Summer School on Novel Quantum Phases and Non-Equilibrium Phenomena in Cold Atomic Gases. 27 August - 7 September, 2007

Transcription

1 Summer School on Novel Quantum Phases and Non-Equilibrium Phenomena in Cold Atomic Gases 27 August - 7 September, 2007 Introduction to the physics of low dimensional systems Thierry Giamarchi University of Geneva

2 T. Giamarchi Systems of reduced dimensionality i

3 A. Iucci (Geneva) C. Kollath (Geneva) M. Zvonarev (Geneva) A. Kleine (Aachen) M. A. Cazalilla (Donostia) V. Cheianov (Lancaster) G. Fiete (Caltech) A. F. Ho (Imperial) W. Hofstetter (Frankfurt) M. Koehl (Cambridge) IPMcCulloch(Aachen) I.P. U. Schollwock (Aachen)

4 BEC in cold atomic gases 2001: Cornell, Ketterle, Wieman 1924: predicted by Bose and Einstein

5 Strong correlations Condensed matter: E cin = E coul Strong correlations!

6 Atoms in a lattice Tunnelling Short range interaction

7 Optical lattices: control kinetic energy Greiner et al. (2002);

8 Quantum simulators! Interactions Statistics Dimensionality ENS, ETH, LENS, Mainz, MIT, NIST, Penn State,

9 But... not so idilic!

10 Confining potential U/J=10 H = r 2 ρ(r) No homogeneous phase!

11 Can change physics drastically M.A. Cazalilla, A. F. Ho, TG, PRL (2005)

12 Probes! Atoms are neutral! n(k) (time of flight) useless for fermions! Need to probe correlations!

13 time-of-flight measurement proposed: Raman spectroscopy -> momentum distribution ->Green s function, Fermi surface München noise measurement: -> density-density correlations Paris periodic lattice modulation Mainz microwave spin-changing transitions density spatially resolved molecule formation binding energy doubly occupied sites Zurich Main z Zurich

14 So why reduced dimensionality? Not easy to realize in condesed matter Effect of interactions at their strongest Novel lph physics!

15 Let us start with 1D

16 Does one dimension exists? Hard to realize in condensed matter Nanotubes Organic conductors Quantum wires Josephson junctions Ladders Edge states in FQHE

17 Free bosons : crash course Free particles: condensation in k=0 state n(k) k

18 Excitations:singleparticles single with momentum k Do interactions change this? n(k) Not much!

19 One dimension is different No individual excitation can exist (only collective ones) Strong quantum fluctuations Continuous symmetry

20 Cold atoms: ideal systems Optical lattices, Chips N 0 ~ 10 to 10 3 atoms M. Greiner et al. PRL (2001) W. Haensel et al. Nature (2002) T. Stoferle et al. PRL (2004)

21 What does 1D means? ξ(k x ) n y = 1 n y =0 U,T, k x

22 Models Continuum: Lattice:

23 Questions How to deal with ihi interactions/quantum i fluctuations What is the new physics in 1D? Change of nature of the «particles» New phases? How to go from 1D to higher dimensions

24 General references on 1D Will follow closely: TG, Quantum physics in one dimension, Oxford (2004) TG, cond-mat/ (Salerno lectures) Emery, V. J. (1979). Highly conducting one dimensional solids, pp Plenum. Solyom, J. (1979). Adv. Phys., 28, 209. Schulz, H. J. (1995). Les Houches LXI pp Elsevier. Voit,,J J. (1995). Rep. Prog. Phys., 58, 977. Gogolin, A. O., Nersesyan, A. A., and Tsvelik, A. M. (1999). Bosonization and Strongly Correlated Systems. Cambridge University Press, Cambridge. M.A. Cazalilla, J. Phys B, 37 S1 (2004)

25 How to study Exact methods (Bethe Ansatz) Exact spectrum; limitedi to very special ilmodels dl Numerics ``Exact special models, size limitations, quantities specific to models Low energy methods

26 Labelling the particles 1D: unique way of labelling lli

27 φ(x) varies slowly CDW q 0 q 2πρ 0

28 θ: superfluid phase Quantum fluctuations All short distance properties: form of the operators. φ, θ: smooth fields

