Quantum Phase Transitions and Exotic Quantum Phase Transitions and Exotic Phases in Metallic HelimagnetsPhases 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

APS March Meeting Denver 2March 2007

I. Ferromagnets versus Helimagnets

Ferromagnets:

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

APS March Meeting Denver 3March 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)

APS March Meeting Denver 4March 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)

APS March Meeting Denver 5March 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

APS March Meeting Denver 6March 2007

II. Phenomenology of MnSi

1. Phase diagram

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

(Pfleiderer et al 1997)

TCP

APS March Meeting Denver 7March 2007

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)

TCP

APS March Meeting Denver 8March 2007

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!

)

(Pfleiderer et al 1997)

TCP

APS March Meeting Denver 9March 2007

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”

(Pfleiderer et al 1997)

(Pfleiderer, Julian, Lonzarich 2001)

TCP

APS March Meeting Denver 10March 2007

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

(Pfleiderer et al 1997)

(Pfleiderer, Julian, Lonzarich 2001)

TCP

APS March Meeting Denver 11March 2007

2. Neutron Scattering

(Pfleiderer et al 2004)

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

APS March Meeting Denver 12March 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)

APS March Meeting Denver 13March 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 > T0 (p)

APS March Meeting Denver 14March 2007

3. Transport Properties

• Non-Fermi-liquid behavior of the resistivity:

APS March Meeting Denver 15March 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

T1.5(K1.5)

ρ(μ

Ωcm

)

APS March Meeting Denver 16March 2007

III. Theory

1. Nature of the Phase Diagram

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

APS March Meeting Denver 17March 2007

III. Theory1. 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)

APS March Meeting Denver 18March 2007

III. Theory1. 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!

APS March Meeting Denver 19March 2007

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:

APS March Meeting Denver 20March 2007

2. Disordered Phase: Interpretation of T0(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:

APS March Meeting Denver 21March 2007

2. Disordered Phase: Interpretation of T0(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:

APS March Meeting Denver 22March 2007

2. Disordered Phase: Interpretation of T0(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 1st order transition (e.g., discontinuity in

the spin susceptibility) should be observable across T0

Borrow an idea from liquid-crystal physics:

APS March Meeting Denver 23March 2007

Proposed phase diagram :

APS March Meeting Denver 24March 2007

Analogy: Blue Phase III in chiral liquid crystals

Proposed phase diagram :

(J. Sethna) (Lubensky & Stark 1996)

APS March Meeting Denver 25March 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)

APS March Meeting Denver 26March 2007

3. Ordered Phase: Nature of the Goldstone mode

Helical ground state:

breaks translational symmetry

soft (Goldstone) mode

APS March Meeting Denver 27March 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)

APS March Meeting Denver 28March 2007

4. Ordered Phase: Specific heat

Internal energy density:

Specific heat: helimagnon contribution

total low-T specific heat

APS March Meeting Denver 29March 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)

APS March Meeting Denver 30March 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 (T3 vs T5)

APS March Meeting Denver 31March 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 (T3 vs T5)

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

APS March Meeting Denver 32March 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 (T3 vs T5)

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

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

APS March Meeting Denver 33March 2007

6. Ordered Phase: Breakdown of hydrodynamics

• Use TDGL theory to study magnetization dynamics:

APS March Meeting Denver 34March 2007

6. Ordered Phase: Breakdown of hydrodynamics

• Use TDGL theory to study magnetization dynamics:

Bloch term damping Langevin force

APS March Meeting Denver 35March 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

APS March Meeting Denver 36March 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, 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)

• 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)

APS March Meeting Denver 37March 2007

IV. Summary

Basic T-p-h phase diagram is understood

APS March Meeting Denver 38March 2007

IV. Summary

Basic T-p-h phase diagram is understood

Possible additional 1st order transition in disordered phase

APS March Meeting Denver 39March 2007

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

APS March Meeting Denver 40March 2007

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

APS March Meeting Denver 41March 2007

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

APS March Meeting Denver 42March 2007

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

APS March Meeting Denver 43March 2007

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?

APS March Meeting Denver 44March 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