1. Electronic transport in one-dimensional wires Akira Furusaki (RIKEN)

2. Aug 14, 2003 Electronic transport in 1D wires 2 Outline Tomonaga-Luttinger (TL) liquid Bosonization Single impurity in a TL liquid Two impurities in a TL liquid linear conductance G Random-matrix approach to transport in disordered wires

3. Aug 14, 2003 Electronic transport in 1D wires 3 1D metals= Tomanaga-Luttinger liquid No single-particle excitations Collective bosonic excitations spin-charge separation charge density fluctuations spin density fluctuations Power-law decay of correlation functions (T=0) tunneling density of states

4. Aug 14, 2003 Electronic transport in 1D wires 4 TL liquids are realized in: Very narrow (single-channel) quantum wires edge states of fractional quantum Hall liquids Carbon nanotubes

5. Aug 14, 2003 Electronic transport in 1D wires 5 Interacting spinless fermions Simplified continuum model kinetic energy short-range repulsive interaction (forward scattering)

6. Aug 14, 2003 Electronic transport in 1D wires 6 Abelian Bosonization Fermions = Bosons in 1D

7. Aug 14, 2003 Electronic transport in 1D wires 7 Electron density

8. Aug 14, 2003 Electronic transport in 1D wires 8 Kinetic energy

9. Aug 14, 2003 Electronic transport in 1D wires 9 Bosonized Hamiltonian TL liquid parameter g g 1: attractive interaction Interacting fermions = free bosons

10. Aug 14, 2003 Electronic transport in 1D wires 10 Correlation functions （ T=0 ） Scaling dimension of is

11. Aug 14, 2003 Electronic transport in 1D wires 11 Single impurity Non-interacting case (free spinless fermions) transmission probability

12. Aug 14, 2003 Electronic transport in 1D wires 12 Current Conductance G changes continuously. no temperature dependence. is a marginal perturbation

13. Aug 14, 2003 Electronic transport in 1D wires 13 Interacting spinless fermions reflection at the barrier potential Hamiltonian free boson + = pinning of charge density wave electric current

14. Aug 14, 2003 Electronic transport in 1D wires 14 Partition function (path integral) effective action for linear: dissipation due to gapless excitations in TL liquid (Caldeira-Leggett: Macroscopic Quantum Coherence) a particle (with coordinate ) moving in a cosine potential with friction

15. Aug 14, 2003 Electronic transport in 1D wires 15 Renormalization-group analysis Weak-potential limit weak perturbation: scaling equation (lowest order): renormalized potential: conductance

16. Aug 14, 2003 Electronic transport in 1D wires 16 Strong-potential limit (weak-tunneling limit) duality transformation [A. Schmid (’83); compact QED by A.M. Polyakov] “dilute instanton (=tunneling) gas” t: tunneling matrix element (fugacity)

17. Aug 14, 2003 Electronic transport in 1D wires 17 scaling equation: renormalized tunneling matrix element: conductance

18. Aug 14, 2003 Electronic transport in 1D wires 18 Flow diagram for transmission probability ( Kane & Fisher, 1992) g<1 (repulsive int.) perfect reflection at T=0 g=1 (free fermions) marginal g>1 (attractive int.) perfect transmission at T=0 1 0 1 Trans. Prob. g

19. Aug 14, 2003 Electronic transport in 1D wires 19 Exact results “Toulouse limit” g=1/2 introduce new fields refermionization quadratic Hamiltonian cf. 2-channel Kondo problem (Emery-Kivelson, 1992)

20. Aug 14, 2003 Electronic transport in 1D wires 20 C onductance at g=1/2 General g The boundary sine-Gordon theory is exactly solvable (Ghoshal & Zamolodchikov, 1994) Bethe ansatz elastic single-quasiparticle S-matrix (Fendley, Ludwig & Saleur, 1995)

21. Aug 14, 2003 Electronic transport in 1D wires 21 Spinful case (electrons) (Furusaki & Nagaosa, 1993; Kane & Fisher, 1992) charge boson: spin boson: Hamiltonian : non-interacting electrons : repulsive interactions : if spin sector has SU(2) symmetry

22. Aug 14, 2003 Electronic transport in 1D wires 22 Weak-potential limit Strong-potential limit (weak-tunneling limit) single-electron tunneling: t RG flow diagram critical surface at intermediate coupling 0 1 0 1 1 Trans. Prob. Trans. Prob.

