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DYNAMICAL SYMMETRIES IN NANOPHYSICS

DYNAMICAL SYMMETRIES IN NANOPHYSICS. K. Kikoin (Tel-Aviv University). PLAN I. Group theory in quantum mechanics. Basic definitions . Symmetry of Hamiltonian. Hidden symmetry of Hamiltonian Dynamical symmetries of multiplet II .

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DYNAMICAL SYMMETRIES IN NANOPHYSICS

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  1. Galitsky-2010 DYNAMICAL SYMMETRIES IN NANOPHYSICS K. Kikoin (Tel-Aviv University) • PLAN • I. • Group theory in quantum mechanics. Basic definitions . • Symmetry of Hamiltonian. • Hidden symmetry of Hamiltonian • Dynamical symmetries of multiplet II. • Nanoobjects: single-electron tunneling through quantum dots and molecules • Dynamical symmetries of few-electron systems • Kondo effect and Kondo-regime in single electron tunneling • Kondo exotics

  2. Galitsky-2010 OMNIPRESENT GROUP THEORY Set of elementsE,A,B,C,… N plus multiplication rule Closeness: Product of two elements AB=C in the set belongs to the set Associativity: (AB)C = A(BC) Existence of a unite element: EA = AE = A Existence of inverse elements: AA-1 = A-1A = E for any A The set without (4) formssemigroup E J K L M N Tutorial example: Multiplication table: Isomorphism: = Group D3 of discrete rotations = Permutation group P123 E J K L M N 1 2 3

  3. Galitsky-2010 GROUP THEORY IN QUANTUM MECHANICS Ingredients: Noether theorem: Any differentiable symmetry of the action has a corresponding conservation law. Homogeneity in time – energy conservation Homogeneity in space: –momentum conservation Isotropy in space: – angular momentum conservation The wave functions belonging to a given eigen energy E transform along a representation of the group G of the Schroedinger equation Wigner theorem: Hψ({x}) = Eψ({x}) HRψ({x}) = ERψ({x}). δr Infinitesimal operators as generators of Lie algebras These operators perform infinitesimal translations and rotations in space time and form the basis if irreducible representations of continuous Lie groups. Example: rotations on a 3D sphere Laplacian Δis invariant undertransformations in Euclidean space. In case of rotation group SO(3) we deal with invariant under rotations on the sphere.

  4. Galitsky-2010 These operator form closed Lie algebra o(3) J2is the Casimir operator, which commutes with unit operator. In general case several Casimir operators may be constructed from group generators. Usually Casimir operators explicitly enter the Hamiltonian. In particular, one may use J2instead of Δ . In some special cases Casimir operators do not enter the Hamiltonian. This is the sign of hidden symmetry Example: electron in a Coulomb potential ~ 1/r . It was noticed (W. Pauli, 1927, V.A. Fock, 1935) that the Schroedinger equation with this potential is in fact 4-dimensional (in {p,ε}-space) . As a result three more generators may be introduced which form a Runge-Lenz vector (L=iħI) Two last equation define two Casimir operators for a group of 4D rotations SO(4) in this specific potential. But the operator N does not enter the Hamiltonian!

  5. Galitsky-2010 SO(4) group of rotations on 4D sphere – example of semisimple group. Let us add one more dimension to our Euclidean 3D world. Time t may play part of his additional coordinate. To describe rotations on 4D sphere one needs three more infinitesimal operators (*) These 6 operators form closed so(4) algebra Two Casimir operators are A linear transformation converts (*) into another basis with commutation relations Triads with commutation relations of this type form ideals of the o(4) algebra. Algebra possessing non-Abelian ideal is qualified as a semisimple algebra. Two Casimir operators in these terms are

  6. Galitsky-2010 Now we are prepared to discussion of dynamical symmetries Representation theory in quantum mechanics implies symmetry operations on matrices. Each symmetry group is realized in some basis forming the irreducible representations. In case of rotation group SO(3) these are spherical functions Ylm( φ,θ). Wigner theorem in these terms means that the matrix representation of Schroedinger equation has a block-diagonal form Δl=1 Δl=2 … … l=0 l=1 l=2 … s s Question: what about off-diagonal blocks ? p p d d … … To find the answer let us return to the physical realization of SO(4) symmetry, namely to rigid quantum rotator l – operators have nonzero matrix elements only within a given block. Each block corresponds to the eigen energy El

  7. Galitsky-2010 z z’ Angular coordinates contain three angles: (φ,θ,α). The latter angle describes precession of rotator axis z’ around cartesian axis z in 3D space. This angle play the same part as the time t in definition of the operator M . Thus, effective dimension of phase space for the states of rigid rotation is D=4. Basis of representations of SO(4) group in angular coordinates is given by hyperspherical functions α Ladder operators Mi act on the spherical functions with given n in the following way: (cl and al are some constants). Thus M-operators unlike L-operators raise (lower) orbital index l , and the set of operators L, M involves both diagonal blocks and off-diagonal blocks with Δl = 1 in the Hamiltonian matrix. One says that the group SO(4) realizes the dynamical symmetry of rigid rotator. Generators of dynamical symmetry group describes transitions between the levels of the supermultiplet, i.e., the levels belonging to different irreducible representations and different eigenstates of the Hamiltonian.

