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First Order vs Second Order Transitions in Quantum Magnets

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

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First Order vs Second Order Transitions in Quantum Magnets

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

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

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

  4. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) Quantum Criticality Workshop Toronto

  5. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2nd order transition at high T, 1st order transition at low T: Quantum Criticality Workshop Toronto

  6. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2nd order transition at high T, 1st order transition at low T: UGe2 (Pfleiderer & Huxley 2002) Quantum Criticality Workshop Toronto

  7. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2nd order transition at high T, 1st order transition at low T: UGe2 ZrZn2 (Pfleiderer & Huxley 2002) (Uhlarz et al 2004) Quantum Criticality Workshop Toronto

  8. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2nd order transition at high T, 1st order transition at low T: UGe2 ZrZn2 MnSi (Pfleiderer & Huxley 2002) (Uhlarz et al 2004) (Pfleiderer et al 1997) Quantum Criticality Workshop Toronto

  9. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ Clean materials all show tricritical point, with 2nd order transition at high T, 1st order transition at low T: ○ Additional evidence: μSR (Uemura et al 2007) UGe2 ZrZn2 MnSi (Pfleiderer & Huxley 2002) (Uhlarz et al 2004) (Pfleiderer et al 1997) Quantum Criticality Workshop Toronto

  10. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ T=0 1st order transition persists in a B-field, ends at quantum critical point. Quantum Criticality Workshop Toronto

  11. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● UGe2, ZrZn2, (MnSi) (clean, pressure tuned) ○ T=0 1st order transition persists in a B-field, ends at quantum critical point. Schematic phase diagram: Quantum Criticality Workshop Toronto

  12. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● URu2-xRexSi2 (disordered, concentration tuned) Quantum Criticality Workshop Toronto

  13. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● URu2-xRexSi2 (disordered, concentration tuned) ○ Disordered material shows a 2nd order transition down to T=0: Bauer et al (2005) Butch & Maple (2008) Quantum Criticality Workshop Toronto

  14. I. Quantum Ferromagnetic Transitions: Experiments ■ Itinerant ferromagnets whose Tc can be tuned to zero: ● URu2-xRexSi2 (disordered, concentration tuned) ○ Disordered material shows a 2nd order transition down to T=0: Bauer et al (2005) Butch & Maple (2008) ○ Observed exponents are not mean-field like (see below) Quantum Criticality Workshop Toronto

  15. 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. Quantum Criticality Workshop Toronto

  16. 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 = f0 – h m + t m2 + u m4 + w m6 Equation of state: h = t m + u m3 + w m5 + … Quantum Criticality Workshop Toronto

  17. 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 = f0 – h m + t m2 + u m4 + w m6 Equation of state: h = t m + u m3 + w m5 + … ■ Landau theory predicts: ● 2nd order transition at t=0 if u<0 ● 1st order transition if u<0 } for both clean and dirty systems Quantum Criticality Workshop Toronto

  18. 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 = f0 – h m + t m2 + u m4 + w m6 Equation of state: h = t m + u m3 + w m5 + … ■ Landau theory predicts: ● 2nd order transition at t=0 if u<0 ● 1st order transition if u<0 ■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe2 u<0 } for both clean and dirty systems Quantum Criticality Workshop Toronto

  19. 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 = f0 – h m + t m2 + u m4 + w m6 Equation of state: h = t m + u m3 + w m5 + … ■ Landau theory predicts: ● 2nd order transition at t=0 if u<0 ● 1st order transition if u<0 ■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe2 u<0 ■ Problems: ● Not universal ● Does not explain the tricritical point ● Observed critical behavior not mean-field like } for both clean and dirty systems Quantum Criticality Workshop Toronto

  20. 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 = f0 – h m + t m2 + u m4 + w m6 Equation of state: h = t m + u m3 + w m5 + … ■ Landau theory predicts: ● 2nd order transition at t=0 if u<0 ● 1st order transition if u<0 ■ Sandeman et al 2003, Shick et al 2004: Band structure in UGe2 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 Quantum Criticality Workshop Toronto

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

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

  23. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f0: Quantum Criticality Workshop Toronto

  24. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f0: ● Contribution to eq. of state: Quantum Criticality Workshop Toronto

  25. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ■ Hertz theory misses effects of soft p-h excitations (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f0: ● Contribution to eq. of state: ● Renormalized mean-field equation of state: (clean, d=3, T=0) Quantum Criticality Workshop Toronto

  26. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ■ In general, Hertz theory misses effects of soft modes (TRK & DB 1996 ff) ● Soft modes (clean case) ● Contribution to f0: ● Contribution to eq. of state: ● Renormalized mean-field equation of state: (clean, d=3, T=0) ● v>0 Transition is generically 1st order! (TRK, T Vojta, DB 1999) Quantum Criticality Workshop Toronto

  27. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point Quantum Criticality Workshop Toronto

  28. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes Quantum Criticality Workshop Toronto

  29. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes ○ fermion dispersion relation md -> md/2 Quantum Criticality Workshop Toronto

  30. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● T>0 gives soft p-h excitations a mass ln m -> ln (m+T) tricritical point ● Quenched disorder G changes ○ fermion dispersion relation md -> md/2 ○ sign of the coefficient Quantum Criticality Workshop Toronto

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

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

  33. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● Phase diagrams: G=0 Quantum Criticality Workshop Toronto

  34. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● Phase diagrams: G=0 T=0 Quantum Criticality Workshop Toronto

  35. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● External field h produces tricritical wings: (DB, TRK, J. Rollbühler 2005) Quantum Criticality Workshop Toronto

  36. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● 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! Quantum Criticality Workshop Toronto

  37. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● 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: δmc ~ -T4/9 Quantum Criticality Workshop Toronto

  38. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● 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: δmc ~ -T4/9 ■ Conclusion:Renormalized mean-field theory explains the experimentally observed phase diagram: (Pfleiderer, Julian, Lonzarich 2001) Quantum Criticality Workshop Toronto

  39. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● 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: δmc ~ -T4/9 ■ 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) Quantum Criticality Workshop Toronto

  40. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● 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: δmc ~ -T4/9 ■ 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) Quantum Criticality Workshop Toronto

  41. II. Quantum Ferromagnetic Transitions: Theory 2. Renormalized mean-field theory ● 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: δmc ~ -T4/9 ■ 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) Quantum Criticality Workshop Toronto

  42. II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ■ Complete theory: Keep all soft modes explicitly: Quantum Criticality Workshop Toronto

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

  44. II. Quantum Ferromagnetic Transitions: Theory 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) Quantum Criticality Workshop Toronto

  45. II. Quantum Ferromagnetic Transitions: Theory 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) Quantum Criticality Workshop Toronto

  46. II. Quantum Ferromagnetic Transitions: Theory 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: Quantum Criticality Workshop Toronto

  47. II. Quantum Ferromagnetic Transitions: Theory 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 q2 term in effective action) Quantum Criticality Workshop Toronto

  48. II. Quantum Ferromagnetic Transitions: Theory 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 q2 term in effective action) ○ mean-field approx for OP + Gaussian approx for fermions renormalized mean-field theory (FP marginally unstable) Quantum Criticality Workshop Toronto

  49. II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ RG analysis for disordered case upper critical dimension is d=4 m q2 is marginal for all 0<d<4, and is the only marginal term Quantum Criticality Workshop Toronto

  50. II. Quantum Ferromagnetic Transitions: Theory 3. Order-parameter fluctuations ○ RG analysis for disordered case upper critical dimension is d=4 m q2 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 Quantum Criticality Workshop Toronto

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