Quantum Phase Transitions and Exotic Phases in the Metallic Helimagnet MnSi
description
Transcript of 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)
Lorentz Center 3August 2006
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)
Lorentz Center 4August 2006
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)
• HHM invariant under rotations, but not under x → - x
• Crystal-field effects ultimately pin helix (very weak)
Lorentz Center 5August 2006
II. Phenomenology of MnSi
1. Phase diagram
• magnetic transition at Tc ≈ 30 K (at ambient pressure)
(Pfleiderer et al 1997)
Lorentz Center 6August 2006
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)
Lorentz Center 7August 2006
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
Lorentz Center 8August 2006
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
Lorentz Center 9August 2006
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
Lorentz Center 10August 2006
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 Ǻ, pinning in (111) d d direction
(Pfleiderer et al 1997)
(Pfleiderer, Julian, Lonzarich 2001)(Pfleiderer et al 2004)
TCP
Lorentz Center 11August 2006
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 Ǻ, pinning in (111) d d direction
• Cubic unit cell lacks inversion symmetry (in agreement with DM)
(Pfleiderer et al 1997)
(Pfleiderer, Julian, Lonzarich 2001)(Pfleiderer et al 2004)(Carbone et al 2005)
TCP
Lorentz Center 12August 2006
2. Neutron Scattering
(Pfleiderer et al 2004)
• Ordered phase shows helical order, see above
Lorentz Center 13August 2006
2. Neutron Scattering
(Pfleiderer et al 2004)
• Ordered phase shows helical order, see above
• Short-ranged helical order persists in the paramagnetic phase below a temperature T0 (p)
Lorentz Center 14August 2006
2. Neutron Scattering
(Pfleiderer et al 2004)
• 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)
Lorentz Center 15August 2006
2. Neutron Scattering
(Pfleiderer et al 2004)
• 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)
Lorentz Center 16August 2006
2. Neutron Scattering
(Pfleiderer et al 2004)
• 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
Lorentz Center 17August 2006
2. Neutron Scattering
(Pfleiderer et al 2004)
• 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
• So far only three data points for T0 (p)
Lorentz Center 18August 2006
3. Transport Properties
• Non-Fermi-liquid behavior of the resistivity:
• Resistivity ρ ~ T 1.5 o over a huge range in parameter space
T(K)
T1.5(K1.5)
T1.5(K1.5)
ρ(μ
Ωcm
)
p = 14.8kbar > pc
ρ(μ
Ωcm
)ρ
(μΩ
cm)
Lorentz Center 19August 2006
III. Theory
1. Nature of the Phase Diagram
Basic features can be understood by approximating the system as a FM
Lorentz Center 20August 2006
III. Theory1. 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 21August 2006
III. Theory1. 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 22August 2006
III. Theory1. 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 23August 2006
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
Lorentz Center 24August 2006
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:
Lorentz Center 25August 2006
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:
Lorentz Center 26August 2006
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:
Lorentz Center 27August 2006
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:
Lorentz Center 28August 2006
Proposed phase diagram :
Lorentz Center 29August 2006
Proposed phase diagram :
Lorentz Center 30August 2006
Analogy: Blue Phase III in chiral liquid crystals
Proposed phase diagram :
(J. Sethna)
Lorentz Center 31August 2006
Analogy: Blue Phase III in chiral liquid crystals
Proposed phase diagram :
(J. Sethna) (Lubensky & Stark 1996)
Lorentz Center 32August 2006
Analogy: Blue Phase III in chiral liquid crystals
Proposed phase diagram :
(J. Sethna) (Lubensky & Stark 1996) (Anisimov et al 1998)
Lorentz Center 33August 2006
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 34August 2006
3. Ordered Phase: Nature of the Goldstone mode
Helical ground state:
breaks translational symmetry
soft (Goldstone) mode
Lorentz Center 35August 2006
3. Ordered Phase: Nature of the Goldstone mode
Helical ground state:
breaks translational symmetry
soft (Goldstone) mode
Phase fluctuations:
Energy: ??
Lorentz Center 36August 2006
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) (1,2,q) cannot cost energy,
yet corresponds to f(x) = 1x + 2y H fluct > 0
cannot depend on
Lorentz Center 37August 2006
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) (1,2,q) cannot cost energy,
yet corresponds to f(x) = 1x + 2y H fluct > 0
cannot depend on
Lorentz Center 38August 2006
anisotropic!
Lorentz Center 39August 2006
anisotropic!
anisotropic dispersion relation (as in chiral liquid crystals)
“helimagnon”
Lorentz Center 40August 2006
anisotropic!
anisotropic dispersion relation (as in chiral liquid crystals)
“helimagnon”
Compare with
ferromagnets (k) ~ k2
antiferromagnets (k) ~ |k|
Lorentz Center 41August 2006
4. Ordered Phase: Specific heat
Internal energy density:
Specific heat: helimagnon contribution
total low-T specific heat
Lorentz Center 42August 2006
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 43August 2006
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)
Lorentz Center 44August 2006
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)
Lorentz Center 45August 2006
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
Lorentz Center 46August 2006
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)
Lorentz Center 47August 2006
6. Ordered Phase: Breakdown of hydrodynamics (T.R. Kirkpatrick & DB, work in progress)
• Use TDGL theory to study magnetization dynamics:
Lorentz Center 48August 2006
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 49August 2006
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
F
Lorentz Center 50August 2006
• 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 51August 2006
IV. Summary
Basic T-p-h phase diagram is understood
Lorentz Center 52August 2006
IV. Summary
Basic T-p-h phase diagram is understood
Possible additional 1st order transition in disordered phase
Lorentz Center 53August 2006
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 54August 2006
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 55August 2006
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 56August 2006
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 57August 2006
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 58August 2006
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