Max Planck-UBC-UTokyo SchoolHongo (2018218)
Thermal Hall effect of magnons
Hosho Katsura(Dept Phys UTokyo)
HK Nagaosa Lee Phys Rev Lett 104 066403 (2010)
Onose et al Science 329 297 (2010)
Ideue et al Phys Rev B 85 134411 (2012)
Related papers
Outline
1 Spin Hamiltonian
bull Exchange and DM interactions
bull Microscopic origins
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
5 Summary
125
Coupling between magnetic moments 225
Classical vs Quantum
bull Dipole-dipole interaction
bull Exchange interaction
Usually too small (lt 1K) to explain transition temperatureshellip
Anisotropies
Direct exchange J lt 0 ferromagnetic (FM)
Super-exchange J gt 0 antiferromagnetic (AFM)
( Si spin at site i )
Spin-orbit int breaks SU(2) symmetry
Dzyaloshinskii-Moriya (DM) int DNOTE) Inversion breaking is necessary
Spin tend to
be orthogonal
(Crude) derivation 325
2-site Hubbard model
Origin of exchange int = electron correlation
Can explain AFM int What about FM int
(Multi-orbital nature Kanamori-Goodenough hellip)
bull Hamiltonian
U
1 2
bull 2nd order perturbation at half-filling
Paulirsquos exclusionuarruarr
uarrdarr
darruarr
darrdarr
Origin of DM interaction (1) 425
Spin-dependent hopping
1 2
Inversion symmetry broken but
Time-reversal symmetry exists
bull Hopping matrix θ=0 reduces to the
spin-independent case
Unitary transformation
One can ``absorbrdquo the spin-dependent hopping
New fermions satisfy the same anti-commutation relations
Number ops remain unchanged
bull Hamiltonian in terms of f
Due to spin-orbit
Origin of DM interaction (2) 525
Effective Hamiltonian
bull How does it look like in original spins
Heisenberg int Dzyaloshinskii-
Moriya (DM) intKaplan-Shekhtman-Aharony
-Entin-Wohlman (KSAE) int
Ex) Prove the relation
Hint express in terms of σs
NOTE) One can eliminate the effect of
the DM interaction if there is no loop
Outline
1 Spin Hamiltonian
2 Elementary excitations
bull What are magnons
bull From spins to bosons
bull Diagonalization of BdG Hamiltonian
3 Hall effect and thermal Hall effect
4 Main results
5 Summary
625
What are magnons 725
FM Heisenberg model in a field
Groundstate
Excitation=NG mode
Elementary excitations -- Intuitive picture --
bull Ground state spins are aligned in the same direction
z coordination number
The picture is classical But in ferromagnets ground state and
1-magnon states are exact eigenstates of the Hamiltonian
Cf) non-relativistic Nambu-Goldstone bosons
Watanabe-Murayama PRL 108 (2012) Hidaka PRL 110 (2013)
1-Magnon eigenstates 825
``Motionrdquo of flipped spin
1 2 N
|igt is not an
eigenstate
Flipped spin hops to
the neighboring sites
Ex） 1DBloch state
is an exact eigenstate with energy E(k)
What about DM int D vector z-axis
Magnon picks up a phase factor
From spins to bosons 925
Holstein-Primakoff transformation bull Bose operators
Number op
bull Spins in terms of b
Obey the commutation relations of spins
Often neglect nonlinear terms
(Good at low temperatures)
bull Magnetic ground state = vacuum of bosons
Sublattice structure
AFM int Approximate 1-magnon state
bull Spins on the other sublattice
One needs to introduce more species for a more complex order
b raises Sz
a lowers Sz
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Outline
1 Spin Hamiltonian
bull Exchange and DM interactions
bull Microscopic origins
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
5 Summary
125
Coupling between magnetic moments 225
Classical vs Quantum
bull Dipole-dipole interaction
bull Exchange interaction
Usually too small (lt 1K) to explain transition temperatureshellip
Anisotropies
Direct exchange J lt 0 ferromagnetic (FM)
Super-exchange J gt 0 antiferromagnetic (AFM)
( Si spin at site i )
Spin-orbit int breaks SU(2) symmetry
