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Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSi

Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSi. Dietrich Belitz, University of Oregon with Ted Kirkpatrick, Achim Rosch, Thomas Vojta, et al. Ferromagnets and Helimagnets II. Phenomenology of MnSi Theory 1. Phase diagram 2. Disordered phase

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Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSi

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  1. Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSi Dietrich Belitz, University of Oregon with Ted Kirkpatrick, Achim Rosch, Thomas Vojta, et al. Ferromagnets and Helimagnets II. Phenomenology of MnSi Theory 1. Phase diagram 2. Disordered phase 3. Ordered phase

  2. I. Ferromagnets versus Helimagnets Ferromagnets: 0 < J ~ exchange interaction (strong) (Heisenberg 1930s) Lorentz Center

  3. I. Ferromagnets versus Helimagnets Ferromagnets: 0 < J ~ exchange interaction (strong) (Heisenberg 1930s) Helimagnets: (Dzyaloshinski 1958, Moriya 1960) c ~ spin-orbit interaction (weak) q ~ c pitch wave number of helix Lorentz Center

  4. I. Ferromagnets versus Helimagnets Ferromagnets: 0 < J ~ exchange interaction (strong) (Heisenberg 1930s) Helimagnets: (Dzyaloshinski 1958, Moriya 1960) c ~ spin-orbit interaction (weak) q ~ c pitch wave number of helix • HHM invariant under rotations, but not under x → - x • Crystal-field effects ultimately pin helix (very weak) Lorentz Center

  5. II. Phenomenology of MnSi • magnetic transition at Tc ≈ 30 K (at ambient pressure) 1. Phase diagram (Pfleiderer et al 1997) Lorentz Center

  6. II. Phenomenology of MnSi • magnetic transition at Tc ≈ 30 K (at ambient pressure) • transition tunable by means of hydrostatic pressure p 1. Phase diagram (Pfleiderer et al 1997) Lorentz Center

  7. II. Phenomenology of MnSi • magnetic transition at Tc ≈ 30 K (at ambient pressure) • transition tunable by means of hydrostatic pressure p • Transition is 2nd order at high T, 1st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane!) 1. Phase diagram TCP (Pfleiderer et al 1997) Lorentz Center

  8. II. Phenomenology of MnSi • magnetic transition at Tc ≈ 30 K (at ambient pressure) • transition tunable by means of hydrostatic pressure p • Transition is 2nd order at high T, 1st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane! ) • In an external field B there are “tricritical wings” 1. Phase diagram TCP (Pfleiderer et al 1997) (Pfleiderer, Julian, Lonzarich 2001) Lorentz Center

  9. II. Phenomenology of MnSi • magnetic transition at Tc ≈ 30 K (at ambient pressure) • transition tunable by means of hydrostatic pressure p • Transition is 2nd order at high T, 1st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane! ) • In an external field B there are “tricritical wings” • Quantum critical point at B≠ 0 1. Phase diagram TCP (Pfleiderer et al 1997) (Pfleiderer, Julian, Lonzarich 2001) Lorentz Center

  10. II. Phenomenology of MnSi • magnetic transition at Tc ≈ 30 K (at ambient pressure) • transition tunable by means of hydrostatic pressure p • Transition is 2nd order at high T, 1st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane! ) • In an external field B there are “tricritical wings” • Quantum critical point at B≠ 0 • Magnetic state is a helimagnet with q≈ 180 Ǻ, pinning in (111) dd direction 1. Phase diagram TCP (Pfleiderer et al 1997) (Pfleiderer et al 2004) (Pfleiderer, Julian, Lonzarich 2001) Lorentz Center

  11. II. Phenomenology of MnSi • magnetic transition at Tc ≈ 30 K (at ambient pressure) • transition tunable by means of hydrostatic pressure p • Transition is 2nd order at high T, 1st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane!) • In an external field B there are “tricritical wings” • Quantum critical point at B≠ 0 • Magnetic state is a helimagnet with q≈ 180 Ǻ, pinning in (111) dd direction • Cubic unit cell lacks inversion symmetry (in agreement with DM) 1. Phase diagram TCP (Pfleiderer et al 1997) (Carbone et al 2005) (Pfleiderer et al 2004) (Pfleiderer, Julian, Lonzarich 2001) Lorentz Center

  12. 2. Neutron Scattering • Ordered phase shows helical order, see above (Pfleiderer et al 2004) Lorentz Center

  13. 2. Neutron Scattering • Ordered phase shows helical order, see above • Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p) (Pfleiderer et al 2004) Lorentz Center

  14. 2. Neutron Scattering • Ordered phase shows helical order, see above • Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p) • Pitch little changed, but axis orientation much more isotropic than in the ordered phase (helical axis essentially de-pinned) (Pfleiderer et al 2004) Lorentz Center

  15. 2. Neutron Scattering • Ordered phase shows helical order, see above • Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p) • Pitch little changed, but axis orientation much more isotropic than in the ordered phase (helical axis essentially de-pinned) • No detectable helical order for T > T0 (p) (Pfleiderer et al 2004) Lorentz Center

