Waves and Quantum Physics on Fractals
Waves and quantum physics on fractals :
From continuous to discrete scaling symmetry Show selected photi http://www.admin.technion.ac.il/pard/mediaarc/showphoto.asp?photol... Eric Akkermans Physics-Technion
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1 of 1 12/12/08 12:46 PM Four lectures
• General introduction - Photons and Quantum Electrodynamics on fractals
• Interplay between topology and discrete scaling symmetry : Quasi-crystals
• Critical behaviour on fractals : BEC and superfluidity
• Efimov physics from geometric and spectral perspectives Benefitted from discussions and collaborations with:
Technion: Elsewhere:
Evgeni Gurevich (KLA-Tencor) Gerald Dunne (UConn.) Dor Gittelman Alexander Teplyaev (UConn.) Ariane Soret (ENS Cachan) Raphael Voituriez (LPTMC, Jussieu) Or Raz Olivier Benichou (LPTMC, Jussieu) Omrie Ovdat Jacqueline Bloch (LPN, Marcoussis) Ohad Shpielberg Dimitri Tanese (LPN, Marcoussis) Alex Leibenzon Florent Baboux (LPN, Marcoussis) Alberto Amo (LPN, Marcoussis) Julien Gabelli (LPS, Orsay) Rafael:
Eli Levy Assaf Barak Amnon Fisher Part 1: General introduction to fractals
• attractive objects - Bear exotic names
Julia sets Anatomy of the Hofstadter butterfly Outline 0 General features ----- square lattice
Half-flux quantum per unit cell ------Dirac spectrum
Honeycomb lattice, graphene, manipulation of Dirac points
Finite systems, edge states
Hofstadter in other contexts
Anatomy of the Hofstadter butterfly M. Azbel (1964) 0 G.H. Wannier Y. Avron, B. Simon
h J. Bellissard, R. Rammal 0 e D. Thouless, Q. Niu TKNN : Thouless, Kohmoto, Nightingale, den Nijs Y. Hatsugai
Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Hofstadter butterfly Douglas Hofstadter, Phys. Rev. B 14 (1976) 2239
Sierpinski carpet Sierpinski gasket Diamond fractals
Figure: Diamond fractals, non-p.c.f., but finitely ramified
Triadic Cantor set
Convey the idea of highly symmetric objects yet with an unusual type of symmetry and a notion of extreme subdivision Fractal : Iterative graph structure
n → ∞
Sierpinski gasket
s ss s
s sss s
s s ss s s Diamond fractals Figure: Diamond fractals, non-p.c.f., but finitely ramified Fractal graph and Euclidean equivalent
d = 1
d = ?
d = 2
Similar graphs but have very distinct properties As opposed to Euclidean spaces characterised by translation symmetry, fractals possess a dilatation symmetry.
Fractals are self-similar objects Fractal ↔ Self-similar
Discrete scaling symmetry Notion of dimension
• Euclidean systems are characterised by a single integer dimension : d =1,2,3,4,…10
• We have learned that the dimension d can be considered a varying parameter : ε -expansion (phase transition, statistical mechanics, quantum field theory)
Yet, there is a single parameter d which enters into all physical quantities On fractal manifolds, the notion of dimension depends on the measured physical quantity.
Dimensions are not necessarily integers To summarise
• Euclidean manifolds : d is a fixed (boring ?) parameter.
• Fields (Stat. Mech., QFT) on Euclidean manifolds : d may be a variable ( ε -expansion) quantity
• Underlying essential idea : spontaneous (continuous) symmetry breaking.
• Fractal manifolds : genuine non integer dimension
• Fields on fractals : d is not variable, but distinct physical quantities are characterised by different dimensions.