Dzyaloshinskii-Moriya (DM) int DNOTE) Inversion breaking is necessary
Spin tend to
be orthogonal
(Crude) derivation 325
2-site Hubbard model
Origin of exchange int = electron correlation
Can explain AFM int What about FM int
(Multi-orbital nature Kanamori-Goodenough hellip)
bull Hamiltonian
U
1 2
bull 2nd order perturbation at half-filling
Paulirsquos exclusionuarruarr
uarrdarr
darruarr
darrdarr
Origin of DM interaction (1) 425
Spin-dependent hopping
1 2
Inversion symmetry broken but
Time-reversal symmetry exists
bull Hopping matrix θ=0 reduces to the
spin-independent case
Unitary transformation
One can ``absorbrdquo the spin-dependent hopping
New fermions satisfy the same anti-commutation relations
Number ops remain unchanged
bull Hamiltonian in terms of f
Due to spin-orbit
Origin of DM interaction (2) 525
Effective Hamiltonian
bull How does it look like in original spins
Heisenberg int Dzyaloshinskii-
Moriya (DM) intKaplan-Shekhtman-Aharony
-Entin-Wohlman (KSAE) int
Ex) Prove the relation
Hint express in terms of σs
NOTE) One can eliminate the effect of
the DM interaction if there is no loop
Outline
1 Spin Hamiltonian
2 Elementary excitations
bull What are magnons
bull From spins to bosons
bull Diagonalization of BdG Hamiltonian
3 Hall effect and thermal Hall effect
4 Main results
5 Summary
625
What are magnons 725
FM Heisenberg model in a field
Groundstate
Excitation=NG mode
Elementary excitations -- Intuitive picture --
bull Ground state spins are aligned in the same direction
z coordination number
The picture is classical But in ferromagnets ground state and
1-magnon states are exact eigenstates of the Hamiltonian
Cf) non-relativistic Nambu-Goldstone bosons
Watanabe-Murayama PRL 108 (2012) Hidaka PRL 110 (2013)
1-Magnon eigenstates 825
``Motionrdquo of flipped spin
1 2 N
|igt is not an
eigenstate
Flipped spin hops to
the neighboring sites
Ex） 1DBloch state
is an exact eigenstate with energy E(k)
What about DM int D vector z-axis
Magnon picks up a phase factor
From spins to bosons 925
Holstein-Primakoff transformation bull Bose operators
Number op
bull Spins in terms of b
Obey the commutation relations of spins
Often neglect nonlinear terms
(Good at low temperatures)
bull Magnetic ground state = vacuum of bosons
Sublattice structure
AFM int Approximate 1-magnon state
bull Spins on the other sublattice
One needs to introduce more species for a more complex order
b raises Sz
a lowers Sz
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Coupling between magnetic moments 225
Classical vs Quantum
bull Dipole-dipole interaction
bull Exchange interaction
Usually too small (lt 1K) to explain transition temperatureshellip
Anisotropies
Direct exchange J lt 0 ferromagnetic (FM)
Super-exchange J gt 0 antiferromagnetic (AFM)
( Si spin at site i )
Spin-orbit int breaks SU(2) symmetry
Dzyaloshinskii-Moriya (DM) int DNOTE) Inversion breaking is necessary
Spin tend to
be orthogonal
(Crude) derivation 325
2-site Hubbard model
Origin of exchange int = electron correlation
Can explain AFM int What about FM int
(Multi-orbital nature Kanamori-Goodenough hellip)
bull Hamiltonian
U
1 2
bull 2nd order perturbation at half-filling
Paulirsquos exclusionuarruarr
uarrdarr
darruarr
darrdarr
Origin of DM interaction (1) 425
Spin-dependent hopping
1 2
Inversion symmetry broken but
Time-reversal symmetry exists
bull Hopping matrix θ=0 reduces to the
spin-independent case
Unitary transformation
One can ``absorbrdquo the spin-dependent hopping
New fermions satisfy the same anti-commutation relations
Number ops remain unchanged
bull Hamiltonian in terms of f
Due to spin-orbit
Origin of DM interaction (2) 525
Effective Hamiltonian
bull How does it look like in original spins
Heisenberg int Dzyaloshinskii-
