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Magnetic phases and critical points of insulators and superconductors

Magnetic phases and critical points of insulators and superconductors. Colloquium article in Reviews of Modern Physics , July 2003, cond-mat/0211005. cond-mat/0109419. Quantum Phase Transitions Cambridge University Press. Talks online: Sachdev . E. E. g. g.

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Magnetic phases and critical points of insulators and superconductors

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  1. Magnetic phases and critical points of insulators and superconductors • Colloquium article in Reviews of Modern Physics, July 2003, cond-mat/0211005. • cond-mat/0109419 Quantum Phase Transitions Cambridge University Press Talks online: Sachdev

  2. E E g g Avoided level crossing which becomes sharp in the infinite volume limit: second-order transition True level crossing: Usually a first-order transition What is a quantum phase transition ? Non-analyticity in ground state properties as a function of some control parameter g

  3. 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.

  4. Outline • Quantum Ising Chain • Coupled Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point • Coupled dimer antiferromagnet in a magnetic field Bose condensation of “triplons” • Magnetic transitions in superconductors Quantum phase transition in a background Abrikosov flux lattice • Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B:Z2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons • Conclusions I. Quantum Ising chain Single order parameter. Multiple order parameters.

  5. I. Quantum Ising Chain 2Jg

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

  7. 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:

  8. 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

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

  10. 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

  11. “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

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

  13. 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).

  14. Outline • Quantum Ising Chain • Coupled Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point • Coupled dimer antiferromagnet in a magnetic field Bose condensation of “triplons” • Magnetic transitions in superconductors Quantum phase transition in a background Abrikosov flux lattice • Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B:Z2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons • Conclusions II. Coupled Dimer Antiferromagnet Single order parameter. Multiple order parameters.

  15. II.Coupled Dimer Antiferromagnet M. P. Gelfand, R. R. P. Singh, and D. A. Huse, Phys. Rev. B40, 10801-10809 (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, 014407 (2002). S=1/2 spins on coupled dimers

  16. Square lattice antiferromagnet Experimental realization: Ground state has long-range magnetic (Neel) order Excitations: 2 spin waves (magnons)

  17. Weakly coupled dimers Paramagnetic ground state

  18. Weakly coupled dimers Excitation: S=1 triplon (exciton, spin collective mode) Energy dispersion away from antiferromagnetic wavevector

  19. Weakly coupled dimers S=1/2 spinons are confinedby a linear potential into a S=1 triplon

  20. d in • cuprates ? Quantum paramagnet T=0 c Neel order N0 Spin gap D 1 Neel state

  21. II.ACoherent state path integral Path integral for quantum spin fluctuations Key ingredient: Spin Berry Phases

  22. II.ACoherent state path integral Path integral for quantum spin fluctuations Key ingredient: Spin Berry Phases

  23. II.ACoherent state path integral See Chapter 13 of Quantum Phase Transitions, S. Sachdev, Cambridge University Press (1999). Path integral for a single spin Action for lattice antiferromagnet n and L vary slowly in space and time

  24. Integrate out L and take the continuum limit Discretize spacetime into a cubic lattice

  25. Integrate out L and take the continuum limit Berry phases can be neglected for coupled dimer antiferromagent (justified later) Discretize spacetime into a cubic lattice Quantum path integral for two-dimensional quantum antiferromagnet Partition function of a classical three-dimensional ferromagnet at a “temperature” g Quantum transition atl=lcis related to classical Curie transition atg=gc

  26. II.BQuantum field theory for critical point l close to lc : use “soft spin” field 3-component antiferromagnetic order parameter Oscillations of about zero (for l < lc ) spin-1 collective mode T=0 spectrum w

  27. Critical coupling Dynamic spectrum at the critical point No quasiparticles --- dissipative critical continuum

  28. Outline • Quantum Ising Chain • Coupled Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point • Coupled dimer antiferromagnet in a magnetic field Bose condensation of “triplons” • Magnetic transitions in superconductors Quantum phase transition in a background Abrikosov flux lattice • Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B:Z2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons • Conclusions Single order parameter. III. Coupled Dimer Antiferromagnet in a magnetic field Multiple order parameters.

