Disordered Quantum Systems
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1 Disordered Quantum Systems Boris Altshuler Physics Department, Columbia University and NEC Laboratories America Collaboration: Igor Aleiner, Columbia University Part 1: Introduction Part 2: BCS + disorder Centre Emile Borel Gaz quantiques 23 avril - 20 juillet 2007
2 Finite size quantum physical systems Atoms Nuclei Molecules... Quantum Dots Cold gas in a trap?
3 e e e e e Quantum Dot 1. Disorder ( impurities) 2. Complex geometry 3. e-e interactions Realizations: Metallic clusters Gate determined confinement in 2D gases (e.g. GaAs/AlGaAs) Carbon nanotubes
4
5 Finite number N of electrons: Ĥ Ψ = α E α Ψ α No interactions between electrons Shrodinger eqn in d dimensions In the presence of the interactions between electrons Shrodinger equation in dn dimensions
6 e e e e e Quantum Dot 1. Disorder ( impurities) 2. Complex geometry 3. e-e interactions for a while Realizations: Metallic clusters Gate determined confinement in 2D gases (e.g. GaAs/AlGaAs) Carbon nanotubes
7 I. Without interactions Random Matrices, Anderson Localization Quantum Chaos
8 e e e e e 1. Disorder ( impurities) 2. Complex geometry How to deal with disorder? Solve the Shrodinger equation exactly Start with plane waves, introduce the mean free path, and... How to take quantum interference into account?
9 Dynamics Integrable Chaotic
10 Classical (h =0) Dynamical Systems with d degrees of freedom Integrable Systems Examples The variables can be separated and the problem reduces to d onedimensional problems 1. A ball inside rectangular billiard; d=2 d integrals of motion Vertical motion can be separated from the horizontal one Vertical and horizontal components of the momentum, are both integrals of motion 2. Circular billiard; d=2 Radial motion can be separated from the angular one Angular momentum and energy are the integrals of motion
11 Classical Dynamical Systems with d degrees of freedom Integrable The variables can be separated d one-dimensional Systems problems d integrals of motion Chaotic Systems Examples Rectangular and circular billiard, Kepler problem,..., 1d Hubbard model and other exactly solvable models,.. The variables can not be separated there is only one integral of motion - energy B Sinai billiard Stadium Kepler problem in magnetic field
12 Chaotic Systems Examples The variables can not be separated there is only one integral of motion - energy B Sinai billiard Stadium Kepler problem in magnetic field Yakov Sinai Leonid Bunimovich Johnnes Kepler
13 Integrable d-dimensional systems Chaotic d-dimensional systems d integrals of motion, d quantum numbers I k k =1, 2,...,d The only conserved quantity is the energy Each eigenstate is characterized only by the eigenvalue of the Hamiltonian Connection with the inverse problem: Q: A: Why original conditions can not be used as the integrals of motion? Not stable
14 Classical Chaos h =0 Nonlinearities Lyapunov exponents Exponential dependence on the original conditions Ergodicity Quantum description of any System with a finite number of the degrees of freedom is a linear problem Shrodinger equation Q: What does it mean Quantum Chaos?
