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
0 < J ~ exchange interaction (strong) (Heisenberg 1930s)
August 2006
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3. I. Ferromagnets versus Helimagnets
0 < J ~ exchange interaction (strong) (Heisenberg 1930s)
Helimagnets:
(Dzyaloshinski 1958,
Moriya 1960)
c ~ spin-orbit interaction (weak)
q ~ c pitch wave number of helix
August 2006
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4. I. Ferromagnets versus Helimagnets
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)
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5. II. Phenomenology of MnSi
1. Phase diagram
magnetic transition at Tc ≈ 30 K (at ambient pressure)
(Pfleiderer et al 1997)
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6. II. Phenomenology of MnSi
1. Phase diagram
magnetic transition at Tc ≈ 30 K (at ambient pressure)
transition tunable by means of hydrostatic pressure p
(Pfleiderer et al 1997)
August 2006
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7. II. Phenomenology of MnSi
1. Phase diagram
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 !)
TCP
(Pfleiderer et al 1997)
August 2006
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8. II. Phenomenology of MnSi
1. Phase diagram
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”
TCP
(Pfleiderer et al 1997)
(Pfleiderer, Julian, Lonzarich 2001)
August 2006
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9. II. Phenomenology of MnSi
1. Phase diagram
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
TCP
(Pfleiderer et al 1997)
(Pfleiderer, Julian, Lonzarich 2001)
August 2006
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10. II. Phenomenology of MnSi
1. Phase diagram
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) d d direction
TCP
(Pfleiderer et al 1997)
(Pfleiderer et al 2004)
(Pfleiderer, Julian, Lonzarich 2001)
August 2006
Lorentz Center

11. II. Phenomenology of MnSi
1. Phase diagram
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) d d direction
Cubic unit cell lacks inversion symmetry (in agreement with DM)
TCP
(Pfleiderer et al 1997)
(Carbone et al 2005)
(Pfleiderer et al 2004)
(Pfleiderer, Julian, Lonzarich 2001)
August 2006
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12. 2. Neutron Scattering Ordered phase shows helical order, see above
(Pfleiderer et al 2004)
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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)
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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)
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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)
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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)
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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)
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18. 3. Transport Properties Non-Fermi-liquid behavior of the resistivity:
p = 14.8kbar > pc
ρ(μΩcm)
T(K)
ρ(μΩcm)
Resistivity ρ ~ T o over a huge range in parameter space
T1.5(K1.5)
ρ(μΩcm)
T1.5(K1.5)
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19. III. Theory 1. Nature of the Phase Diagram
Basic features can be understood by approximating the system as a FM
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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)
August 2006
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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
August 2006
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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,
(2005)
Critical behavior at QCP determined exactly!
(Hertz theory is valid due to B > 0)
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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, (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
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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)
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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
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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
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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
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28. Proposed phase diagram :
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29. Proposed phase diagram :
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30. Proposed phase diagram :
Analogy: Blue Phase III in chiral liquid crystals
(J. Sethna)
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31. Proposed phase diagram :
Analogy: Blue Phase III in chiral liquid crystals
(J. Sethna)
(Lubensky & Stark 1996)
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32. Proposed phase diagram :
Analogy: Blue Phase III in chiral liquid crystals
(J. Sethna)
(Lubensky & Stark 1996) (Anisimov et al 1998)
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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)
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34. 3. Ordered Phase: Nature of the Goldstone mode
Helical ground state:
breaks translational symmetry
soft (Goldstone) mode
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35. 3. Ordered Phase: Nature of the Goldstone mode
Helical ground state:
breaks translational symmetry
soft (Goldstone) mode
Phase fluctuations:
Energy: ??
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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
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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
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38. anisotropic!
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39. anisotropic!
anisotropic dispersion relation (as in chiral liquid crystals)
“helimagnon”
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40. anisotropic!
anisotropic dispersion relation (as in chiral liquid crystals)
“helimagnon”
Compare with
ferromagnets w(k) ~ k2
antiferromagnets (k) ~ |k|
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41. 4. Ordered Phase: Specific heat
Internal energy density:
Specific heat: helimagnon contribution
total low-T specific heat
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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
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43. 5. Ordered Phase: Relaxation times and resistivity
Quasi-particle relaxation time: /t(T) ~ T 3/ stronger than FL T 2 contribution!
(hard to measure)
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44. 5. Ordered Phase: Relaxation times and resistivity
Quasi-particle relaxation time: /t(T) ~ T 3/ stronger than FL T 2 contribution!
(hard to measure)
Resistivity: r(T) ~ T 5/ weaker than QP relaxation time,
cf. phonon case (T3 vs T5)
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45. 5. Ordered Phase: Relaxation times and resistivity
Quasi-particle relaxation time: /t(T) ~ T 3/ stronger than FL T 2 contribution!
(hard to measure)
Resistivity: r(T) ~ T 5/ weaker than QP relaxation time,
cf. phonon case (T3 vs T5)
r(T) = r2 T 2 + r5/2 T 5/ total low-T resistivity
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46. 5. Ordered Phase: Relaxation times and resistivity
Quasi-particle relaxation time: /t(T) ~ T 3/ stronger than FL T 2 contribution!
(hard to measure)
Resistivity: r(T) ~ T 5/ weaker than QP relaxation time,
cf. phonon case (T3 vs T5)
r(T) = r2 T 2 + r5/2 T 5/ total low-T resistivity
Experiment: r (T→ 0) ~ T (more analysis needed)
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47. 6. Ordered Phase: Breakdown of hydrodynamics
(T.R. Kirkpatrick & DB, work in progress)
Use TDGL theory to study magnetization dynamics:
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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
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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
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50. Strictly speaking, helimagnetic order is not stable at T > 0
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 loop integral more singular at T > 0 than at T = 0 !
All renormalizations are finite at T = 0 !
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51. IV. Summary Basic T-p-h phase diagram is understood August 2006
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52. IV. Summary Basic T-p-h phase diagram is understood
Possible additional 1st order transition in disordered phase
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53. IV. Summary Basic T-p-h phase diagram is understood
Possible additional 1st order transition in disordered phase
Helimagnons predicted in ordered phase; lead to T2 term in specific heat
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54. IV. Summary Basic T-p-h phase diagram is understood
Possible additional 1st order transition in disordered phase
Helimagnons predicted in ordered phase; lead to T2 term in specific heat
NFL quasi-particle relaxation time predicted in ordered phase
August 2006
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55. IV. Summary Basic T-p-h phase diagram is understood
Possible additional 1st order transition in disordered phase
Helimagnons predicted in ordered phase; lead to T2 term in specific heat
NFL quasi-particle relaxation time predicted in ordered phase
Resistivity in ordered phase is FL-like with T5/2 correction
August 2006
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56. IV. Summary Basic T-p-h phase diagram is understood
Possible additional 1st order transition in disordered phase
Helimagnons predicted in ordered phase; lead to T2 term in specific heat
NFL quasi-particle relaxation time predicted in ordered phase
Resistivity in ordered phase is FL-like with T5/2 correction
Hydrodynamic description of ordered phase breaks down
August 2006
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57. IV. Summary Basic T-p-h phase diagram is understood
Possible additional 1st order transition in disordered phase
Helimagnons predicted in ordered phase; lead to T2 term in specific heat
NFL quasi-particle relaxation time predicted in ordered phase
Resistivity in ordered phase is FL-like with T5/2 correction
Hydrodynamic description of ordered phase breaks down
Main open question: Origin of T3/2 resistivity in disordered phase?
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58. 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
August 2006
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