Simulations of strongly correlated electron systems using cold atoms

Transcription

1 Simulations of strongly correlated electron systems using cold atoms Eugene Demler Harvard University Main collaborators: Anatoli Polkovnikov Ehud Altman Daw-Wei Wang Vladimir Gritsev Adilet Imambekov Ryan Barnett Mikhail Lukin Harvard/Boston University Harvard/Weizmann Harvard/Tsing-Hua University Harvard Harvard Harvard/Caltech Harvard

2 Strongly correlated electron systems

3 Conventional solid state materials Bloch theorem for non-interacting electrons in a periodic potential

4 Consequences of the Bloch theorem B V H Metals I d E F E F Insulators and Semiconductors First semiconductor transistor

5 Conventional solid state materials Electron-phonon and electron-electron interactions are irrelevant at low temperatures k y k x k F Landau Fermi liquid theory: when frequency and temperature are smaller than E F electron systems are equivalent to systems of non-interacting fermions Ag Ag Ag

6 Non Fermi liquid behavior in novel quantum materials UCu 3.5 Pd 1.5 Andraka, Stewart, PRB 47:3208 (93) Violation of the Wiedemann-Franz law in high Tc superconductors Hill et al., Nature 414:711 (2001) CeCu 2 Si 2. Steglich et al., Z. Phys. B 103:235 (1997)

7 Puzzles of high temperature superconductors Unusual normal state Resistivity, opical conductivity, Lack of sharply defined quasiparticles, Nernst effect Maple, JMMM 177:18 (1998) Mechanism of Superconductivity High transition temperature, retardation effect, isotope effect, role of elecron-electron and electron-phonon interactions Competing orders Role of magnetsim, stripes, possible fractionalization

8 Applications of quantum materials: High Tc superconductors

9 Applications of quantum materials: Ferroelectric RAM V FeRAM in Smart Cards Non-Volatile Memory High Speed Processing

10 Modeling strongly correlated systems using cold atoms

11 Bose-Einstein condensation Cornell et al., Science 269, 198 (1995) Ultralow density condensed matter system Interactions are weak and can be described theoretically from first principles

12 New Era in Cold Atoms Research Focus on Systems with Strong Interactions Feshbach resonances Rotating systems Low dimensional systems Atoms in optical lattices Systems with long range dipolar interactions

13 Feshbach resonance and fermionic condensates Greiner et al., Nature 426:537 (2003); Ketterle et al., PRL 91: (2003) Ketterle et al., Nature 435, (2005)

14 One dimensional systems 1D confinement in optical potential Weiss et al., Science (05); Bloch et al., Esslinger et al., One dimensional systems in microtraps. Thywissen et al., Eur. J. Phys. D. (99); Hansel et al., Nature (01); Folman et al., Adv. At. Mol. Opt. Phys. (02) Strongly interacting regime can be reached for low densities

15 Atoms in optical lattices Theory: Jaksch et al. PRL (1998) Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004); and many more

16 Strongly correlated systems Electrons in Solids Atoms in optical lattices Simple metals Perturbation theory in Coulomb interaction applies. Band structure methods wotk Strongly Correlated Electron Systems Band structure methods fail. Novel phenomena in strongly correlated electron systems: Quantum magnetism, phase separation, unconventional superconductivity, high temperature superconductivity, fractionalization of electrons

17 New Era in Cold Atoms Research Focus on Systems with Strong Interactions Goals Resolve long standing questions in condensed matter physics (e.g. origin of high temperature superconductivity) Resolve matter of principle questions (e.g. existence of spin liquids in two and three dimensions) Study new phenomena in strongly correlated systems (e.g. coherent far from equilibrium dynamics)

18 Outline Introduction. Cold atoms in optical lattices. Bose Hubbard model Two component Bose mixtures Quantum magnetism. Competing orders. Fractionalized phases Fermions in optical lattices Pairing in systems with repulsive interactions. High Tc mechanism Boson-Fermion mixtures Polarons. Competing orders Interference experiments with fluctuating BEC Analysis of correlations beyond mean-field Moving condensates in optical lattices Non equilibrium dynamics of interacting many-body systems Emphasis: detection and characterzation of many-body states