15 N N E α RANDOM MATRICES ensemble of Hermitian matrices with random matrix element N - spectrum (set of eigenvalues) ν ε ( ) ( )... α δ ε E α - density of states - ensemble averaging Gaussian ensembles (matrix elements are normally distributed) N ν ( ε ) Wigner Semicircle ε
16 E α s N N P () s RANDOM MATRICES 1 δ1 Eα+ 1 Eα = ν... E α + 1 α E δ 1 ensemble of Hermitian matrices with random matrix element α - spectrum (set of eigenvalues) - mean level spacing - ensemble averaging Noncrossing - spacing between nearest neighbors rule Spectral statistics N - distribution function of spacings between the nearest neighbors Spectral Rigidity P ( s = 0) = 0 Level repulsion P ( ) β s << 1 s β= 1, 2, 4
17 Noncrossing rule (theorem) Suggested by Hund (Hund F Phys. v.40, p.742) Justified by von Neumann & Wigner (v. Neumann J. & Wigner E Phys. Zeit. v.30, p.467).... Usually textbooks present a simplified version of the justification due to Teller (Teller E., 1937 J. Phys. Chem ). Arnold V. I., 1972 Funct. Anal. Appl.v. 6, p.94 Mathematical Methods of Classical Mechanics (Springer-Verlag: New York), Appendix 10, 1989
18 N N RANDOM MATRICES ensemble of Hermitian matrices with random matrix element N Dyson Ensembles Matrix elements real complex 2 2 matrices Ensemble orthogonal unitary simplectic β 1 2 4
19 Hˆ P s s 0: ( ) 0 Reason for when H11 H12 = H12 H22 E E = H H + H ( ) small small small 1. The assumption is that the matrix elements are statistically independent. Therefore probability of two levels to be degenerate vanishes. 2. If H 12 is real (orthogonal ensemble), then for s to be small two statistically independent variables ((H 22 - H 11 ) and H 12 ) should be small and thus Ps () s β = 1
20 Hˆ P s s 0: ( ) 0 Reason for when H11 H12 = H12 H22 E E = H H + H ( ) small small small 1. The assumption is that the matrix elements are statistically independent. Therefore probability of two levels to be degenerate vanishes. 2. If H 12 is real (orthogonal ensemble), then for s to be small two statistically independent variables ((H 22 - H 11 ) and H 12 ) should be small and thus Ps () s β = 1 3. Complex H 12 (unitary ensemble) both Re(H 12 ) and Im(H 12 ) are statistically independent three independent 2 random variables should be small Ps () s β = 2
21 Wigner-Dyson; GOE Poisson Gaussian Orthogonal Ensemble Unitary β=2 Orthogonal β=1 Simplectic β=4 Poisson completely uncorrelated levels
22 RANDOM MATRICES N N ensemble of Hermitian matrices with random matrix element N No conservation laws no quantum numbers except the energy
23 N N matrices with random matrix elements. N Spectral Rigidity Level repulsion ( ) P s<< 1 s β β = 1,2,4 Dyson Ensembles Realizations Matrix elements Ensemble β real complex orthogonal unitary 1 2 T-inv potential broken T-invariance (e.g., by magnetic field) 2 2 matrices simplectic 4 T-inv, but with spinorbital coupling
24 ATOMS NUCLEI Main goal is to classify the eigenstates in terms of the quantum numbers For the nuclear excitations this program does not work N. Bohr, Nature 137 (1936) 344.
25 ATOMS Main goal is to classify the eigenstates in terms of the quantum numbers NUCLEI E.P. Wigner (Ann.Math, v.62, 1955) For the nuclear excitations this program does not work Study spectral statistics of a particular quantum system a given nucleus
26 P(s) Particular nucleus 166 Er N. Bohr, Nature 137 (1936) 344. P(s) s Spectra of several nuclei combined (after spacing) rescaling by the mean level
27 ATOMS NUCLEI Main goal is to classify the eigenstates in terms of the quantum numbers For the nuclear excitations this program does not work E.P. Wigner (Ann.Math, v.62, 1955) Random Matrices Ensemble Ensemble averaging Study spectral statistics of a particular quantum system a given nucleus Atomic Nuclei Particular quantum system Spectral averaging (over α) Nevertheless Statistics of the nuclear spectra are almost exactly the same as the Random Matrix Statistics
28 Q: Why the random matrix theory (RMT) works so well for nuclear spectra
29 Q: Original answer: Why the random matrix theory (RMT) works so well for nuclear spectra These are systems with a large number of degrees of freedom, and therefore the complexity is high Later it became clear that there exist very simple systems with as many as 2 degrees of freedom (d=2), which demonstrate RMT - like spectral statistics
30 Chaotic Systems Examples The variables can not be separated there is only one integral of motion - energy B Sinai billiard Stadium Kepler problem in magnetic field Yakov Sinai Leonid Bunimovich Johnnes Kepler
31 Integrable d-dimensional systems Chaotic d-dimensional systems d integrals of motion, d quantum numbers I k k =1, 2,...,d The only conserved quantity is the energy Each eigenstate is characterized only by the eigenvalue of the Hamiltonian Connection with the inverse problem: Q: A: Why original conditions can not be used as the integrals of motion? Not stable
32 h 0 Bohigas Giannoni Schmit conjecture
33 h 0 Bohigas Giannoni Schmit conjecture
