a cavity whose frequency spectrum is fractal.

15 Density of modes ρ(ω) : Discrete scaling symmetry Minicourse 2 - Tomorrow

16 Operators and fields on fractal manifolds Operators are often expressed by local differential equations relating the space-time behaviour of a field Ex. Wave equation 2 u t 2 = Δu Such local equations cannot be defined on a fractal 16

17 But operators are essential quantities for physics Quantum transport in fractal structures : e.g., networks, waveguides,... electrons, photons Density of states Scattering matrix (transmission/reflection) 17

18 But operators are essential quantities for physics Quantum fields on fractals, e.g., fermions (spin 1/2), photons (spin 1) - canonical quantisation (Fourier modes) - path integral quantisation : path integrals, Brownian motion. curved space QFT or quantum gravity Scaling symmetry (renormalisation group) - critical behaviour. 18

19 Recent new ideas >2000 Maths. Michel Lapidus Bob Strichartz Jun Kigami 19

20 Intermezzo : heat and waves

21 From classical diffusion to wave propagation Important relation between classical diffusion and wave propagation on a manifold. Expresses the idea that it is possible to measure and characterise a manifold using waves (eigenvalue spectrum of the Laplace operator) geometry curvature volume dimension Differential operator propagating probe physically: Laplacian Spectral data Heat kernel Zeta function

22 Use propagating waves/particles to probe : spectral information: density of states, transport, heat kernel,... geometric information: dimension, volume, boundaries, shape,... 22

23 Use propagating waves/particles to probe : spectral information: density of states, transport, heat kernel,... geometric information: dimension, volume, boundaries, shape,... Mathematical physics 1910 Lorentz: why is the Jeans radiation law only dependent on the volume? 1911 Weyl : relation between asymptotic eigenvalues and dimension/volume Kac : can one hear the shape of a drum? 23

24 Important examples Heat equation Wave equation Schr. equation. u t = Δu 2 u t 2 = Δu i u t = Δu u x,t ( ) = dµ y ( )P t x, y ( )u y,0 ( ) P t ( x, y) = xe D x( 0)=x,x( t)=y (i) t 0 x 2 dτ Brownian motion P t ( x, y) 1 t d 2 n a n (x, y)t n Heat kernel expansion P t ( x, y) (#) e (i)s classical (x,y,t ) geodesics Gutzwiller - instantons 24

25 Important examples Heat equation Wave equation Schr. equation. u t = Δu 2 u t 2 = Δu i u t = Δu ( ) = dµ y u x,t ( )P ( x, y )u y,0 t ( ) P t ( x, y) = xe D x( 0)=x,x( t)=y (i) t 0 x 2 dτ Brownian motion P t ( x, y) 1 t d 2 n a n (x, y)t n Heat kernel expansion P t ( x, y) (#) e (i)s classical (x,y,t ) geodesics Gutzwiller - instantons 25

26 Important examples Heat equation Wave equation Schr. equation. u t = Δu 2 u t 2 = Δu i u t = Δu ( ) = dµ y u x,t ( )P ( x, y )u y,0 t ( ) P t ( x, y ) = xe D x ( 0 )=x,x ( t )=y (i) t 0 x 2 dτ Brownian motion P t ( x, y ) 1 t d 2 n a n (x, y)t n Heat kernel expansion P t ( x, y) (#) e (i)s classical (x,y,t ) geodesics Gutzwiller - instantons 26

27 Important examples Heat equation Wave equation Schr. equation. u t = Δu 2 u t 2 = Δu i u t = Δu ( ) = dµ y u x,t ( )P ( x, y )u y,0 t ( ) P t ( x, y ) = xe D x ( 0 )=x,x ( t )=y (i) t 0 x 2 dτ Brownian motion P t ( x, y ) 1 t d 2 n a n (x, y)t n Heat kernel expansion P t ( x, y) (#) e (i)s classical (x,y,t ) geodesics Gutzwiller - instantons 27

28 Spectral functions P t λ ( x, y) = y e Δt x = ψ λ (y) ψ λ (x)e λt Z(t) =Tre Δt = dx x e Δt x = e λt λ Heat kernel ζ Z ( s) 1 Γ s ( ) 0 dtt s 1 Z ( t) Mellin transform ζ Z ( s) = Tr 1 Δ = 1 s λ s λ Small t behaviour of Z(t) poles of ζ ( Z s) Weyl expansion 28

