Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSi

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 lt J exchange interaction (strong)
(Heisenberg 1930s)
3
I. Ferromagnets versus Helimagnets
Ferromagnets
0 lt J exchange interaction (strong)
(Heisenberg 1930s)
Helimagnets
(Dzyaloshinski 1958, Moriya 1960)
c spin-orbit interaction (weak) q c pitch
wave number of helix
4
I. Ferromagnets versus Helimagnets
Ferromagnets
0 lt 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)

5
II. Phenomenology of MnSi

• 1. Phase diagram

• magnetic transition at Tc 30 K (at ambient
  pressure)
  
  
  
  

(Pfleiderer et al 1997)
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)
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)
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)
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)
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 ?,
  pinning in (111) d d direction

TCP
(Pfleiderer et al 1997)
(Pfleiderer et al 2004)
(Pfleiderer, Julian, Lonzarich 2001)
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 ?,
  pinning in (111) d d direction
• Cubic unit cell lacks inversion symmetry (in
  agreement with DM)

TCP
(Pfleiderer et al 1997)
(Pfleiderer et al 2004)
(Carbone et al 2005)
(Pfleiderer, Julian, Lonzarich 2001)
12
2. Neutron Scattering

• Ordered phase shows helical order, see above

(Pfleiderer et al 2004)
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)
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)
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 gt T0 (p)

(Pfleiderer et al 2004)
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 gt T0 (p)
• T0 (p) originates close to TCP

(Pfleiderer et al 2004)
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 gt T0 (p)
• T0 (p) originates close to TCP
• So far only three data points for T0 (p)

(Pfleiderer et al 2004)
18
3. Transport Properties

• Non-Fermi-liquid behavior of the resistivity

p 14.8kbar gt pc
?(µOcm)
T(K)
?(µOcm)

• Resistivity ? T 1.5
  o over a huge
  range in parameter space

T1.5(K1.5)
?(µOcm)
T1.5(K1.5)
19
III. Theory
1. Nature of the Phase Diagram

• Basic features can be understood by
  approximating the system as a FM

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)

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

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 gt 0)

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

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)
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

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

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

28
Proposed phase diagram
29
Proposed phase diagram
30
Proposed phase diagram
Analogy Blue Phase III in chiral liquid crystals
(J. Sethna)
31
Proposed phase diagram
Analogy Blue Phase III in chiral liquid crystals
(J. Sethna)
(Lubensky Stark 1996)
32
Proposed phase diagram
Analogy Blue Phase III in chiral liquid crystals
(J. Sethna)
(Lubensky Stark 1996) (Anisimov
et al 1998)
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)
34
3. Ordered Phase Nature of the Goldstone mode
Helical ground state
breaks translational
symmetry
soft (Goldstone) mode




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

??




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 gt 0






cannot depend on
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 gt 0






cannot depend on
38
anisotropic!
39
anisotropic!
anisotropic dispersion relation (as in chiral
liquid crystals)

helimagnon
40
anisotropic!
anisotropic dispersion relation (as in chiral
liquid crystals)

helimagnon
Compare with
ferromagnets w(k) k2

antiferromagnets ?(k) k
41
4. Ordered Phase Specific heat
Internal energy density
Specific heat

helimagnon contribution


total low-T specific heat




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


43
5. Ordered Phase Relaxation times and resistivity
Quasi-particle relaxation time 1/t(T)
T 3/2 stronger
than FL T 2 contribution!

(hard to
measure)




44
5. Ordered Phase Relaxation times and resistivity
Quasi-particle relaxation time 1/t(T)
T 3/2 stronger
than FL T 2 contribution!

(hard to
measure) Resistivity
r(T) T 5/2
weaker than QP relaxation time,


cf. phonon case (T3 vs T5)




45
5. Ordered Phase Relaxation times and resistivity
Quasi-particle relaxation time 1/t(T)
T 3/2 stronger
than FL T 2 contribution!

(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






46
5. Ordered Phase Relaxation times and resistivity
Quasi-particle relaxation time 1/t(T)
T 3/2 stronger
than FL T 2 contribution!

(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)




47
6. Ordered Phase Breakdown of hydrodynamics
(T.R. Kirkpatrick DB, work in progress)

• Use TDGL theory to study magnetization dynamics

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
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
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 gt 0
• In practice, cz is predicted to
  change linearly with T, by 10 from T0 to T10K
• 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 -gt
  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 gt 0 than at T 0 !
• All renormalizations are finite at T 0 !

51
IV. Summary

• Basic T-p-h phase diagram is understood

52
IV. Summary

• Basic T-p-h phase diagram is understood
• Possible additional 1st order transition in
  disordered phase

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

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

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

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

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?

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