19 Atoms in optical lattices. Bose Hubbard model

20 Bose Hubbard model U t tunneling of atoms between neighboring wells repulsion of atoms sitting in the same well

21 Bose Hubbard model. Mean-field phase diagram µ U N=3 n + 1 N=2 Mott Mott Superfluid M.P.A. Fisher et al., PRB40:546 (1989) N=1 Mott 0 Superfluid phase Weak interactions Mott insulator phase Strong interactions

22 Bose Hubbard model Set. Hamiltonian eigenstates are Fock states 2 4 µ U

23 Bose Hubbard Model. Mean-field phase diagram µ U 4 2 N=3 n + 1 N=2 Mott Mott Superfluid N=1 Mott 0 Mott insulator phase Particle-hole excitation Tips of the Mott lobes

24 Gutzwiller variational wavefunction Normalization Interaction energy Kinetic energy z number of nearest neighbors

25 Phase diagram of the 1D Bose Hubbard model. Quantum Monte-Carlo study Batrouni and Scaletter, PRB 46:9051 (1992)

26 Optical lattice and parabolic potential µ U 4 N=3 n N=2 MI SF N=1 MI 0 Jaksch et al., PRL 81:3108 (1998)

27 Superfluid to Insulator transition Greiner et al., Nature 415:39 (2002) µ U Mott insulator Superfluid n 1 t/u

28 Time of flight experiments Quantum noise interferometry of atoms in an optical lattice Second order coherence

29 Second order coherence in the insulating state of bosons. Hanburry-Brown-Twiss experiment Theory: Altman et al., PRA 70:13603 (2004) Experiment: Folling et al., Nature 434:481 (2005)

30 Hanburry-Brown-Twiss stellar interferometer

31 Hanburry-Brown-Twiss interferometer

32 Second order coherence in the insulating state of bosons Bosons at quasimomentum expand as plane waves with wavevectors First order coherence: Oscillations in density disappear after summing over Second order coherence: Correlation function acquires oscillations at reciprocal lattice vectors

33 Second order coherence in the insulating state of bosons. Hanburry-Brown-Twiss experiment Theory: Altman et al., PRA 70:13603 (2004) Experiment: Folling et al., Nature 434:481 (2005)

34 Effect of parabolic potential on the second order coherence Experiment: Spielman, Porto, et al., Theory: Scarola, Das Sarma, Demler, cond-mat/ Width of the correlation peak changes across the transition, reflecting the evolution of Mott domains

35 Width of the noise peaks

36 Interference of an array of independent condensates Hadzibabic et al., PRL 93: (2004) Smooth structure is a result of finite experimental resolution (filtering)

37 Extended Hubbard Model - on site repulsion - nearest neighbor repulsion Checkerboard phase: Crystal phase of bosons. Breaks translational symmetry

38 Extended Hubbard model. Mean field phase diagram van Otterlo et al., PRB 52:16176 (1995) Hard core bosons. Supersolid superfluid phase with broken translational symmetry

39 Extended Hubbard model. Quantum Monte Carlo study Hebert et al., PRB 65:14513 (2002) Sengupta et al., PRL 94: (2005)

40 Dipolar bosons in optical lattices Goral et al., PRL88: (2002)

41 How to detect a checkerboard phase Correlation Function Measurements

42 Magnetism in condensed matter systems

43 Ferromagnetism Magnetic needle in a compass Magnetic memory in hard drives. Storage density of hundreds of billions bits per square inch.

