1 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 Δϕ ij between |ψ(ξ i )> and |ψ(ξ j )> 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 = Δϕ 12 + Δϕ 23 + Δϕ 31 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.

2 2 Consider Evolution of a system when adiabatic theorem holds (discrete spectrum, no degeneracy, slow changes) Adiabatic theorem and Berry phase

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 a n removes all other states!

5 5 Professor Sir Michael Berry

6 C

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

8 8 Limit:

9 99 Discrete (Pancharatnam) Continuous limit (Berry) Berry’s connection The Pancharatnam formulation is the most useful e.g. in numerics. Among the Applications: Molecular Aharonov-Bohm effect Wannier-Stark ladders in solid state physics Polarization of solids Pumping Trajectory C is in parameter space: one needs at least 2 parameters.

Vector Potential Analogy 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 the field as well, such that 10

The last term vanishes, 11 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

The m,n indices refer to adiabatic eigenstates of H ; the m=n term actually vanishes (vector product of a vector with itself). It is useful to make the Berry conections appearing here more explicit, by taking the gradient of the Schroedinger equation in parameter space: 12 Taking the scalar product with an orthogonal a m Formula for the curvature (alias B) A nontrivial topology of parameter space is associated to the Berry phase, and degeneracies lead to singular lines or surfaces

13 Ballistic conductor between contacts W left electroderight electrode Quantum Transport in nanoscopic devices Ballistic conduction - no resistance. V=RI in not true If all lengths are small compared to the electron mean free path the transport is ballistic (no scattering, no Ohm law). This occurs in experiments with Carbon Nanotubes (CNT), nanowires, Graphene,… A graphene nanoribbon field-effect transistor (GNRFET) from Wikipedia This makes problems a lot easier (if interactions can be neglected). In macroscopic conductors the electron wave functions that can be found by using quantum mechanics for particles moving in an external potential.

14 Fermi level right electrode Fermi level left electrode Particles lose coherence when travelling a mean free path because of scattering. Dissipative events obliterate the microscopic motion of the electrons. For nanoscopic objects we can do without the theory of dissipation (Caldeira-Leggett (1981). See Altland-Simons- Condensed Matter Field Theory page 130)

If V is the bias, eV= difference of Fermi levels across the junction, How long does it take for an electron to cross the device? If V is the bias, eV= difference of Fermi levels across the junction, How long does it take for an electron to cross the device? This quantum can be measured! 15 W left electroderight electrode junction with M conduction modes, i.e. bands of the unbiased hamiltonian at the Fermi level

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

Current-Voltage Characteristics J(V) of a junction : Landauer formula(1957) Current-Voltage Characteristics J(V) of a junction : Landauer formula(1957) 17 Phenomenological description of conductance at a junction Rolf Landauer Stutgart 1927-New York 1999

18 Phenomenological description of conductance at a junction More general formulation, describing the propagation inside a device. Quantum system leads with Fermi energy E F, Fermi function f(  ), density of states 

19 Quantum system

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

22 Microscopic current operator device J

23 Microscopic current operator device J

Partitioned approach (Caroli 1970, Feuchtwang 1976): fictitious unperturbed biased system with left and right parts that obey special boundary conditions: allows to treat electron-electron and phonon interaction by Green’s functions. device this is a perturbation (to be treated at all orders = left-right bond 24 Drawback: separate parts obey strange bc and do not exist. =pseudo-Hamiltonian connecting left and right Pseudo-Hamiltonian Approach

25 Simple junction-Static current-voltage characteristics J U=0 (no bias) no current Left wire DOS Right wire DOS no current U=2 current U=1

26 Static current-voltage characteristics: example J