1. First Order vs Second Order Transitions in Quantum Magnets I. Quantum Ferromagnetic Transitions: Experiments II. Theory 1. Conventional (mean-field) theory 2. Renormalized mean-field theory 3. Effects of flucuations III. Other Transitions Dietrich Belitz, University of Oregon Ted Kirkpatrick, University of Maryland

2. Quantum Criticality Workshop Toronto 2 Sep 2008 I. Quantum Ferromagnetic Transitions: Experiments

3. Quantum Criticality Workshop Toronto 3 Sep 2008 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose T c can be tuned to zero:

4. Quantum Criticality Workshop Toronto 4 Sep 2008 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose T c can be tuned to zero: ● UGe 2, ZrZn 2, (MnSi) (clean, pressure tuned)

5. Quantum Criticality Workshop Toronto 5 Sep 2008 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose T c can be tuned to zero: ● UGe 2, ZrZn 2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2 nd order transition at high T, 1 st order transition at low T:

6. Quantum Criticality Workshop Toronto 6 Sep 2008 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose T c can be tuned to zero: ● UGe 2, ZrZn 2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2 nd order transition at high T, 1 st order transition at low T: (Pfleiderer & Huxley 2002) UGe 2

7. Quantum Criticality Workshop Toronto 7 Sep 2008 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose T c can be tuned to zero: ● UGe 2, ZrZn 2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2 nd order transition at high T, 1 st order transition at low T: (Pfleiderer & Huxley 2002) UGe 2 ZrZn 2 (Uhlarz et al 2004)

8. Quantum Criticality Workshop Toronto 8 Sep 2008 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose T c can be tuned to zero: ● UGe 2, ZrZn 2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2 nd order transition at high T, 1 st order transition at low T: (Pfleiderer & Huxley 2002) UGe 2 ZrZn 2 MnSi (Pfleiderer et al 1997) (Uhlarz et al 2004)

9. Quantum Criticality Workshop Toronto 9 Sep 2008 I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose T c can be tuned to zero: ● UGe 2, ZrZn 2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2 nd order transition at high T, 1 st order transition at low T: ○ Additional evidence: μSR (Uemura et al 2007) (Pfleiderer & Huxley 2002) UGe 2 ZrZn 2 MnSi (Pfleiderer et al 1997) (Uhlarz et al 2004)

10. Quantum Criticality Workshop Toronto 10 Sep 2008

11. Quantum Criticality Workshop Toronto 11 Sep 2008

12. Quantum Criticality Workshop Toronto 12 Sep 2008

13. Quantum Criticality Workshop Toronto 13 Sep 2008 Bauer et al (2005) Butch & Maple (2008)

14. Quantum Criticality Workshop Toronto 14 Sep 2008 Bauer et al (2005) Butch & Maple (2008) ○ Observed exponents are not mean-field like (see below)

15. Quantum Criticality Workshop Toronto 15 Sep 2008 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones.

16. Quantum Criticality Workshop Toronto 16 Sep 2008 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. ■ Landau free energy density: f = f 0 – h m + t m 2 + u m 4 + w m 6 Equation of state: h = t m + u m 3 + w m 5 + …

17. Quantum Criticality Workshop Toronto 17 Sep 2008 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. ■ Landau free energy density: f = f 0 – h m + t m 2 + u m 4 + w m 6 Equation of state: h = t m + u m 3 + w m 5 + … ■ Landau theory predicts: ● 2 nd order transition at t=0 if u<0 ● 1 st order transition if u<0 } for both clean and dirty systems

18. Quantum Criticality Workshop Toronto 18 Sep 2008 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. ■ Landau free energy density: f = f 0 – h m + t m 2 + u m 4 + w m 6 Equation of state: h = t m + u m 3 + w m 5 + … ■ Landau theory predicts: ● 2 nd order transition at t=0 if u<0 ● 1 st order transition if u<0 ■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe 2 u<0 } for both clean and dirty systems

19. Quantum Criticality Workshop Toronto 19 Sep 2008 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. ■ Landau free energy density: f = f 0 – h m + t m 2 + u m 4 + w m 6 Equation of state: h = t m + u m 3 + w m 5 + … ■ Landau theory predicts: ● 2 nd order transition at t=0 if u<0 ● 1 st order transition if u<0 ■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe 2 u<0 ■ Problems: ● Not universal ● Does not explain the tricritical point ● Observed critical behavior not mean-field like } for both clean and dirty systems