44 Stoner model of ferromagnetism Spontaneous spin polarization decreases interaction energy but increases kinetic energy of electrons Mean-field criterion I N(0) = 1 I interaction strength N(0) density of states at the Fermi level

45 Antiferromagnetism Maple, JMMM 177:18 (1998) High temperature superconductivity in cuprates is always found near an antiferromagnetic insulating state

46 Antiferromagnetism Antiferromagnetic Heisenberg model AF = S = t = ( - ) ( + ) ( + ) AF = S t Antiferromagnetic state breaks spin symmetry. It does not have a well defined spin

47 Spin liquid states Alternative to classical antiferromagnetic state: spin liquid states Properties of spin liquid states: fractionalized excitations topological order gauge theory description Systems with geometric frustration?

48 Spin liquid behavior in systems with geometric frustration Kagome lattice Pyrochlore lattice SrCr 9-x Ga 3+x O 19 Ramirez et al. PRL (90) Broholm et al. PRL (90) Uemura et al. PRL (94) ZnCr 2 O 4 A 2 Ti 2 O 7 Ramirez et al. PRL (02)

49 Engineering magnetic systems using cold atoms in an optical lattice

50 Spin interactions using controlled collisions Experiment: Mandel et al., Nature 425:937(2003) Theory: Jaksch et al., PRL 82:1975 (1999)

51 Effective spin interaction from the orbital motion. Cold atoms in Kagome lattices Santos et al., PRL 93:30601 (2004) Damski et al., PRL 95:60403 (2005)

52 Two component Bose mixture in optical lattice Example:. Mandel et al., Nature 425:937 (2003) t t Two component Bose Hubbard model

53 Quantum magnetism of bosons in optical lattices Kuklov and Svistunov, PRL (2003) Duan et al., PRL (2003) Ferromagnetic Antiferromagnetic

54 Exchange Interactions in Solids antibonding bonding Kinetic energy dominates: antiferromagnetic state Coulomb energy dominates: ferromagnetic state

55 Two component Bose mixture in optical lattice. Mean field theory + Quantum fluctuations Hysteresis Altman et al., NJP 5:113 (2003) 1 st order 2 nd order line

56 Coherent spin dynamics in optical lattices Widera et al., cond-mat/ atoms in the F=2 state

57 How to observe antiferromagnetic order of cold atoms in an optical lattice?

58 Second order coherence in the insulating state of bosons. Hanburry-Brown-Twiss experiment Theory: Altman et al., PRA 70:13603 (2004) See also Bach, Rzazewski, PRL 92: (2004) Experiment: Folling et al., Nature 434:481 (2005) See also Hadzibabic et al., PRL 93: (2004)

59 Probing spin order of bosons Correlation Function Measurements

60 Engineering exotic phases Optical lattice in 2 or 3 dimensions: polarizations & frequencies of standing waves can be different for different directions YY ZZ Example: exactly solvable model Kitaev (2002), honeycomb lattice with H = J x σ ix σ x j + J y σ iy σ y z j + J z σ iz σ j <i, j> x <i, j> y <i, j> z Can be created with 3 sets of standing wave light beams! Non-trivial topological order, spin liquid + non-abelian anyons those has not been seen in controlled experiments

61 Fermionic atoms in optical lattices Pairing in systems with repulsive interactions. Unconventional pairing. High Tc mechanism

62 Fermionic atoms in a three dimensional optical lattice Kohl et al., PRL 94:80403 (2005)

63 Fermions with attractive interaction Hofstetter et al., PRL 89: (2002) U t t Highest transition temperature for Compare to the exponential suppresion of Tc w/o a lattice

64 Reaching BCS superfluidity in a lattice Turning on the lattice reduces the effective atomic temperature K in NdYAG lattice 40 K 6 Li Li in CO 2 lattice Superfluidity can be achived even with a modest scattering length

65 Fermions with repulsive interactions U t Possible d-wave pairing of fermions t

66 High temperature superconductors Superconducting Tc 93 K Picture courtesy of UBC Superconductivity group Hubbard model minimal model for cuprate superconductors P.W. Anderson, cond-mat/ After many years of work we still do not understand the fermionic Hubbard model

