Subir Sachdev Harvard University

Transcription

1 Quantum phase transitions of correlated electrons and atoms Subir Sachdev Harvard University Course at Harvard University: Physics 268r Classical and Quantum Phase Transitions. MWF 10 in Jefferson 256 First meeting: Feb 1. Quantum Phase Transitions Cambridge University Press

2 Outline A. Magnetic quantum phase transitions in dimerized Mott insulators Landau-Ginzburg-Wilson (LGW) theory D. Boson-vortex duality Breakdown of the LGW paradigm B. Mott insulators with spin S=1/2 per unit cell 1. Berry phases and the mapping to a compact U(1) gauge theory 2. Valence-bond-solid (VBS) order in the paramagnet; 3. Mapping to hard-core bosons at half-filling C. The superfluid-insulator transition of bosons in lattices Multiple order parameters in quantum systems

3 A. Magnetic quantum phase transitions in dimerized Mott insulators: Landau-Ginzburg-Wilson (LGW) theory: Second-order phase transitions described by fluctuations of an order parameter associated with a broken symmetry

4 TlCuCl 3 M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, cond-mat/

5 Coupled Dimer Antiferromagnet M. P. Gelfand, R. R. P. Singh, and D. A. Huse, Phys. Rev. B 40, (1989). N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 4529 (1994). J. Tworzydlo, O. Y. Osman, C. N. A. van Duin, J. Zaanen, Phys. Rev. B 59, 115 (1999). M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, (2002). S=1/2 spins on coupled dimers H = < ij> J ij S i S j 0 λ 1 J λ J

6

7 λ close to 0 Weakly coupled dimers

8 λ close to 0 Weakly coupled dimers 1 = 2 ( ) Paramagnetic ground state = 0, ϕ = 0 Si

9 λ close to 0 Weakly coupled dimers 1 = 2 ( ) Excitation: S=1 triplon

10 λ close to 0 Weakly coupled dimers 1 = 2 ( ) Excitation: S=1 triplon

11 λ close to 0 Weakly coupled dimers 1 = 2 ( ) Excitation: S=1 triplon

12 λ close to 0 Weakly coupled dimers 1 = 2 ( ) Excitation: S=1 triplon

13 λ close to 0 Weakly coupled dimers 1 = 2 ( ) Excitation: S=1 triplon

14 λ close to 0 Weakly coupled dimers 1 = 2 ( ) Excitation: S=1 triplon (exciton, spin collective mode) Energy dispersion away from antiferromagnetic wavevector ε spin gap p = + c p x x y y 2 c p

15 TlCuCl 3 triplon N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B (2001). For quasi-one-dimensional systems, the triplon linewidth takes kte / the exact universal value 1.20 kt B B at low T = K. Damle and S. Sachdev, Phys. Rev. B 57, 8307 (1998) This result is in good agreement with observations in CsNiCl 3 (M. Kenzelmann, R. A. Cowley, W. J. L. Buyers, R. Coldea, M. Enderle, and D. F. McMorrow Phys. Rev. B 66, (2002)) and Y 2 NiBaO 5 (G. Xu, C. Broholm, G. Aeppli, J. F. DiTusa, T.Ito, K. Oka, and H. Takagi, preprint).

16 Coupled Dimer Antiferromagnet

17 λ close to 1 Weakly dimerized square lattice

18 λ close to 1 Weakly dimerized square lattice Excitations: 2 spin waves (magnons) ε = c p + c p p x x y y Ground state has long-range spin density wave (Néel) order at wavevector K= (π,π) ϕ 0 Si spin density wave order parameter: ϕ = ηi ; ηi = ± 1 on two sublattices S

19 TlCuCl 3 J. Phys. Soc. Jpn 72, 1026 (2003)

20 T=0 λ c = (3) M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, (2002) Néel state ϕ 0 Quantum paramagnet ϕ = 0 λ 1 λ c Pressure in TlCuCl 3 The method of bond operators (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) provides a quantitative description of spin excitations in TlCuCl 3 across the quantum phase transition (M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, Phys. Rev. Lett. 89, (2002))

21 LGW theory for quantum criticality Landau-Ginzburg-Wilson theory: write down an effective action for the antiferromagnetic order parameter ϕ by expanding in powers of ϕ and its spatial and temporal derivatives, while preserving all symmetries of the microscopic Hamiltonian S ϕ 2 1 ( ) 2 1 ( ) 2 ( ) 2 u ( 2 d xdτ ) 2 = xϕ + 2 τ ϕ + λc λ ϕ + ϕ 2 c 4!