  8. Galitsky-2010 SO(4) symmetry in spin systems: singlet-triplet multiplet formed by two electrons 4 states form spinor of 4th rank 1 s 0 -1 R S=0 S=1 These operators were invented by J. Hubbard s T … Spin 1 operators S generate o(3) algebra and generators Sz, S+, S- describe rotations on a 3D sphere in spin space. Adding vector R (allowing singlet-triplet transitions) we extend the rotation group from SO(3) to SO(4).

  9. Galitsky-2010 Commutation relations are the same as for operators L and M in quantum rotator problem. Casimir operators: Electron spins are not independent in SO(4) group: they are constrained by the second Casimir operator. When the dynamical symmetry becomes actual? In stationary state spin is conserved as well as angular momentum, and we are satisfied with conventional (Wigner’s) symmetry of the Hamiltonian. Dynamical symmetry is activated in interaction with external fields (electromagnetic field etc) and in interaction with environment. The latter possibility is realized in nanophysics.

  10. One more physical example: SO(5) algebra for Wannier excitons Galitsky-2010 for three vectors S,R,Mand scalar A T-exciton S-Exciton Singlet GS A M Commutation relations [Si,Sk] = ie ijkSk, [Si,Rk] = ie ijkSk, [Ri,Rk] = ie ijkSk, [Si,Mk] = ie ijkMk, [Mi,Mk] = ie ijkSk, [Ri,Mk] = iAΔ ik 1 s 0 -1 R Kinematics: three Casimir operators S2 + R2 + M2 + A2 = 4

  11. Galitsky-2010 This construction may be generalized for higher groups SO(n) Group generators are combined in vectors (V) and scalars (A) Multiplet consists of singlets (S) and triplets (T) In some physical problems group generators form tensors (e.g. spin systems with uniaxial magnetic anistropy). In the latter case the dynamical symmetry SU(3) is realized by one vector and one tensor.

  12. Galitsky-2010 A little more aboutSU(n)groups in a context of dynamical symmetries Mathematically SU(n) is a group of unitary matrices of n-th rank. In nanophysics one frequently deals with the groups SU(3) and SU(4). SU(3) group describes all interlevel transitions in a three-level system. Its generators are so called Gell-Mann matrices which are the generalization of su(2) Pauli matrices σ. The GM matrices may be combined in three triads: 3x3 Only eight of these matrices may be used as linearly independent generators of SU(3) group. σ+ σ- σ3

  13. Galitsky-2010 The simplest physical realization of SU(4) symmetry group is a two-well potential with one electron inside. Electron is characterized by spin with two projections (up and down) and its position in the wells (left and right). Both quantum numbers are realized by means of Pauli matrices σ and τ (spin and pseudospin, resp.). 15 generators of SU(4) group are constructed as (*) These generators describe all positions of electron in the potential relief and all possible transitions between the quantum states. If one is interested only in spin conserving transitions between two wells, but the spin flips in a given well are allowed, the matrix (*) reduces to In this case the dynamical symmetry is reduced: Now we are well equipped for investigation of dynamical symmetries in quantum dots and related nano-objects.

  14. Galitsky-2010 Planar quantum dots

  15. Galitsky-2010 Coulomb blockade due to noticeable capacitive energy of QD.

  16. Galitsky-2010 Coulomb windows diagram: tunnel conductance G as a function of eV, Vg G = dI/dV odd even odd even Single electron tunneling

  17. Galitsky-2010 In a Coulomb window direct tunneling is suppressed by Coulomb blockade and only cotunneling processes are possible. These processes are accompanied by creation of electron- hole pairs either in the same lead or in different leads. The latter process is responsible for charge transport. The electron spin is not conserved in cotunneling processes. σ’ σ (a) (b) σ σ’ (c) (d)

  18. CONVENTIONAL KONDO COTUNNELING:theory Zero energy spin reversal peak ~δ(0) develops into resonance due to dynamical screening by electron-hole pairs from continuum (Kondo cloud) • N=1 E+U TK ~ D0 exp(-1/2νJ(0)) E Kondo Hamiltonian H = J (S s) S=1/2 Galitsky-2010

  19. Galitsky-2010 How does Kondo effect in quantum dots looks experimentally: G = dI/dV – tunneling conductance a b

  20. Galitsky-2010 KONDO SCREENING Undersreened screened overscreened (two-channel) N N N N N-1 N-1 N-1 N-1 reduced spin+ free electrons Kondo singlet + free electrons (FL) . . . non-Fermi liquid (NFL) Magnetic response: const - (T/TK)2 1/T + const T a