29 Hamiltonian

30 Luttinger liquid concept How much is perturbative Nothing provided d the correct u,k are used (Haldane) Low energy properties: Luttinger liquid id (fermions, bosons, spins )

31 Luttinger parameters u: velocity of collective excitations K: dimensionless parameter 0 U 1 K Tonks gas

32 Correlations

33 Condensate? No true condensate (K 0)

34 Finite temperature Conformal theory β χ

35 Check for the powerlaws A. Schwartz et al. PRB 58 Z. Yao et al. Nature B (1998) (1999)

36 Cold atoms? Difficult! (trap trap!) Need probes!

37 Quantum depletion of condensate P. Bouyer et al. (2004) T. Stoferle et al. (2004) Good qualitative ti agreement

38 Tonks limit U= : spinless fermions Not tfor rn(k) : ψ F ψ B B. Paredes et al Nature (2004) T. Kinoshita et al. Science (2004) M. Kohl et al. PRL (2004)

39 Systems with «spins»

40 Same treatmentt t Luttinger liquid ρ Φ ρ Φ More convenient 1 1 ρ = ( ρ + ρ ) σ = ( ρ ρ 2 2 ) H H + H = H + H kin ρ σ =

41 H int = U ρ ρ i = U ( ρ + σ )( ρ σ ) = U ( ρρ σσ ) H = H + H ρ σ ( u, K ) ρ ρ ( u, K ) σ σ Charge excitations Spin excitations Charge-spin separation

42 Spin-Charge Separation Spin Charge Spinon Holon

43 Spin-Charge Separation higher D? Spin Charge Energy increases with spin-charge separation Confinement of spin-charge: «quasiparticle»

44 Can one observe spin-charge separation? Condensed matter: difficult One serious experiment: Yacoby et al.

45 Two component Bosons A. Kleine et al. cond-mat/ e.g. 87 Rb (two hyperfine states)

46 t-dmr study

47 Parameters and Spectral functions Spectral function by bosonization: A. Iucci, G. Fiete, TG PRB 75, (2007))

48 Other effects for bosons?

49 Ferromagnetism (M. Zvonarev, V. Cheianov, TG cond-mat/ ) Excitations: k 2 not k Not a LL!!

50

51 Mott transition

52 Lattice: Mott transition Costs U Quantum phase transition SU MI U

53 Mott transition and cold atoms Superfluid Mott Insulator Lase r Lase r Superfluid to Mott insulator transition in a 3D optical lattice [M Greiner et al. Nature, 415 (2002)]

54 How to treat? Incommensurate: Q 2 π ρ 0 Commensutate: Q = 2 πρ 0

55 S dxdττ = [ 2πK Competition 2 1 ( φ( x, τ )) + u( φ( x, τ )) u τ x 2 ] Beresinskii- Kosterlitz-Thouless transition K=2

56 Lattice Mott insulator: φ is locked

57 Mott insulator: φ is locked TG, Physica B (97) Gap in the excitation spectrum Density is fixed T. Kuhner et al. PRB (2000) n(k)

58 Important for 1D J 0 ``trivial MI J À V but K < 2 MI!!

59 Disorder

60 Disorder and quantum systems disorder less important?? No!! (Anderson localization): interferences

61 Bosons Free bosons: pathological (rare events)

62 How to treat TG + H. J. Schulz PRB (1988) ``Two fourier components of disorder

63 Forward scatting (q 0) Random (smooth) chemical potential No localization Can break commensurate phases

64 Backward scattering (q 2 π ρ 0 ) Relevant for K < 3/2 Localized d!!

65 Bose glass phase 1D : TG + H. J. Schulz PRB (1988) Superfluid Localized (Bose glass) transition for K < 3/2 BKT like transition Higher dimensions: M.P.A. Fisher et al. PRB (1989) Bose glass also exists continuous transition

66 Weak disorder (strong interactions) is enough in 1D Localized for K < 3/2 even if V μ Quantum effect: destructive interferences

67 Bosons 1D Luttinger liquid physics New phases : Mott insulator/bose glass Good qualitative agreement with exp. Correlation functions!

