Introduction to Mesoscopics. Boris Altshuler Princeton University, NEC Laboratories America,

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Introduction to Mesoscopics Boris Altshuler Princeton University, NEC Laboratories America,

ORIGINS E.P. Wigner, Conference on Neutron Physics by Time of Flight, November 1956 P.W. Anderson, Absence of Diffusion in Certain Random Lattices ; Phys.Rev., 1958, v.109, p.1492 L.D. Landau, Fermi-Liquid Theory Zh. Exp. Teor. Fiz.,1956, v.30, p.1058 J. Bardeen, L.N. Cooper & J. Schriffer, Theory of Superconductivity ; Phys.Rev., 1957, v.108, p.1175.

Part 1 Without interactions Random Matrices, Anderson Localization, and Quantum Chaos

N N E δ RANDOM MATRIX THEORY ensemble of Hermitian matrices with random matrix element N - spectrum (set of eigenvalues) 1 E +1 E - mean level spacing... - ensemble averaging s P ( s) E 1 E + - spacing between nearest neighbors δ 1 Spectral Rigidity - distribution function of nearest neighbors spacing between P ( s = 0) = 0 Level repulsion P ( ) β s << 1 s β= 1, 2, 4

Dyson Ensembles and Hamiltonian systems Matrix elements real complex Ensemble orthogonal unitary β 1 2 2 2 matrices simplectic 4

Wigner-Dyson; GOE Poisson Gaussian Orthogonal Ensemble Poisson completely uncorrelated levels

Wigner-Dyson; GOE Poisson Gaussian Orthogonal Ensemble Unitary β=2 Orthogonal β=1 Simplectic β=4 Poisson completely uncorrelated levels

RANDOM MATRICES N N matrices with random matrix elements. N Dyson Ensembles Matrix elements Ensemble β realization 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

Hˆ P s s 0: ( ) 0 Reason for when H11 H12 = H12 H22 E E = H H + H ( ) 2 2 2 1 22 11 12 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

Dyson Ensembles and Hamiltonian systems Matrix elements Ensemble β realization 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

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: Random Matrices Ensemble Ensemble averaging Study spectral statistics of a particular quantum system a given nucleus Atomic Nuclei Particular quantum system Spectral averaging (over ) Nevertheless Statistics of the nuclear spectra are almost exactly the same as the Random Matrix Statistics

P(s) Particular nucleus 166 Er s P(s) Spectra of several nuclei combined (after rescaling by the mean level spacing)

Q: Why the random matrix theory (RMT) works so well for nuclear spectra Original answer: 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

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 2. Circular billiard; d=2 Radial motion can be separated from the angular one Vertical and horizontal components of the momentum, are both integrals of motion Angular momentum and energy are the integrals of motion

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

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?

h 0 Bohigas Giannoni Schmit conjecture Chaotic classical analog Wigner- Dyson spectral statistics No quantum numbers except energy

Q: What does it mean Quantum Chaos? Two possible definitions Chaotic classical analog Wigner - Dyson-like spectrum

Classical Integrable? Quantum Poisson Chaotic? Wigner- Dyson

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 (1958) At strong enough disorder all eigenstates are localized in space

f = 3.04 GHz f = 7.33 GHz Anderson Insulator Anderson Metal

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 White noise potential Lattice models Anderson model Lifshits model

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

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

Zharekeschev & Kramer. Exact diagonalization of the Anderson model Disorder W

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

One-particle problem (Thouless,( 1972) 1. Mean level spacing L energy δ 1 = 1/ν L d Energy scales 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) dimensionless Thouless conductance g = E T / δ 1 g = Gh/e 2

Thouless Conductance and One-particle Spectral Statistics 0 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

Scaling theory of Localization (Abrahams, Anderson, Licciardello and Ramakrishnan 1979) g = E / δ Dimensionless Thouless g = Gh/e T 1 conductance 2 L = 2L = 4L = 8L... without quantum corrections E T L -2 δ 1 L -d 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

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

Conductance g 100 100 100 Anderson model cube

Anderson transition in terms of pure level statistics P(s)

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

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!?)