• Underlying essential idea : discrete scaling symmetry. • 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 stack of dielectric layers of 2 types A, B
Fibonacci sequence F B; F A; F ⎡F F ⎤ : 1 = 2 = j≥3 = ⎣ j−2 j−1 ⎦
Defines a cavity whose mode spectrum is fractal. Density of modes ρ(ω) :
Discrete scaling symmetry
Existence of a fractal behaviour (discrete scaling symmetry) may be the expression of a genuine specific symmetry (next lecture). Why studying fractals in physics ? Fractals or the skill of playing with dimensions
Fractals define a very useful testing ground for dimensionality dependent physical problems since distinct physical properties are characterised by different (usually non integer) dimensions. Some examples : • Anderson localization phase transition : exists for
• Bose-Einstein condensation
• Mermin-Wagner theorem (Superfluidity d ≥ 2 )
• Levy flights-Percolation (quantum and classical)
• Recurrence properties of random walks d ≥ 2
• Renormalisability d = 2 and d = 4 special
• Quantum and classical phase transitions-Existence of topological defects... The meaning of the critical dependence upon dimensionality is not always clear :
Is it a geometric, spectral, transport,… feature ?
In this context, the fractal paradigm is interesting since it removes the degenerate role of dimension by assigning a different dimension to distinct physical properties. The meaning of the critical dependence upon dimensionality is not always clear :
Is it a geometric, spectral, transport,… feature ?
In this context, the fractal paradigm is interesting since it removes the degenerate role of dimension by assigning a different dimension to distinct physical properties. Topological aspects
• We have discussed geometrical aspects : dimensions of manifolds, spectra,…
• Additional essential information is provided by topological properties.
Example: Some d=2 surfaces cannot transform one into another by means of continuous transformations. This is expressed by a constraint a.k.a a topological invariant.
≠
20 Euler-Poincare characteristics
χ (S) = 2(1− h) h : number of holes
χ T2 = 0 χ (S2 ) = 2 ( ) Euler : χ (S) = V − E + F V=# of vertices ; E = # of edges and F = # of faces
21 • Although the sphere and the torus have the same dimension, these two manifolds have distinguishable properties when it comes to topology namely defining fields and operators (e.g., the Laplace operator − Δ which measure the energy cost to adapt a field to a specific manifold).
• General theory of operators defined on manifolds proposes a systematic framework to account for the connexion between (fields + operators) and topology of a manifold : Chern classes/numbers
• Topology of fractals is at a much earlier stage : difficulty in defining operators on fractals.
• Important progresses : gap labeling theorem (Bellissard ’82), aspects of the QHE and Quasi-crystals (Lectures 2+3). χSierpinski → − ∞ 22 Part 2 :Fractal dimensions
A working definition of a fractal
23 Hausdorff geometric dimension
An iterative structure : Sierpinski gasket
n → ∞
At each step of the iteration, the fractal graph is n characterised by its length scale L n = 2 L and the number n of bonds (mass) M n = 3 M .
Scaling of these dimensionless quantities allows to define a (mass) fractal dimension :
lnM n Hausdorff geometric dimension ⎯n⎯→∞⎯→ dh lnLn
~1.585 < 2 24 The Cantor set (disconnected zero measure)
n L M n = 2 M n Ln = 3 L
lnM n ln2 ⎯n⎯→∞⎯→ dh = lnLn ln 3
Alternatively, define the mass density m ( L ) of the triadic Cantor set
The mass density observed after a 2 m L = m L magnification by a factor 3 is ( 3) ( )
L 2 m( 3) = m(L) ⇔2 m(L) = m(3L)
25 Scanned by CamScanner Electric dimension : ς
Scaling of the equivalent electric resistance R(L)
Kirchhoff’s laws
Electric Sierpinski network
R(L) ∼ Lς
ln5 ς = 3 ln2
The electric dimension ς is different from d = ln 3 h ln2 26 Classical diffusion - walk dimension
On an Euclidean manifold, we write the mean square displacement
while on a fractal,
where is the anomalous walk dimension.
