Meir-WinGreen Formula Consider a quantum dot ( a nano conductor, modeled for example by an Anderson model) connected with quantum wires Quantum dot U

Consider a quantum dot ( a nano conductor, modeled for example by an Anderson model) connected with wires Quantum dot U where L,R refers to the left and right electrodes. Due to small size, charging energy U is important. If one electron jumps into it, the arrival of a second electron is hindered (Coulomb blockade)

This has been used for weak V also in the presence of strong U. Meir and WinGreen have shown, using the Keldysh formalism, that the current through the quantum dot is given in terms of the local retarded Green’s function for electrons of spin s at the dot by This has been used for weak V also in the presence of strong U. 3 3

General partition-free framework and rigorous Time-dependent current formula Partitioned approach has drawbacks: it is different from what is done experimentally, and L and R subsystems not physical, due to specian boundary conditions. It is best to include time-dependence! 4 Interactions can be included by Keldysh formalism, (now also by time-dependent density functional) 4 4

Time-dependent Quantum Transport System is in equilibrium until at time t=0 blue sites are shifted to V and J starts device J 5 5 5

Use of Green’s functions

Rigorous Time-dependent current formula derived by equation of motion or Keldysh method Note: Occupation numbers refer to H before the time dependence sets in. System remembers initial conditions!

Current-Voltage characteristics In the 1980 paper I have shown how one can obtain the current-voltage characteristics by a long-time asyptotic development. Recently Stefanucci and Almbladh have shown that the characteristics for non-interacting systems agree with Landauer 8

Long-Time asymptotics and current-voltage characteristics are the same as in the earlier partitioned approach 9

In addition one can study transient phenomena Transient current Current in the bond from site 0 to -1 asymptote 10

Example: M. Cini E.Perfetto C. Ciccarelli G. Stefanucci and S. Bellucci, PHYSICAL REVIEW B 80, 125427 2009

M. Cini E. Perfetto C. Ciccarelli G. Stefanucci and S M. Cini E.Perfetto C. Ciccarelli G. Stefanucci and S. Bellucci, PHYSICAL REVIEW B 80, 125427 2009

Retardation + relativistic effects totally to be invented! G. Stefanucci and C.O. Almbladh (Phys. Rev 2004) extended to TDDFT LDA scheme TDDFT LDA scheme not enough for hard correlation effects: Josephson effect would not arise Keldysh diagrams should allow extension to interacting systems, but this is largely unexplored. Retardation + relativistic effects totally to be invented!

Magnetic effects in quantum transport Michele Cini, Enrico Perfetto and Gianluca Stefanucci Dipartimento di Fisica, Universita’ di Roma Tor Vergata and LNF, INFN, Roma, Italy ,PHYSICAL REVIEW B 81, 165202 (2010) 14 14

Quantum ring connected to leads in asymmetric way current Tight-binding model Current excited by bias  magnetic moment. How to compute ring magnetic moment and copuling to magnetic field? (important e.g. for induction effects) 15 15 15

State-of-the-art calculation of connected ring magnetic moment J1 J2 J3 J4 J5 J7 J6 State-of-the-art calculation of connected ring magnetic moment this is arbitrary and physically unsound. 16 16 16

problems with the standard approach Isolated ring: vortex current excited by B  magnetic moment Insert flux f by Peierls Phases: NN S h1 h2 h3 h4 h5 h7 h6 h5exp(ia5) h6exp(ia6) h1exp(ia1) h2exp(ia2) h4exp(ia4) h3exp(ia3) h7exp(ia7) Connected ring: current excited by E  magnetic moment. Bias current 17 17

Gauges Physics does not change c a b Probe flux, vanishes eventually Insert flux f by Peierls Phases: NN S c a b Probe flux, vanishes eventually All real orbitals, all hoppings= t Gauges Blue orbital picks phase a , previous bond  t e ia, following bond  t e-ia Physics does not change 18 18

c a b NN Insert flux f by Peierls Phases: S counted counterclockwise 19 19

Thought experiment: Local mechanical measurement of ring magnetic moment. Atomic force microscope A commercial Atomic Force Microscope setup (Wikipedia) The information is gathered by "feeling" the surface with a mechanical probe. Piezoelectric elements that facilitate tiny but accurate and precise movements on (electronic) command enable the very precise scanning.  The atom at the apex of the "senses" individual atoms on the underlying surface when it forms incipient chemical bonds. Thus one can measure a torque, or a force. System also performs self-measurement (induction effects) 20 20 20

Quantum theory of Magnetic moments of ballistic Rings   Quantum theory of Magnetic moments of ballistic Rings 21 21 21

Green’s function formalism Wires accounted for by embedding self-energy This is easily worked out Explicit formula: 22 22 22

Density of States of wires 2 -2 1 Density of States of wires 2 -2 1 no current U=2 U=0 (no bias) no current Left wire DOS Right wire U=1 current 23 23 23

2 -2 1 U 1.0 0.0 0.5 1.5 0.04 -0.02 Slope=0 for U=0 Cini Michele, Enrico Perfetto and Gianluca Stefanucci, Phys.Rev. B 81, 165202-1 (2010) 24 24 24

2 -2 1 Ring conductance vanishes by quantum interference (no laminar current at small U) Slope=0 for U=0 0.0 0.5 1.0 1.5 2.0 U 0.04 -0.04 0.0 25 25 25

2 -2 1 Slope=0 for U=0 -0.04 0.04 0.0 0.0 0.5 1.0 1.5 2.0 U 26 26 26

2 -2 1 Slope=0 for U=0 0.1 0.0 -0.1 1.0 1.5 2.0 0.0 0.5 U 27 27 27

Law: the linear response current in the ring is always laminar and produces no magnetic moment The circulating current which produces the magnetic moment is localized and does not shift charge from one lead to the other, contrary to semiclassical formula. 28 28

Quantized adiabatic particle transport (Thouless Phys. Rev. B 27,6083 (1983) ) Consider a 1d insulator with lattice parameter a; electronic Hamiltonian Consider a slow perturbation with the same spatial periodicity as H whici ia also periodic in time with period T , such that the Fermi level remains in the gap. This allows adiabaticity. The perturbed H has two parameters

A B

A B

Niu and Thouless have shown that weak perturbations, interactions and disorder cannot change the integer.