20. Quantum Criticality Workshop Toronto 20 Sep 2008 II. Quantum Ferromagnetic Transitions: Theory 1. Conventional (= mean-field) theory ■ Hertz 1976: Mean-field theory correctly describes T=0 transition for d>1 in clean systems, and for d>0 in disordered ones. ■ Landau free energy density: f = f 0 – h m + t m 2 + u m 4 + w m 6 Equation of state: h = t m + u m 3 + w m 5 + … ■ Landau theory predicts: ● 2 nd order transition at t=0 if u<0 ● 1 st order transition if u<0 ■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe 2 u<0 ■ Problems: ● Not universal ● Does not explain the tricritical point ● Observed critical behavior not mean-field like ■ Conclusion: Conventional theory not viable } for both clean and dirty systems

21. Quantum Criticality Workshop Toronto 21 Sep 2008 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff)

22. Quantum Criticality Workshop Toronto 22 Sep 2008 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case)

23. Quantum Criticality Workshop Toronto 23 Sep 2008 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f 0 :

24. Quantum Criticality Workshop Toronto 24 Sep 2008 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f 0 : ● Contribution to eq. of state:

25. Quantum Criticality Workshop Toronto 25 Sep 2008 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f 0 : ● Contribution to eq. of state: ● Renormalized mean-field equation of state: (clean, d=3, T=0)

26. Quantum Criticality Workshop Toronto 26 Sep 2008 2. Renormalized mean-field theory ■ In general, Hertz theory misses effects of soft modes (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f 0 : ● Contribution to eq. of state: ● Renormalized mean-field equation of state: (clean, d=3, T=0) ● v>0 Transition is generically 1 st order! (TRK, T Vojta, DB 1999)

27. Quantum Criticality Workshop Toronto 27 Sep 2008 ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point

28. Quantum Criticality Workshop Toronto 28 Sep 2008 ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes

29. Quantum Criticality Workshop Toronto 29 Sep 2008 ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes ○ fermion dispersion relation m d -> m d/2

30. Quantum Criticality Workshop Toronto 30 Sep 2008 ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes ○ fermion dispersion relation m d -> m d/2 ○ sign of the coefficient

31. Quantum Criticality Workshop Toronto 31 Sep 2008 ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes ○ fermion dispersion relation m d -> m d/2 ○ sign of the coefficient Renormalized mean-field equation of state: (disordered, d=3, T=0)

32. Quantum Criticality Workshop Toronto 32 Sep 2008 ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes ○ fermion dispersion relation m d -> m d/2 ○ sign of the coefficient Renormalized mean-field equation of state: (disordered, d=3, T=0) ● v>0 Transition is 2 nd order with non-mean-field (and non-classical) exponents: β=2, δ=3/2, etc.

33. Quantum Criticality Workshop Toronto 33 Sep 2008 ● Phase diagrams: G=0

34. Quantum Criticality Workshop Toronto 34 Sep 2008 ● Phase diagrams: G=0 T=0

35. Quantum Criticality Workshop Toronto 35 Sep 2008 ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005)

36. Quantum Criticality Workshop Toronto 36 Sep 2008 ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works!

37. Quantum Criticality Workshop Toronto 37 Sep 2008 ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Mean-field exponents: β=1/2, δ=3, z=3 Magnetization at QCP: δm c ~ -T 4/9

38. Quantum Criticality Workshop Toronto 38 Sep 2008 ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Mean-field exponents: β=1/2, δ=3, z=3 Magnetization at QCP: δm c ~ -T 4/9 ■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram: (Pfleiderer, Julian, Lonzarich 2001)

39. Quantum Criticality Workshop Toronto 39 Sep 2008 ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Mean-field exponents: β=1/2, δ=3, z=3 Magnetization at QCP: δm c ~ -T 4/9 ■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram: ■ Remarks: ● Landau theory with a TCP also produces tricritical wings (Griffiths 1970) (Pfleiderer, Julian, Lonzarich 2001)

40. Quantum Criticality Workshop Toronto 40 Sep 2008 ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Mean-field exponents: β=1/2, δ=3, z=3 Magnetization at QCP: δm c ~ -T 4/9 ■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram: ■ Remarks: ● Landau theory with a TCP also produces tricritical wings (Griffiths 1970) ● So far no OP fluctuations have been considered (Pfleiderer, Julian, Lonzarich 2001)

41. Quantum Criticality Workshop Toronto 41 Sep 2008 ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) ● h>0 gives soft modes a mass, ln m -> ln (m+h) Hertz theory works! Mean-field exponents: β=1/2, δ=3, z=3 Magnetization at QCP: δm c ~ -T 4/9 ■ Conclusion: Renormalized mean-field theory explains the experimentally observed phase diagram: ■ Remarks: ● Landau theory with a TCP also produces tricritical wings (Griffiths 1970) ● So far no OP fluctuations have been considered ● More generally, Hertz theory works if field conjugate the OP does not change the soft-mode spectrum (DB, TRK, T Vojta 2002) (Pfleiderer, Julian, Lonzarich 2001)

