Dual vortex theory of doped antiferromagnets

Transcription

1 Dual vortex theory of doped antiferromagnets Physical Review B 71, and (2005), cond-mat/ , cond-mat/ Leon Balents (UCSB) Lorenz Bartosch (Harvard) Anton Burkov (Harvard) Predrag Nikolic (Harvard) Subir Sachdev (Harvard) Krishnendu Sengupta (Saha Institute, India) Talk online at

2 Experiments on the cuprate superconductors show: Proximity to insulating ground states with density wave order at carrier density δ=1/8 Vortex/anti-vortex fluctuations for a wide temperature range in the normal state

3 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).

4 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).

5 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

6 Experiments on the cuprate superconductors show: Proximity to insulating ground states with density wave order at carrier density δ=1/8 Vortex/anti-vortex fluctuations for a wide temperature range in the normal state Needed: A quantum theory of transitions between superfluid/supersolid/insulating phases at fractional filling, and a deeper understanding of the role of vortices

7 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. 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. These modulations may be be viewed as as strong-coupling analogs of of Friedel oscillations in in a Fermi liquid.

8 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).

9 Outline I. Superfluid-insulator transition of bosons II. Vortices as elementary quasiparticle excitations of the superfluid III. The vortex flavor space IV. Theory of doped antiferromagnets: doping a VBS insulator V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor VI. Predictions for experiments on the cuprates

10 I. Superfluid-insulator transition of bosons

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

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

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

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

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

16 Bosons at filling fraction f = 1/2 Ψ 0 Weak interactions: superfluidity

17 Bosons at filling fraction f = 1/2 Ψ 0 Weak interactions: superfluidity

18 Bosons at filling fraction f = 1/2 Ψ 0 Weak interactions: superfluidity

19 Bosons at filling fraction f = 1/2 Ψ 0 Weak interactions: superfluidity

20 Bosons at filling fraction f = 1/2 Ψ 0 Weak interactions: superfluidity

21 Bosons at filling fraction f = 1/2 Ψ = 0 Strong interactions: insulator

22 Bosons at filling fraction f = 1/2 Ψ = 0 Strong interactions: insulator

23 Bosons at filling fraction f = 1/2 Ψ = 0 Strong interactions: insulator Insulator has density wave order

24 Superfluid-insulator transition of bosons at generic filling fraction f The transition is characterized by multiple distinct order parameters (boson condensate, density-wave order..) Traditional (Landau-Ginzburg-Wilson) view: Such a transition is first order, and there are no precursor fluctuations of the order of the insulator in the superfluid.

25 Superfluid-insulator transition of bosons at generic filling fraction f The transition is characterized by multiple distinct order parameters (boson condensate, density-wave order...) Traditional (Landau-Ginzburg-Wilson) view: Such a transition is first order, and there are no precursor fluctuations of the order of the insulator in the superfluid. Recent theories: Quantum interference effects can render such transitions second order, and the superfluid does contain precursor CDW fluctuations. T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

26 Outline I. Superfluid-insulator transition of bosons II. Vortices as elementary quasiparticle excitations of the superfluid III. The vortex flavor space IV. Theory of doped antiferromagnets: doping a VBS insulator V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor VI. Predictions for experiments on the cuprates

27 II. Vortices as elementary quasiparticle excitations of the superfluid Magnus forces, duality, and point vortices as dual electric charges

28 Excitations of the superfluid: Vortices

29 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?

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

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32 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) 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);

33 Outline I. Superfluid-insulator transition of bosons II. Vortices as elementary quasiparticle excitations of the superfluid III. The vortex flavor space IV. Theory of doped antiferromagnets: doping a VBS insulator V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor VI. Predictions for experiments on the cuprates

34 III. The vortex flavor space Vortices carry a flavor index which encodes the density-wave order of the proximate insulator

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36 Bosons at rational filling fraction f=p/q Quantum mechanics of the vortex particle in a periodic potential with f flux quanta per unit cell Space group symmetries of vortex Hamiltonian: Magnetic space group: TT = e TT 2πif x y y x R T R= T ; R T R= T ; R = y x x y ; The low energy vortex states must form a representation of this algebra