29 Spectral functions P t λ ( x, y) = y e Δt x = ψ λ (y) ψ λ (x)e λt Z(t) =Tre Δt = dx x e Δt x = e λt λ Heat kernel ζ Z ( s) 1 Γ s ( ) 0 dtt s 1 Z ( t) Mellin transform ζ Z ( s) = Tr 1 Δ = 1 s λ s λ Small t behaviour of Z(t) poles of ζ ( Z s) Weyl expansion 29

30 Spectral functions P t λ ( x, y) = y e Δt x = ψ λ (y) ψ λ (x)e λt Z(t) =Tre Δt = dx x e Δt x = e λt λ Heat kernel ζ Z ( s) 1 Γ s ( ) 0 ( ) dtt s 1 Z t ζ Z Return probability ( s) = Tr 1 Δ = 1 s λ s λ Mellin transform Small t behaviour of Z(t) poles of ζ ( Z s) Weyl expansion 30

31 Spectral functions P t λ ( x, y) = y e Δt x = ψ λ (y) ψ λ (x)e λt Z(t) =Tre Δt = dx x e Δt x = e λt λ Heat kernel ζ Z ( s) 1 Γ s ( ) 0 dtt s 1 Z ( t) Mellin transform ζ Z ( s) = Tr 1 Δ = 1 s λ s λ Small t behaviour of Z(t) poles of ζ ( Z s) Weyl expansion 31

32 Spectral functions P t λ ( x, y) = y e Δt x = ψ λ (y) ψ λ (x)e λt Z(t) =Tre Δt = dx x e Δt x = e λt λ Heat kernel ζ Z ( s) 1 Γ s ( ) 0 dtt s 1 Z ( t) Mellin transform ζ Z ( s) = Tr 1 Δ = 1 s λ s λ Small t behaviour of Z(t) poles of ζ ( Z s) 32

33 Spectral functions P t λ ( x, y) = y e Δt x = ψ λ (y) ψ λ (x)e λt Z(t) =Tre Δt = dx x e Δt x = e λt λ Heat kernel ζ Z ( s) 1 Γ s ( ) 0 dtt s 1 Z ( t) Mellin transform ζ Z ( s) = Tr 1 Δ = 1 s λ s λ Small t behaviour of Z(t) poles of ζ ( Z s) Weyl expansion 33

34 The heat kernel is related to the density of states of the Laplacian There are Laplace transform of each other: From the Weyl expansion, it is possible to obtain the density of states.

35 How does it work? Diffusion (heat) equation in d=1 whose spectral solution is P t ( x, y) = 1 ( 4π Dt ) 1 2 e ( x y)2 4 Dt Probability of diffusing from x to y in a time t. In d space dimensions: P t ( x, y) = 1 ( 4π Dt ) d 2 e ( x y)2 4 Dt We can characterise the spatial geometry by watching how the heat flows. The heat kernel t is Z d ( ) Z d ( t) = d d xp ( t x, x)= Volume Vol. 4π Dt ( ) d 2 access the volume of the manifold

36 How does it work? Diffusion (heat) equation in d=1 whose spectral solution is P t ( x, y) = 1 ( 4π Dt ) 1 2 e ( x y)2 4 Dt Probability of diffusing from x to y in a time t. In d space dimensions: P t ( x, y) = 1 ( 4π Dt ) d 2 e ( x y)2 4 Dt We can characterise the spatial geometry by watching how the heat flows. The heat kernel t is Z d ( ) Z d ( t) = d d xp ( t x, x)= Volume Vol. 4π Dt ( ) d 2 access the volume of the manifold

37 How does it work? Diffusion (heat) equation in d=1 whose spectral solution is P t ( x, y) = 1 ( 4π Dt ) 1 2 e ( x y)2 4 Dt Probability of diffusing from x to y in a time t. In d space dimensions: P t ( x, y) = 1 ( 4π Dt ) d 2 e ( x y)2 4 Dt We can characterise the spatial geometry by watching how the heat flows. The heat kernel t is Z d ( ) Z d ( t) = d d xp ( t x, x)= Volume Vol. 4π Dt ( ) d 2 access the volume of the manifold

38 How does it work? Diffusion (heat) equation in d=1 whose spectral solution is P t ( x, y) = 1 ( 4π Dt ) 1 2 e ( x y)2 4 Dt Probability of diffusing from x to y in a time t. In d space dimensions: P t ( x, y) = 1 ( 4π Dt ) d 2 e ( x y)2 4 Dt We can characterise the spatial geometry by watching how the heat flows. The heat kernel t is Z d ( ) Z d ( t) = d d xp ( t x, x)= Volume Vol. 4π Dt ( ) d 2 volume of the manifold