42. Quantum Criticality Workshop Toronto 42 Sep 2008 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly:

43. Quantum Criticality Workshop Toronto 43 Sep 2008 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations

44. Quantum Criticality Workshop Toronto 44 Sep 2008 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations ● two divergent time scales: ○ critical time scale z=3 (clean) or z=4 (disordered) ○ fermionic time scale z=1 (clean) or z=2 (disordered)

45. Quantum Criticality Workshop Toronto 45 Sep 2008 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations ● two divergent time scales: ○ critical time scale z=3 (clean) or z=4 (disordered) ○ fermionic time scale z=1 (clean) or z=2 (disordered) ● Construct coupled field theory for both fields (DB, TRK, S.L. Sessions, M.T. Mercaldo 2001)

46. Quantum Criticality Workshop Toronto 46 Sep 2008 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations ● two divergent time scales: ○ critical time scale z=3 (clean) or z=4 (disordered) ○ fermionic time scale z=1 (clean) or z=2 (disordered) ● Construct coupled field theory for both fields (DB, TRK, S.L. Sessions, M.T. Mercaldo 2001) ● Analysis at various levels:

47. Quantum Criticality Workshop Toronto 47 Sep 2008 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations ● two divergent time scales: ○ critical time scale z=3 (clean) or z=4 (disordered) ○ fermionic time scale z=1 (clean) or z=2 (disordered) ● Construct coupled field theory for both fields (DB, TRK, S.L. Sessions, M.T. Mercaldo 2001) ● Analysis at various levels: ○ Gaussian approx Hertz theory (FP unstable with respect to m q 2 term in effective action)

48. Quantum Criticality Workshop Toronto 48 Sep 2008 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: ● OP fluctuations m(x,Ω) p-h fluctuations ● two divergent time scales: ○ critical time scale z=3 (clean) or z=4 (disordered) ○ fermionic time scale z=1 (clean) or z=2 (disordered) ● Construct coupled field theory for both fields (DB, TRK, S.L. Sessions, M.T. Mercaldo 2001) ● Analysis at various levels: ○ Gaussian approx Hertz theory (FP unstable with respect to m q 2 term in effective action) ○ mean-field approx for OP + Gaussian approx for fermions renormalized mean-field theory (FP marginally unstable)

49. Quantum Criticality Workshop Toronto 49 Sep 2008 ○ RG analysis for disordered case upper critical dimension is d=4 m q 2 is marginal for all 0<d<4, and is the only marginal term

50. Quantum Criticality Workshop Toronto 50 Sep 2008 ○ RG analysis for disordered case upper critical dimension is d=4 m q 2 is marginal for all 0<d<4, and is the only marginal term log terms in critical behavior (cf. Wegner 1970s) e.g., correlation length

51. Quantum Criticality Workshop Toronto 51 Sep 2008 ○ RG analysis for disordered case upper critical dimension is d=4 m q 2 is marginal for all 0<d<4, and is the only marginal term log terms in critical behavior (cf. Wegner 1970s) e.g., correlation length ○ 4-ε expansion does not work! Flow eqs depend singularly on the subdominant time scale: where w = ratio of time scales

52. Quantum Criticality Workshop Toronto 52 Sep 2008 ○ RG analysis for disordered case upper critical dimension is d=4 m q 2 is marginal for all 0<d<4, and is the only marginal term log terms in critical behavior (cf. Wegner 1970s) e.g., correlation length ○ 4-ε expansion does not work! Flow eqs depend singularly on the subdominant time scale: where w = ratio of time scales NB: One-loop (or any finite-loop) order yields misleading results Infinite resummation logs

53. Quantum Criticality Workshop Toronto 53 Sep 2008 ○ Comparison with experiments: Butch & Maple (2008)

54. Quantum Criticality Workshop Toronto 54 Sep 2008 ○ Comparison with experiments: ▫ δ close to 3/2, effectively x-dependent agrees with theory! (3/2 + logs) Butch & Maple (2008)

55. Quantum Criticality Workshop Toronto 55 Sep 2008 ○ Comparison with experiments: ▫ δ close to 3/2, effectively x-dependent agrees with theory! (3/2 + logs) ▫ γ → 0, x-over to 1 st order?? (Should go the other way: 1 st to 2 nd !) Butch & Maple (2008)