Weak enough hopping - Localization - Poisson Strong hopping - transition to Wigner-Dyson 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!?)

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

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 Line (surface) of constant energy Ideal billiard localization in the momentum space Poisson x L y p x Deformation or smooth random potential delocalization in the momentum space Wigner-Dyson

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

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

D.Poilblanc, T.Ziman, J.Bellisard, F.Mila & G.Montambaux Europhysics Letters, v.22, p.537, 1993 J exchange 1D t-j model on a periodic chain 1d t-j model t forbidden hopping J=t J=2t J=5t N=16; one hole

Part 2 Disorder/Chaos + Interactions Zero-Dimensional Fermi-liquid

What does it mean - non-fermi liquid? What does it mean Fermi liquid?

Fermi Liquid Fermi statistics Low temperatures Not too strong interactions Fermi Liquid Translation invariance What does it mean?

Fermi statistics Low temperatures Not too strong interactions Translation invariance It means that Fermi Liquid 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, χ gµ are all equal to the one-particle density of states. These quantities are different in a Fermi liquid B

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 momenta 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.

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

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

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 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

Zero DimensionalFermi Liquid 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

Ν Ν Random Matrices Ν The same statistics of the random spectra and oneparticle wave functions (eigenvectors) Quantum Dots with dimensionless conductance g g

Two-Body Interactions,σ> Set of one particle states. σ and label correspondingly spin and orbit. Hˆ 0 = ε a +, σ a, σ Hˆ int =, β, γ, δ σ, σ M βγδ a +, σ a + β, σ a γ, σ a δ, σ ε -one-particle orbital energies M βγδ -interaction matrix elements Nuclear Physics ε M βγδ are taken from the shell model are assumed to be random Quantum Dots ε M βγδ RANDOM; Wigner-Dyson statistics????????

Hˆ int =, β, γ, δ σ, σ M βγδ a Matrix Elements +, σ a + β, σ a γ, σ a δ, σ Matrix Elements Mβγδ Diagonal -,β,γ,δ are equal pairwise =γ and β=δ or =δ and β=γ or =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

Toy model: U M r ( r ) ( r ) βγδ 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 r r ( r ) ψ ( r ) ψ ( r ) ψ ( r ) β γ r δ r ( r ψ ) one-particle eigenfunctions ψ electron wavelength x Ψ (x) is a random function that rapidly oscillates ψ (x) 2 ψ (x) 2 0 0 as long as T-invariance is preserved

In the limit M g ββ = ψ λ ν ( ) r r r drψ 2 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 r 2 r ( r ) ψ ( r ) 2 1 volume β M ββ = λδ 1 More general: finite range interaction potential U( r r ) M ββ = λ ν ψ r 1 () 2 ψ β r 2 () 2 U r ( 1 r 2 )dr 1 d r r 2 The same conclusion

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

There are only three operators, which are quadratic in the fermion operators,, and invariant under RM transformations: a + a + + + + + = = =,,,,,,,,,, ˆ ˆ ˆ σ σ σ σ σ σ σ σ σ σ a a T a a S a a n 2 2 1 2 1 1 r total number of particles total spin????

Charge conservation (gauge invariance) -no ˆ T or T ˆ + only T ˆ T ˆ + Invariance under rotations in spin space -no S ˆ only S ˆ 2 Therefore, in a very general case H ˆ = evˆ n + E n ˆ 2 + JS ˆ 2 + λ T ˆ + T ˆ. int c BCS Only three coupling constants describe all of the effects of e-e interactions

In a very general case only three coupling constants describe all effects of electron-electron interactions: Hˆ Hˆ = int ε n = evnˆ + + Hˆ E c nˆ int 2 + JSˆ 2 + λ BCS Tˆ + Tˆ. 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

In a very general case only three coupling constants describe all effects of electron-electron interactions: Hˆ Hˆ = int ε n = evnˆ + + Hˆ E c nˆ int 2 + JSˆ 2 + λ BCS Tˆ + Tˆ. 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