39 Boundary terms- Hearing the shape of a drum Mark Kac (1966) 0 L

40 Boundary terms- Hearing the shape of a drum Mark Kac (1966) 0 L

41 Boundary terms- Hearing the shape of a drum Mark Kac (1966) 0 L Poisson formula

42 Boundary terms- Hearing the shape of a drum Mark Kac (1966) 0 L Poisson formula

43 Boundary terms- Hearing the shape of a drum Mark Kac (1966) 0 L Poisson formula Weyl expansion (1d)

44 Boundary terms- Hearing the shape of a drum Mark Kac (1966) 0 L Poisson formula Weyl expansion (2d) : Z d=2 (t) Vol. 4πt L 4 1 4πt

45 Boundary terms- Hearing the shape of a drum Mark Kac (1966) 0 L Poisson formula Weyl expansion (2d) : Z d=2 (t) Vol. 4πt L 4 1 4πt bulk

46 Boundary terms- Hearing the shape of a drum Mark Kac (1966) 0 L Poisson formula Weyl expansion (2d) : Z d=2 (t) Vol. 4πt L 4 1 4πt bulk sensitive to boundary

47 Boundary terms- Hearing the shape of a drum Mark Kac (1966) 0 L Poisson formula Weyl expansion (2d) : Z d=2 (t) Vol. 4πt L 4 1 4πt bulk sensitive to boundary integral of bound. curvature

48 -function ζ ζ Z s λ s λ ( ) = Tr 1 Δ s = 1 has a simple pole at so that,

49 How does it work on a fractal? Differently No access to the eigenvalue spectrum but we know how to calculate the Heat Kernel. Z(t) =Tre Δt = dx x e Δt x = e λt λ and thus, the density of states,

50 How does it work on a fractal? Differently No simple access to the eigenvalue spectrum but we know how to calculate the heat kernel. Z(t) =Tre Δt = dx x e Δt x = e λt λ and thus, the density of states,

51 More precisely, is the total length upon iteration of the elementary step which has poles at

52 s n = d s 2 + 2iπn d w lna Infinite number of complex poles : complex fractal dimensions. They control the behaviour of the heat kernel which exhibits oscillations. K t K leading t Z diamond ( t) t t A new fractal dimension : spectral dimension d s

53 Notion of spectral volume

54 From the previous expression we obtain Z ( t) Consider for simplicity, namely n = 1 s 1 = d s 2 + 2iπ d w lna d s 2 + iδ to compare with

55 From the previous expression we obtain Z ( t) Consider for simplicity, namely n = 1 s 1 = d s 2 + 2iπ d w lna d s 2 + iδ so that to compare with

56 From the previous expression we obtain Z ( t) Consider for simplicity, namely n = 1 s 1 = d s 2 + 2iπ d w lna d s 2 + iδ so that Spectral volume to compare with

57 From the previous expression we obtain Z ( t) Consider for simplicity, namely n = 1 s 1 = d s 2 + 2iπ d w lna d s 2 + iδ so that Spectral volume to compare with Z d ( t) = d d xp ( t x, x)= Volume Vol. 4π Dt ( ) d 2

58 Spectral volume? Geometric volume described by the Hausdorff dimension is large (infinite) Spectral volume is the finite volume occupied by the modes Numerical solution of Maxwell eqs. in the Sierpinski gasket (courtesy of S.F. Liew and H. Cao, Yale)

59 Spectral volume? Geometric volume described by the Hausdorff dimension is large (infinite) Spectral volume is the finite volume occupied by the modes Numerical solution of Maxwell eqs. on the Sierpinski gasket

$Physical application : Thermodynamics of photons on fractals$

60 Physical application : Thermodynamics of photons on fractals Electromagnetic field in a waveguide fractal structure. How to measure the spectral volume?

61 The radiating fractal blackbody Equation of state at thermodynamic equilibrium relating pressure, volume and internal energy: In an enclosure with a perfectly reflecting surface there can form standing electromagnetic waves analogous to tones of an organ pipe; we shall confine our attention to very high overtones. Jeans asks for the energy in the frequency interval dν... It is here that there arises the mathematical problem to prove that the number of sufficiently high overtones that lies in the interval ν to ν+dν is independent of the shape of the enclosure and is simply proportional to its volume. H. Lorentz, 1910

