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

2. APS March Meeting Denver 2 March 2007 I. Ferromagnets versus Helimagnets Ferromagnets: 0 < J ~ exchange interaction (strong) (Heisenberg 1930s)

3. APS March Meeting Denver 3 March 2007 I. Ferromagnets versus Helimagnets Ferromagnets: Helimagnets: 0 < J ~ exchange interaction (strong) (Heisenberg 1930s) c ~ spin-orbit interaction (weak) q ~ c pitch wave number of helix (Dzyaloshinski 1958, Moriya 1960)

4. APS March Meeting Denver 4 March 2007 I. Ferromagnets versus Helimagnets Ferromagnets: Helimagnets: 0 < J ~ exchange interaction (strong) (Heisenberg 1930s) c ~ spin-orbit interaction (weak) q ~ c pitch wave number of helix (Dzyaloshinski 1958, Moriya 1960) Crystal-field effects ultimately pin helix (very weak)

5. APS March Meeting Denver 5 March 2007 I. Ferromagnets versus Helimagnets Ferromagnets: Helimagnets: 0 < J ~ exchange interaction (strong) (Heisenberg 1930s) c ~ spin-orbit interaction (weak) q ~ c pitch wave number of helix (Dzyaloshinski 1958, Moriya 1960) Crystal-field effects ultimately pin helix (very weak) Examples: MnSi, FeGe

6. APS March Meeting Denver 6 March 2007 II. Phenomenology of MnSi 1. Phase diagram magnetic transition at T c ≈ 30 K (at ambient pressure) (Pfleiderer et al 1997) TCP

7. APS March Meeting Denver 7 March 2007 II. Phenomenology of MnSi 1. Phase diagram magnetic transition at T c ≈ 30 K (at ambient pressure) transition tunable by means of hydrostatic pressure p (Pfleiderer et al 1997) TCP

8. APS March Meeting Denver 8 March 2007 II. Phenomenology of MnSi 1. Phase diagram magnetic transition at T c ≈ 30 K (at ambient pressure) transition tunable by means of hydrostatic pressure p Transition is 2 nd order at high T, 1 st order at low T t tricritical point at T ≈ 10 K (no QCP in T-p plane!) (Pfleiderer et al 1997) TCP

9. APS March Meeting Denver 9 March 2007 II. Phenomenology of MnSi 1. Phase diagram magnetic transition at T c ≈ 30 K (at ambient pressure) transition tunable by means of hydrostatic pressure p Transition is 2 nd order at high T, 1 st 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” (Pfleiderer et al 1997) (Pfleiderer, Julian, Lonzarich 2001) TCP

10. APS March Meeting Denver 10 March 2007 II. Phenomenology of MnSi 1. Phase diagram magnetic transition at T c ≈ 30 K (at ambient pressure) transition tunable by means of hydrostatic pressure p Transition is 2 nd order at high T, 1 st 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 (Pfleiderer et al 1997) (Pfleiderer, Julian, Lonzarich 2001) TCP

11. APS March Meeting Denver 11 March 2007 2. Neutron Scattering (Pfleiderer et al 2004) Magnetic state is a helimagnet with q ≈ 180 Ǻ, pinning in (111) direction

12. APS March Meeting Denver 12 March 2007 2. Neutron Scattering (Pfleiderer et al 2004) Magnetic state is a helimagnet with q ≈ 180 Ǻ, pinning in (111) direction 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 (helical axis essentially de-pinned)

13. APS March Meeting Denver 13 March 2007 2. Neutron Scattering (Pfleiderer et al 2004) Magnetic state is a helimagnet with q ≈ 180 Ǻ, pinning in (111) direction 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 > T 0 (p)

14. APS March Meeting Denver 14 March 2007 3. Transport Properties Non-Fermi-liquid behavior of the resistivity:

15. APS March Meeting Denver 15 March 2007 3. Transport Properties Non-Fermi-liquid behavior of the resistivity: Over a huge range in parameter space, the resistivity behaves as ρ ~ T 1.5 o T 1.5 (K 1.5 ) ρ ( μΩ cm)

