Transcript Breakdown of the Standard Model

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.
I. Ferromagnets and Helimagnets
II. Phenomenology of MnSi
III. Theory
1. Phase diagram
2. Disordered phase
3. Ordered phase
I. Ferromagnets versus Helimagnets
Ferromagnets:
0 < J ~ exchange interaction (strong)
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(Heisenberg 1930s)
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I. Ferromagnets versus Helimagnets
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
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I. Ferromagnets versus Helimagnets
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)
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II. Phenomenology of MnSi
1. Phase diagram
• magnetic transition at Tc ≈ 30 K (at ambient pressure)
(Pfleiderer et al 1997)
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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)
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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 !)
TCP
(Pfleiderer et al 1997)
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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”
TCP
(Pfleiderer et al 1997)
(Pfleiderer, Julian, Lonzarich 2001)
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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”
TCP
• Quantum critical point at B ≠ 0
(Pfleiderer et al 1997)
(Pfleiderer, Julian, Lonzarich 2001)
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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”
TCP
• Quantum critical point at B ≠ 0
• Magnetic state is a helimagnet with q ≈ 180 Ǻ, pinning in (111) d
d direction
(Pfleiderer et al 2004)
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(Pfleiderer et al 1997)
(Pfleiderer, Julian, Lonzarich 2001)
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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”
TCP
• 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)
• Cubic unit cell lacks inversion symmetry (in agreement with DM)
(Carbone et al 2005)
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(Pfleiderer et al 2004)
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(Pfleiderer, Julian, Lonzarich 2001)
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2. Neutron Scattering
• Ordered phase shows helical order, see above
(Pfleiderer et al 2004)
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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)
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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)
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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)
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(Pfleiderer et al 2004)
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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)
• T0 (p) originates close to TCP
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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)
• T0 (p) originates close to TCP
• So far only three data points for T0 (p)
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3. Transport Properties
• Non-Fermi-liquid behavior of the resistivity:
ρ(μΩcm)
p = 14.8kbar > pc
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)
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III. Theory
1. Nature of the Phase Diagram
 Basic features can be understood by approximating the system as a FM
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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)
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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
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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)
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 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
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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)
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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:
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• Chirality parameter c acts as external field conjugate to chiral OP
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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
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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
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Proposed phase
diagram :
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Proposed phase
diagram :
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Proposed phase
diagram :
Analogy: Blue Phase III in chiral liquid crystals
(J. Sethna)
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Proposed phase
diagram :
Analogy: Blue Phase III in chiral liquid crystals
(J. Sethna)
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(Lubensky & Stark 1996)
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Proposed phase
diagram :
Analogy: Blue Phase III in chiral liquid crystals
(J. Sethna)
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(Lubensky & Stark 1996)
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(Anisimov et al 1998)
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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)
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3. Ordered Phase: Nature of the Goldstone mode
Helical ground state:
breaks translational symmetry
soft (Goldstone) mode
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3. Ordered Phase: Nature of the Goldstone mode
Helical ground state:
breaks translational symmetry
soft (Goldstone) mode
Phase fluctuations:
Energy:
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??
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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
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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
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anisotropic!
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anisotropic!
anisotropic dispersion relation (as in chiral liquid crystals)
“helimagnon”
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anisotropic!
anisotropic dispersion relation (as in chiral liquid crystals)
“helimagnon”
Compare with
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ferromagnets
w(k) ~ k2
antiferromagnets
w(k) ~ |k|
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4. Ordered Phase: Specific heat
Internal energy density:
Specific heat:
helimagnon contribution
total low-T specific heat
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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
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5. Ordered Phase: Relaxation times and resistivity
Quasi-particle relaxation time:
1/t(T) ~ T 3/2
stronger than FL T 2 contribution!
(hard to measure)
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5. Ordered Phase: Relaxation times and resistivity
Quasi-particle relaxation time:
1/t(T) ~ T 3/2
stronger than FL T 2 contribution!
(hard to measure)
Resistivity:
r(T) ~ T 5/2
weaker than QP relaxation time,
cf. phonon case (T3 vs T5)
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5. Ordered Phase: Relaxation times and resistivity
Quasi-particle relaxation time:
1/t(T) ~ T 3/2
stronger than FL T 2 contribution!
(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
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total low-T resistivity
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5. Ordered Phase: Relaxation times and resistivity
Quasi-particle relaxation time:
1/t(T) ~ T 3/2
stronger than FL T 2 contribution!
(hard to measure)
Resistivity:
r(T) ~ T 5/2
weaker than QP relaxation time,
cf. phonon case (T3 vs T5)
Experiment:
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r(T) = r2 T 2 + r5/2 T 5/2
total low-T resistivity
r (T→ 0) ~ T 2
(more analysis needed)
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6. Ordered Phase: Breakdown of hydrodynamics
(T.R. Kirkpatrick & DB, work in progress)
• Use TDGL theory to study magnetization dynamics:
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6. Ordered Phase: Breakdown of hydrodynamics
(T.R. Kirkpatrick & DB, work in progress)
• Use TDGL theory to study magnetization dynamics:
Bloch term
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damping
Langevin force
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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:
c
• One-loop correction to c :
F
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• The elastic coefficients
and
, and the transport coefficients
and
singular corrections at one-loop order due to mode-mode coupling effects:
all acquire
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 ?
• coth in FD theorem
1-loop integral more singular at T > 0 than at T = 0 !
• All renormalizations are finite 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)
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IV. Summary
 Basic T-p-h phase diagram is understood
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IV. Summary
 Basic T-p-h phase diagram is understood
 Possible additional 1st order transition in disordered phase
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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
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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
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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
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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
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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?
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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
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