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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