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Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSi

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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

Ferromagnets:

0 < J ~ exchange interaction (strong) (Heisenberg 1930s)

Lorentz Center

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

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

- magnetic transition at Tc ≈ 30 K (at ambient pressure)

1. Phase diagram

(Pfleiderer et al 1997)

Lorentz Center

- 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

- 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

- 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

- 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

- magnetic transition at Tc ≈ 30 K (at ambient pressure)
- transition tunable by means of hydrostatic pressure p
- 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

- magnetic transition at Tc ≈ 30 K (at ambient pressure)
- transition tunable by means of hydrostatic pressure p
- 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

2. Neutron Scattering

- Ordered phase shows helical order, see above

(Pfleiderer et al 2004)

Lorentz Center

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

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

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

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

2. Neutron Scattering

- Ordered phase shows helical order, see above
- Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p)
- 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

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

III. Theory

1. Nature of the Phase Diagram

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

Lorentz Center

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

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

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

- 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

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

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

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

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

Proposed phase diagram :

Lorentz Center

Proposed phase diagram :

Lorentz Center

Proposed phase diagram :

Analogy: Blue Phase III in chiral liquid crystals

(J. Sethna)

Lorentz Center

Proposed phase diagram :

Analogy: Blue Phase III in chiral liquid crystals

(J. Sethna)

(Lubensky & Stark 1996)

Lorentz Center

Proposed phase diagram :

Analogy: Blue Phase III in chiral liquid crystals

(J. Sethna)

(Lubensky & Stark 1996) (Anisimov et al 1998)

Lorentz Center

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

3. Ordered Phase: Nature of the Goldstone mode

Helical ground state:

breaks translational symmetry

soft (Goldstone) mode

Lorentz Center

3. Ordered Phase: Nature of the Goldstone mode

Helical ground state:

breaks translational symmetry

soft (Goldstone) mode

Phase fluctuations:

Energy: ??

Lorentz Center

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

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

anisotropic!

Lorentz Center

anisotropic!

anisotropic dispersion relation (as in chiral liquid crystals)

“helimagnon”

Lorentz Center

anisotropic!

anisotropic dispersion relation (as in chiral liquid crystals)

“helimagnon”

Compare with

ferromagnets w(k) ~ k2

antiferromagnets (k) ~ |k|

Lorentz Center

4. Ordered Phase: Specific heat

Internal energy density:

Specific heat: helimagnon contribution

total low-T specific heat

Lorentz Center

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

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

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

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

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

6. Ordered Phase: Breakdown of hydrodynamics

(T.R. Kirkpatrick & DB, work in progress)

- Use TDGL theory to study magnetization dynamics:

Lorentz Center

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

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

- 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

IV. Summary

- Basic T-p-h phase diagram is understood

Lorentz Center

IV. Summary

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

Lorentz Center

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

Lorentz Center

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

Lorentz Center

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

Lorentz Center

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

Lorentz Center

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?

Lorentz Center

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

Lorentz Center