Classifying two-dimensional superfluids: why there is more to cuprate superconductivity than the condensation of charge -2e Cooper pairs

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Classifying two-dimensional superfluids: why there is more to cuprate superconductivity than the condensation of charge -2e Cooper pairs cond-mat/0408329,

1 Classifying two-dimensional superfluids: why there is more to cuprate superconductivity than the condensation of charge -2e Cooper pairs cond-mat/ , cond-mat/ , and to appear Leon Balents (UCSB) Lorenz Bartosch (Yale) Anton Burkov (UCSB) Subir Sachdev (Yale) Krishnendu Sengupta (Toronto) Talk online: Sachdev

2 Outline I. Bose-Einstein condensation and superfluidity V. The cuprate superconductors The quantum order of the superconducting state: evidence for vortex flavors II. Vortices in the superfluid Magnus forces, duality, and point vortices as dual electric charges III. The superfluid-mott insulator quantum phase transition IV. Vortices in superfluids near the superfluid-insulator quantum phase transition The Hofstadter Hamiltonian and vortex flavors

3 I. Bose-Einstein condensation and superfluidity

4 Superfluidity/superconductivity occur in: liquid 4 He metals Hg, Al, Pb, Nb, Nb 3 Sn.. liquid 3 He neutron stars cuprates La 2-x Sr x CuO 4, YBa 2 Cu 3 O 6+y. M 3 C 60 ultracold trapped atoms MgB 2

5 The Bose-Einstein condensate: A macroscopic number of bosons occupy the lowest energy quantum state Such a condensate also forms in systems of fermions, where the bosons are Cooper pairs of fermions: k y ( 2 2 Ψ = k )( ) x ky k x Pair wavefunction in cuprates: S = 0

6 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)

7 Superflow: The wavefunction of the condensate Superfluid velocity Ψ Ψe v s = m iθ r ( ) θ (for non-galilean invariant superfluids, the co-efficient of θ is modified)

8 II. Vortices in the superfluid Magnus forces, duality, and point vortices as dual electric charges

9 Excitations of the superfluid: Vortices

10 Observation of quantized vortices in rotating 4 He E.J. Yarmchuk, M.J.V. Gordon, and R.E. Packard, Observation of Stationary Vortex Arrays in Rotating Superfluid Helium, Phys. Rev. Lett. 43, 214 (1979).

11 Observation of quantized vortices in rotating ultracold Na J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of Vortex Lattices in Bose-Einstein Condensates, Science 292, 476 (2001).

12 Quantized fluxoids in YBa 2 Cu 3 O 6+y J. C. Wynn, D. A. Bonn, B.W. Gardner, Yu-Ju Lin, Ruixing Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Phys. Rev. Lett. 87, (2001).

13 Excitations of the superfluid: Vortices Central question: In two dimensions, we can view the vortices as point particle excitations of the superfluid. What is the quantum mechanics of these particles?

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

16 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)

17 III. The superfluid-mott insulator quantum phase transition

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

19 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).

20 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).

21 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

22 Superfluid-insulator quantum phase transition at T=0 M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

$Bosons at filling fraction f = 1 Weak interactions: superfluidity Strong interactions: Mott insulator which$

23 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).

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

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

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

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

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

29 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)

30 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)

31 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)

32 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)

33 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)

34 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)

35 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)

36 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

37 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

38 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

39 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

40 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

41 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

42 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

43 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

44 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

45 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

$Insulating phases of bosons at filling fraction f = 1/2 = 1 2 ( + ) Charge density wave (CDW) order Valence bond solid (VBS) order Valence bond solid (VBS) order Can define a common CDW/VBS order$

46 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

47 IV. Vortices in superfluids near the superfluid-insulator quantum phase transition The Hofstadter Hamiltonian and vortex flavors

48 Upon approaching the insulator, the phase of the condensate becomes uncertain. Vortices cost less energy and vortex-antivortex pairs proliferate. The quantum mechanics of vortices plays a central role in the superfluid-insulator quantum phase transition.

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

50 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.