34 h 0 Bohigas Giannoni Schmit conjecture Chaotic classical analog Wigner- Dyson spectral statistics No quantum numbers except energy
35 Q: What does it mean Quantum Chaos? Two possible definitions Chaotic classical analog Wigner - Dyson-like spectrum
36 Classical Integrable? Quantum Poisson Chaotic? Wigner- Dyson
37 Poisson to Wigner-Dyson crossover
38 Poisson to Wigner-Dyson crossover Important example: quantum particle subject to a random potential disordered conductor Scattering centers, e.g., impurities e
39 Poisson to Wigner-Dyson crossover Important example: quantum particle subject to a random potential disordered conductor Scattering centers, e.g., impurities e As well as in the case of Random Matrices (RM) there is a luxury of ensemble averaging. The problem is much richer than RM theory There is still a lot of universality. Anderson localization (1956) At strong enough disorder all eigenstates are localized in space
40 f = 3.04 GHz f = 7.33 GHz Anderson Insulator Anderson Metal
41 Poisson to Wigner-Dyson crossover Important example: quantum particle subject to a random potential disordered conductor Scattering centers, e.g., impurities e Models of disorder: Randomly located impurities Ur ( r ) = ur r r i i ( )
42 Poisson to Wigner-Dyson crossover Important example: quantum particle subject to a random potential disordered conductor Scattering centers, e.g., impurities Models of disorder: Randomly located impurities i r White noise potential ur ( ) λδ r ( ) λ 0 cim e Ur ( r ) = ur r r i ( ) Anderson model tight-binding model with onsite disorder Lifshits model tight... tight-binding model with offdiagonal disorder
43 Anderson Model Lattice - tight binding model Onsite energies ε i - random j i Hopping matrix elements I ij I ij I = I i and j are nearest -W < ε <W neighbors i ij uniformly distributed Anderson Transition 0 otherwise I < I c Insulator All eigenstates are localized Localization length ξ I > I c Metal There appear states extended all over the whole system
44 Localization of single-electron wave-functions: extended localized
45 Localization of single-electron wave-functions: extended d=1; All states are localized d=2; All states are localized d>2; Anderson transition localized
46 Anderson Transition I < I c Insulator All eigenstates are localized Localization length ξ I > I c Metal There appear states extended all over the whole system The eigenstates, which are localized at different places will not repel each other Any two extended eigenstates repel each other Poisson spectral statistics Wigner Dyson spectral statistics
47 Zharekeschev & Kramer. Exact diagonalization of the Anderson model 2 2m + WU r ε α ψ α r ( r ) ( r ) = 0 Disorder W
48 Q: What does it mean Quantum Chaos? Two possible definitions Chaotic classical analog? Wigner - Dyson-like spectrum Are the two definitions equivalent? Maybe not because of the localization!
49 Quantum particle in a random potential Quantum particle in a random potential (Thouless, 1972) Energy scales 1. Mean level spacing L energy δ 1 = 1/ν L d 2. Thouless energy E = hd/l 2 T D is the diffusion const δ 1 L is the system size; d is the number of dimensions E T has a meaning of the inverse diffusion time of the traveling through the system or the escape rate. (for open systems) g = E T / δ 1 dimensionless Thouless conductance g = Gh/e 2
50 0 Thouless Conductance and One-particle Spectral Statistics 1 g Localized states Insulator Poisson spectral statistics Extended states Metal Wigner-Dyson spectral statistics Ν Ν Random Matrices Ν The same statistics of the random spectra and oneparticle wave functions (eigenvectors) Quantum Dots with Thouless conductance g g
51 Scaling theory of Localization (Abrahams, Anderson, Licciardello and Ramakrishnan 1979) g = E T / δ 1 Dimensionless Thouless conductance L = 2L = 4L = 8L... without quantum corrections E T L -2 δ 1 L -d g = Gh/e 2 E T E T E T E T δ 1 δ 1 δ 1 δ 1 g g g g d( log g) d( log L) =β() g
52 β(g) d log g β - function = β ( g) d log L 1 unstable fixed point g c 1 3D 2D g -1 1D Metal insulator transition in 3D All states are localized for d=1,2
53 Conductance g
54 Anderson transition in terms of pure level statistics P(s)
55 Square billiard Integrable All chaotic systems resemble each other. Sinai billiard Chaotic Disordered localized All integrable systems are integrable in their own way Disordered extended
56 Disordered Systems: E > δ T 1 ; g > 1 E < δ T 1 ; g < 1 Anderson metal; Wigner-Dyson spectral statistics Anderson insulator; Poisson spectral statistics Q: Is it a generic scenario for the? Wigner-Dyson to Poisson crossover Speculations Consider an integrable system. Each state is characterized by a set of quantum numbers. It can be viewed as a point in the space of quantum numbers. The whole set of the states forms a lattice in this space. A perturbation that violates the integrability provides matrix elements of the hopping between different sites (Anderson model!?)