  21. Kondo cotunnelling through DQD: N=2 Two electrons always form spinSinglet and spin Triplet 1 0 T -1 S R Usually ground state is S, but the S/T level crossing is possible under certain conditions. Unconventional features of Kondo tunneling may be observed when T S Generalized Kondo Hamiltonian H= J1(S s) + J2(R s) S=1 plus singlet Galitsky-2010

  22. Galitsky-2010 General view on emergence of dynamical symmetries C O N T I N U U M δE Flow RG?

  23. Galitsky-2010 Dynamical symmetry alters (or arises) in the course of contraction of the energy scale δE. Let us look how does this mechanism work in case of Kondo effect. Even conventional Kondo effect (N=1) may be described in terms of dynamical symmetries. Basic Hamiltonian is that of the Anderson model describing cotunneling through a quantum dot. E εd +U 0 2D0 εd The Renormalization Group procedure is applied to this Hamiltonian. In the course of energy scale reduction the dot level εd shifts ~ ln (D0 -D). When the renormalized level crosses the boundary D the first stage of RG procedure terminates. We are now in a regime of localized spin described by the Kondo Hamiltonian H = J (S s) with J ~ W2/ εd , and in the second stage of scaling procedure the coupling constant J is renormalized. The scaling equation reads D0 0 D (a) (b) with Its solution is J(T) turns to infinity (reaches the stable fixed point) at T=TK

  24. Galitsky-2010 Symmetric Hubbard parabola E The same procedure in terms of dynamical symmetries: Kinematic scheme of possible spin and charge transitions N= 0 1 2 0, 2 1 0 0 Spin Charge All these transitions in 4-level system are described by generators of SU(4) group. This is the basic symmetry of Hubbard atom with varying N =0,1,2

  25. Galitsky-2010 Asymmetric Hubbard parabola E If E(2) – E(1) >> E(0) – E(1), the doubly occupied state is first to be integrated out in the process if RG procedure and we remain with SU(3) symmetry 0 N= 0 1 2 1 0 Spin Charge Transition from Anderson model to exchange model means elimination of empty state and eventually we come to the scheme containing only spin states and describing SU(2) Kondo effect.

  26. Galitsky-2010 This construction is only rephrasing of the well known procedure which means not too much from practical point of view, but dynamical symmetries are deeply involved in Kondo effect at even occupation . Within the energy interval δE ~ the exchange (cotunneling) Hamiltonian is H= J1(S s) + J2(R s) and both parameters J1 and J2are subject to RG renormalization. The system of scaling flow equations Solution of this system gives for TK as a function of exchange gap 0 - ΔTS Maximum TK is reached at ΔTS = 0, where the singlet and triplet states are degenerate. The sign and the magnitude of this gap may be controlled by the gate voltage vg , so this effect is experimentally checkable.

  27. -1 • Magnetic field induced(!)Kondo effect ET 0 S ES 1 B It was the first example of Kondo effect due to dynamical symmetry of DQD D. Kobden et al, Nature, 408, 342 (2000) M. Pustilnik, Y. Avishai & KK, PRL, 84, 1756 (2000) Galitsky-2010

  28. Galitsky-2010 Intermezzo: other types of quantum dots. Vertical dots with parabolic confinement potential “Fock – Darwin” atoms with cylindrical symmetry

  29. Galitsky-2010 “Periodic table” for vertical dots Transition between the oscillator levels are described by SO(2,1) two-dimensional Lorenz group.

  30. Galitsky-2010 Endofullerene Fullerene quantum dots Nanotube quantum dots

  31. Galitsky-2010 Double and triple quantum dots Ring-like dots Self-assembled semiconductor quantum dots Excitonic atom

  32. Galitsky-2010 Basic configurations of double quantum dots s W V r l W d (a) (b) (c) (d)

  33. Galitsky-2010 … triple quantumdots (a) (b) (c) (d) W V (e) (f) (g) (h) (i)

  34. Galitsky-2010 …vertical quantumdots (a) (b) (c) (d)

  35. Galitsky-2010 Kondo effect and Aharonov – Bohm interference in “which path” geometry Two paths have different chirality when turning the magnetic flux from the left and from the right. As a result two partial waves meet in the dot 3 with phase difference Ф W V The interference effect ~ 1 – may partially or completele suppress Kondo tunneling

  36. Galitsky-2010 Exotic Kondo effect where spin and charge variables change their role. Triple quantum dot in parallel configuration. Central dot is smaller than side dots, and the Coulomb blockade is stronger. εc + Qc s Qic l r εi + Qic c W εi d εc

  37. Galitsky-2010 Two electrons, singlet ground state with S = 0 Spin reversal is impossible in a singlet state, but position of the level have changed (pseudospin reversal). This is possible realization of two-channel Kondo effect, where spin enumerates channels and position plays part of spin!

  38. Galitsky-2010 CONCLUDING REMARKS In artificial nanoobjects various types of dynamical symmetries may be realized and used in practical (?) applications Y. Avishai (Israel) Y. Oreg (Israel) M. Kiselev (Italy) T. Kuzmenko (Israel) R. Shekhter (Sweden) M. Wegewijs (Germany)

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