H ˆ = H ˆ 0 + H ˆ int ˆ H 0 = ε n H ˆ = evˆ n + E n ˆ 2 + JS ˆ 2 + λ T ˆ + T ˆ. int c BCS Only one-particle part of the Hamiltonian,, contains randomness ˆ H 0

H ˆ = H ˆ 0 + H ˆ int ˆ H 0 = ε n H ˆ = evˆ n + E n ˆ 2 + JS ˆ 2 + λ T ˆ + T ˆ. int c BCS E c J determines the charging energy (Coulomb blockade) describes the spin exchange interaction λ BCS determines effect of superconducting-like pairing

Example 1: Coulomb Blockade V g gate voltage gate source drain QD current valley peak

0.08 0.06 2 0.04 /h) 6.2 6.0 5.8 5.6 5.4 5.2 B = 30 mt B = -30 mt -400-350 -300-250 V g (mv) 1 µm 0.02 g (e 0-300 -280-260 -240-220 V g (mv) Coulomb Blockade Peak Spacing Patel, et al. PRL 80 4522 (1998) (Marcus Lab)

Example 2: Spontaneous Magnetization 1. Disorder ( impurities) 2. Complex geometry 3. e-e interactions chaotic one-particle motion e e e e e What is the spin of the Quantum Dot in the ground state

How to measure the Magnetization motion of the Coulomb blockade peaks in the parallel magnetic field

In the presence of magnetic field ˆ int H = ε n + r JSˆ 2 r + B r Sˆ Scaling: the probability to find a ground state at a given magnetic field, B, with a given spin, S, depends on the combination rather than on B and J separately B X = J + gµ B 2S

Probability to observe a triplet state as a function of the parameter X - results of the calculation based on the universal Hamiltonian with the RM oneparticle states The rest exact diagonalization for Hubbard clusters with disorder. No adjustable parameters

H ˆ = H ˆ 0 + H ˆ int ˆ H 0 = ε n H ˆ = evˆ n + E n ˆ 2 + JS ˆ 2 + λ T ˆ + T ˆ. int c BCS 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

CONCLUSIONS Anderson localization provides a generic scenario for the transition between chaotic and integrable behavior. One-particle chaos + moderate interaction of the electrons a to a rather simple Hamiltonian of the system, which can be called Zero-dimensional Fermi liquid. The main parameter that justifies this description is the Thouless conductance, which is supposed to be large Excitations are characterized by their one-particle energy, charge and spin, but not by their momentum. These excitations have the lifetime, which is proportional to the Thouless conductance, i.e., is long. This approach allows to describe Coulomb blockade (renormalization of the compressibility), as well as the substantial renormalization of the magnetic susceptibility and effects of superconducting pairing

BCS Hamiltonian Finite systems Hˆ = ε n E ˆ ˆ ˆ + cn + JS + T T 2 2 + ˆ λbcs. ˆT = a a + + +,, commute with each other does not commute with K.E. ˆ ˆ+ H = εn + λbcst Tˆ.

ˆ ˆ+ H = εn + λbcst Tˆ. () (β) double occupied empty (γ) (δ) single occupied

ˆ ˆ+ H = εn + λbcst Tˆ. () (β) (γ) (δ) double occupied empty single occupied + + a a a a β β Ĥ int () mixes and + + a a a a at the same time = 0 β β (β) (γ,δ) This single-occupied states are not effected by the interaction. They are blocked The Hilbert space is separated into two independent Hilbert subspaces Blocking effect. (V.G.Soloviev, 1961)

One-particle energy BCS interaction The energy is measured relative to the Fermi level