57 Does Anderson localization provide a generic scenario for the Wigner- Dyson to Poisson crossover Q:? Consider an integrable system. Each state is characterized by a set of quantum numbers. It can be viewed as a point in the space of quantum numbers. The whole set of the states forms a lattice in this space. A perturbation that violates the integrability provides matrix elements of the hopping between different sites (Anderson model!?) Weak enough hopping - Localization - Poisson Strong hopping - transition to Wigner-Dyson
58 The very definition of the localization is not invariant one should specify in which space the eigenstates are localized. Level statistics is invariant: Poissonian statistics eigenfunctions basis where the are localized Wigner -Dyson statistics basis the eigenfunctions are extended
59 Example 1 Low concentration of donors Higher donor concentration Doped semiconductor Electrons are localized on donors Poisson Electronic states are extended Wigner-Dyson e L x Example 2 Rectangular billiard Lattice in the momentum space p y Two integrals of motion p x πn = ; L x p y πm = L x L y Line (surface) of constant energy Ideal billiard localization in the momentum space Poisson p x Deformation or smooth random potential delocalization in the momentum space Wigner-Dyson
60 Localization and diffusion in the angular momentum space a ε ε > 0 R Chaotic stadium ε 0 Integrable circular billiard Angular momentum is the integral of motion h = 0 ; ε << 1 Diffusion in the angular momentum space 5 2 D ε Poisson ε=0.01 g=0.012 Wigner-Dyson ε=0.1 g=4
61 D.Poilblanc, T.Ziman, J.Bellisard, F.Mila & G.Montambaux Europhysics Letters, v.22, p.537, 1993 H V V = t = 0 0 i, σ 1D Hubbard Model on a periodic chain ( + + c ) i, σ ci+ 1, σ + ci+ 1, σ ci, σ + U ni, σ ni, σ + V Hubbard model extended Hubbard model integrable nonintegrable i, σ Onsite interaction i, σ, σ n i, σ n i+ 1, σ n. neighbors interaction 12 sites 3 particles Zero total spin Total momentum π/6 U=4 V=0 U=4 V=4
62 Finite number N of electrons: Ĥ Ψ = α E No interactions between electrons Shrodinger eqn in d dimensions Integrable system each energy is conserved Poissonian many-body spectrum In the presence of the interactions between electrons Shrodinger eqn in dn dimensions α Ψ α
63 Finite number N of electrons: Ĥ Ψ = α No interactions between electrons Shrodinger eqn in d dimensions Integrable system each energy is conserved Poissonian many-body spectrum In the presence of the interactions between electrons Shrodinger eqn in dn dimensions Q: Can interaction between the particles drive this system into chaos and make it ergodic? E α Ψ Random Matrics statistics of nuclear spectra α
64 II. With interactions Fermi Liquid and Disorder Zero Dimensional Fermi Liquid
65 Fermi Liquid Fermi statistics Low temperatures Not too strong interactions Fermi Liquid Translation invariance What does it mean?
66 Fermi statistics Low temperatures Not too strong interactions Translation invariance It means that 1. Excitations are similar to the excitations in a Fermi-gas: a) the same quantum numbers momentum, spin ½, charge e b) decay rate is small as compared with the excitation energy 2. Substantial renormalizations. For example, in a Fermi gas n μ, γ = c T, are all equal to the one-particle density of states. These quantities are different in a Fermi liquid χ g μ B Fermi Liquid
67 Signatures of the Fermi - Liquid state?! 1. Resistivity is proportional to T 2 : L.D. Landau & I.Ya. Pomeranchuk To the properties of metals at very low temperatures ; Zh.Exp.Teor.Fiz., 1936, v.10, p.649 The increase of the resistance caused by the interaction between the electrons is proportional to T 2 and at low temperatures exceeds the usual resistance, which is proportional to T 5. the sum of the moments of the interaction electrons can change by an integer number of the periods of the reciprocal lattice. Therefore the momentum increase caused by the electric field can be destroyed by the interaction between the electrons, not only by the thermal oscillations of the lattice.
