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Quantum phase transitions of correlated electrons and atoms

Quantum phase transitions of correlated electrons and atoms. Physical Review B 71 , 144508 and 144509 (2005), cond-mat/0502002. Leon Balents (UCSB) Lorenz Bartosch (Yale) Anton Burkov (UCSB) Subir Sachdev (Yale) Krishnendu Sengupta (Toronto).

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Quantum phase transitions of correlated electrons and atoms

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  1. Quantum phase transitions of correlated electrons and atoms Physical Review B 71, 144508 and 144509 (2005), cond-mat/0502002 Leon Balents (UCSB) Lorenz Bartosch (Yale) Anton Burkov (UCSB) Subir Sachdev (Yale) Krishnendu Sengupta (Toronto)

  2. Why study quantum phase transitions ? T Quantum-critical • Theory for a quantum system with strong correlations: describe phases on either side of gc by expanding in deviation from the quantum critical point. Important property of ground state at g=gc : temporal and spatial scale invariance; characteristic energy scale at other values of g: gc g • Critical point is a novel state of matter without quasiparticle excitations • Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures.

  3. Outline • The Quantum Ising chain • The superfluid-Mott insulator quantum phase transition • The cuprate superconductors Superfluids proximate to finite doping Mott insulators with VBS order ? • Vortices in the superfluid • Vortices in superfluids near the superfluid-insulator quantum phase transition The “quantum order” of the superconducting state: evidence for vortex flavors

  4. I. Quantum Ising Chain

  5. I. Quantum Ising Chain 2Jg

  6. Full Hamiltonian leads to entangled states at g of order unity

  7. Experimental realization LiHoF4

  8. Lowest excited states: Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p p Entire spectrum can be constructed out of multi-quasiparticle states Weakly-coupled qubits Ground state:

  9. S. Sachdev and A.P. Young, Phys. Rev. Lett.78, 2220 (1997) Weakly-coupled qubits Quasiparticle pole Three quasiparticle continuum ~3D Structure holds to all orders in 1/g

  10. Lowest excited states: domain walls Coupling between qubits creates new “domain-wall” quasiparticle states at momentum p p Strongly-coupled qubits Ground states:

  11. S. Sachdev and A.P. Young, Phys. Rev. Lett.78, 2220 (1997) Strongly-coupled qubits Two domain-wall continuum ~2D Structure holds to all orders in g

  12. “Flipped-spin” Quasiparticle weight Z A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994) g gc Ferromagnetic moment N0 P. Pfeuty Annals of Physics, 57, 79 (1970) g gc Excitation energy gap D g gc Entangled states at g of order unity

  13. Critical coupling No quasiparticles --- dissipative critical continuum

  14. Quasiclassical dynamics Quasiclassical dynamics P. Pfeuty Annals of Physics, 57, 79 (1970) S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).

  15. II. The superfluid-Mott insulator quantum phase transition

  16. Bose condensation Velocity distribution function of ultracold 87Rb atoms M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science269, 198 (1995)

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

  18. 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, Nature415, 39 (2002).

  19. Superfluid-insulator quantum phase transition at T=0 V0=10Er V0=3Er V0=0Er V0=7Er V0=13Er V0=14Er V0=16Er V0=20Er

  20. 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, Nature415, 39 (2002).

  21. Bosons at filling fraction f= 1 Weak interactions: superfluidity

  22. Bosons at filling fraction f= 1 Weak interactions: superfluidity

  23. Bosons at filling fraction f= 1 Weak interactions: superfluidity

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

  25. Bosons at filling fraction f= 1 Strong interactions: insulator

  26. Bosons at filling fraction f= 1/2 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  27. Bosons at filling fraction f= 1/2 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  28. Bosons at filling fraction f= 1/2 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  29. Bosons at filling fraction f= 1/2 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  30. Bosons at filling fraction f= 1/2 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  31. Bosons at filling fraction f= 1/2 Strong interactions: insulator C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  32. Bosons at filling fraction f= 1/2 Strong interactions: insulator C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  33. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  34. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  35. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  36. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  37. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  38. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  39. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  40. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  41. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  42. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  43. Insulating phases of bosons at filling fraction f= 1/2 Valence bond solid (VBS) order Valence bond solid (VBS) order Charge density wave (CDW) order C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

  44. Superfluid-insulator transition of bosons at generic filling fraction f The transition is characterized by multiple distinct order parameters (boson condensate, VBS/CDW 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.

  45. Superfluid-insulator transition of bosons at generic filling fraction f The transition is characterized by multiple distinct order parameters (boson condensate, VBS/CDW 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 VBS/CDW fluctuations. 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).

  46. III. The cuprate superconductors Superfluids proximate to finite doping Mott insulators with VBS order ?

  47. La2CuO4 La O Cu

  48. La2CuO4 Mott insulator: square lattice antiferromagnet

  49. La2-dSrdCuO4 Superfluid: condensate of paired holes

  50. Many experiments on the cuprate superconductors show: • Tendency to produce modulations in spin singlet observables at wavevectors (2p/a)(1/4,0) and (2p/a)(0,1/4). • Proximity to a Mott insulator at hole density d =1/8 with long-range charge modulations at wavevectors (2p/a)(1/4,0) and (2p/a)(0,1/4).

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