67 Positive U Hubbard model Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995) Antiferromagnetic insulator D-wave superconductor

68 Second order correlations in the BCS superfluid n(k) n(r ) k F BCS k n(r) BEC n( r, r') n( r) n( r') n( r, r) Ψ = BCS 0

69 Momentum correlations in paired fermions Greiner et al., PRL 94: (2005)

70 Fermion pairing in an optical lattice Second Order Interference In the TOF images Normal State Superfluid State measures the Cooper pair wavefunction One can identify unconventional pairing

71 Boson Fermion mixtures Fermions interacting with phonons. Polarons. Competing orders

72 Boson Fermion mixtures Experiments: ENS, Florence, JILA, MIT, Rice, BEC Bosons provide cooling for fermions and mediate interactions. They create non-local attraction between fermions Charge Density Wave Phase Periodic arrangement of atoms Non-local Fermion Pairing P-wave, D-wave,

73 Boson Fermion mixtures Phonons : Bogoliubov (phase) mode 4 Effective fermion- phonon interaction ω/e F Fermion- phonon vertex Similar to electron-phonon systems q/k F

74 Boson Fermion mixtures in 1d optical lattices Cazalila et al., PRL (2003); Mathey et al., PRL (2004) Spinless fermions Spin ½ fermions Note: Luttinger parameters can be determined using correlation function measurements in the time of flight experiments. Altman et al. (2005)

75 BF mixtures in 2d optical lattices Wang, Lukin, Demler, PRA (1972) 40K -- 87Rb 40K -- 23Na (a) (b) =1060nm =765.5nm =1060 nm

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77 Systems of cold atoms with strong interactions and correlations Goals Resolve long standing questions in condensed matter physics (e.g. origin of high temperature superconductivity) Resolve matter of principle questions (e.g. existence of spin liquids in two and three dimensions) Study new phenomena in strongly correlated systems Interference experiments with fluctuating BEC Analysis of high order correlation functions in low dimensional systems Moving condensates in optical lattices Non equilibrium dynamics of interacting many-body systems

78 Interference experiments with fluctuating BEC Analysis of high order correlation functions in low dimensional systems

79 Interference of two independent condensates Andrews et al., Science 275:637 (1997)

80 Interference of two independent condensates r r 1 r+d d 2 Clouds 1 and 2 do not have a well defined phase difference. However each individual measurement shows an interference pattern

81 Interference of one dimensional condensates d Experiments: Schmiedmayer et al., Nature Physics (2005) x 1 Amplitude of interference fringes,, contains information about phase fluctuations within individual condensates x 2 x y

82 Interference amplitude and correlations Polkovnikov, Altman, Demler, PNAS (2006) L For identical condensates Instantaneous correlation function

83 Interference between Luttinger liquids Luttinger liquid at T=0 K Luttinger parameter L For non-interacting bosons and For impenetrable bosons and Luttinger liquid at finite temperature Analysis of can be used for thermometry

84 Rotated probe beam experiment For large imaging angle,, θ Luttinger parameter K may be extracted from the angular dependence of

85 Interference between two-dimensional BECs at finite temperature. Kosteritz-Thouless transition

86 Interference of two dimensional condensates Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/ Gati, Oberthaler, et al., cond-mat/ L x L y L x Probe beam parallel to the plane of the condensates

87 Interference of two dimensional condensates. Quasi long range order and the KT transition L x L y Theory: Polkovnikov, Altman, Demler, PNAS (2006) Above KT transition Below KT transition

88 Experiments with 2D Bose gas Haddzibabic et al., Nature (2006) z Time of flight x Typical interference patterns low temperature higher temperature

89 Experiments with 2D Bose gas Hadzibabic et al., Nature (2006) z x integration over x axis z 0.4 Contrast after integration integration over x axis integration z 0.2 middle T low T high T D x over x axis z integration distance D x (pixels)

90 Integrated contrast Experiments with 2D Bose gas Hadzibabic et al., Nature (2006) 0.4 low T middle T 0.2 high T fit by: 0.5 C Dx ~ [ g1(0, x) ] dx ~ 1 Dx Dx Exponent α 2α integration distance D x if g 1 (r) decays exponentially with : if g 1 (r) decays algebraically or exponentially with a large : high T low T central contrast Sudden jump!?

