Current and noise in chiral and non chiral Luttinger liquids. Thierry Martin Université de la Méditerranée Centre de Physique Théorique, Marseille

Transcription

1 Current and noise in chiral and non chiral Luttinger liquids Thierry Martin Université de la Méditerranée Centre de Physique Théorique, Marseille

2 Outline: Luttinger liquids 101 Transport in Mesoscopic Physics: current and noise Edge states in the QHE Calculation of current and noise: fractionalization Transport in Non chiral Luttinger Liquids 1) Electron injection 2) Two terminal transport 3) Experimental evidence of charge fractionalization («funny charges» in Luttinger liquids)

3 One dimensional wires exist in nature!

4 Luttinger 101 E F Tight binding Hamiltonian for fermions Jordan Wigner transfo boson Amplitude/phase Continuum limit Canonical relation

5 Charge density Integrate: Luttinger Liquid Hamiltonian density Velocity of collective excitations Interaction parameter Compressibility Implies that in the absence of interactions

6 Fermion operator: bring an electron from infinity. mixture of and conjugate of Creates a kink at x

7 Tunneling in a Luttinger liquid (add an e- at x with energy E), n eigenstates e- Green s function Calculated with Lagrangian electron «not welcome» in a Luttinger liquid

8 (PRL01) Difference between tunneling in the bulk/ end of a nanotube

9 2LL s : Transmission through a Barrier 1) large barrier: 2 semi infinite LL with Tunnel current Direction Remove Add I(V) characteristics and differential conductance High temperature conductance

10 Transmission through a Barrier 2) Weak barrier (delta function) Backscattered current Fourier component of the potential at 2k F Conductance Duality between weak and strong backscattering situations Beyond perturbation theory?.rg

11 Renormalization group analysis (map of Q. Brownian Motion Pb) strong backscattering Integrate out fast degrees of freedom weak backscattering

12 Quantum transport: Open systems, with a bias imposed. Need to calculate the current operator in the absence of equilibrium Scattering theory (non interacting case mostly) with large e- reservoirs (Landauer Buttiker) Hamiltonian approach: use non-eq technique (Keldysh): Nozières Bethe ansatz applicable also out of equilibrium (Saleur, Andrei ) Mostly current, but

13 NOISE 102 «the noise is the signal» (R. Landauer) Ambiguity: symmerize or not-symmetrize noise? important (not important) at «high» («low») frequencies

14 Johnson-Nyquist noise and Shot noise Johnson-Nyquist noise for equilibrium circuit information about resistance & temperature just disturbance Harry Nyquist ( : U.S.) Shot noise in a vacuum tube Electrons are emitted by thermal agitation (the «best» thermometer for measuring electron temperature)

15 noise power simple way to measure the charge of electron Annals der Physik (1918) Walter Schottky ( : Germany) classical picture of current Electrons are emitted Independently from each other: Poissonian process.

16 Wave packet approach to noise

17 Shot noise reduction: Weizmann group (Reznikov PRL 95)

18 Test of the noise reduction factor (Kumar PRL96) Also Ruitenbeek s group, break junctions.

19 Conductors connected to several leads: prerequisite The Hanbury-Brown and Twiss experiment

20 Electron analogs of the Hanbury Brown and Twiss experiment S 23 = - (4e 2 /h)tr[s 21 s 21* s 31 s 3 * 1 ] (arbitrary transmission) CORRELATIONS < 0 Thermal source

21 Finite frequency noise? Emission and absorption PRL07 ( j.a coupling, Lesovik ) Also: Capacitive coupling Kouvenhoven,Deblock (Science, PRL 2000)

22 Noise in chiral Luttinger liquids Poissonian noise in the FQHE Edge states: Wen s hydrodynamical picture Hall droplet, filling factor unknown still

23 Electrical potential at the edge 2D density Quadratic Hamiltonian for edge excitations Filling factor enters here

24 Quantize? choose Continuity equation Kac-Moody commutation relations

25 Introduce fermion creation operator Introduce bosonic field regularization Recall From Kac-Moody

26 Conclusion: fermion operator Do they anticommute??? Yes! Use BCA identity Fermion if Laughlin fractions only!

27 Action? Lagrangian choose Chiral Luttinger liquid action

28 Thermal Green s function: Yet we need Greens function in real time (Now: bla bla bla Keldysh contour )

29 Keldysh 103 Heisenberg Interaction representation «dictionary» Evolution Adiabatic approximation Recover time ordered product, Wick and all the good stuff.