56. Quantum Criticality Workshop Toronto 56 Sep 2008 ○ Comparison with experiments: ▫ δ close to 3/2, effectively x-dependent agrees with theory! (3/2 + logs) ▫ γ → 0, x-over to 1 st order?? (Should go the other way: 1 st to 2 nd !) ▫ β ≈ 0.8 with no x-dependence, ?? Butch & Maple (2008)

57. Quantum Criticality Workshop Toronto 57 Sep 2008 ○ Comparison with experiments: ▫ δ close to 3/2, effectively x-dependent agrees with theory! (3/2 + logs) ▫ γ → 0, x-over to 1 st order?? (Should go the other way: 1 st to 2 nd !) ▫ β ≈ 0.8 with no x-dependence, ?? ○ Needed: ▫ Analysis of width of asymptotic region ▫ Analysis of x-overs to pre-asymptotic region, and to clean behavior Butch & Maple (2008)

58. Quantum Criticality Workshop Toronto 58 Sep 2008 ○ RG analysis for clean case upper critical dimension is d=3

59. Quantum Criticality Workshop Toronto 59 Sep 2008 ○ RG analysis for clean case upper critical dimension is d=3 ○ 3-ε expansion to 1-loop order suggests 2 nd order transition is possible in certain parameter regimes (fluctuation-induced 2 nd order: u driven negative is counteracted by couplings at loop level).

60. Quantum Criticality Workshop Toronto 60 Sep 2008 ○ RG analysis for clean case upper critical dimension is d=3 ○ 3-ε expansion to 1-loop order suggests 2 nd order transition is possible in certain parameter regimes (fluctuation-induced 2 nd order: u driven negative is counteracted by couplings at loop level). This analysis is suspect due to the problems with the ε-expansion! More work is needed.

61. Quantum Criticality Workshop Toronto 61 Sep 2008 4. Summary of quantum ferromagnetic transitions ■ Renormalized mean-field theory explains the phase diagram, and the qualitative disorder dependence (1 st vs 2 nd order)..

62. Quantum Criticality Workshop Toronto 62 Sep 2008 4. Summary of quantum ferromagnetic transitions ■ Renormalized mean-field theory explains the phase diagram, and the qualitative disorder dependence (1 st vs 2 nd order). ■ External magnetic field restores QCP in clean case. Here, Hertz theory works!

63. Quantum Criticality Workshop Toronto 63 Sep 2008 4. Summary of quantum ferromagnetic transitions ■ Renormalized mean-field theory explains the phase diagram, and the qualitative disorder dependence (1 st vs 2 nd order). ■ External magnetic field restores QCP in clean case. Here, Hertz theory works! ■ For disordered systems, exotic critical behavior is predicted. Experiments are now available, analysis is needed!

64. Quantum Criticality Workshop Toronto 64 Sep 2008 4. Summary of quantum ferromagnetic transitions ■ Renormalized mean-field theory explains the phase diagram, and the qualitative disorder dependence (1 st vs 2 nd order). ■ External magnetic field restores QCP in clean case. Here, Hertz theory works! ■ For disordered systems, exotic critical behavior is predicted. Experiments are now available, analysis is needed! ■ Role of fluctuations in clean systems needs to be investigated.

65. Quantum Criticality Workshop Toronto 65 Sep 2008 III. Some Other Transitions 1. Metamagnetic transitions. ■ Some quantum FMs show metamagnetic transitions: ● UGe 2 (Pfleiderer & Huxley 2002)

66. Quantum Criticality Workshop Toronto 66 Sep 2008 III. Some Other Transitions 1. Metamagnetic transitions. ■ Some quantum FMs show metamagnetic transitions: ● UGe 2 (Pfleiderer & Huxley 2002) ● Sr 3 Ru 2 O 7 (e.g., Grigera et al 2004) (“hidden order”) Possibly a Pomeranchuk instability (Ho & Schofield 2008)

67. Quantum Criticality Workshop Toronto 67 Sep 2008 III. Some Other Transitions 1. Metamagnetic transitions. ■ Some quantum FMs show metamagnetic transitions: ● UGe 2 (Pfleiderer & Huxley 2002) ● Sr 3 Ru 2 O 7 (e.g., Grigera et al 2004) (“hidden order”) Possibly a Pomeranchuk instability (Ho & Schofield 2008) ■ Another example of a restored ferromagnetic QCP: Critical behavior at a ○ metamagnetic end point. Is Hertz theory valid? (magnons!)