91 Experiments with 2D Bose gas Hadzibabic et al., Nature (2006) Exponent α c.f. Bishop and Reppy T (K) high T low T central contrast He experiments: universal jump in the superfluid density Ultracold atoms experiments: jump in the correlation function. KT theory predicts α=1/4 just below the transition

92 Experiments with 2D Bose gas. Proliferation of thermal vortices Haddzibabic et al., Nature (2006) 30% 20% Fraction of images showing at least one dislocation Exponent α % 0 high T low T central contrast central contrast The onset of proliferation coincides with α shifting to 0.5! Z. Hadzibabic et al., Nature (2006)

93 Interference between two interacting one dimensional Bose liquids Full distribution function of the interference amplitude Gritsev, Altman, Demler, Polkovnikov, cond-mat/

94 Higher moments of interference amplitude is a quantum operator. The measured value of will fluctuate from shot to shot. Can we predict the distribution function of? L Higher moments Changing to periodic boundary conditions (long condensates) Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)

95 Impurity in a Luttinger liquid Expansion of the partition function in powers of g Partition function of the impurity contains correlation functions taken at the same point and at different times. Moments of interference experiments come from correlations functions taken at the same time but in different points. Euclidean invariance ensures that the two are the same

96 Relation between quantum impurity problem and interference of fluctuating condensates Normalized amplitude of interference fringes Distribution function of fringe amplitudes Relation to the impurity partition function Distribution function can be reconstructed from using completeness relations for the Bessel functions

97 Bethe ansatz solution for a quantum impurity can be obtained from the Bethe ansatz following Zamolodchikov, Phys. Lett. B 253:391 (91); Fendley, et al., J. Stat. Phys. 79:799 (95) Making analytic continuation is possible but cumbersome Interference amplitude and spectral determinant is related to the single particle Schroedinger equation Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999) Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001) Spectral determinant

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99 Evolution of the distribution function Probability P(x) K=1 K=1.5 K=3 K=5 Narrow distribution for. Approaches Gumble distribution. Width Wide Poissonian distribution for x

100 From interference amplitudes to conformal field theories correspond to vacuum eigenvalues of Q operators of CFT Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999 When K>1, is related to Q operators of CFT with c<0. This includes 2D quantum gravity, nonintersecting loop model on 2D lattice, growth of random fractal stochastic interface, high energy limit of multicolor QCD, 2D quantum gravity, non-intersecting loops on 2D lattice Yang-Lee singularity

101 Moving condensates in optical lattices Non equilibrium coherent dynamics of interacting many-body systems

102 Atoms in optical lattices. Bose Hubbard model Theory: Jaksch et al. PRL 81:3108(1998) Experiment: Kasevich et al., Science (2001) Greiner et al., Nature (2001) Cataliotti et al., Science (2001) Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004),

103 Equilibrium superfluid to insulator transition µ U Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98) Experiment: Greiner et al. Nature (01) Mott insulator Superfluid n 1 t/u

104 Moving condensate in an optical lattice. Dynamical instability Theory: Niu et al. PRA (01), Smerzi et al. PRL (02) Experiment: Fallani et al. PRL (04) v Related experiments by Eiermann et al, PRL (03)

105 This discussion: How to connect the dynamical instability (irreversible, classical) to the superfluid to Mott transition (equilibrium, quantum) p π/2 Unstable Stable??? This discussion SF MI U/J Possible experimental sequence: p??? U/t SF MI

106 Superconductor to Insulator transition in thin films Bi films d Superconducting films of different thickness Marcovic et al., PRL 81:5217 (1998)