30 perform analytic continuation Keldysh Green matrix for bosonic field Times can be attached to either contour

31 Transport with the simplest scatterer: A quantum Point Contact, yet in FQHE conditions Tunneling between two counter propagating edges

32 Tunnel Hamiltonian Quasiparticle operator Klein factor, not needed here Current operator 1/ν quasi part = 1 e-

33 Average current Lowest order perturbation theory, O(Γ 2 ) Green for R Green for L

34 Average current (cont) Like most LL situations, no absolute congergence. Convergence is insured by Finite Voltage oscillatory factor

35 Average current (cont) Result is cutoff dependent

36 Backscattering NOISE: Real time correlator Lowest order Again, QP conservation

37 Real time correlator (cont) Low frequencies Fano factor gives fractional charge

38 Duality: similar to Kane Fisher RG analysis strong backscattering Integrate out fast degrees of freedom weak backscattering

39 Fendley Ludwig Saleur (PRL 04 05) Exact solution from boundary sine Gordon model «Folding» «integrable model» (Zamolodchikov 2 )

40 Wave packet approach to noise

41 Scattering Probability of current kinks antikinks noise QP e- Describes crossover between weak and strong backscattering

42 Actual measurement: Hanbury Brown and Twiss Geometry to eliminate amplifyier noise I 3 =I 0 =νe 2 /h I 5 =I I 2 =I-I B S=<I B (I-I B )>

43 Experiments: De Picciotto (Heiblum) Saminadayar (Glattli) I(V) characteristics: confirms 1/3 filling at point contact

44 Kane PRL 1994, Chamon PRB1995 Depiciotto, Nature 1997, Saminadayar PRL 1997)

45 Temperature crossover

46 How to detect these funny charges In NON-CHIRAL Luttinger Liquids? Interaction parameter Theoretical proposals: g=k! 3 terminal, electron injection + Shottky noise analysis (no leads included) 2 terminal, including leads, finite frequency 3 terminal with leads, finite frequency Experiment-Theory (Havard-Yale consortium ) 3 terminal, momentum controlled injection detection via conductance.

47 Noise it non chiral Luttinger liquids Electron injection in an infinite nanotube Current Noise autocorrelation Noise cross correlation «funny charges» (Entanglement)

48 Electron injection in an infinite nanotube: Current, Noise autocorrelation Noise cross correlations «funny charges» Metallic nanotubes

49 Hamiltonian for 4 modes LL K c+ 1 Fermion operator

50 Green s functions Integrate out either field

51 In our problem, these are not the only Green s functions!

52 Tip: fermionic, or «boring» non interacting semi infinite Luttinger liquid No interaction parameter

53 Tunneling Hamiltonian

54 Tunneling current Recover zero bias anomaly

55 Tunneling noise No surprise: recover Schottky s formula Same zero bias anomaly Poissonian process of electrons injected on a Luttinger liquid boring!

56 More interesting: Current and noise in a section of the nanotube Usual convention: Current «SHOULD» be measured AWAY from the injection location

57 Noise + noise cross-correlations: 2nd order in the tunneling amplitude Usual situation for non-interacting Fermions: 4th order. (Opposite sign with HBT convention) HERE, POSITIVE CORRELATIONS FOR AN INTERACTING FERMIONIC SYSTEM!!!

58 Interpretation: anomalous charges:

59 Entanglement? Symptom: positive cross correlations, as in NS forks e- injected at x=0 «Triplet» wavefunction! Q + Q - > + Q - Q + > Chiral fields

60 What about contacts? Do they spoil everything? Fermi Liquid Almost! No signatures of funny charges at zero frequency, with Fermi contacts TO BE CONTINUED

61 Two terminal noise predictions in LL Detection of charges («fractionalization») Problem with leads DC current (Safi Schulz, Maslov Stone, Ponomarenko PRB s 95 ) LL with an inhomogeneous interaction parameter K G=e 2 /h Multiple Andreev reflections at boundaries.