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38 Vortices in a superfluid near a Mott insulator at filling f=p/q Simplest representation of magnetic space group by 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

39 Vortices in a superfluid near a Mott insulator at filling f=p/q The wavefunction of the vortices in flavor ϕ space characterizes the density-wave order Density-wave order: Status of space group symmetry determined by 2π p density operators ρq at wavevectors Qmn = mn, q T x ρ q iπmnf * 2πi mf ρmn = e ϕ ϕ + ne = 1 ρ e iqixˆ : Q Q ; y : R: ρ Q ρ T RQ ( ) ( ) ρ Q ρ e Q ( ) iqi yˆ

40 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 orientation of the vortex in flavor space implies a particular configuration of density-wave order in its vicinity.

41 Mott insulators obtained by condensing vortices at 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 Mott insulators obtained by condensing vortices at f = 1/4, 3/4 a b unit cells; q, q, ab, a b q all integers

43 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 orientation of the vortex in flavor space implies a particular configuration of VBS order in its vicinity.

44 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 orientation of the vortex in flavor space implies a particular configuration of VBS order in its vicinity. i Any pinned vortex must pick an orientation in flavor space: this induces a halo of VBS order in its vicinity

45 Outline I. Superfluid-insulator transition of bosons II. Vortices as elementary quasiparticle excitations of the superfluid III. The vortex flavor space IV. Theory of doped antiferromagnets: doping a VBS insulator V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor VI. Predictions for experiments on the cuprates

46 IV. Theory of doped antiferromagnets: doping a VBS insulator

47 (B.1) Phase diagram of doped antiferromagnets g = parameter controlling strength of quantum fluctuations in a semiclassical theory of the destruction of Neel order La 2 CuO 4 Neel order

48 (B.1) Phase diagram of doped antiferromagnets g or VBS order La 2 CuO 4 Neel order N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

49 (B.1) Phase diagram of doped antiferromagnets g or VBS order Upon viewing down spins as hard-core bosons, these phases map onto superfluid and VBS phases of bosons at f=1/2 considered La 2 CuO 4 earlier Neel order N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

50 (B.1) Phase diagram of doped antiferromagnets g or VBS order La 2 CuO 4 Neel order N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

51 (B.1) Phase diagram of doped antiferromagnets g or Dual vortex theory of of for interplay between VBS order and d-wave superconductivity VBS order La 2 CuO 4 Neel order Hole density δ

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57 (B.1) Phase diagram of doped antiferromagnets g VBS order δ = 1 32 La 2 CuO 4 Neel order Hole density δ

58 (B.1) Phase diagram of doped antiferromagnets g VBS order δ = 1 16 La 2 CuO 4 Neel order Hole density δ

59 (B.1) Phase diagram of doped antiferromagnets g VBS order δ = 1 8 La 2 CuO 4 Neel order Hole density δ

60 (B.1) Phase diagram of doped antiferromagnets VBS order g d-wave superconductivity above a critical δ La 2 CuO 4 Neel order Hole density δ

61 Outline I. Superfluid-insulator transition of bosons II. Vortices as elementary quasiparticle excitations of the superfluid III. The vortex flavor space IV. Theory of doped antiferromagnets: doping a VBS insulator V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor VI. Predictions for experiments on the cuprates

62 V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor

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65 A finite effective mass m v ~ Λ v where Λ~ is a high energy cutoff 2 F

66 C 1 sub-ohmic damping with 2 v = v F Universal function of v F

67 C Bardeen-Stephen viscous drag with 2 2 v = v F Universal function of v F

68 C Bardeen-Stephen viscous drag with 2 2 v = v F Universal function of v F Effect of nodal quasiparticles on vortex dynamics is relatively innocuous.

69 Outline I. Superfluid-insulator transition of bosons II. Vortices as elementary quasiparticle excitations of the superfluid III. The vortex flavor space IV. Theory of doped antiferromagnets: doping a VBS insulator V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor VI. Predictions for experiments on the cuprates

70 VI. Predictions for experiments on the cuprates

71 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).

72 Measuring the inertial mass of a vortex

73 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.

74 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. 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. These modulations may be be viewed as as strong-coupling analogs of of Friedel oscillations in in a Fermi liquid.