68. Quantum Criticality Workshop Toronto 68 Sep 2008 2. Partial order transition in MnSi. ■ MnSi is a weak helimagnet with a complicated phase diagram

69. Quantum Criticality Workshop Toronto 69 Sep 2008 2. Partial order transition in MnSi. ■ MnSi is a weak helimagnet with a complicated phase diagram ■ Some features can be explained by approximating MnSi as a FM, while others cannot. Neutron scattering shows “partial order” in the PM phase (Pfleiderer et al 2006, Uemura et al 2007): Magnetic state is a helimagnet with 2π/q ≈ 180 Ǻ, pinning in (111) direction

70. Quantum Criticality Workshop Toronto 70 Sep 2008 2. Partial order transition in MnSi. ■ MnSi is a weak helimagnet with a complicated phase diagram ■ Some features can be explained by approximating MnSi as a FM, while others cannot. Neutron scattering shows “partial order” in the PM phase: Short-ranged helical order persists in the paramagnetic phase below a temperature T 0 (p). Pitch little changed, but axis orientation much more isotropic than in the ordered phase. Slow dynamics.

71. Quantum Criticality Workshop Toronto 71 Sep 2008 2. Partial order transition in MnSi. ■ MnSi is a weak helimagnet with a complicated phase diagram ■ Some features can be explained by approximating MnSi as a FM, while others cannot. Neutron scattering shows “partial order” in the PM phase: No detectable helical order for T > T0 (p)

72. Quantum Criticality Workshop Toronto 72 Sep 2008 2. Partial order transition in MnSi. ■ Theory: Chiral OP in analogy to the theory of Blue Phase III or Blue Fog in liquid crystals

73. Quantum Criticality Workshop Toronto 73 Sep 2008 2. Partial order transition in MnSi. ■ Theory: Chiral OP in analogy to the theory of Blue Phase III or Blue Fog in liquid crystals 1 st order transition from a chiral gas (PM phase) to a chiral liquid (partial order phase, “blue quantum fog”) (S. Tewari, DB, TRK 2006)

74. Quantum Criticality Workshop Toronto 74 Sep 2008 2. Partial order transition in MnSi. ■ Theory: Chiral OP in analogy to the theory of Blue Phase III or Blue Fog in liquid crystals 1 st order transition from a chiral gas (PM phase) to a chiral liquid (partial order phase, “blue quantum fog”) (S. Tewari, DB, TRK 2006) ■ Alternative explanations: Analogies to crystalline blue phases (Binz et al 2006, Fischer, Shah, Rosch 2008)

75. Quantum Criticality Workshop Toronto 75 Sep 2008 3. Quantum critical point in an inhomogeneous ferromagnet. ■ Consider a FM with a linearly position dependent electron density (can be achieved by bending a metallic plate)

76. Quantum Criticality Workshop Toronto 76 Sep 2008 3. Quantum critical point in an inhomogeneous ferromagnet. ■ Consider a FM with a linearly position dependent electron density (can be achieved by bending a metallic plate) ■ Magnetization is inhomogeneous, but goes to zero uniformly at a QCP (DB, TRK, R. Saha 2007):

77. Quantum Criticality Workshop Toronto 77 Sep 2008 3. Quantum critical point in an inhomogeneous ferromagnet. ■ Consider a FM with a linearly position dependent electron density (can be achieved by bending a metallic plate) ■ Magnetization is inhomogeneous, but goes to zero uniformly at a QCP (DB, TRK, R. Saha 2007):

78. Quantum Criticality Workshop Toronto 78 Sep 2008 3. Quantum critical point in an inhomogeneous ferromagnet. ■ Consider a FM with a linearly position dependent electron density (can be achieved by bending a metallic plate) ■ Magnetization is inhomogeneous, but goes to zero uniformly at a QCP (DB, TRK, R. Saha 2007): ■ NB: Mean-field exponents (another example where Hertz theory works!)

79. Quantum Criticality Workshop Toronto 79 Sep 2008 3. Quantum critical point in an inhomogeneous ferromagnet. ■ Consider a FM with a linearly position dependent electron density (can be achieved by bending a metallic plate) ■ Magnetization is inhomogeneous, but goes to zero uniformly at a QCP (DB, TRK, R. Saha 2007): ■ NB: Mean-field exponents (another example where Hertz theory works!) ■ Open problem: Non-equilibrium behavior

80. Quantum Criticality Workshop Toronto 80 Sep 2008 Acknowledgments Ted Kirkpatrick Maria-Teresa Mercaldo Rajesh Narayanan Jörg Rollbühler Achim Rosch Ronojoy Saha Sharon Sessions Sumanta Tewari John Toner Thomas Vojta Peter Böni Christian Pfleiderer Aspen Center for Physics KITP at UCSB Lorentz Center Leiden National Science Foundation