Moriya (DM) intKaplan-Shekhtman-Aharony
-Entin-Wohlman (KSAE) int
Ex) Prove the relation
Hint express in terms of σs
NOTE) One can eliminate the effect of
the DM interaction if there is no loop
Outline
1 Spin Hamiltonian
2 Elementary excitations
bull What are magnons
bull From spins to bosons
bull Diagonalization of BdG Hamiltonian
3 Hall effect and thermal Hall effect
4 Main results
5 Summary
625
What are magnons 725
FM Heisenberg model in a field
Groundstate
Excitation=NG mode
Elementary excitations -- Intuitive picture --
bull Ground state spins are aligned in the same direction
z coordination number
The picture is classical But in ferromagnets ground state and
1-magnon states are exact eigenstates of the Hamiltonian
Cf) non-relativistic Nambu-Goldstone bosons
Watanabe-Murayama PRL 108 (2012) Hidaka PRL 110 (2013)
1-Magnon eigenstates 825
``Motionrdquo of flipped spin
1 2 N
|igt is not an
eigenstate
Flipped spin hops to
the neighboring sites
Ex） 1DBloch state
is an exact eigenstate with energy E(k)
What about DM int D vector z-axis
Magnon picks up a phase factor
From spins to bosons 925
Holstein-Primakoff transformation bull Bose operators
Number op
bull Spins in terms of b
Obey the commutation relations of spins
Often neglect nonlinear terms
(Good at low temperatures)
bull Magnetic ground state = vacuum of bosons
Sublattice structure
AFM int Approximate 1-magnon state
bull Spins on the other sublattice
One needs to introduce more species for a more complex order
b raises Sz
a lowers Sz
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
(Crude) derivation 325
2-site Hubbard model
Origin of exchange int = electron correlation
Can explain AFM int What about FM int
(Multi-orbital nature Kanamori-Goodenough hellip)
bull Hamiltonian
U
1 2
bull 2nd order perturbation at half-filling
Paulirsquos exclusionuarruarr
uarrdarr
darruarr
darrdarr
Origin of DM interaction (1) 425
Spin-dependent hopping
1 2
Inversion symmetry broken but
Time-reversal symmetry exists
bull Hopping matrix θ=0 reduces to the
spin-independent case
Unitary transformation
One can ``absorbrdquo the spin-dependent hopping
New fermions satisfy the same anti-commutation relations
Number ops remain unchanged
bull Hamiltonian in terms of f
Due to spin-orbit
Origin of DM interaction (2) 525
Effective Hamiltonian
bull How does it look like in original spins
Heisenberg int Dzyaloshinskii-
Moriya (DM) intKaplan-Shekhtman-Aharony
-Entin-Wohlman (KSAE) int
Ex) Prove the relation
Hint express in terms of σs
NOTE) One can eliminate the effect of
the DM interaction if there is no loop
Outline
1 Spin Hamiltonian
2 Elementary excitations
bull What are magnons
bull From spins to bosons
bull Diagonalization of BdG Hamiltonian
3 Hall effect and thermal Hall effect
4 Main results
5 Summary
625
What are magnons 725
FM Heisenberg model in a field
Groundstate
Excitation=NG mode
Elementary excitations -- Intuitive picture --
bull Ground state spins are aligned in the same direction
z coordination number
The picture is classical But in ferromagnets ground state and
1-magnon states are exact eigenstates of the Hamiltonian
Cf) non-relativistic Nambu-Goldstone bosons
Watanabe-Murayama PRL 108 (2012) Hidaka PRL 110 (2013)
1-Magnon eigenstates 825
``Motionrdquo of flipped spin
1 2 N
|igt is not an
eigenstate
Flipped spin hops to
the neighboring sites
Ex） 1DBloch state
is an exact eigenstate with energy E(k)
What about DM int D vector z-axis
Magnon picks up a phase factor
From spins to bosons 925
Holstein-Primakoff transformation bull Bose operators
Number op
bull Spins in terms of b
Obey the commutation relations of spins
Often neglect nonlinear terms
(Good at low temperatures)
bull Magnetic ground state = vacuum of bosons
