Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSi

Quantum Phase Transitions and Exotic Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSiPhases in the Metallic Helimagnet MnSi

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,

Thomas Vojta, et al.

Lorentz Center 2August 2006

I. Ferromagnets versus Helimagnets

Ferromagnets:

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

Ferromagnets:

Helimagnets:

c ~ spin-orbit interaction (weak)q ~ c pitch wave number of helix

(Dzyaloshinski 1958,

Moriya 1960)

Ferromagnets:

Helimagnets:

c ~ spin-orbit interaction (weak)q ~ c pitch wave number of helix

(Dzyaloshinski 1958,

Moriya 1960)

• HHM invariant under rotations, but not under x → - x

• Crystal-field effects ultimately pin helix (very weak)

1. Phase diagram

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

(Pfleiderer et al 1997)

1. Phase diagram

• transition tunable by means of hydrostatic pressure p

1. Phase diagram

• 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

• 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”(Pfleiderer et al 1997)

(Pfleiderer, Julian, Lonzarich 2001)

1. Phase diagram

• In an external field B there are “tricritical wings”

• Quantum critical point at B ≠ 0(Pfleiderer et al 1997)

(Pfleiderer, Julian, Lonzarich 2001)

1. Phase diagram

• Quantum critical point at B ≠ 0

• Magnetic state is a helimagnet with q ≈ 180 Ǻ, pinning in (111) d d direction

(Pfleiderer, Julian, Lonzarich 2001)(Pfleiderer et al 2004)

1. Phase diagram

• 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

• 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)

(Pfleiderer, Julian, Lonzarich 2001)(Pfleiderer et al 2004)(Carbone et al 2005)

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)

•Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p)

• T0 (p) originates close to TCP

• So far only three data points for T0 (p)

3. Transport Properties

• Non-Fermi-liquid behavior of the resistivity:

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

T1.5(K1.5)

p = 14.8kbar > pc

III. Theory

1. Nature of the Phase Diagram

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

III. Theory1. Nature of the Phase Diagram

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

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)

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,

m (pc,Hc,T) ~ -T 4/9

2. Disordered Phase: Interpretation of T0(p)

Basic idea: Liquid-gas-type phase transition with chiral order parameter

(cf. Lubensky & Stark 1996)

Borrow an idea from liquid-crystal physics:

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

• 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

Proposed phase diagram :

Analogy: Blue Phase III in chiral liquid crystals

(J. Sethna)

(J. Sethna) (Lubensky & Stark 1996)

(J. Sethna) (Lubensky & Stark 1996) (Anisimov et al 1998)

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)

3. Ordered Phase: Nature of the Goldstone mode

Helical ground state:

breaks translational symmetry

soft (Goldstone) mode

Phase fluctuations:

Energy: ??

Phase fluctuations:

Energy: ??

NO! rotation (0,0,q) (1,2,q) cannot cost energy,

yet corresponds to f(x) = 1x + 2y H fluct > 0

cannot depend on

Phase fluctuations:

Energy: ??

NO! rotation (0,0,q) (1,2,q) cannot cost energy,

yet corresponds to f(x) = 1x + 2y H fluct > 0

cannot depend on

anisotropic!

anisotropic dispersion relation (as in chiral liquid crystals)

“helimagnon”

anisotropic!

anisotropic dispersion relation (as in chiral liquid crystals)

“helimagnon”

Compare with

ferromagnets (k) ~ k2

antiferromagnets (k) ~ |k|

4. Ordered Phase: Specific heat

Internal energy density:

Specific heat: helimagnon contribution

total low-T specific heat

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

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 (T3 vs T5)

(hard to measure)

(T) = 2 T 2 + 5/2 T 5/2 total low-T resistivity

(hard to measure)

(T) = 2 T 2 + 5/2 T 5/2 total low-T resistivity

Experiment: (T→ 0) ~ T 2 (more analysis needed)

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

• Bare magnetic response function:

helimagnon frequency

damping coefficient

• Fluctuation-dissipation theorem:

• One-loop correction to

• 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 !

IV. Summary

Basic T-p-h phase diagram is understood

IV. Summary

Possible additional 1st order transition in disordered phase

IV. Summary

Helimagnons predicted in ordered phase; lead to T2 term in specific heat

IV. Summary

NFL quasi-particle relaxation time predicted in ordered phase

IV. Summary

Resistivity in ordered phase is FL-like with T5/2 correction

IV. Summary

Hydrodynamic description of ordered phase breaks down

IV. Summary

Hydrodynamic description of ordered phase breaks down

Main open question: Origin of T3/2 resistivity in disordered phase?

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

Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSi

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