23. Aug 14, 2003 Electronic transport in 1D wires 23 External leads (Fermi-liquid reservoir) (Maslov & Stone, 1994) Tomonaga-Luttinger liquid: Fermi-liquid leads: Action Current I vs Electric field E dc conductance is not renormalized by the e-e interaction if the wire is connected to Fermi-liquid reservoirs

24. Aug 14, 2003 Electronic transport in 1D wires 24 Weak e-e interactions (Matveev, Yue & Glazman, 1993) small parameter: V(q): Fourier transform of interaction potential scaling equation for the transmission probability lowest order in but exact in conductance

25. Aug 14, 2003 Electronic transport in 1D wires 25 Coulomb interactions (Nagaosa & Furusaki, 1994; Fabrizio, Gogolin & Scheidel, 1994) : width of a quantum wire scaling equation for tunneling conductance stronger suppression than power law

26. Aug 14, 2003 Electronic transport in 1D wires 26 Experiments on tunneling Edge states in FQHE (Chang, Pfeiffer & West, 1996) tunneling between a Fermi liquid and edge state [Fig. 1 & Fig. 2 of PRL 77, 2538 (1996) were shown in the lecture]

27. Aug 14, 2003 Electronic transport in 1D wires 27 Single-wall carbon nanotubes Yao, Postma, Balents & Dekker, Nature 402, 273 (1999) [Fig. 1 and Fig. 3 were shown in the lecture.] Segment I & II: bulk tunneling Across the kink: end-to-end tunneling exp:

28. Aug 14, 2003 Electronic transport in 1D wires 28 Resonant Tunneling (Double barriers) Non-interacting case transmission amplitude: t has maximum when resonance (symmetric barrier) symmetric case backscattering is irrelevant asymmetric case backscattering is marginal = single impurity 0 d x LR

29. Aug 14, 2003 Electronic transport in 1D wires 29 When life time of discrete levels Conductance if coherent tunneling if incoherent sequential tunneling peak width

30. Aug 14, 2003 Electronic transport in 1D wires 30 Resonant tunneling in TL liquids Spinless fermions Hamiltonian gate voltage Current Excess charge in [0, d ] is massive

31. Aug 14, 2003 Electronic transport in 1D wires 31 Weak-potential limit (Kane & Fisher, 1992) effective action for single-barrier problem scaling equation if (symmetric) and (on resonance) gg1 1/4

32. Aug 14, 2003 Electronic transport in 1D wires 32 Resonance line shape symmetric ¼ < g < 1 is the only relevant operator, on resonance universal line shape peak width not Lorentzian

33. Aug 14, 2003 Electronic transport in 1D wires 33 Weak-tunneling limit (Furusaki & Nagaosa, 1993; Furusaki,1998) Off resonance process is not allowed at low T virtual tunneling On resonance sequential tunneling life time due to tunneling through a barrier peak width

34. Aug 14, 2003 Electronic transport in 1D wires 34 Phase diagram at T=0 Symmetric barriers Asymmetric barriers g 1 g 0 1 11/21/4 Transmission probability 01 1

35. Aug 14, 2003 Electronic transport in 1D wires 35 T > 0 Weak potential Weak tunneling sequential tunneling

36. Aug 14, 2003 Electronic transport in 1D wires 36 Experiments on resonant tunneling in TL liquids Auslaender et al., Phys. Rev. Lett. 84, 1764 (2000)

37. Aug 14, 2003 Electronic transport in 1D wires 37 Carbon nanotubes Postma et al., Science 293, 76 (2001)

38. Aug 14, 2003 Electronic transport in 1D wires 38 Summary In 1D e-e interaction is crucial Tomonaga-Luttinger liquid Repulsive e-e interaction backward potential scattering is relevant power-law suppression of tunnel density of states Problems nontrivial fixed points at intermediate coupling Resonant-tunneling experiment?