Sublattice structure
AFM int Approximate 1-magnon state
bull Spins on the other sublattice
One needs to introduce more species for a more complex order
b raises Sz
a lowers Sz
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Origin of DM interaction (1) 425
Spin-dependent hopping
1 2
Inversion symmetry broken but
Time-reversal symmetry exists
bull Hopping matrix θ=0 reduces to the
spin-independent case
Unitary transformation
One can ``absorbrdquo the spin-dependent hopping
New fermions satisfy the same anti-commutation relations
Number ops remain unchanged
bull Hamiltonian in terms of f
Due to spin-orbit
Origin of DM interaction (2) 525
Effective Hamiltonian
bull How does it look like in original spins
Heisenberg int Dzyaloshinskii-
Moriya (DM) intKaplan-Shekhtman-Aharony
-Entin-Wohlman (KSAE) int
Ex) Prove the relation
Hint express in terms of σs
NOTE) One can eliminate the effect of
the DM interaction if there is no loop
Outline
1 Spin Hamiltonian
2 Elementary excitations
bull What are magnons
bull From spins to bosons
bull Diagonalization of BdG Hamiltonian
3 Hall effect and thermal Hall effect
4 Main results
5 Summary
625
What are magnons 725
FM Heisenberg model in a field
Groundstate
Excitation=NG mode
Elementary excitations -- Intuitive picture --
bull Ground state spins are aligned in the same direction
z coordination number
The picture is classical But in ferromagnets ground state and
1-magnon states are exact eigenstates of the Hamiltonian
Cf) non-relativistic Nambu-Goldstone bosons
Watanabe-Murayama PRL 108 (2012) Hidaka PRL 110 (2013)
1-Magnon eigenstates 825
``Motionrdquo of flipped spin
1 2 N
|igt is not an
eigenstate
Flipped spin hops to
the neighboring sites
Ex） 1DBloch state
is an exact eigenstate with energy E(k)
What about DM int D vector z-axis
Magnon picks up a phase factor
From spins to bosons 925
Holstein-Primakoff transformation bull Bose operators
Number op
bull Spins in terms of b
Obey the commutation relations of spins
Often neglect nonlinear terms
(Good at low temperatures)
bull Magnetic ground state = vacuum of bosons
Sublattice structure
AFM int Approximate 1-magnon state
bull Spins on the other sublattice
One needs to introduce more species for a more complex order
b raises Sz
a lowers Sz
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Origin of DM interaction (2) 525
Effective Hamiltonian
bull How does it look like in original spins
Heisenberg int Dzyaloshinskii-
Moriya (DM) intKaplan-Shekhtman-Aharony
-Entin-Wohlman (KSAE) int
Ex) Prove the relation
Hint express in terms of σs
NOTE) One can eliminate the effect of
the DM interaction if there is no loop
Outline
1 Spin Hamiltonian
2 Elementary excitations
bull What are magnons
bull From spins to bosons
bull Diagonalization of BdG Hamiltonian
3 Hall effect and thermal Hall effect
4 Main results
5 Summary
625
What are magnons 725
FM Heisenberg model in a field
Groundstate
Excitation=NG mode
Elementary excitations -- Intuitive picture --
bull Ground state spins are aligned in the same direction
z coordination number
The picture is classical But in ferromagnets ground state and
1-magnon states are exact eigenstates of the Hamiltonian
Cf) non-relativistic Nambu-Goldstone bosons
Watanabe-Murayama PRL 108 (2012) Hidaka PRL 110 (2013)
1-Magnon eigenstates 825
``Motionrdquo of flipped spin
1 2 N
|igt is not an
eigenstate
Flipped spin hops to
the neighboring sites
Ex） 1DBloch state
is an exact eigenstate with energy E(k)
What about DM int D vector z-axis
Magnon picks up a phase factor
From spins to bosons 925
Holstein-Primakoff transformation bull Bose operators
Number op
bull Spins in terms of b
Obey the commutation relations of spins
Often neglect nonlinear terms