Another fractal dimension distinct from d h and ς
Related through the Einstein relation Continuous vs. discrete scale invariance
In all previous cases, we have found that exponents are determined by a scaling relation: f (a x) = b f (x)
If this relation is satisfied ∀ b ( a ) ∈ ! , the system has a continuous scale invariance
General solution (by direct inspection) f (x) = C xα lnb with α = (does not need to be an integer) lna
Power laws are signature of scale invariance
28 Discrete scale invariance
Instead of f (a x) = b f (x), ∀∀(ba(,ba))∈! for fractals, we have a weaker version of scale invariance, discrete scale invariance, i.e., f (a x) = b f (x), with fixed (a,b)
α ⎛ ln x⎞ whose general solution is f (x) = x G ⎝⎜ lna⎠⎟ lnb with α = (could be integer) lna where G ( u + 1 ) = G ( u ) is a periodic function of period unity Generalizes to scaling equations : f (a x) = b f (x)+ g(x), with fixed (a,b)
29 Relation between the two cases : discrete vs. continuous
d = 1
m(2L) = 2 m(L) ∀∀(ba(,ba))∈!
m(3L) = 2 m(L) (a,b) = (3,2)
d = 2 m(2L) = 3m(L) (a,b) = (2,3)
Both satisfy f ( a x ) = b f ( x ) but with fixed ( a , b ) for the fractals. 30 Complex fractal exponents and oscillations
α ⎛ ln x⎞ For a discrete scale invariance, f (x) = x G ⎝⎜ lna⎠⎟ and G ( u + 1 ) = G ( u ) is a periodic function of period unity
∞ 2πn α +i lna Fourier expansion: f (x) = ∑ cn x n=−∞ The scaling quantity f ( x ) is characterised by an infinite set of complex valued exponents, 2πn d =α + i n lna
The existence of such an infinite set is sometimes taken as a definition of an underlying fractal structure.
31 A convenient transform more adapted to discrete scaling invariance (Fourier/Laplace)
1 ∞ Mellin or ζ -transform : ζ s ≡ dxxs−1 f x f ( ) ∫ ( ) s ∈! Γ(s) 0
for the scaling relation f (a x) = b f (x)+ g(x), with fixed (a,b)
s ba ζ g (s) ζ f (s) = s provided g x is not a scaling function ba −1 ( )
The behaviour of f ( x ) is driven by the poles of ζ f ( s ), namely, lnb 2iπn bas −1 = 0 ⇔ s = − + n lna lna Inverse Mellin transform gives immediately:
lnb ⎛ ln x⎞ f (x) = x lna G⎜ ⎟ with G(u +1) = G(u) ⎝ lna⎠ 32 Part 3: Operators and fields on fractal manifolds
Operators are often expressed by local differential equations relating the space-time behaviour of a field
∂2 u Ex. Wave equation = Δu ∂t 2
Such local equations cannot be defined on a fractal
Figure: Diamond fractals, non-p.c.f., but finitely ramified
33 But operators are essential quantities for physics!
• Quantum transport in fractal structures : e.g., networks, waveguides, ...
electrons, photons dependence on temperature, on external fields (E, B) • Density of states • Scattering matrix (transmission/reflection)
34 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.
35 Recent new ideas >2000
Maths.
Michel Lapidus Bob Strichartz
Jun Kigami
36 Intermezzo : heat and waves From classical diffusion to wave propagation
There• is an importantGeneralities relation between on fractals classical diffusion and wave propagation on a manifold.
Many self-similar (fractal) structures in nature and many ways to model them: It expressesA random walkthis in profound free space or idea on a periodic that it lattice is possible etc. to measureFractals andprovide characterise a useful testing aground manifold to investigate using properties waves, of more disordered preciselyclassical with or quantum the eigenvalue systems, renormalization spectrum group of andthe phase Laplacian transitions , operator.gravitational systems and quantum field theory. Motivation: Differential operator geometry “propagating probe” curvature Spectral data volume Heat kernel dimension physically: Zeta function Laplacian
How does it look on a fractal ? Even the simplest question, e.g. dimension, are surprisingly different ! Mathematical physics
Use propagating physical waves/particles to probe geometry • spectral information: density of states, transport, heat kernel, ... • geometric information: dimension, volume, boundaries, shape, ...
1910 Lorentz: why is the Jeans radiation law only dependent on the volume ? 1911 Weyl : relation between asymptotic eigenvalues and dimension/ volume.