62 The radiating fractal blackbody Equation of state at thermodynamic equilibrium relating pressure, volume and internal energy: In an enclosure with a perfectly reflecting surface there can form standing electromagnetic waves analogous to tones of an organ pipe; we shall confine our attention to very high overtones. Jeans asks for the energy in the frequency interval dν... It is here that there arises the mathematical problem to prove that the number of sufficiently high overtones that lies in the interval ν to ν+dν is independent of the shape of the enclosure and is simply proportional to its volume. Spectral volume? H. Lorentz, 1910

proach: count modes in momentum space Usual approach : count modes in momentum space thermal equilibrium: Calculate the partition (generating) is a dimensionless equation of statefunction function z

radiation in a largeenergy volume pressure internal Mode decomposition of the field: Mode decomposition of the field ensional integer-valued vector-elementary momentum space cells ction (generating

63 proach: count modes in momentum space Usual approach : count modes in momentum space thermal equilibrium: Calculate the partition (generating) is a dimensionless equation of statefunction function z (T,V ) for a blackbody of 1 a large volume Black-body radiation P V = in U d is a dimensionless function large volume V in dimension d ( 2π )d V e decomposition of the volume field: Black-body radiation in a largeenergy volume pressure internal Mode decomposition of the field: Mode decomposition of the field ensional integer-valued vector-elementary momentum space cells ction (generating function) hat nz V ln Z(T, V ) d-dimensional integer-valued vector-elementary momentum space cells so that U= ln Z with β = 1k T B is the photon thermal wavelength. is the photon thermal wavelength. Lβ β c 1 =thermal (photon T wavelength)

64 Thermodynamics : so that (The exact expression of Q is unimportant) Stefan-Boltzmann is a consequence of Adiabatic expansion

65 Thermodynamics : so that (The exact expression of Q is unimportant) Stefan-Boltzmann is a consequence of Adiabatic expansion

66 Thermodynamics : so that (The exact expression of Q is unimportant) Stefan-Boltzmann is a consequence of Adiabatic expansion

67 On a fractal there is no notion of Fourier mode decomposition. Dimensions of momentum and position spaces are usually different : problem with the conventional formulation in terms of phase space cells. Volume of a fractal is usually infinite. Nevertheless, is the spectral volume.

68 On a fractal there is no notion of Fourier mode decomposition. Dimensions of momentum and position spaces are usually different : problem with the conventional formulation in terms of phase space cells. Volume of a fractal is usually infinite. Nevertheless, is the spectral volume.

69 On a fractal there is no notion of Fourier mode decomposition. Dimensions of momentum and position spaces are usually different : problem with the conventional formulation in terms of phase space cells. Volume of a fractal is usually infinite. Nevertheless, is the spectral volume.

$On a fractal there is no notion of Fourier mode decomposition.$

70 On a fractal there is no notion of Fourier mode decomposition. Dimensions of momentum and position spaces are usually different : problem with the conventional formulation in terms of phase space cells. Volume of a fractal is usually infinite. Nevertheless, is the spectral volume.

71 Re-phrase the thermodynamic problem in terms of heat kernel and zeta function.

72 Partition function of equilibrium quantum radiation ln z( T,V ) = 1 2 lndet M V 2 τ + 2 c2 Δ Looks (almost) like a bona fide wave equation but proper time. This expression does not rely on mode decomposition. Rescale by L β βc

73 Partition function of equilibrium quantum radiation ln z( T,V ) = 1 2 lndet M V 2 u 2 +L β 2 Δ M : circle of radius L β βc Spatial manifold (fractal) Thermal equilibrium of photons on a spatial manifold V at temperature T is described by the (scaled) wave equation on M V

74 can be rewritten ln z( T,V ) = 1 2 lndet M V 2 u 2 +L β 2 Δ ln z( T,V ) = dτ τ f ( τ )Tr e τ L 2 β Δ V ( ) Z L β2 τ Heat kernel Large volume limit (a high temperature limit) Weyl expansion: Z ( L β2 τ ) V ( 4π L β2 τ ) d 2

75 can be rewritten ln z( T,V ) = 1 2 lndet M V 2 u 2 +L β 2 Δ ln z( T,V ) = dτ τ f ( τ )Tr e τ L 2 β Δ V Large volume limit (a high temperature limit) Weyl expansion: Z ( L β2 τ ) V ( 4π L β2 τ ) d 2

76 can be rewritten ln z( T,V ) = 1 2 lndet M V 2 u 2 +L β 2 Δ ln z( T,V ) = dτ τ f ( τ )Tr e τ L 2 β Δ V Large volume limit (a high temperature limit) Weyl expansion: Z ( L β2 τ ) V ( 4π L β2 τ ) d 2