68 Signatures of the Fermi - Liquid state?! 1. Resistivity is proportional to T 2 : L.D. Landau & I.Ya. Pomeranchuk To the properties of metals at very low temperatures ; Zh.Exp.Teor.Fiz., 1936, v.10, p.649 Umklapp electron electron scattering dominates the charge transport (?!) 2. Jump in the momentum distribution function at T=0. 2a. Pole in the one-particle Green function G r ( ε, p) = iε n Z ξ r ( p) n( p r ) Fermi liquid = 0<Z<1 (?!) p F p
69 Momentum Landau Fermi - Liquid theory r p Momentum distribution Total energy Quasiparticle energy n E ξ r ( p) { n( p) } r r r ( p) δe δn( p) Landau f-function f r r ( p, p ) δξ ( p) δn( p ) r r Q:? Can Fermi liquid survive without the momenta Does it make sense to speak about the Fermi liquid state in the presence of a quenched disorder
70 Q: Does it make sense to speak about the Fermi? liquid state in the presence of a quenched disorder 1. Momentum is not a good quantum number the momentum uncertainty is inverse proportional to the elastic mean free path, l. The step in the momentum distribution function is broadened by this uncertainty
71 Q: Does it make sense to speak about the Fermi? liquid state in the presence of a quenched disorder 1. Momentum is not a good quantum number the momentum uncertainty is inverse proportional to the elastic mean free path, l. The step in the momentum distribution function is broadened by this uncertainty n( p r ) 2. Neither resistivity nor its temperature dependence is determined by the umklapp processes and thus does not behave as T 2 3. Sometimes (e.g., for random quenched magnetic field) the disorder averaged oneparticle Green function even without interactions does not have a pole as a function of the energy, ε. The residue, Z, makes no sense. Nevertheless even in the presence of the disorder I. Excitations are similar to the excitations in a disordered Fermi-gas. II. Small decay rate III. Substantial renormalizations ~ p F h l p
72 e e e e e Quantum Dot 1. Disorder ( impurities) } 2. Complex geometry 3. e-e interactions chaotic one-particle motion Realizations: Metallic clusters Gate determined confinement in 2D gases (e.g. GaAs/AlGaAs) Carbon nanotubes
73 Finite Thouless System energy E T ε << E T def 0D At the same time, we want the typical energies, ε, to exceed the mean level spacing, δ 1 : δ << ε << g >> 1 1 E T E T δ 1
74 Two-Body Interactions α,σ> Set of one particle states. σ and α label correspondingly spin and orbit. Hˆ 0 = α ε α a + α, σ a α, σ Hˆ int = α, β, γ, δ σ, σ M αβγδ a + α, σ a + β, σ a γ, σ a δ, σ ε α -one-particle orbital energies M αβγδ -interaction matrix elements Nuclear Physics ε α M αβγδ are taken from the shell model are assumed to be random Quantum Dots ε α M αβγδ RANDOM; Wigner-Dyson statistics????????