1966 Kac : can one hear the shape of a drum ? 39 Important examples
∂u • Heat equation = Δ u ∂t ∂2 u • Wave equation = Δu u(x,t) = dµ(y)Pt (x, y)u(y,0) ∂t 2 ∫ ∂u Schr. equation. i = Δu ∂t
t −(i)∫ x!2 dτ P x, y = xe 0 t ( ) ∫ D Brownian motion x(0)=x,x(t)=y
1 n Pt (x, y) ∼ d ∑ an (x, y)t Heat kernel expansion t 2 n
−(i)Sclassical (x,y,t ) Pt (x, y) ∼ ∑ (#)e Gutzwiller - instantons geodesics 40 Spectral functions
−Δt ∗ −λt Pt (x, y) = y e x = ∑ψ λ (y)ψ λ (x)e λ
Z(t) =Tre−Δt = ∫ dx x e−Δt x = ∑ e−λt Heat kernel λ
1 ∞ ζ s ≡ dtt s−1 Z t Z ( ) ∫ ( ) Mellin transform Γ(s) 0
1 1 s Tr ζ Z ( ) = s = ∑ s Δ λ λ
ζ s Small t behaviour of Z(t) ⇔ poles of Z ( )
Weyl expansion 41 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 thus possible to obtain the density of states.
How does it work ?
Diffusion (heat) equation in d=1
2 (x−y) − 1 4 Dt whose spectral solution is Pt (x, y) = 1 e (4π Dt) 2 Probability of diffusing from x to y in a time t.
2 (x−y) − In d space dimensions: 1 4 Dt Pt (x, y) = d e (4π Dt) 2
We can characterise the “spatial geometry” by watching how the heat flows.
The heat kernel Z d ( t ) is
d Volume access the volume Zd (t) = d xPt (x, x)= d ∫ 2 Vol. (4π Dt) of the manifold Boundary terms- Hearing the shape of a drum Mark Kac (1966)
0 L
Poisson formula
Weyl expansion (1d) Boundary terms- Hearing the shape of a drum Mark Kac (1966)
0 L
Poisson formula
Weyl expansion (2d) : Vol. L 1 1 Z (t) ∼ − + +… d=2 4πt 4 4πt 6
integral of bound. sensitive to boundary bulk curvature 1 1 s Tr ζ -functionZeta functionζ Z ( ) = s = ∑ s Δ λ λ
has a simple pole at so that, 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, More precisely, we have,
where is the total length upon iteration of the elementary step
The whole calculation !
which has poles at d 2iπn s s ⇔ n = + 2 dw lna
Infinite number of complex poles : complex fractal dimensions. They control the behaviour of the heat kernel which exhibits oscillations!
K!t"#Kleading!t" 1.0
0.8
Zdiamond (t) 0.6 1.005 Figure: Diamond fractals, non-p.c.f., but finitely ramified 1.000 0.4 0.995 0.990 0.985 0.2 0.980 t 2.10!4 10!3 2.10!3 t 0.00 0.05 0.10 0.15 0.20 0.25
A new fractal dimension : spectral dimension ds Another surprise : Notion of spectral volume From the previous expression we obtain Z (t)
ds 2iπ ds Consider for simplicity n = 1 , namely s1 = + ≡ + iδ 2 dw lna 2
so that Spectral volume
to compare with d Volume Zd (t) = d xPt (x, x)= d ∫ 2 Vol. (4π Dt) 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) Part 4: Physical application. Thermodynamics of photons on fractals
Quantisation of the electromagnetic field in a waveguide fractal structure.