77 can be rewritten ln z( T,V ) = 1 2 lndet M V 2 u 2 +L β 2 Δ ln z( T,V ) = dτ τ f ( τ )Tr e τ L 2 β Δ V ( ) Z L β2 τ Heat kernel Large volume limit (a high temperature limit) Weyl expansion: Z ( L β2 τ ) V ( 4π L β2 τ ) d 2

78 can be rewritten ln z( T,V ) = 1 2 lndet M V 2 u 2 +L β 2 Δ ln z( T,V ) = dτ τ f ( τ )Tr e τ L 2 β Δ V ( ) Z L β2 τ Heat kernel Large volume limit (a high temperature limit) Weyl expansion: Z ( L β2 τ ) V ( 4π L β2 τ ) d 2

79 ln z( T,V ) = dτ τ f ( τ )Tr e τ L 2 β Δ V + Weyl expansion ln z T,V ( ) V L β d

ln z( T,V ) = 1 2 0 dτ τ f ( τ )Tr e τ L 2 β Δ V

80 ln z( T,V ) = dτ τ f ( τ )Tr e τ L 2 β Δ V ln z( T,V ) V + Weyl expansion d L β Thermodynamics : Thermodynamics measures the spectral volume so that (The exact expression of Q is unimportant)

81 ln z( T,V ) = dτ τ f ( τ )Tr e τ L 2 β Δ V + Weyl expansion ln z T,V ( ) V L β d Thermodynamics measures the spectral volume

82 On a fractal Z ( L β2 τ ) V s ( ) f lnτ ( 4π L β2 τ ) d s 2 Thermodynamic equation of state for a fractal manifold Thermodynamics measures the spectral volume and the spectral dimension.

83 On a fractal Spectral volume Z ( L β2 τ ) V s ( ) f lnτ ( 4π L β2 τ ) d s 2 Thermodynamic equation of state for a fractal manifold Thermodynamics measures the spectral volume and the spectral dimension.

84 On a fractal Spectral volume Z ( L β2 τ ) V s ( ) f lnτ ( 4π L β2 τ ) d s 2 Spectral dimension Thermodynamic equation of state for a fractal manifold Thermodynamics measures the spectral volume and the spectral dimension.

85 On a fractal Spectral volume Z ( L β2 τ ) V s ( ) f lnτ ( 4π L β2 τ ) d s 2 Spectral dimension Thermodynamic equation of state for a fractal manifold Thermodynamics measures the spectral volume and the spectral dimension.

86 Summary Significant progress in understanding and computing the asymptotic behaviour (Weyl) of heat kernels on fractals. Thermodynamics is directly related to the heat kernel (partition function) - fractal blackbody - importance of the spectral volume. Phase transitions on fractals : scaling/hyperscaling relations are modified on fractals (dependence on distinct fractal dimensions).

87 Summary Significant progress in understanding and computing the asymptotic behaviour (Weyl) of heat kernels on fractals. Thermodynamics is directly related to the heat kernel (partition function) - fractal blackbody - importance of the spectral volume. Phase transitions on fractals : scaling/hyperscaling relations are modified on fractals (dependence on distinct fractal dimensions).

88 Summary Significant progress in understanding and computing the asymptotic behaviour (Weyl) of heat kernels on fractals. Thermodynamics is directly related to the heat kernel (partition function) - fractal blackbody - importance of the spectral volume. Phase transitions on fractals : scaling/hyperscaling relations are modified on fractals (dependence on distinct fractal dimensions).

89 Non gaussian fixed points (limit cycles) - Harris criterion : fractal geometry is a specific type of disorder similar to quasicrystals. Off-diagonal long range order - superfluidity (Mermin, Wagner, Coleman theorem) - Non diagonal Green s function. Applications to other problems : quantum phase transitions - quantum Einstein gravity,

90 Non gaussian fixed points (limit cycles) - Harris criterion : fractal geometry is a specific type of disorder similar to quasicrystals. Off-diagonal long range order - superfluidity (Mermin, Wagner, Coleman theorem) - Non diagonal Green s function. Applications to other problems : quantum phase transitions - quantum Einstein gravity,

91 Non gaussian fixed points (limit cycles) - Harris criterion : fractal geometry is a specific type of disorder similar to quasicrystals. Off-diagonal long range order - superfluidity (Mermin, Wagner, Coleman theorem) - Non diagonal Green s function. Applications to other problems : quantum phase transitions - quantum Einstein gravity,

92 Thank you for your attention.

Waves and quantum physics on fractals :

$Waves and quantum physics on fractals :$ Waves and quantum physics on fractals : From continuous to discrete scaling symmetry Eric Akkermans Physics-Technion I S R A T I O N E L S C I E N C E F O U N D A Four lectures General introduction - Photons