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Topologic quantum phases

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  1. Topologic quantum phases Pancharatnam phase The Indian physicist S. Pancharatnam in 1956 introduced the concept of a geometrical phase. Let H(ξ ) be an Hamiltonian which depends from some parameters, represented by ξ ; let |ψ(ξ )> be the ground state. Compute the phase difference Δϕij between |ψ(ξ i)> and |ψ(ξj)> defined by This is gauge dependent and cannot have any physical meaning. Now consider 3 points ξ and compute the total phase γ in a closed circuit ξ1 → ξ2 → ξ3 → ξ1; remarkably, γ = Δϕ12 + Δϕ23 + Δϕ31 is gauge independent! Indeed, the phase of any ψ can be changed at will by a gauge transformation, but such arbitrary changes cancel out in computing γ. This clearly holds for any closed circuit with any number of ξ. Therefore γ is entitled to have physical meaning. There may be observables that are not given by Hermitean operators. 1 1

  2. Adiabatic theorem and Berry phase ConsiderEvolution of a systemwhenadiabatictheoremholds (discrete spectrum, no degeneracy, slow changes) 2 2

  3. To find the Berry phase, we start from the expansion on instantaneous basis 3

  4. Negligible because second order (derivative is small, in a small amplitude) Now, scalar multiplication by an removes all other states! 4

  5. 5 5 Professor Sir Michael Berry

  6. C 6

  7. Relation of Berry toPancharatnamphases C Idea: discretize path C assuming regular variation of phase and compute Pancharatnam phase differences of neighboring ‘sites’. 7 7

  8. Limit: 8 8

  9. Continuous limit (Berry) Discrete (Pancharatnam) Berry’s connection The Pancharatnamformulationis the mostuseful e.g. in numerics. Trajectory C is in parameterspace: oneneedsatleast 2 parameters. Among the Applications: MolecularAharonov-Bohmeffect Wannier-Starkladders in solid state physics Polarization of solids Pumping 9 9 9

  10. VectorPotentialAnalogy One naturally writes introducing a sort of vector potential (which depends on n, however). The gauge invariance arises in the familiar way, that is, if we modify the basis with and the extra term, being a gradient in R space, does not contribute. The Berry phase is real since We prefer to work with a manifestly real and gauge independent integrand; going on with the electromagnetic analogy, we introduce thefieldas well, such that 10

  11. The last term vanishes, To avoid confusion with the electromagnetic field in real space one often speaks about the Berry connection and the gauge invariant antisymmetric curvature tensor with components 11

  12. Formula for the curvature (alias B) The m,n indicesrefertoadiabaticeigenstatesof H ; the m=ntermactuallyvanishes (vectorproductof a vectorwithitself). Itisusefultomake the Berry conectionsappearinghere more explicit, bytaking the gradientof the Schroedingerequation in parameterspace: Taking the scalar product with an orthogonal am A nontrivial topology of parameter space is associated to the Berry phase, and degeneracies lead to singular lines or surfaces 12

  13. W left electrode right electrode Quantum Transport in nanoscopic devices Ballisticconduction - no resistance. V=RI in nottrue Ballistic conductor between contacts Ifalllengths are small compared to the electron mean free path the transportisballistic (no scattering, no Ohm law). Thisoccurs in experiments with Carbon Nanotubes (CNT), nanowires, Graphene,… Thismakesproblems a loteasier (ifinteractions can be neglected). In macroscopicconductors the electron wavefunctionsthat can be found by using quantum mechanics for particlesmoving in an externalpotential. A graphenenanoribbon field-effect transistor (GNRFET) from Wikipedia

  14. Fermi level left electrode Particleslosecoherencewhentravelling a mean free pathbecause of scattering . Dissipative events obliterate the microscopicmotion of the electrons. For nanoscopicobjectswe can do without the theory of dissipation (Caldeira-Leggett (1981). SeeAltland-Simons- CondensedMatter Field Theorypage 130) Fermi level right electrode

  15. W left electrode right electrode junctionwith M conductionmodes, i.e. bands of the unbiasedhamiltonianat the Fermi level IfV is the bias, eV= difference of Fermi levelsacross the junction, How long doesit take for an electron to cross the device? This quantum can be measured! 15

  16. B.J. Van Wees experiment (prl 1988) A negative gate voltage depletes and narrows down the constriction progressively Conductance is indeed quantized in units 2e2/h 16

  17. Current-Voltage Characteristics J(V) of a junction : Landauerformula(1957) Rolf Landauer Stutgart 1927-New York 1999 Phenomenologicaldescription of conductanceat a junction

  18. Quantum system J Phenomenologicaldescription of conductanceat a junction More general formulation, describing the propagation inside a device. leadswith Fermi energy EF, Fermi function f(e), density of statesr(e)

  19. Quantum system J 19

  20. This scheme was introduced phenomenologically by Landauer but later confirmed by rigorous quantum mechanical calculations for non-interacting models. 20

  21. Microscopiccurrentoperator device J 22

  22. Microscopiccurrentoperator device J 23

  23. device =pseudo-Hamiltonian connecting left and right Pseudo-Hamiltonian Approach Partitionedapproach (Caroli 1970, Feuchtwang 1976): fictitiousunperturbedbiasedsystem with left and right partsthatobey special boundaryconditions: allows to treat electron-electron and phononinteraction by Green’sfunctions. thisis a perturbation (to be treatedatallorders = left-right bond Drawback: separate parts obey strange bc and do not exist. 24

  24. U=1 current Simple junction-Staticcurrent-voltagecharacteristics J 1 U=0 (no bias) U=2 -2 2 Left wire DOS 0 Right wire DOS no current no current

  25. Static current-voltage characteristics: example J 26