(Good at low temperatures)
bull Magnetic ground state = vacuum of bosons
Sublattice structure
AFM int Approximate 1-magnon state
bull Spins on the other sublattice
One needs to introduce more species for a more complex order
b raises Sz
a lowers Sz
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Outline
1 Spin Hamiltonian
2 Elementary excitations
bull What are magnons
bull From spins to bosons
bull Diagonalization of BdG Hamiltonian
3 Hall effect and thermal Hall effect
4 Main results
5 Summary
625
What are magnons 725
FM Heisenberg model in a field
Groundstate
Excitation=NG mode
Elementary excitations -- Intuitive picture --
bull Ground state spins are aligned in the same direction
z coordination number
The picture is classical But in ferromagnets ground state and
1-magnon states are exact eigenstates of the Hamiltonian
Cf) non-relativistic Nambu-Goldstone bosons
Watanabe-Murayama PRL 108 (2012) Hidaka PRL 110 (2013)
1-Magnon eigenstates 825
``Motionrdquo of flipped spin
1 2 N
|igt is not an
eigenstate
Flipped spin hops to
the neighboring sites
Ex） 1DBloch state
is an exact eigenstate with energy E(k)
What about DM int D vector z-axis
Magnon picks up a phase factor
From spins to bosons 925
Holstein-Primakoff transformation bull Bose operators
Number op
bull Spins in terms of b
Obey the commutation relations of spins
Often neglect nonlinear terms
(Good at low temperatures)
bull Magnetic ground state = vacuum of bosons
Sublattice structure
AFM int Approximate 1-magnon state
bull Spins on the other sublattice
One needs to introduce more species for a more complex order
b raises Sz
a lowers Sz
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
What are magnons 725
FM Heisenberg model in a field
Groundstate
Excitation=NG mode
Elementary excitations -- Intuitive picture --
bull Ground state spins are aligned in the same direction
z coordination number
The picture is classical But in ferromagnets ground state and
1-magnon states are exact eigenstates of the Hamiltonian
Cf) non-relativistic Nambu-Goldstone bosons
Watanabe-Murayama PRL 108 (2012) Hidaka PRL 110 (2013)
1-Magnon eigenstates 825
``Motionrdquo of flipped spin
1 2 N
|igt is not an
eigenstate
Flipped spin hops to
the neighboring sites
Ex） 1DBloch state
is an exact eigenstate with energy E(k)
What about DM int D vector z-axis
Magnon picks up a phase factor
From spins to bosons 925
Holstein-Primakoff transformation bull Bose operators
Number op
bull Spins in terms of b
Obey the commutation relations of spins
Often neglect nonlinear terms
(Good at low temperatures)
bull Magnetic ground state = vacuum of bosons
Sublattice structure
AFM int Approximate 1-magnon state
bull Spins on the other sublattice
One needs to introduce more species for a more complex order
b raises Sz
a lowers Sz
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
1-Magnon eigenstates 825
``Motionrdquo of flipped spin
1 2 N
|igt is not an
eigenstate
Flipped spin hops to
the neighboring sites
Ex） 1DBloch state
is an exact eigenstate with energy E(k)
What about DM int D vector z-axis
Magnon picks up a phase factor
From spins to bosons 925
Holstein-Primakoff transformation bull Bose operators
Number op
bull Spins in terms of b
Obey the commutation relations of spins
Often neglect nonlinear terms
(Good at low temperatures)
bull Magnetic ground state = vacuum of bosons
Sublattice structure
AFM int Approximate 1-magnon state
bull Spins on the other sublattice
One needs to introduce more species for a more complex order
b raises Sz
a lowers Sz
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
From spins to bosons 925
Holstein-Primakoff transformation bull Bose operators
Number op
bull Spins in terms of b
Obey the commutation relations of spins