  16. 2. Neutron Scattering • Ordered phase shows helical order, see above • Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p) • Pitch little changed, but axis orientation much more isotropic than in the ordered phase (helical axis essentially de-pinned) • No detectable helical order for T > T0 (p) • T0 (p)originates close to TCP (Pfleiderer et al 2004) Lorentz Center

  17. 2. Neutron Scattering • Ordered phase shows helical order, see above • Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p) • Pitch little changed, but axis orientation much more isotropic than in the ordered phase (helical axis essentially de-pinned) • No detectable helical order for T > T0 (p) • T0 (p)originates close to TCP • So far only three data points for T0 (p) (Pfleiderer et al 2004) Lorentz Center

  18. 3. Transport Properties • Non-Fermi-liquid behavior of the resistivity: p = 14.8kbar > pc ρ(μΩcm) T(K) ρ(μΩcm) • Resistivity ρ ~T 1.5 o over a huge range in parameter space T1.5(K1.5) ρ(μΩcm) T1.5(K1.5) Lorentz Center

  19. III. Theory 1. Nature of the Phase Diagram • Basic features can be understood by approximating the system as a FM Lorentz Center

  20. III. Theory 1. Nature of the Phase Diagram • Basic features can be understood by approximating the system as a FM • Tricritical point due to many-body effects (coupling of fermionic soft modes to magnetization) Quenched disorder suppresses the TCP, restores a quantum critical point! DB, T.R. Kirkpatrick, T. Vojta, PRL 82, 4707 (1999) Lorentz Center

  21. III. Theory 1. Nature of the Phase Diagram • Basic features can be understood by approximating the system as a FM • Tricritical point due to many-body effects (coupling of fermionic soft modes to magnetization) Quenched disorder suppresses the TCP, restores a quantum critical point! DB, T.R. Kirkpatrick, T. Vojta, PRL 82, 4707 (1999) NB: TCP can also follow from material-specific band-structure effects (Schofield et al), but the many-body mechanism is generic Lorentz Center

  22. III. Theory 1. Nature of the Phase Diagram • Basic features can be understood by approximating the system as a FM • Tricritical point due to many-body effects (coupling of fermionic soft modes to magnetization) Quenched disorder suppresses the TCP, restores a quantum critical point! DB, T.R. Kirkpatrick, T. Vojta, PRL 82, 4707 (1999) NB: TCP can also follow from material-specific band-structure effects (Schofield et al), but the many-body mechanism is generic • Wings follow from existence of tricritical point DB, T.R. Kirkpatrick, J. Rollbühler, PRL 94, 247205 (2005) Critical behavior at QCP determined exactly! (Hertz theory is valid due to B > 0) Lorentz Center

  23. Example of a more general principle: Hertz theory is valid if the field conjugate to the order parameter does not change the soft-mode structure (DB, T.R. Kirkpatrick, T. Vojta, Phys. Rev. B 65, 165112 (2002)) Here, B field already breaks a symmetry no additional symmetry breaking by the conjugate field mean-field critical behavior with corrections due to DIVs in particular, d m (pc,Hc,T) ~ -T 4/9 Lorentz Center

  24. 2. Disordered Phase: Interpretation of T0(p) Borrow an idea from liquid-crystal physics: Basic idea: Liquid-gas-type phase transition with chiral order parameter (cf. Lubensky & Stark 1996) Lorentz Center

  25. 2. Disordered Phase: Interpretation of T0(p) Borrow an idea from liquid-crystal physics: Basic idea: Liquid-gas-type phase transition with chiral order parameter (cf. Lubensky & Stark 1996) Important points: • Chirality parameter c acts as external field conjugate to chiral OP Lorentz Center

  26. 2. Disordered Phase: Interpretation of T0(p) Borrow an idea from liquid-crystal physics: Basic idea: Liquid-gas-type phase transition with chiral order parameter (cf. Lubensky & Stark 1996) Important points: • Chirality parameter c acts as external field conjugate to chiral OP • Perturbation theory Attractive interaction between OP fluctuations! • Condensation of chiral fluctuations is possible Lorentz Center

  27. 2. Disordered Phase: Interpretation of T0(p) Borrow an idea from liquid-crystal physics: Basic idea: Liquid-gas-type phase transition with chiral order parameter (cf. Lubensky & Stark 1996) Important points: • Chirality parameter c acts as external field conjugate to chiral OP • Perturbation theory Attractive interaction between OP fluctuations! • Condensation of chiral fluctuations is possible • Prediction: Feature characteristic of 1st order transition (e.g., discontinuity in • the spin susceptibility) should be observable across T0 Lorentz Center

  28. Proposed phase diagram : Lorentz Center

  29. Proposed phase diagram : Lorentz Center

  30. Proposed phase diagram : Analogy: Blue Phase III in chiral liquid crystals (J. Sethna) Lorentz Center

  31. Proposed phase diagram : Analogy: Blue Phase III in chiral liquid crystals (J. Sethna) (Lubensky & Stark 1996) Lorentz Center