22 S φα Quantum field theory for critical point λ close to λ c : use soft spin field 1 u = d xdτ φ + φ + λ λ φ + φ 2 4! (( ) 2 c ( ) 2 ( ) ) ( ) 2 α τ α α α b x c 3-component antiferromagnetic order parameter Oscillations of φ α about zero (for λ < λ c ) spin-1 collective mode Im χ ( p, ω ) ε p = + c p T=0 spectrum = λ λ c c ω

23 Dynamic spectrum at the critical point Critical coupling ( λ = λ c ) ( p ω ) Im χ, ( ω ) ~ c p (2 η )/2 cp ω No quasiparticles --- dissipative critical continuum

24 Outline A. Magnetic quantum phase transitions in dimerized Mott insulators Landau-Ginzburg-Wilson (LGW) theory D. Boson-vortex duality Breakdown of the LGW paradigm B. Mott insulators with spin S=1/2 per unit cell 1. Berry phases and the mapping to a compact U(1) gauge theory 2. Valence-bond-solid (VBS) order in the paramagnet; 3. Mapping to hard-core bosons at half-filling C. The superfluid-insulator transition of bosons in lattices Multiple order parameters in quantum systems

25 B. Mott insulators with spin S=1/2 per unit cell: 1. Berry phases and the mapping to a compact U(1) gauge theory.

26 Recall: dimerized Mott insulators λ c = (3) M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, (2002) Néel state ϕ 0 Quantum paramagnet ϕ = 0 λ 1 λ c Pressure in TlCuCl 3

27 Mott insulator with two S=1/2 spins per unit cell

28 Mott insulator with one S=1/2 spin per unit cell

29 Mott insulator with one S=1/2 spin per unit cell Ground state has Neel order with ϕ 0

30 Mott insulator with one S=1/2 spin per unit cell Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange. The strength of this perturbation is measured by a coupling g. Small g ground state has Neel order with ϕ 0 Large g paramagnetic ground state with ϕ = 0

31 Mott insulator with one S=1/2 spin per unit cell Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange. The strength of this perturbation is measured by a coupling g. Small g ground state has Neel order with ϕ 0 Large g paramagnetic ground state with ϕ = 0

32 LGW theory for such a quantum transition S ϕ 2 1 ( ) 2 1 ( ) 2 2 u ( 2 d xdτ ) 2 = xϕ + r 2 τ ϕ + ϕ + ϕ 2 c 4! What is the state with ϕ = 0? The field theory predicts that this state has no broken symmetries and has a stable S=1 quasiparticle excitation (the triplon)

33 Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries

34 Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries Liquid of valence bonds has fractionalized S=1/2 excitations

35 Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries Liquid of valence bonds has fractionalized S=1/2 excitations

36 Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries Liquid of valence bonds has fractionalized S=1/2 excitations

37 Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries Liquid of valence bonds has fractionalized S=1/2 excitations

38 Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries Liquid of valence bonds has fractionalized S=1/2 excitations

39 Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries Liquid of valence bonds has fractionalized S=1/2 excitations

40 Quantum theory for destruction of Neel order Ingredient missing from LGW theory: Spin Berry Phases Coherent state path integral for a single spin Z = Tr ( H[ S] / T e ) ( is A d d H S ) D N τ δ N 1 exp ( τ) τ τ N τ ( ) ( 2 = ) ( ) τ ( ) Aτ τ dτ = Oriented area of triangle on surface of unit sphere bounded ( τ) ( τ + dτ) 0 by N, N,and a fixed reference N See Chapter 13 of Quantum Phase Transitions, S. Sachdev, Cambridge University Press (1999).