Often neglect nonlinear terms
(Good at low temperatures)
bull Magnetic ground state = vacuum of bosons
Sublattice structure
AFM int Approximate 1-magnon state
bull Spins on the other sublattice
One needs to introduce more species for a more complex order
b raises Sz
a lowers Sz
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
are
ev of
Diagonalization of Hamiltonian 1025
Quadratic form of bosons
bull Ferromagnetic case
h Δ NtimesN matrices
Problem reduces to the diagonalization of h
Most easily done in k-space (Fourier tr)
bull AFM (or more general) case
Para-unitary
Transformation leaves the boson commutations unchanged
Involved procedure See eg Colpa Physica 93A 327 (1978)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
bull Hall effect and Berry curvature
bull Anomalous and thermal Hall effects
bull General formulation
4 Main results
5 Summary
1125
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
x
y
Hall effect and Berry curvature 1225
Quantum Hall effect (2D el Gas)
TKNN formulaInteger n is a topological number
bull Bloch wave function
bull Berry connection
bull Berry curvature
Chern number Kubo formula relates
Chern and σxy
PRL 49 (1982)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Anomalous Hall effect 1325
QHE without net magnetic fieldbull Onsagerrsquos reciprocal relation
Time-reversal symmetry (TRS)
must be broken for nonzero σxy
bull Haldanersquos model (PRL 61 2015 (1988) Nobel prize 2016)
Local magnetic field can break TRS
nn real and nnn complex hopping
Integer QHE without Landau levels
Spontaneous symmetry breaking
magnetization
Itinerant electrons in ferromagnets
(i) Intrinsic and (ii) extrinsic origins
Anomalous velocity by Berry curvature in (i)
TRS can be broken by magnetic ordering
bull Anomalous Hall effect Review Nagaosa et al RMP 82 1539 (2010)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Thermal Hall effect 1425
Thermal current
Cs are matrix in general
bull Wiedemann-Franz lawUniversal for weakly
interacting electrons
Righi-Leduc effectTransverse temperature gradient is
produced in response to heat current
In itinerant electron systems
from Wiedemann-Franz
What about Mott insulators Hall effect without Lorentz force
Berry curvature plays the role of magnetic field
Onsager relation
Absence of J
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
General formulation 1525
TKNN-like formula for bosons
bull Earlier work- Fujimoto PRL 103 047203 (2009)
- HK Nagaosa amp Lee PRL 104 066403 (2010)
- Onose et al Science 329 297 (2010)Δ energy separation
Bose
distribution
bull Bloch wf Berry curvature Still well defined for
1-magnon Hamiltonian
without paring term
Terms due to the orbital motion of magnon are missinghellip
bull Modified linear-response theory
- Matsumoto amp Murakami PRL 106 197202 PRB 84 184406 (2011)
Universally applicable to (free) bosonic systemsMagnons phonons triplons photons () hellip NOTE) No quantization
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Outline
1 Spin Hamiltonian
2 Elementary excitations
3 Hall effect and thermal Hall effect
4 Main results
bull Kagome-lattice FM
bull Pyrochlore FM
bull Comparison of theory and experimement
5 Summary
1625
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Magnon Hall effect 1725
Theory
Experiment
Magnons do not have charge They do not feel Lorentz force
Nevertheless they exhibit thermal Hall effect (THE)
Keys
1 TRS is broken spontaneously in FM
2 DM interaction leads to Berry curvature ne 0
Magnon THE was indeed observed in FM insulators Onose et al Science 329 297 (2010)
NOTE) Original theory concerned the effect of scalar chirality
Lu2V2O7
0 50 100 150
0
05
1
15
0
02
04
06
08
1
(a)kxy 03Tkxy 7TM 03TM 7T
kxy
(10
-3W
Km
)
Magnetization (m