  32. Proposed phase diagram : Analogy: Blue Phase III in chiral liquid crystals (J. Sethna) (Lubensky & Stark 1996) (Anisimov et al 1998) Lorentz Center

  33. Other proposals: • Superposition of spin spirals with different wave vectors (Binz et al 2006), see following talk. • Spontaneous skyrmion ground state (Roessler et al 2006) • Stabilization of analogs to crystalline blue phases (Fischer & Rosch 2006, see poster) (NB: All of these proposals are also related to blue-phase physics) Lorentz Center

  34. 3. Ordered Phase: Nature of the Goldstone mode Helical ground state: breaks translational symmetry soft (Goldstone) mode Lorentz Center

  35. 3. Ordered Phase: Nature of the Goldstone mode Helical ground state: breaks translational symmetry soft (Goldstone) mode Phase fluctuations: Energy: ?? Lorentz Center

  36. 3. Ordered Phase: Nature of the Goldstone mode Helical ground state: breaks translational symmetry soft (Goldstone) mode Phase fluctuations: Energy: ?? NO! rotation (0,0,q) (a1,a2,q) cannot cost energy, yet corresponds to f(x) = a1x + a2y H fluct > 0 cannot depend on Lorentz Center

  37. 3. Ordered Phase: Nature of the Goldstone mode Helical ground state: breaks translational symmetry soft (Goldstone) mode Phase fluctuations: Energy: ?? NO! rotation (0,0,q) (a1,a2,q) cannot cost energy, yet corresponds to f(x) = a1x + a2y H fluct > 0 cannot depend on Lorentz Center

  38. anisotropic! Lorentz Center

  39. anisotropic! anisotropic dispersion relation (as in chiral liquid crystals) “helimagnon” Lorentz Center

  40. anisotropic! anisotropic dispersion relation (as in chiral liquid crystals) “helimagnon” Compare with ferromagnets w(k) ~ k2 antiferromagnets (k) ~ |k| Lorentz Center

  41. 4. Ordered Phase: Specific heat Internal energy density: Specific heat: helimagnon contribution total low-T specific heat Lorentz Center

  42. 4. Ordered Phase: Specific heat Internal energy density: Specific heat: helimagnon contribution total low-T specific heat Experiment: (E. Fawcett 1970, C. Pfleiderer unpublished) Caveat: Looks encouraging, but there is a quantitative problem, observed T2 may be accidental Lorentz Center

  43. 5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/t(T) ~ T3/2stronger than FL T 2contribution! (hard to measure) Lorentz Center

  44. 5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/t(T) ~ T3/2stronger than FL T 2contribution! (hard to measure) Resistivity: r(T) ~ T 5/2 weaker than QP relaxation time, cf. phonon case (T3 vs T5) Lorentz Center

  45. 5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/t(T) ~ T3/2stronger than FL T 2contribution! (hard to measure) Resistivity: r(T) ~ T 5/2 weaker than QP relaxation time, cf. phonon case (T3 vs T5) r(T) = r2 T 2 + r5/2 T 5/2 total low-T resistivity Lorentz Center

  46. 5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/t(T) ~ T3/2stronger than FL T 2contribution! (hard to measure) Resistivity: r(T) ~ T 5/2 weaker than QP relaxation time, cf. phonon case (T3 vs T5) r(T) = r2 T 2 + r5/2 T 5/2 total low-T resistivity Experiment: r (T→ 0) ~ T 2 (more analysis needed) Lorentz Center

  47. 6. Ordered Phase: Breakdown of hydrodynamics (T.R. Kirkpatrick & DB, work in progress) • Use TDGL theory to study magnetization dynamics: Lorentz Center

  48. 6. Ordered Phase: Breakdown of hydrodynamics (T.R. Kirkpatrick & DB, work in progress) • Use TDGL theory to study magnetization dynamics: Bloch term damping Langevin force Lorentz Center

  49. 6. Ordered Phase: Breakdown of hydrodynamics (T.R. Kirkpatrick & DB, work in progress) • Use TDGL theory to study magnetization dynamics: • Bare magnetic response function: • helimagnon frequency • damping coefficient • Fluctuation-dissipation theorem: • One-loop correction to c : c F Lorentz Center

  50. The elastic coefficients and , and the transport coefficients and all acquire singular corrections at one-loop order due to mode-mode coupling effects: • Strictly speaking, helimagnetic order is not stable at T > 0 • In practice, cz is predicted to change linearly with T, by ~10% from T=0 to T=10K • Analogous to situation in smectic liquid crystals (Mazenko, Ramaswamy, Toner 1983) • What happens to these singularities at T = 0 ? • Special case of a more general problem: As T -> 0, classical mode-mode coupling effects die (how?), whereas new quantum effects appear (e.g., weak localization and related effects) • coth in FD theorem 1-loop integral more singular at T > 0 than at T = 0 ! • All renormalizations are finite at T = 0 ! Lorentz Center

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