BCS Hamiltonian; no single-occupied states n one-particle levels; one-particle energies ε ˆ BCS H = ε a a λ a a a a + + +, σ, σ BCS,, β, β, 0 n 1 0, β n 1 σ =, β Anderson spin chain ˆ 1 T = 1+ a a Tˆ = a a T ˆ = a z + + + +, σ, σ,,,, 2 σ =, a SU 2 algebra Hˆ = ε Tˆ λ Tˆ Tˆ = ε Tˆ λ Lˆ Lˆ z + z BCS BCS p p BCS 0 n 1 0 β n 1 0 n 1 Lˆ ˆ ± ± T +

Anderson spin chain ˆ 1 T = 1+ a a Tˆ = a a Tˆ = a a z + + + +, σ, σ,,,, 2 σ =, r Lˆ r ˆ T Hˆ = ε Tˆ λ Tˆ Tˆ = ε Tˆ λ Lˆ Lˆ z + z BCS BCS p p BCS 0 n 1 0 β n 1 0 n 1 + ˆ T + ˆz T 1 ˆ λ BCS Nˆ 0 ε Superconducting order parameter Total number of the particles N ˆ, ˆ 0

Anderson spin chain ˆ 1 T = 1+ a a Tˆ = a a Tˆ = a a z + + + + β, σ β, σ,,,, 2 σ =, Hˆ = ε Tˆ λ Tˆ Tˆ = ε Tˆ λ Lˆ Lˆ z + z BCS BCS p p BCS 0 n 1 0 β n 1 0 n 1 + Tˆ z Nˆ Total number of the particles For a fixed number of the particles (closed system we can add the term gn 2 =const to the hamiltonian: 0 n 1 ( ˆ) 2 Hˆ ε Tˆz λ L BCS BCS r rˆ rˆ L T

0 n 1 ( ˆ) 2 Hˆ ε Tˆz λ L BCS BCS r rˆ rˆ L T Integrable model Richardson solution Bethe Ansatz 1 2 1 + = λ BCS 0 β n 1E Eβ 0 β n 1E 2ε β E = 0 n 1 E n equations for the parameters E p How to describe dynamics in the time domain?

0 β n 1 β β Gaudin Magnets rˆ rˆ ˆ TT β 2 ˆz H = AT = 0,1,2,..., n 1 ε ε Hˆ, Hˆ = β 0 Hˆ 0 Hamiltonian ˆ H β > 0 intgrals of motion BCS Hamiltonian 0 n 1 ε ˆ ˆ H = H BCS + const λ BCS = 2 A

Overhauser interaction of electronic spins in quantum dots e

Overhauser interaction of electronic spins in quantum dots B e

Overhauser interaction of electronic spins in quantum dots N N N N N N N N N N e N N More than 10 6 of nuclear spins per electron Excange interaction of the electronic spin with the nuclear ones. Collective effect of the nuclei on the electron is pretty strong. No interaction between the nuclear spins rˆ S0 spin of the electron ˆ ˆ ˆ ˆ z HeN = γ SS r r rˆ 0 BS r 0 γ ( ) 2 S > 0 nuclear spins ψ r the electron w. f. 1 n 1 ψ B ( ) magnetic field z

Overhauser interaction of electronic spins in quantum dots N N N N N N N N N N e N N ˆ ˆ Hˆ = SS BSˆ en rˆ rˆ 2TT = z γ r r 0 0 Hˆ β 1 n 1 0 β n 1, β ε εβ Central spin problem Hˆ en ˆ γ rˆ S rˆ T = H0 ε0 Gaudin problem 2 B A ε AT ˆz

ˆ ˆ Hˆ = SS BSˆ en rˆ rˆ 2TT = z γ r r 0 0 Hˆ β 1 n 1 0 β n 1, β ε εβ Central spin problem Hˆ en ˆ γ rˆ S = H0 ε0 B Gaudin problem rˆ T Idea: Can we consider the classical dynamics of these Hamiltonians? Can it be described explicitly? What are connections between the classical and quantum dynamics? Substitute quantum spin operators by classical vectors 2 A ε rˆ T rˆ L r s r J AT ˆz

Barankov, Levitov & Spivak cond-mat/0312053 Yuzbashyan, BA,Kuznetsov & Enolskii cond-mat/0407501