BV
)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Role of DM interaction 1825
Kagome modelDM vectors
MOF material Cu(1-3 bdc)
Bosonic ver of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))
Scalar chirality order there ( ) DM
Nonzero Berry curvature is expected to be nonzero
FM exchange int bw Cu2+ moments
- Hirschberger et al PRL 115 106603 (2015)
- Chisnell et al PRL 115 147201 (2015)
Nonzero THE responseSign change consistent with theories
Mook Heng amp Mertig PRB 89 134409 (2014)
Lee Han amp Lee PRB 91 125413 (2015)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Pyrochlore ferromagnet Lu2V2O71925
V4+ (t2g)1 S=12
bull Origin of FM orbital pattern
Polarized neutron diffraction
(Ichikawa et al JPSJ 74 (lsquo03))
bull Trigonal crystal field
0
02
04
06
08
1Lu2V2O7
H || [111]H=01T
M(m
BV
))
A
100
101
102
103
104
Resis
tivity(
cm
) B
50 100 150
05
1
15
0
kxx(W
Km
)
T(K)
C
02 04 06 08 1
02
04
06
08
1
0
H || [100] H || [111] H || [110]
m0H (T)
M (m
BV
)
T=5KD
10 20
02
04
06
08
0
T15
(K15
)
C (
Jm
olK
) 0T 5T 9T
H||[111]
E
Y Onose et al Science 329 297 (lsquo10)
Isotropic
Magnon amp phonon
Highly
resistive
Tc=70K
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Observed thermal Hall conductivity 2025
-5 0 5
20K
Magnetic Field (T)
-5 0 5
30K
-5 0 5
-2
-1
0
1
2 40K
50K
-5 0 5
10K
60K70K
-2
-1
0
1
2 80Kk
xy (
10
-3 W
Km
)Lu2V2O7 H||[100]
Anomalous Related to TRS breaking
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Model Hamiltonian 2125
bull Allowed DM vectors
Elhajal et al PRB 71 Kotov et al PRB 72 (2005)
bull Stability of FM gs against DM
FM Heisenberg + DM
Spin-wave HamiltonianOnly is important
Band structure ( )
bull Hamiltonian in k-space
Λ 4x4 matrix 4 bands
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Comparison of theory and experiment 2225
Formula (at H=0+)
Berry curvature around k=0
can be obtained analytically
Fitting
The fit yields |DJ| ~ 038
- Explains the observed isotropy
- DJ is the only fitting parameter
Chern Fennie amp Tchernyshov PRB 74 (2006)
Reasonable
Cf) DJ ~ 019 in pyrochlore AFM CdCr2O4
Observed in other pyrochlore FM insulators
Ho2V2O7 DJ ~ 007 In2Mn2O7 DJ ~ -002
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
What about other lattices 2325
Provskite-like latticesbull Absence of THE in La2NiMnO6 and YTiO3
YTiO3 S=12 Tc=30K
- Ideal cubic perovskite No DM
- In reality itrsquos distorted nonzero DM
Flux pattern staggered
Berry curvature is zero
because of pseudo TRS in
Whatrsquos the reason
bull Presence of THE in BiMnO3
Tc ~100K- The origin is unclearhellip
May be due to complex
orbital order
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Summary2425
Thermal Hall effect in FM insulators
bull Mechanism
Heat current is carried by magnons
Driven by Berry curvature due to DM int
bull TKNN-like formula
bull Observation in pyrochlore FMs
Lu2V2O7 Ho2V2O7 In2Mn2O7 Consistency
- Below FM transition
- Isotropy of
- Reasonable DJ
Agreement is excellent
Mysteries
- Nonzero in BiMnO3
- Effect of int between magnons
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
Other directions Thermal Hall effects of bosonic particles
bull Phonon(Exp) Strohm Rikken Wyder PRL 95 155901 (2005)
(Theory) Sheng Sheng Ting PRL 96 155901 (2006)
Kagan Maksimov PRL 100 145902 (2008)
bull Triplon (Theory) Romhaacutenyi Penc Ganesh Nat Comm 6 6805 (2015)
bull Photon (Theory) Ben-Abdallah PRL 116 084301 (2016)
2525
Topological magnon physics
bull Dirac magnon
Honeycomb Fransson Black-Schaffer Balatsky PRB 94 075401 (2016)
bull Weyl magnonPyrochlore AFM F-Y Li et al Nat Comm 7 12691 (2016)
Pyrochlore FM Mook Henk Mertig PRL 117 157204 (2016)
bull Topological magnon insulators
Nakata Kim Klinovaja Loss PRB 96 224414 (2017)