41 Quantum theory for destruction of Neel order Ingredient missing from LGW theory: Spin Berry Phases A isa e

42 Quantum theory for destruction of Neel order Ingredient missing from LGW theory: Spin Berry Phases A isa e

43 Quantum theory for destruction of Neel order

44 Quantum theory for destruction of Neel order Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a

45 Quantum theory for destruction of Neel order Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a Recall ϕa = 2ηaS a ϕa= (0,0,1) in classical Neel state; η ± 1 on two square sublattices ; a ϕ 0 ϕ a ϕa+µ ( µ = x, y, τ )

46 Quantum theory for destruction of Neel order Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a Recall ϕa = 2ηaS a ϕa= (0,0,1) in classical Neel state; ηa ± 1 on two square sublattices ; Aa µ half oriented area of spherical triangle formed by ϕ, ϕ, and an arbitrary reference point ϕ a a+ µ 0 ϕ 0 2A aµ ϕ a ϕ a+µ S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

47 Quantum theory for destruction of Neel order Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a Recall ϕa = 2ηaS a ϕa= (0,0,1) in classical Neel state; ηa ± 1 on two square sublattices ; Aa µ half oriented area of spherical triangle formed by ϕ, ϕ, and an arbitrary reference point ϕ a a+ µ 0 ϕ 0 ϕ a ϕ a+µ S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

48 Quantum theory for destruction of Neel order Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a Recall ϕa = 2ηaS a ϕa= (0,0,1) in classical Neel state; ηa ± 1 on two square sublattices ; Aa µ half oriented area of spherical triangle formed by ϕ, ϕ, and an arbitrary reference point ϕ a a+ µ 0 γ a ϕ 0 ϕ a ϕ a+µ S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

49 Quantum theory for destruction of Neel order Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a Recall ϕa = 2ηaS a ϕa= (0,0,1) in classical Neel state; ηa ± 1 on two square sublattices ; Aa µ half oriented area of spherical triangle formed by ϕ, ϕ, and an arbitrary reference point ϕ a a+ µ 0 ϕ 0 γ a γ a + µ ϕ a ϕ a+µ S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

50 Quantum theory for destruction of Neel order Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a Recall ϕa = 2ηaS a ϕa= (0,0,1) in classical Neel state; ηa ± 1 on two square sublattices ; Aa µ half oriented area of spherical triangle formed by ϕ, ϕ, and an arbitrary reference point ϕ a a+ µ 0 2A 2A γ + γ aµ aµ a+ µ a Change in choice of ϕ0 is like a gauge transformation γ a ϕ 0 ϕ a γ a + µ ϕ a+µ S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

51 Quantum theory for destruction of Neel order Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a Recall ϕa = 2ηaS a ϕa= (0,0,1) in classical Neel state; ηa ± 1 on two square sublattices ; Aa µ half oriented area of spherical triangle formed by ϕ, ϕ, and an arbitrary reference point ϕ a a+ µ 0 2A 2A γ + γ aµ aµ a+ µ a Change in choice of ϕ0 is like a gauge transformation γ a ϕ 0 ϕ a γ a + µ ϕ a+µ The area of the triangle is uncertain modulo 4π, and the action has to be invariant under A A + 2 aµ aµ π S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

52 Quantum theory for destruction of Neel order Ingredient missing from LGW theory: Spin Berry Phases exp i ηa a a A τ static Sum of Berry phases of all spins on the square lattice. = exp i Jaµ Aaµ a, µ with "current" J aµ of charges ± 1 on sublattices

53 Quantum theory for destruction of Neel order Partition function on cubic lattice ( 2 ) 1 Z = dϕδ ϕ 1exp ϕ ϕ + i η A a a a a a+ µ a aτ g a, µ a Modulus of weights in partition function: those of a classical ferromagnet at a temperature g Small g ground state has Neel order with ϕ 0 Large g paramagnetic ground state with ϕ = 0 Berry phases lead to large cancellations between different time histories need an effective action for Aa µ at large g S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

54 Simplest large g effective action for the A aµ 1 ( ) Z = daaµ exp cos 2 µ Aa ν ν Aaµ + i ηaaa τ a, 2e µ a 2 2 with e ~ g This is compact QED in 3 spacetime dimensions with static charges ± 1 on two sublattices. Analysis by a duality mapping shows that this gauge theory has valence bond solid (VBS) order in the ground state for all e N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990). S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

55 B. Mott insulators with spin S=1/2 per unit cell: 1. Berry phases and the mapping to a compact U(1) gauge theory. 2. Valence bond solid (VBS) order in the paramagnet.