62 Absence of renormalization of conductance? Confirmed by radiative boundary condition approach (Egger Grabert PRL96/PRB98) Interaction effects in the presence of impurity (solution at g=1/2 refermionization) (also, perturbative treatment of impurity, Safi )

63 «Funny charges» in Luttinger liquids (many contributors!!! Safi, Degiovanni, Pham, Imura ) Q 1 =(1+g)/2 Q 2 =(1-g)/2 Andreev like reflections at the boundaries 2 terminal 3 terminal

64 PRL04,PRB 05 current symmetrized noise Time of flight frequency

65 Method: Keldysh approach Partition function impurity applied bias source term Result: perturbation theory (no time order)

66 Backscattering current Oscillatory behavior Normalized current: dependence on g Voltage phase shift at the impurity Also: Dependence on the position (of center) of impurity

67 Analytical results recovered for (e- wave packet well localized in time/space) Leading order, indep of L «old» result High temperature No oscillations, thermal crossover

68 Finite frequency noise No impurity No impurity: equilibrium noise Position of the impurity No oscillations Fluctuation dissipation theorem Zero point Impurity needed for charge detection!

69 Non-equilibrium noise Nonlocal conductivity of clean system Voltage, field correlators Cusps at Sharp contrast with LL Not connected to leads, Where power law sigularities

70 Non-equilibrium noise: detection of Luttinger liquid parameter Central result Interaction dependent reflection coefficient, from I(V) Frequency average of Fano factor gives g!

71 Probe the effects of the leads only Noise: Comparison interacting and non interacting wires.

72 «Injection of e- the return» Finite frequency noise cross correlations In the presence of leads

73 Finite frequency noise Unsymmetrized correlator Tunneling density of states Propagation along the nanotube

74 Here is Greens functions include reflections at the location of the inhomogeneities of the LL

75 Autocorrelation noise in an infinite nanotube Non-interacting case (dashed): singularity (Yang, Lesovik) Interacting case (full) Auto and cross correlations are proportional

76 Cross correlation in a nanotube with leads: two time scales 1) Injection (voltage) time time spread of electron wave packet (Lesovik Levitov) 2) Time of flight

77 Cross correlations in a nanotube with leads Several round trips No round trips

78 Detection of anomalous charges? Specify frequency associated with peaks Ratio of cross correlations to autocorrelation noise

79 Measurement of Current assymmetry Two terminal conductance

80 Chemical potentials for left and right moving charge modes Upper wires chemical potential Right moving current LL parameter Specified by charge density

81 Linear response regime: relation between chemical potentials Assume symmetric (left /right) system Two terminal conductance Three terminal conductance Injected current Fraction of e- transmitted to R 1?

82 Two terminal conductance as a function of B Vary e- density in 1D wire Find (always)

83 Measure Interaction Parameter Independently (charge velocity) Conclusion: Experimental findings are consistent with charge fractionalization

84 Acknowledgements Rolf Landauer, Daniel Loss, Gordey Lesovik, Ines Safi, Adeline Crépieux, Alex Zazunov, Thibaut Jonckheere Students: Julien Torres, Rodolphe Guyon, Andrei Lebedev, Marjorie Creux