75 0 Thouless Conductance and One-particle Quantum Mechanics 1 g Localized states Insulator Poisson spectral statistics Extended states Metal Wigner-Dyson spectral statistics Ν Ν Random Matrices Ν The same statistics of the random spectra and oneparticle wave functions (eigenvectors) Quantum Dots with dimensionless conductance g g
76 Hˆ int = α, β, γ, δ σ, σ M αβγδ a Matrix Elements + α, σ a + β, σ a γ, σ a δ, σ Matrix Elements Mαβγδ Diagonal - α,β,γ,δ are equal pairwise α=γ and β=δ or α=δ and β=γ or α=β and γ=δ Offdiagonal - otherwise It turns out that in the limit g Diagonal matrix elements are much bigger than the offdiagonal ones M >> diagonal M offdiagonal Diagonal matrix elements in a particular sample do not fluctuate - selfaveraging
77 Toy model: U M r ( r ) ( r ) αβγδ Short range e-e interactions λ r = λ is dimensionless coupling constant δ ν ν is the electron density of states λ = ν r drψ α r r ( r ) ψ ( r ) ψ ( r ) ψ ( r ) β γ r δ r ( r ψ ) α one-particle eigenfunctions ψ α electron wavelength x Ψ α (x) is a random function that rapidly oscillates ψ α (x) 2 ψ α (x) as long as T-invariance is preserved
78 In the limit M g αβαβ = ψ α λ ν ( ) r r r drψ 2 More general: Diagonal matrix elements are much bigger than the offdiagonal ones M >> diagonal M offdiagonal Diagonal matrix elements in a particular sample do not fluctuate - selfaveraging α r 2 r ( r ) ψ ( r ) 2 1 volume β M αβαβ = λδ 1 More general: finite range interaction potential U r r ( ) M αβαβ = λ ν ψ α r 1 () 2 ψ β r 2 () 2 U r ( 1 r 2 )dr 1 d r r 2 The same conclusion
79 Universal (Random Matrix) limit - Random Matrix symmetry of the correlation functions: All correlation functions are invariant under arbitrary orthogonal transformation: ~ r r r r r r r r ( r ) dro ν ( ) ψ ( ) ψ, μ 1 μ 1 ν 1 = ν r dro 1 r r r r ( r r ) O η ( r r ) = δ δ ( r r ) μη ν,, μ 1 ν 1 r r
80 There are only three operators, which are quadratic in the fermion operators a +, a, and invariant under RM transformations: ˆn= a a ασ, + ασ, ασ, total number of particles r Ŝ = a σ a ασ,, σ α, σ σ, σ α, σ total spin ˆK = a a α, α, α????
81 Charge conservation ˆ ˆ (gauge invariance) -nok K + Kˆ K ˆ or only + Invariance under rotations in spin space -no S ˆ only S ˆ 2 Therefore, in a very general case = ˆ 2 + int c BCS Hˆ evnˆ E nˆ JS λ Kˆ + K. ˆ Only three coupling constants describe all of the effects of e-e interactions
82 In a very general case only three coupling constants describe all effects of electron-electron interactions: Hˆ = ε n + Hˆ α α α int Hˆ evnˆ E nˆ JSˆ Kˆ + K. ˆ = λ int c BCS I.L. Kurland, I.L.Aleiner & B.A., 2000 See also P.W.Brouwer, Y.Oreg & B.I.Halperin, 1999 H.Baranger & L.I.Glazman, 1999 H-Y Kee, I.L.Aleiner & B.A., 1998
83 In a very general case only three coupling constants describe all effects of electron-electron interactions: Hˆ = ε n + Hˆ α α α int Hˆ evnˆ E nˆ JSˆ Kˆ + K. ˆ = λ int c BCS For a short range interaction with a coupling constant λ E c = λδ 1 2 J = 2λδ 1 λ = λδ ( 2 β) BCS 1 where δ 1 is the one-particle mean level spacing
84 H ˆ = H ˆ 0 + H ˆ int ˆ H 0 = ε α n α α = ˆ 2 + int c BCS Hˆ evnˆ E nˆ JS λ Kˆ + K. ˆ Only one-particle part of the Hamiltonian,, contains randomness ˆ H 0
85 H ˆ = H ˆ 0 + H ˆ int ˆ H 0 = ε α n α α = ˆ 2 + int c BCS Hˆ evnˆ E nˆ JS λ Kˆ + K. ˆ E c J determines the charging energy (Coulomb blockade) describes the spin exchange interaction λ BCS determines effect of superconducting-like pairing
86 H ˆ = H ˆ 0 + H ˆ int ˆ H 0 = ε α n α α = ˆ 2 + int c BCS Hˆ evnˆ E nˆ JS λ Kˆ + K. ˆ I. Excitations are similar to the excitations in a disordered Fermi-gas. II. Small decay rate III. Substantial renormalizations Isn t it a Fermi liquid? Fermi liquid behavior follows from the fact that different wave functions are almost uncorrelated
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