51 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 ;

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

52 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. T : ; T : e x 2πif ϕ ϕ + 1 y ϕ ϕ 1 R: ϕ q q m= 1 ϕ e m 2πimf

53 Vortices in a superfluid near a Mott insulator at filling f=p/q i The excitations of the superfluid are described by the quantum mechanics of q flavors of low energy vortices moving in zero dual "magnetic" field. i The Mott insulator is a Bose-Einstein condensate of vortices, with ϕ 0 i Any set of values of ϕ breaks the space group symmetry, and the orientation of the vortex condensate, ϕ, in flavor space determines the CDW/VBS order.

54 Mott insulators obtained by condensing vortices Spatial structure of insulators for q=2 (f=1/2) = 1 2 ( + )

55 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

56 V. The cuprate superconductors The quantum order of the superconducting state: evidence for vortex flavors

57 La Superconductivity of holes, of density δ, moving on the square lattice of Cu sites. O Cu

58 Experiments on the cuprate superconductors show: Tendency to produce density wave order near wavevectors (2π/a)(1/4,0) and (2π/a)(0,1/4). Proximity to a Mott insulator at hole density δ =1/8 with long-range density wave order at wavevectors (2π/a)(1/4,0) and (2π/a)(0,1/4). Vortex/anti-vortex fluctuations for a wide temperature range in the normal state

59 The cuprate superconductor Ca 2-x Na x CuO 2 Cl 2 Multiple order parameters: superfluidity and density wave. Phases: Superconductors, Mott insulators, and/or supersolids T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, Nature 430, 1001 (2004).

Distinct experimental charcteristics of underdoped cuprates at T > T c Measurements of Nernst effect are well explained by a model of a liquid of vortices and anti-vortices N. P.

60 Distinct experimental charcteristics of underdoped cuprates at T > T c Measurements of Nernst effect are well explained by a model of a liquid of vortices and anti-vortices N. P. Ong, Y. Wang, S. Ono, Y. Ando, and S. Uchida, Annalen der Physik 13, 9 (2004). Y. Wang, S. Ono, Y. Onose, G. Gu, Y. Ando, Y. Tokura, S. Uchida, and N. P. Ong, Science 299, 86 (2003).

61 LDOS of Bi 2 Sr 2 CaCu 2 O 8+δ at 100 K. M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science, 303, 1995 (2004). Distinct experimental charcteristics of underdoped cuprates at T > T c STM measurements observe density modulations with a period of 4 lattice spacings

62 Pinned vortices in the superfluid Any pinned vortex breaks the space group symmetry, and so has a preferred orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion. In the cuprates, assuming boson density=density of Cooper pairs we have ρ = 7/16, and q= 16 (both models in part B yield this value of q). So MI modulation must have period a b with 16/ a, 16/ b, and ab/16 all integers.

Vortex-induced LDOS of Bi 2 Sr 2 CaCu 2 O 8+δ integrated from 1meV to 12meV at 4K 7 pa b Vortices have halos with LDOS modulations at a period 4 lattice spacings 0 pa 100Å J. Hoffman, E. W.

63 Vortex-induced LDOS of Bi 2 Sr 2 CaCu 2 O 8+δ integrated from 1meV to 12meV at 4K 7 pa b Vortices have halos with LDOS modulations at a period 4 lattice spacings 0 pa 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).

64 Measuring the inertial mass of a vortex

65 Measuring the inertial mass of a vortex Preliminary estimates for the BSCCO experiment: Inertial vortex mass 10m Vortex magnetoplasmon frequency ν m v e p 1 THz = 4 mev Future experiments can directly detect vortex zero point motion by looking for resonant absorption at this frequency. Vortex oscillations can also modify the electronic density of states.

66 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.

Dual vortex theory of doped antiferromagnets

Dual vortex theory of doped antiferromagnets Physical Review B 71, 144508 and 144509 (2005), cond-mat/0502002, cond-mat/0511298 Leon Balents (UCSB) Lorenz Bartosch (Harvard) Anton Burkov (Harvard) Predrag