How to measure the spectral volume physical application: thermodynamics on a fractal Akkermans, GD, Teplyaev, 2010
usual approach:Blackbody count radiation modes from in amomentum fractal or space thermodynamics without phase space thermal equilibrium: Equationequation of state of atstate thermodynamic equilibrium relating1 pressure, volume and PV = U dE T internal energy: d = V ⇥2 d⇥ 2 2c3 pressure volume internal energy
In an enclosure with a perfectly reflecting surface there can form standing electromagnetic waves analogouspartition to tones function of an organ pipe;(generating we shall confine function our attention) to veryln high( T,Vovertones.) Jeans asks for the energy in the frequency interval dν ... It is here that there arisesZ the mathematical problem to prove that the number of sufficiently high overtones that lies in the interval ν to ν+dν is independent1 ⇥ ofln the shape of the enclosure and⇥ ln is simply proportional to its volume. 1 P = Z U = Z H. Lorentz, = 1910 ⇥V ⇥ T Spectral volume ? usual approach: count modes in momentum space Usual approach : count modes in momentum space
Calculatethermal the partition equilibrium: (generating) equationis a dimensionless of state function function z ( T , V ) for a blackbody1 of Black-body radiationPV = in Ua large volume large volume in dimensiond is a dimensionless functiond V d (2π ) V Mode decompositionpressure of the volumeBlack-bodyfield: radiationinternal in a large energy volume
Mode decompositionMode decomposition of the of the field field: d-dimensional integer-valued vector-elementary momentum space cells partition function (generating function) ln (T,V ) d-dimensional integer-valued vector-elementary momentumZ space cells so that with L ≡ β!c 1 ⇥ ln so that ⇥ ln β 1 P = Z U = Z (photon = thermal ⇥V ⇥ 1 T β = k T wavelength) is the photon thermalis thewavelength. photon thermal wavelength.B Thermodynamics :
so that (The exact expression of Q is unimportant)
Stefan-Boltzmann is a consequence of
Adiabatic expansion 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”. But we can re-phrase the thermodynamic problem in terms of heat kernel and zeta function ! Partition function of quantum radiation at equilibrium- General formulation - General geometric shape
2 1 ⎛ ∂ 2 ⎞ ln z(T,V ) = − lnDetM ×V + c Δ 2 ⎝⎜ ∂τ 2 ⎠⎟
Looks (almost) like a bona fide wave equation but proper time.
This expression does not rely on mode decomposition, but results from the thermodynamic equilibrium (Keldysh-Schwinger).
Rescale by Lβ ≡ β!c Partition function of quantum radiation at equilibrium- General formulation - General geometric shape
2 1 ⎛ ∂ 2 ⎞ ln z(T,V ) = − lnDetM ×V +Lβ Δ 2 ⎝⎜ ∂u2 ⎠⎟
M: circle of radius L ≡ β!c β Spatial manifold (fractal) Matsubara frequencies
Thermal equilibrium of photons on a spatial manifold V at temperature T is described by the (scaled) wave equation on M ×V 2 1 ⎛ ∂ 2 ⎞ ln z(T,V ) = − lnDetM ×V +Lβ Δ 2 ⎝⎜ ∂u2 ⎠⎟ can be rewritten
∞ 1 dτ −τ L2 Δ ln z T,V = f τ Tr e β ( ) ∫ ( ) V 2 0 τ
2 Z (Lβ τ ) Heat kernel
Large volume limit (a high temperature limit)
Weyl expansion: 2 V Z Lβ τ ∼ d ( ) 2 2 (4π Lβ τ ) ∞ 1 dτ −τ L2 Δ ln z T,V = f τ Tr e β ( ) ∫ ( ) V 2 0 τ
V ln z T,V + Weyl expansion ( ) ∼ d ⇒ Lβ
Thermodynamics measures the spectral volume On a fractal… Spectral volume
V Z L2 τ ∼ s f lnτ ( β ) ds ( ) 2 2 (4π Lβ τ )
Spectral dimension
Thermodynamic equation of state for a fractal manifold
Thermodynamics measures the spectral volume and the spectral dimension. Something a bit weird… Looks like, on a fractal, coordinate and momentum spaces involve different dimensions…
• Euclidean manifold : coordinate space has dimension d
Nothing but the expression of the uncertainty principle (existence of a Fourier transform).
• Fractal manifold : coordinate space has dimension dh
momentum space has dimension ds
uncertainty principle ? Thank you for your attention.