56 Another possible state, with ϕ = 0, is the valence bond solid (VBS)

57 Another possible state, with ϕ = 0, is the valence bond solid (VBS) Ψ vbs Such a state breaks the symmetry of rotations by nπ / 2 about lattice sites, and has Ψvbs 0, where Ψvbs is the VBS order parameter iarctan( r ) vbs () i S j r Ψ = i iisje ij

58 Another possible state, with ϕ = 0, is the valence bond solid (VBS) Ψ vbs Such a state breaks the symmetry of rotations by nπ / 2 about lattice sites, and has Ψvbs 0, where Ψvbs is the VBS order parameter iarctan( r ) vbs () i S j r Ψ = i iisje ij

59 Another possible state, with ϕ = 0, is the valence bond solid (VBS) Ψ vbs Such a state breaks the symmetry of rotations by nπ / 2 about lattice sites, and has Ψvbs 0, where Ψvbs is the VBS order parameter iarctan( r ) vbs () i S j r Ψ = i iisje ij

60 Another possible state, with ϕ = 0, is the valence bond solid (VBS) Ψ vbs Such a state breaks the symmetry of rotations by nπ / 2 about lattice sites, and has Ψvbs 0, where Ψvbs is the VBS order parameter iarctan( r ) vbs () i S j r Ψ = i iisje ij

61 Another possible state, with ϕ = 0, is the valence bond solid (VBS) Ψ vbs Such a state breaks the symmetry of rotations by nπ / 2 about lattice sites, and has Ψvbs 0, where Ψvbs is the VBS order parameter iarctan( r ) vbs () i S j r Ψ = i iisje ij

62 Another possible state, with ϕ = 0, is the valence bond solid (VBS) Ψ vbs Such a state breaks the symmetry of rotations by nπ / 2 about lattice sites, and has Ψvbs 0, where Ψvbs is the VBS order parameter iarctan( r ) vbs () i S j r Ψ = i iisje ij

63 Another possible state, with ϕ = 0, is the valence bond solid (VBS) Ψ vbs Such a state breaks the symmetry of rotations by nπ / 2 about lattice sites, and has Ψvbs 0, where Ψvbs is the VBS order parameter iarctan( r ) vbs () i S j r Ψ = i iisje ij

64 Another possible state, with ϕ = 0, is the valence bond solid (VBS) Ψ vbs Such a state breaks the symmetry of rotations by nπ / 2 about lattice sites, and has Ψvbs 0, where Ψvbs is the VBS order parameter iarctan( r ) vbs () i S j r Ψ = i iisje ij

65 Another possible state, with ϕ = 0, is the valence bond solid (VBS) Ψ vbs Such a state breaks the symmetry of rotations by nπ / 2 about lattice sites, and has Ψvbs 0, where Ψvbs is the VBS order parameter iarctan( r ) vbs () i S j r Ψ = i iisje ij

66 Another possible state, with ϕ = 0, is the valence bond solid (VBS) Ψ vbs Such a state breaks the symmetry of rotations by nπ / 2 about lattice sites, and has Ψvbs 0, where Ψvbs is the VBS order parameter iarctan( r ) vbs () i S j r Ψ = i iisje ij

67 The VBS state does have a stable S=1 quasiparticle excitation Ψ vbs 0, ϕ = 0

68 The VBS state does have a stable S=1 quasiparticle excitation Ψ vbs 0, ϕ = 0

69 The VBS state does have a stable S=1 quasiparticle excitation Ψ vbs 0, ϕ = 0

70 The VBS state does have a stable S=1 quasiparticle excitation Ψ vbs 0, ϕ = 0

71 The VBS state does have a stable S=1 quasiparticle excitation Ψ vbs 0, ϕ = 0

72 The VBS state does have a stable S=1 quasiparticle excitation Ψ vbs 0, ϕ = 0

73 Ordering by quantum fluctuations

74 Ordering by quantum fluctuations

75 Ordering by quantum fluctuations

76 Ordering by quantum fluctuations

77 Ordering by quantum fluctuations

78 Ordering by quantum fluctuations

79 Ordering by quantum fluctuations

80 Ordering by quantum fluctuations

81 Ordering by quantum fluctuations

82 ( 2 ) 1 Z = dϕδ ϕ 1exp ϕ ϕ + i η A a a a a a+ µ a aτ g a, µ a or 0 Neel order ϕ 0? Bond order Ψ vbs 0 Not present in LGW theory of ϕ order g

83 LGW theory of multiple order parameters F = F Ψ + F + F [ ] int [ ] [ ϕ] vbs vbs ϕ int vbs 2 4 Fvbs Ψ vbs = r1 Ψ vbs + u1 Ψ vbs + [ ] 2 4 Fϕ ϕ = r2 ϕ + u2 ϕ F = v Ψ ϕ + Distinct symmetries of order parameters permit couplings only between their energy densities

84 ϕ LGW theory of multiple order parameters First order transition Ψ vbs ϕ Neel order VBS order Coexistence g Ψ vbs Neel order VBS order g ϕ Neel order "disordered" Ψ vbs VBS order g

85 ϕ LGW theory of multiple order parameters First order transition Ψ vbs ϕ Neel order VBS order Coexistence g Ψ vbs Neel order VBS order g ϕ Neel order "disordered" Ψ vbs VBS order g

86 B. Mott insulators with spin S=1/2 per unit cell: 1. Berry phases and the mapping to a compact U(1) gauge theory. 2. Valence bond solid (VBS) order in the paramagnet. 3. Mapping to hard-core bosons at half-filling.