  29. H Evolution of phase diagram in a magnetic field T=0 SDW Pressure, exchange constant,…. Quantum critical point Both states are insulators

  30. Bose-Einstein condensation of Sz=1 triplon Effect of a field on paramagnet Energy of zero momentum triplon states D 0 H

  31. III. Phase diagram in a magnetic field. H gmBH = D SDW Spin singlet state with a spin gap 1/l 1 Tesla = 0.116 meV Related theory applies to double layer quantum Hall systems at n=2

  32. III. Phase diagram in a magnetic field. H gmBH = D SDW Spin singlet state with a spin gap 1/l 1 Tesla = 0.116 meV Related theory applies to double layer quantum Hall systems at n=2

  33. III. Phase diagram in a magnetic field. H gmBH = D SDW Spin singlet state with a spin gap 1/l 1 Tesla = 0.116 meV Related theory applies to double layer quantum Hall systems at n=2

  34. III. Phase diagram in a magnetic field. M D H

  35. III. Phase diagram in a magnetic field. 1 M At very large H, magnetization saturates D H

  36. III. Phase diagram in a magnetic field. 1 M 1/2 Respulsive interactions between triplons can lead to magnetization plateau at any rational fraction D H

  37. Quantum transitions in and out of plateau are Bose-Einstein condensations of “extra/missing” triplons III. Phase diagram in a magnetic field. 1 M 1/2 D H

  38. Outline • Quantum Ising Chain • Coupled Dimer Antiferromagnet A. Coherent state path integral B. Quantum field theory near critical point • Coupled dimer antiferromagnet in a magnetic field Bose condensation of “triplons” • Magnetic transitions in superconductors Quantum phase transition in a background Abrikosov flux lattice • Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins, bond order and confined spinons in d=2 Class B:Z2 gauge theory: non-collinear spins, RVB, visons, topological order, and deconfined spinons • Conclusions Single order parameter. IV. Magnetic transitions in superconductors Multiple order parameters.

  39. T=0 SDW Pressure, carrier concentration,…. Quantum critical point We have so far considered the case where both states are insulators

  40. T=0 SC+SDW SC Pressure, carrier concentration,…. Quantum critical point Now both sides have a “background” superconducting (SC) order

  41. Interplay of SDW and SC order in the cuprates ky Insulator • /a 0 /a kx T=0 phases of LSCO Néel SC SC+SDW SDW 0 0.02 0.055  ~0.12-0.14 (additional commensurability effects near d=0.125) • J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science278, 1432 (1997).S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999) S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Rev. B 63, 172501 (2001).

  42. Interplay of SDW and SC order in the cuprates T=0 phases of LSCO ky • • Insulator /a • • 0 /a kx Néel SC SC+SDW SDW 0 0.02 0.055  ~0.12-0.14 (additional commensurability effects near d=0.125) • J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science278, 1432 (1997).S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999) S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Rev. B 63, 172501 (2001).

  43. Interplay of SDW and SC order in the cuprates T=0 phases of LSCO ky Superconductor with Tc,min =10 K • • • /a • 0 /a kx Néel SC SC+SDW SDW 0 0.02 0.055  ~0.12-0.14 (additional commensurability effects near d=0.125) • J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science278, 1432 (1997).S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999) S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Rev. B 63, 172501 (2001).

  44. Collinear magnetic (spin density wave) order Collinear spins

  45. Interplay of SDW and SC order in the cuprates Use simplest assumption of a direct second-order quantum phase transition between SC and SC+SDW phases T=0 phases of LSCO ky Superconductor with Tc,min =10 K • • • /a • 0 /a kx Néel SC SC+SDW SDW 0 0.02 0.055  ~0.12-0.14

  46. Magnetic transition in a d-wave superconductor Otherwise, new theory of coupled excitons and nodal quasiparticles L. Balents, M.P.A. Fisher, C. Nayak, Int. J. Mod. Phys. B 12, 1033 (1998).

  47. Coupling to the S=1/2 Bogoliubov quasiparticles of the d-wave superconductor Trilinear “Yukawa” coupling is prohibited unless ordering wavevector is fine-tuned. Magnetic transition in a d-wave superconductor Similar terms present in action for SDW ordering in the insulator

  48. Neutron scattering measurements of dynamic spin correlations of the superconductor (SC) in a magnetic field B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason, and A. Schröder, Science291, 1759 (2001).

  49. Neutron scattering measurements of dynamic spin correlations of the superconductor (SC) in a magnetic field B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason, and A. Schröder, Science291, 1759 (2001). D. P. Arovas, A. J. Berlinsky, C. Kallin, and S.-C. Zhang, Phys. Rev. Lett.79, 2871 (1997) proposed static magnetism localized within vortex cores, but signal was much larger than anticipated.

  50. Dominant effect of magnetic field: Abrikosov flux lattice

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