16. APS March Meeting Denver 16 March 2007 III. Theory 1. Nature of the Phase Diagram  Basic features can be understood by approximating the system as a FM

17. APS March Meeting Denver 17 March 2007 III. Theory 1. Nature of the Phase Diagram  Basic features can be understood by approximating the system as a FM  Tricritical point due to fluctuation effects (coupling of fermionic soft modes to magnetization) DB, T.R. Kirkpatrick, T. Vojta, PRL 82, 4707 (1999)

18. APS March Meeting Denver 18 March 2007 III. Theory 1. Nature of the Phase Diagram  Basic features can be understood by approximating the system as a FM  Tricritical point due to fluctuation effects (coupling of fermionic soft modes to magnetization) DB, T.R. Kirkpatrick, T. Vojta, PRL 82, 4707 (1999)  Wings follow from existence of tricritical point DB, T.R. Kirkpatrick, J. Rollbühler, PRL 94, 247205 (2005) Critical behavior at QCP determined exactly!

19. APS March Meeting Denver 19 March 2007 2. Disordered Phase: Interpretation of T 0 (p) Basic idea: Liquid-gas-type phase transition with chiral order parameter (cf. Lubensky & Stark 1996) Borrow an idea from liquid-crystal physics:

20. APS March Meeting Denver 20 March 2007 2. Disordered Phase: Interpretation of T 0 (p) 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 Borrow an idea from liquid-crystal physics:

21. APS March Meeting Denver 21 March 2007 2. Disordered Phase: Interpretation of T 0 (p) 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 Borrow an idea from liquid-crystal physics:

22. APS March Meeting Denver 22 March 2007 2. Disordered Phase: Interpretation of T 0 (p) 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 1 st order transition (e.g., discontinuity in the spin susceptibility) should be observable across T 0 Borrow an idea from liquid-crystal physics:

23. APS March Meeting Denver 23 March 2007 Proposed phase diagram :

24. APS March Meeting Denver 24 March 2007 Analogy: Blue Phase III in chiral liquid crystals Proposed phase diagram : (J. Sethna) (Lubensky & Stark 1996)

25. APS March Meeting Denver 25 March 2007 Other proposals :  Superposition of spin spirals with different wave vectors (Binz et al 2006)  Spontaneous skyrmion ground state (Roessler et al 2006)  Stabilization of analogs to crystalline blue phases (Fischer & Rosch 2006, Fischer et al 2007) (NB: All of these proposals are also related to blue-phase physics)

26. APS March Meeting Denver 26 March 2007 3. Ordered Phase: Nature of the Goldstone mode Helical ground state: breaks translational symmetry soft (Goldstone) mode

27. APS March Meeting Denver 27 March 2007 3. Ordered Phase: Nature of the Goldstone mode Helical ground state: breaks translational symmetry soft (Goldstone) mode Rotational symmetry anisotropic dispersion relation “helimagnon” (cf. chiral liquid crystals)

28. APS March Meeting Denver 28 March 2007 4. Ordered Phase: Specific heat Internal energy density: Specific heat: helimagnon contribution total low-T specific heat

29. APS March Meeting Denver 29 March 2007 5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/  (T) ~ T 3/2 stronger than FL T 2 contribution! (hard to measure)

30. APS March Meeting Denver 30 March 2007 5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/  (T) ~ T 3/2 stronger than FL T 2 contribution! (hard to measure) Resistivity:  (T) ~ T 5/2 weaker than QP relaxation time, cf. phonon case (T 3 vs T 5 )

31. APS March Meeting Denver 31 March 2007 5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/  (T) ~ T 3/2 stronger than FL T 2 contribution! (hard to measure) Resistivity:  (T) ~ T 5/2 weaker than QP relaxation time, cf. phonon case (T 3 vs T 5 )  (T) =  2 T 2 +  5/2 T 5/2 total low-T resistivity