87

88 Outline A. Magnetic quantum phase transitions in dimerized Mott insulators Landau-Ginzburg-Wilson (LGW) theory D. Boson-vortex duality Breakdown of the LGW paradigm B. Mott insulators with spin S=1/2 per unit cell 1. Berry phases and the mapping to a compact U(1) gauge theory 2. Valence-bond-solid (VBS) order in the paramagnet; 3. Mapping to hard-core bosons at half-filling C. The superfluid-insulator transition of bosons in lattices Multiple order parameters in quantum systems

89 B. Superfluid-insulator transition 1. Bosons in a lattice at integer filling

90 Bose condensation Velocity distribution function of ultracold 87 Rb atoms M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science 269, 198 (1995)

91 Apply a periodic potential (standing laser beams) to trapped ultracold bosons ( 87 Rb)

92 Momentum distribution function of bosons Bragg reflections of condensate at reciprocal lattice vectors M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

93 Superfluid-insulator quantum phase transition at T=0 V 0 =0E r V 0 =3E r V 0 =7E r V 0 =10E r V 0 =13E r V 0 =14E r V 0 =16E r V 0 =20E r

94 Bosons at filling fraction f = 1 Weak interactions: superfluidity Strong interactions: Mott insulator which preserves all lattice symmetries M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

95 Bosons at filling fraction f = 1 Ψ 0 Weak interactions: superfluidity

96 Bosons at filling fraction f = 1 Ψ 0 Weak interactions: superfluidity

97 Bosons at filling fraction f = 1 Ψ 0 Weak interactions: superfluidity

98 Bosons at filling fraction f = 1 Ψ 0 Weak interactions: superfluidity

99 Bosons at filling fraction f = 1 Ψ = 0 Strong interactions: insulator

100 The Superfluid-Insulator transition Degrees of freedom: Bosons, i j j j j, hopping between the sites, j, of a lattice, with short-range repulsive interactions. U H = t bb- µ n + nj ( n j 1) + 2 ij Boson Hubbard model n b b b j j j j M.PA. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher Phys. Rev. B 40, 546 (1989). For small U/t, ground state is a superfluid BEC with superfluid density density of bosons

101 What is the ground state for large U/t? Typically, the ground state remains a superfluid, but with superfluid density density of bosons The superfluid density evolves smoothly from large values at small U/t, to small values at large U/t, and there is no quantum phase transition at any intermediate value of U/t. (In systems with Galilean invariance and at zero temperature, superfluid density=density of bosons always, independent of the strength of the interactions)

102 What is the ground state for large U/t? Incompressible, insulating ground states, with zero superfluid density, appear at special commensurate densities n j = 3 t U n = 7/2 Ground state has density wave order, which j spontaneously breaks lattice symmetries

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104

105 B. Superfluid-insulator transition 2. Bosons in a lattice at fractional filling L. Balents, L. Bartosch, A. Burkov, S. Sachdev, K. Sengupta, Physical Review B 71, and (2005), cond-mat/ , and cond-mat/

106 Bosons at filling fraction f = 1/2 Ψ 0 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

107 Bosons at filling fraction f = 1/2 Ψ 0 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

108 Bosons at filling fraction f = 1/2 Ψ 0 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

109 Bosons at filling fraction f = 1/2 Ψ 0 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

110 Bosons at filling fraction f = 1/2 Ψ 0 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

111 Bosons at filling fraction f = 1/2 Ψ = 0 Strong interactions: insulator C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

112 Bosons at filling fraction f = 1/2 Ψ = 0 Strong interactions: insulator C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

113 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

114 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

115 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

116 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

117 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

118 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

119 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

120 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

121 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

122 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

123 Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order using a generalized "density" ρ All insulators have Ψ = 0 and ρ 0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Q i. ( r ) ρ e = Qr Q Q

124 Ginzburg-Landau-Wilson approach to multiple order parameters: [ ] F = F [ ] sc Ψ sc + F charge ρ Q + F 2 4 F Ψ = r Ψ + u Ψ + sc sc 1 sc 1 sc int sc 2 4 Fcharge ρq = r2 ρq + u2 ρq + F 2 = v Ψ ρ + Q 2 int Distinct symmetries of order parameters permit couplings only between their energy densities S. Sachdev and E. Demler, Phys. Rev. B 69, (2004).

