Talk online at sachdev.physics.harvard

1. Quantum criticality: where are we and where are we going ? Subir Sachdev Harvard University Talk online at http://sachdev.physics.harvard.edu

2. Outline • Density-driven phase transitions A. Fermions with repulsive interactions B. Bosons with repulsive interactions C. Fermions with attractive interactions • Magnetic transitions of Mott insulators A. Dimerized Mott insulators – Landau-Ginzburg- Wilson theory • B. S=1/2 per unit cell: deconfined quantum criticality • Transitions of the Kondo lattice A. Large Fermi surfaces – Hertz theory B. Fractional Fermi liquids and gauge theory

3. I. Density driven transitions Non-analytic change in a conserved density (spin) driven by changes in chemical potential (magnetic field)

4. 1.A Fermions with repulsive interactions Density

5. 1.A Fermions with repulsive interactions • Characteristics of this ‘trivial’ quantum critical point: • No “order parameter”. “Topological” characterization in the existence of the Fermi surface in one state. • No transition at T > 0. • Characteristic crossovers at T > 0, between quantum criticality, and low T regimes. • Strong T-dependent scaling in quantum critical regime, with response functions scaling universally as a function of kz/T and w/T, where z is the dynamic critical exponent.

6. 1.A Fermions with repulsive interactions Characteristics of this ‘trivial’ quantum critical point: Quantum critical: Particle spacing ~ de Broglie wavelength T Classical Boltzmann gas Fermi liquid

7. 1.A Fermions with repulsive interactions Characteristics of this ‘trivial’ quantum critical point: d < 2 u d > 2 u • d > 2 – interactions are irrelevant. Critical theory is the spinful free Fermi gas. • d < 2 – universal fixed point interactions. In d=1 critical theory is the spinless free Fermi gas

8. 1.B Bosons with repulsive interactions d < 2 u d > 2 u • Describes field-induced magnetization transitions in spin gap compounds • Critical theory in d =1 is also the spinless free Fermi gas. • Properties of the dilute Bose gas in d >2 violate hyperscaling and depend upon microscopic scattering length (Yang-Lee). Magnetization

9. 1.C Fermions with attractive interactions d > 2 -u BEC of paired bound state Weak-coupling BCS theory • Universal fixed-point is accessed by fine-tuning to a Feshbach resonance. • Density onset transition is described by free fermions for weak-coupling, and by (nearly) free bosons for strong coupling. The quantum-critical point between these behaviors is the Feshbach resonance. P. Nikolic and S. Sachdev cond-mat/0609106

10. 1.C Fermions with attractive interactions Free fermions Free bosons detuning P. Nikolic and S. Sachdev cond-mat/0609106

11. 1.C Fermions with attractive interactions detuning Universal theory of gapless bosons and fermions, with decay of boson into 2 fermions relevant for d < 4 P. Nikolic and S. Sachdev cond-mat/0609106

12. 1.C Fermions with attractive interactions detuning Quantum critical point at m=0, n=0, forms the basis of a theory which describes ultracold atom experiments, including the transitions to FFLO and normal states with unbalanced densities P. Nikolic and S. Sachdev cond-mat/0609106

13. Outline • Density-driven phase transitions A. Fermions with repulsive interactions B. Bosons with repulsive interactions C. Fermions with attractive interactions • Magnetic transitions of Mott insulators A. Dimerized Mott insulators – Landau-Ginzburg- Wilson theory • B. S=1/2 per unit cell: deconfined quantum criticality • Transitions of the Kondo lattice A. Large Fermi surfaces – Hertz theory B. Fractional Fermi liquids and gauge theory

14. 2.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

15. TlCuCl3 M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, cond-mat/0309440.

16. 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

18. Weakly coupled dimers

19. Weakly coupled dimers Paramagnetic ground state

20. Weakly coupled dimers Excitation: S=1 triplon

21. Weakly coupled dimers Excitation: S=1 triplon

22. Weakly coupled dimers Excitation: S=1 triplon

23. Weakly coupled dimers Excitation: S=1 triplon

24. Weakly coupled dimers Excitation: S=1 triplon

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

26. TlCuCl3 “triplon” N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001).

27. Coupled Dimer Antiferromagnet

28. l close to 1 Weakly dimerized square lattice

29. l Weakly dimerized square lattice close to 1 Excitations: 2 spin waves (magnons) Ground state has long-range spin density wave (Néel) order at wavevector K= (p,p)

30. TlCuCl3 J. Phys. Soc. Jpn72, 1026 (2003)

31. lc = 0.52337(3)M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002) Pressure in TlCuCl3 T=0 Quantum paramagnet Néel state 1

32. LGW theory for quantum criticality

33. 2.A. Magnetic quantum phase transitions in Mott insulators with S=1/2 per unit cell Deconfined quantum criticality

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

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

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

37. 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.

38. 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.

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

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

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

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

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

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

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

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

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

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

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

50. LGW theory of multiple order parameters Distinct symmetries of order parameters permit couplings only between their energy densities