68 Many open points Confining potential Dynamics Disorder Etc

69 To higher dimensions... And beyond...

70 Interaction effects vary enormously with dimension Dimensional Crossover 2D,3D 1D ET E,T

71 Even more interesting : lower dimensional phase gapped T low d gapped large d massless Quantum phase transition : Quantum phase transition : deconfinement t

72 Generic scenario for many systems Questions : Nature and position of the transition? Physical properties in the critical regime Impact of the low-d phase on the massless phase

73 Spin systems Singlet phase of dimers (zero dimensional) J is irrelevant (gapped phase) Break the gap: H m TG and A. M. Tsvelik PRB (1999) BEC Gapped h

74 Deconfinement 600 Δ c TMTTF 2 PF 6 Activation energ gy, T* (K) Δ a C IC Spin-Peierls 60 SDW SDW T* Pressure (kbar) TG Physica B (97); Chem Rev (2004); condmat/ PAuban-Senzier P. Auban-Senzier, D. Jérome, C. Carcel and J.M. Fabre J de Physique IV, (2004) D. Jaccard et al., J. Phys. C, 13 L89 (2001)

75 Deconfinement TSt T. Stoferle frl et al. PRL (2004) 1D Mott insulator 1D physics (Luttinger Liquids)

76 Bosons [ 87 Rb] A. F. Ho, M. A. Cazalilla, TG PRL (2003) M. A. Cazalilla, A. F. Ho, TG, NJP (2006)

77 Mott vs. Josephson R R Josephson coupling: delocalizes atoms Mott potential: localizes atoms

78 Methods RG Gives phase boundary Mean Field

79 Mapping on spin chain Critical properties Universality class of 4d XY model

80 Phase and amplitude mode

81 GP: only Goldstone mode How to recoverer amplitude mode

82 Phase diagram transverse hopping J/μ μ Anisotropic 3D -2 SF (BEC) D MI D MI Array of atomic quantum dots repulsion (γ = + ) (γ = 8) (γ = 3.5) (γ = 2) K

83 Experiments T. Stoferle et al. PRL (2004)

84 Shaking of the lattice TS T. Stoferle et al. PRL (2004) 3D superfluid Mott ins. A. Iucci, M.A. Cazalilla, AF Ho, TG, PRA 73, R (2006); C. Kollath, A. Iucci, TG, Not W. Hofstetter, so simple U. Schollwock, PRL! (06)

85 Fermions [ 6 Li or 40 K] M.A. Cazalilla, A. F. Ho, TG, PRL (2005)

86 Fermionic tubes 2 different hoppings t (optical lattice) Local linteraction ti U(Feshbach hr resonnance) N = N

87 RG Strong coupling -k F +k F M.A. Cazalilla, AF Ho, TG, PRL (2005); cond-mat/

88 1D: phase diagram repulsion Falikov-Kimball attraction Spin gap: Raman transitions

89 Trap is good (for once)!

90 Coupled tubes with Spin gap β α

91 AF Order μ 1 μ 2 Triplet superconductivity (repulsive interactions)

92 Low dimensions Optical Lattices: possibility to study crossover from one to three dimensional physics Boson tubes: deconfinement transition from 1D Mott insulator to 3D superfluid Fermions tubes: spin gap due to spin-dependent hopping; triplet superconductivity for repulsive interactions

93 Shaking of lattice. Efficient measurement to probe phases; efficient theory to compare to. Effects of trap Out of equilibrium i

94 But to observe all that... Probes would be good..

95 time-of-flight flight measurement proposed: Raman spectroscopy -> momentum distribution ->Green s function, Fermi surface noise measurement: -> density-density correlations München Paris periodic lattice modulation Mainz microwave spin-changing transitions density spatially resolved molecule formation binding energy doubly occupied sites Zurich Main z Zurich

96 Need local probes! STM CAT C. Kollath, M. Koehl, TG cond-mat/ ; physics world (2007).

97 Conclusions Cold atoms/condensed matter: complementary Cold atoms: quantum simulators Tunability and local interactions. Ideal to explore low dimensional i physics. Inhomogeneous phases Probes

98 The sky is the limit! Let s have fun!