32. APS March Meeting Denver 32 March 2007 5. Ordered Phase: Relaxation times and resistivity Quasi-particle relaxation time: 1/  (T) ~ T 3/2 stronger than FL T 2 contribution! (hard to measure) Resistivity:  (T) ~ T 5/2 weaker than QP relaxation time, cf. phonon case (T 3 vs T 5 )  (T) =  2 T 2 +  5/2 T 5/2 total low-T resistivity Experiment:  (T→ 0) ~ T 2 (more analysis needed)

33. APS March Meeting Denver 33 March 2007 6. Ordered Phase: Breakdown of hydrodynamics Use TDGL theory to study magnetization dynamics:

34. APS March Meeting Denver 34 March 2007 6. Ordered Phase: Breakdown of hydrodynamics Use TDGL theory to study magnetization dynamics: Bloch term damping Langevin force

35. APS March Meeting Denver 35 March 2007 6. Ordered Phase: Breakdown of hydrodynamics Use TDGL theory to study magnetization dynamics: Bare magnetic response function: helimagnon frequency damping coefficient One-loop correction to 

36. APS March Meeting Denver 36 March 2007 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, c z 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) At T = 0, all renormalizations are finite! (Special answer to a more general question: As T -> 0, classical mode-mode coupling effects die (how?), while new quantum mode-mode coupling effects may appear)

37. APS March Meeting Denver 37 March 2007 IV. Summary  Basic T-p-h phase diagram is understood

38. APS March Meeting Denver 38 March 2007 IV. Summary  Basic T-p-h phase diagram is understood  Possible additional 1 st order transition in disordered phase

39. APS March Meeting Denver 39 March 2007 IV. Summary  Basic T-p-h phase diagram is understood  Possible additional 1 st order transition in disordered phase  Helimagnons predicted in ordered phase; lead to T 2 term in specific heat

40. APS March Meeting Denver 40 March 2007 IV. Summary  Basic T-p-h phase diagram is understood  Possible additional 1 st order transition in disordered phase  Helimagnons predicted in ordered phase; lead to T 2 term in specific heat  NFL quasi-particle relaxation time predicted in ordered phase

41. APS March Meeting Denver 41 March 2007 IV. Summary  Basic T-p-h phase diagram is understood  Possible additional 1 st order transition in disordered phase  Helimagnons predicted in ordered phase; lead to T 2 term in specific heat  NFL quasi-particle relaxation time predicted in ordered phase  Resistivity in ordered phase is FL-like with T 5/2 correction

42. APS March Meeting Denver 42 March 2007 IV. Summary  Basic T-p-h phase diagram is understood  Possible additional 1 st order transition in disordered phase  Helimagnons predicted in ordered phase; lead to T 2 term in specific heat  NFL quasi-particle relaxation time predicted in ordered phase  Resistivity in ordered phase is FL-like with T 5/2 correction  Hydrodynamic description of ordered phase breaks down

43. APS March Meeting Denver 43 March 2007 IV. Summary  Basic T-p-h phase diagram is understood  Possible additional 1 st order transition in disordered phase  Helimagnons predicted in ordered phase; lead to T 2 term in specific heat  NFL quasi-particle relaxation time predicted in ordered phase  Resistivity in ordered phase is FL-like with T 5/2 correction  Hydrodynamic description of ordered phase breaks down  Main open question: Origin of T 3/2 resistivity in disordered phase?

44. APS March Meeting Denver 44 March 2007 Acknowledgments Ted Kirkpatrick Rajesh Narayanan Jörg Rollbühler Achim Rosch Sumanta Tewari John Toner Thomas Vojta Peter Böni Christian Pfleiderer Aspen Center for Physics KITP at UCSB Lorentz Center Leiden National Science Foundation