125 Ψ sc Superconductor Ψ sc Superconductor Predictions of LGW theory First order transition Coexistence (Supersolid) ρ Q Charge-ordered insulator ρ Q r 1 2 r Charge-ordered insulator r r 1 2 Ψ sc Superconductor " Disordered " ( topologically ordered) Ψ = 0, ρ = 0 sc Q ρ Q Charge-ordered insulator r r 1 2

126 Ψ sc Superconductor Ψ sc Superconductor Predictions of LGW theory First order transition Coexistence (Supersolid) ρ Q Charge-ordered insulator ρ Q r 1 2 r Charge-ordered insulator r r 1 2 Ψ sc Superconductor " Disordered " ( topologically ordered) Ψ = 0, ρ = 0 sc Q ρ Q Charge-ordered insulator r r 1 2

127 Outline A. Magnetic quantum phase transitions in dimerized Mott insulators Landau-Ginzburg-Wilson (LGW) theory D. Boson-vortex duality Breakdown of the LGW paradigm B. Mott insulators with spin S=1/2 per unit cell 1. Berry phases and the mapping to a compact U(1) gauge theory 2. Valence-bond-solid (VBS) order in the paramagnet; 3. Mapping to hard-core bosons at half-filling C. The superfluid-insulator transition of bosons in lattices Multiple order parameters in quantum systems

128 D. Boson-vortex duality 1. Bosons in a lattice at integer filling

129 Bosons at density f = 1 Weak interactions: superfluidity Strong interactions: Mott insulator which preserves all lattice symmetries M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

130 Approaching the transition from the insulator (f=1) Excitations of the insulator: Particles ~ ψ Holes ~ ψ

131 Approaching the transition from the superfluid (f=1) Excitations of the superfluid: (A) Spin waves

132 Approaching the transition from the superfluid (f=1) Excitations of the superfluid: (B) Vortices vortex

133 Approaching the transition from the superfluid (f=1) Excitations of the superfluid: (B) Vortices E vortex

134 Approaching the transition from the superfluid (f=1) Excitations of the superfluid: Spin wave and vortices

135 Dual theories of the superfluid-insulator transition (f=1) Excitations of the superfluid: Spin wave and vortices C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

136 A vortex in the vortex field is the original boson

137 A vortex in the vortex field is the original boson The wavefunction of a vortex acquires a phase of 2π each time the vortex encircles a boson Current of ϕ boson vortex e i2 π

138 D. Boson-vortex duality 2. Bosons in a lattice at fractional filling f L. Balents, L. Bartosch, A. Burkov, S. Sachdev, K. Sengupta, Physical Review B 71, and (2005), cond-mat/ , and cond-mat/

139 Boson-vortex duality Current of ϕ boson vortex e i2 π The wavefunction of a vortex acquires a phase of 2π each time the vortex encircles a boson Strength of magnetic field on vortex field ϕ = density of bosons = f flux quanta per plaquette C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988); M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);

140 In ordinary fluids, vortices experience the Magnus Force F M F = mass density of air i velocity of ball i circulation M ( ) ( ) ( )

141

142 Dual picture: The vortex is a quantum particle with dual electric charge n, moving in a dual magnetic field of strength = h (number density of Bose particles)

143 A 3 A 4 A 1 where f is the boson filling fraction. A 2 A 1 +A 2 +A 3 +A 4 = 2π f

144 Bosons at filling fraction f = 1 At f=1, the magnetic flux per unit cell is 2π, and the vortex does not pick up any phase from the boson density. The effective dual magnetic field acting on the vortex is zero, and the corresponding component of the Magnus force vanishes.