107 Dynamical instability Classical limit of the Hubbard model. Discreet Gross-Pitaevskii equation Current carrying states Linear stability analysis: States with p>p/2 are unstable unstable unstable Amplification of density fluctuations r

108 Dynamical instability for integer filling Order parameter for a current carrying state Current GP regime. Maximum of the current for. When we include quantum fluctuations, the amplitude of the order parameter is suppressed decreases with increasing phase gradient

109 Dynamical instability for integer filling ρ s (p) sin(p) p π/2 I(p) p * Condensate momentum p/π Vicinity of the SF-I quantum phase transition. Classical description applies for SF U/J MI Dynamical instability occurs for

110 Dynamical instability. Gutzwiller approximation Wavefunction Time evolution We look for stability against small fluctuations 0.5 unstable Phase diagram. Integer filling p/π d=1 d=2 d=3 0.2 stable U/U c

111 Optical lattice and parabolic trap. Gutzwiller approximation Center of Mass Momentum Time N=1.5 N=3 The first instability develops near the edges, where N=1 U=0.01 t J=1/4 Gutzwiller ansatz simulations (2D)

112 Beyond semiclassical equations. Current decay by tunneling phase j phase j phase j Current carrying states are metastable. They can decay by thermal or quantum tunneling Thermal activation Quantum tunneling

113 Decay of current by quantum tunneling phase j phase Quantum phase slip j Escape from metastable state by quantum tunneling. WKB approximation S classical action corresponding to the motion in an inverted potential.

114 Decay rate from a metastable state. Example 2 τ 1 dx + ( ) 0 0 c 2m dτ S dτ ε x bx ε p p

115 For d>1 we have to include transverse directions. Need to excite many chains to create a phase slip J J cos p, J = J Longitudinal stiffness is much smaller than the transverse. The transverse size of the phase slip diverges near a phase slip. We can use continuum approximation to treat transverse directions

116 Weakly interacting systems. Gross-Pitaevskii regime. Decay of current by quantum tunneling p π/2 SF U/J MI Fallani et al., PRL (04) Quantum phase slips are strongly suppressed in the GP regime

117 Strongly interacting regime. Vicinity of the SF-Mott transition Close to a SF-Mott transition we can use an effective relativistivc GL theory (Altman, Auerbach, 2004) p π/2 SF U/J M I Metastable current carrying state: 1 This state becomes unstable at p c = ξ 3 corresponding to the maximum of the current: ψ = 1 p ( ξ ) I p ψ = p 1 p. 2 2 ip x ξ e ξ

118 Strongly interacting regime. Vicinity of the SF-Mott transition Decay of current by quantum tunneling p π/2 SF U/J MI Action of a quantum phase slip in d=1,2,3 - correlation length Strong broadening of the phase transition in d=1 and d=2 is discontinuous at the transition. Phase slips are not important. Sharp phase transition

119 Decay of current by quantum tunneling

120 0.5 unstable p/π d=1 stable d=2 d= U/U c

121 Decay of current by thermal activation phase j phase Thermal phase slip j E Escape from metastable state by thermal activation

122 Thermally activated current decay. Weakly interacting regime E Thermal phase slip Activation energy in d=1,2,3 Thermal fluctuations lead to rapid decay of currents Crossover from thermal to quantum tunneling

123 Decay of current by thermal fluctuations Phys. Rev. Lett. (2004)

124 Conclusions We understand well: electron systems in semiconductors and simple metals. Interaction energy is smaller than the kinetic energy. Perturbation theory works We do not understand: strongly correlated electron systems in novel materials. Interaction energy is comparable or larger than the kinetic energy. Many surprising new phenomena occur, including high temperature superconductivity, magnetism, fractionalization of excitations Ultracold atoms have energy scales of 10-6 K, compared to 10 4 K for electron systems. However, by engineering and studying strongly interacting systems of cold atoms we should get insights into the mysterious properties of novel quantum materials Our big goal is to develop a general framework for understanding strongly correlated systems. This will be important far beyond AMO and condensed matter

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