85 THE END

86 References: J. B. Johnson, Phys. Rev. B 29, 367 (1927); H. Nyquist, Phys. Rev. 32, 110 (1928). W. Schottky, Ann. Phys. (Leipzig) 57, 541 (1918). R. Landauer, Physica D38, 226 (1987). R. Landauer and Th. Martin, Physica B 175, 167 (1991); 182, 288 (1992). Th. Martin and R. Landauer, Phys. Rev. B 45, 1742 (1992). G. B. Lesovik, JETP Lett. 49, 594 (1989). B. Yurke and G. P. Kochlanski, Phys. Rev. B 41, 8141 (1990). M. Büttiker, Phys. Rev. Lett. 65, 2901 (1990); Phys. Rev. B 46, (1992). C. Caroli, R. Combescot, P. Nozières, and D. Saint-James, J. Phys. C 4, 916 (1971). J. Rammer and H. Smith, Rev. Mod. Phys. 58, (1986) R. Hanbury Brown and R. Q. Twiss, Nature (London) 177, 27 (1956); R. Hanbury Brown and R. Q. Twiss, Proc. Royal. Soc. London Ser. A 242, 300 (1957); ibid. 243, 291 (1957). V. A. Khlus, Sov. Phys. JETP (1987) (Zh. EKsp. Teor. Fiz. 93, 2179 (1987)). M. Reznikov, M. Heiblum, H. Shtrikman and D. Mahalu, Phys. Rev. Lett. 75, 3340 (1995). A. Kumar, L. Saminadayar, D. C. Glattli, Y. Jin and B. Etienne, Phys. Rev. Lett. 76, 2778 (1996). G. B. Lesovik and R. Loosen, Pis ma Zh. Eksp. Theor. Fiz. 65, 280 (1997) [JETP Lett. 65, 295 (1997)]. R. Deblock, E. Onac, L. Gurevich and L. Kouvenhoven, Science 301, 203 (2003). S.R. Eric Yang, Solid State Commun. 81, 375 (1992). C. L. Kane and M. P. A. Fisher Phys. Rev. Lett. 68, 1220 (1992); C. L. Kane and M. P. A. Fisher Phys. Rev. B 46, (1992). D. C. Tsui, H. L. Stormer and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). C. de C. Chamon, D. E. Freed and X.-G. Wen, Phys. Rev. B 51, 2363 (1995). X.G. Wen, Phys. Rev. B 43, (1991); Phys. Rev. Lett. 64, 2206 (1990); X.G. Wen, Int. J. Mod. Phys. B 6, 1711 (1992); Adv. Phys. 44, 405 (1995).

87 C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 72, 724 (1994). C. de C. Chamon, D. E. Freed and X. G. Wen, Phys. Rev. B 53, 4033 (1996). L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne, Phys. Rev. Lett. 79, 2526 (1997). R. de-picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu, Nature (London) 389, 162 (1997). M. Reznikov, R. de-picciotto, T. G. Griffiths, M. Heiblum, V. Umansky, Nature (London) 399, 238 (1999). P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. Lett. 75, 2196 (1995). P. Fendley and H. Saleur Phys. Rev. B 54, (1996) I. Safi, P. Devillard, and T. Martin Phys. Rev. Lett. 86, 4628 (2001). C. L. Kane and M. P. A. Fisher Phys. Rev. B 67, (2003) I. Safi, Ann. Phys. Fr. 22, 463 (1997). K.-V. Pham, M. Gabay, and P. Lederer, Phys. Rev. B 61, (2000). A. Crépieux, R. Guyon, P. Devillard, and T. Martin, Phys. Rev. B 67, (2003). I. Safi and H. Schulz, Phys. Rev. B 52, (1995); D. Maslov and M. Stone, Phys. Rev. B 52, 5539 (1995); V.V. Ponomarenko, Phys. Rev. B 52, 8666 (1995). A. Lebedev, A. Crépieux, and T. Martin, Phys. Rev. B 71, (2005). B. Trauzettel, I. Safi, F. Dolcini, and H. Grabert, Phys. Rev. Lett. 92, (2004); F. Docini,. Trauzettel, I. Safi and H. Grabert, Phys. Rev. B 71, (2005)

88

89 How to measure information about statistics? Hanbury-Brown and Twiss in the fractional quantum Hall effect?

90 Edge states Hamiltonian Quasiparticle Hamiltonian Tunnel Hamiltonian between edges Klein factor

91 Klein factors 101 fractional statistics Tunneling operator for single edge If tunneling paths do not cross

92 Consequence for Klein factors: fractional statistics Antisymmetric matrix

93 Bosonization for Klein factors Green s function for bosonic fields

94 Real time correlator Explicit expression from zero temperature Green s function

95 Real time correlator: Slow decay of tempemoral correlations

96 Zero frequency noise correlations:

97 Relevance of other tunneling operators: General tunneling operator Renormalization group eq. Bare tunneling terms relevant for ν<1 Other operators relevant for ν<1/3