145 Bosons at rational filling fraction f=p/q The low energy vortex states must form a representation of this algebra Quantum mechanics of the vortex particle in a periodic potential with f flux quanta per unit cell Space group symmetries of Hofstadter Hamiltonian: T, T : Translations by a lattice spacing in the x, y directions x R y : Rotation by 90 degrees. Magnetic space group: TT = e TT 2πif x y y x R T R= T ; R T R= T ; R = y x x y ;

146 Vortices in a superfluid near a Mott insulator at filling f=p/q Hofstadter spectrum of the quantum vortex particle with field operator ϕ At filling f = p/ q, there are q species of vortices, ϕ (with =1 q), associated with q degenerate minima in the vortex spectrum. These vortices realize the smallest, q-dimensional, representation of the magnetic algebra. The q vortices form a projective representation of the space group T : ; T : e x 2πif ϕ ϕ + 1 y ϕ ϕ 1 R: ϕ q q m= 1 ϕ e m 2πimf

147 Boson-vortex duality The q vortices characterize both ϕ superconducting and density wave orders Superconductor insulator : ϕ = 0 ϕ 0

148 Boson-vortex duality The q vortices characterize both ϕ superconducting and density wave orders Density wave order: Status of space group symmetry determined by 2π p density operators ρq at wavevectors Qmn = mn, q T q iπmnf * 2πi mf ρmn = e ϕ ϕ + ne = 1 iqixˆ x : ρq ρqe ; Ty : R: ρ ( Q) ρ( RQ) ρ Q ρ e Q ( ) iqi yˆ

149 Field theory with projective symmetry Degrees of freedom: q ϕ complex vortex fields 1 non-compact U(1) gauge field A µ

150 Ψ sc Superconductor ϕ = 0, ρ mn = 0 Field theory with projective symmetry Fluctuation-induced, weak, first order transition r ρ Q Charge-ordered insulator ϕ 0, ρ mn 0 r 1 2

151 Ψ sc Superconductor ϕ = 0, ρ mn = 0 Ψ sc Superconductor ϕ = 0, ρ mn = 0 Field theory with projective symmetry Fluctuation-induced, weak, first order transition Supersolid ϕ = 0, ρ mn 0 r ρ Q Charge-ordered insulator ϕ 0, ρ mn 0 r r 1 2 ρ Q Charge-ordered insulator ϕ 0, ρ mn 0 r 1 2

152 Ψ sc Superconductor ϕ = 0, ρ mn = 0 Ψ sc Superconductor ϕ = 0, ρ mn = 0 Ψ sc Superconductor ϕ = 0, ρ mn = 0 Field theory with projective symmetry Fluctuation-induced, weak, first order transition Supersolid ϕ = 0, ρ mn 0 r ρ Q Charge-ordered insulator ϕ 0, ρ mn 0 r r 1 2 ρ Q Charge-ordered insulator ϕ 0, ρ mn 0 Second order transition r 1 2 r ρ Q Charge-ordered insulator ϕ 0, ρ mn 0 r 1 2

153 Field theory with projective symmetry Spatial structure of insulators for q=2 (f=1/2) = 1 ( + ) 2 ( r ) e i. All insulating phases have density-wave order ρ = ρ Qr with ρ 0 Q Q Q

154 Field theory with projective symmetry Spatial structure of insulators for q=4 (f=1/4 or 3/4) a b unit cells; q, q, ab, a b q all integers

155 Field theory with projective symmetry 2π p Density operators ρq at wavevectors Qmn = mn, q q iπmnf * 2πi mf ρmn = e ϕ ϕ + ne = 1 ( ) Each pinned vortex in the superfluid has a halo of density wave order over a length scale the zero-point quantum motion of the vortex. This scale diverges upon approaching the insulator

156 Vortex-induced LDOS of Bi 2 Sr 2 CaCu 2 O 8+δ integrated from 1meV to 12meV at 4K 7 pa 0 pa b Vortices have halos with LDOS modulations at a period 4 lattice spacings 100Å J. Hoffman E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002). Prediction of VBS order near vortices: K. Park and S. Sachdev, Phys. Rev. B 64, (2001).

157 Superfluids near Mott insulators The Mott insulator has average Cooper pair density, f = p/q per site, while the density of the superfluid is close (but need not be identical) to this value Vortices with flux h/(2e) come in in multiple (usually q) q) flavors The lattice space group acts in in a projective representation on on the vortex flavor space. These flavor quantum numbers provide a distinction between superfluids: they constitute a quantum order Any pinned vortex must chose an an orientation in in flavor space. This necessarily leads to to modulations in in the local density of of states over the spatial region where the vortex executes its its quantum zero point motion.