The Story of SmB6 - Quantum Condensed Matter Journal Clubqcmjc/talk_slides/QCMJC... · 10/9/2014...
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The Story of SmB6Quantum Condensed Matter Journal Club
Adhip Agarwala
Department of PhysicsIndian Institute of Science
October 9, 2014
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 1 / 32
About half of these total papers in last 4 years.and about 30 of these in this year !
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 3 / 32
About half of these total papers in last 4 years.
and about 30 of these in this year !
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 3 / 32
About half of these total papers in last 4 years.and about 30 of these in this year !
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 3 / 32
but first things first..
DISCLAIMERThe speaker not an expert!
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 4 / 32
but first things first..
DISCLAIMERThe speaker not an expert!
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 4 / 32
Major References
End States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014
Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 5 / 32
Major References
End States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014
Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 5 / 32
Major References
End States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 5 / 32
Outline
The StageI Electronic Configuration and Structure.I The early R vs T Measurement on SmB6
Act II Anderson Impurity ModelI Kondo EffectI Kondo Lattice
Act III 1D Topological Insulator
Act IIII 1D Topological Kondo InsulatorI Some understanding of the experiments.I Comments on 3D TKI.
Conclusions
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 6 / 32
Outline
The StageI Electronic Configuration and Structure.I The early R vs T Measurement on SmB6
Act II Anderson Impurity ModelI Kondo EffectI Kondo Lattice
Act III 1D Topological Insulator
Act IIII 1D Topological Kondo InsulatorI Some understanding of the experiments.I Comments on 3D TKI.
Conclusions
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 6 / 32
Outline
The StageI Electronic Configuration and Structure.I The early R vs T Measurement on SmB6
Act II Anderson Impurity ModelI Kondo EffectI Kondo Lattice
Act III 1D Topological Insulator
Act IIII 1D Topological Kondo InsulatorI Some understanding of the experiments.I Comments on 3D TKI.
Conclusions
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 6 / 32
Outline
The StageI Electronic Configuration and Structure.I The early R vs T Measurement on SmB6
Act II Anderson Impurity ModelI Kondo EffectI Kondo Lattice
Act III 1D Topological Insulator
Act IIII 1D Topological Kondo InsulatorI Some understanding of the experiments.I Comments on 3D TKI.
Conclusions
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 6 / 32
Outline
The StageI Electronic Configuration and Structure.I The early R vs T Measurement on SmB6
Act II Anderson Impurity ModelI Kondo EffectI Kondo Lattice
Act III 1D Topological Insulator
Act IIII 1D Topological Kondo InsulatorI Some understanding of the experiments.I Comments on 3D TKI.
Conclusions
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 6 / 32
Outline
The StageI Electronic Configuration and Structure.I The early R vs T Measurement on SmB6
Act II Anderson Impurity ModelI Kondo EffectI Kondo Lattice
Act III 1D Topological Insulator
Act IIII 1D Topological Kondo InsulatorI Some understanding of the experiments.I Comments on 3D TKI.
Conclusions
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 7 / 32
And..
Electronic ConfigurationSm = [Xe] 4f 6 6s2
B = [He] 2s2 2p1
Sm is in a mixed valence state with valency 2.7+
Review, HasanAdhip Agarwala (IISc) The Story of SmB6 October 9, 2014 9 / 32
R − T MeasurementsFor SmB6
(Allen et al., PRB 20, 12(1979).)
Metal?at ≈ 300KAg ≈ 10−6Ωcm,Hg ≈ 10−4Ωcm
(Ziman, Theory ofTransport in Metals, 1960)
Insulator?at ≈ 20KSi ≈ 104Ωcm,glass ≈ 1012Ωcm
(Fritzsche, J. Phys. Chem.Solids 6 69, 1958)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 10 / 32
R − T MeasurementsFor SmB6
(Allen et al., PRB 20, 12(1979).)
Metal?
at ≈ 300KAg ≈ 10−6Ωcm,Hg ≈ 10−4Ωcm
(Ziman, Theory ofTransport in Metals, 1960)
Insulator?at ≈ 20KSi ≈ 104Ωcm,glass ≈ 1012Ωcm
(Fritzsche, J. Phys. Chem.Solids 6 69, 1958)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 10 / 32
R − T MeasurementsFor SmB6
(Allen et al., PRB 20, 12(1979).)
Metal?at ≈ 300KAg ≈ 10−6Ωcm,Hg ≈ 10−4Ωcm
(Ziman, Theory ofTransport in Metals, 1960)
Insulator?at ≈ 20KSi ≈ 104Ωcm,glass ≈ 1012Ωcm
(Fritzsche, J. Phys. Chem.Solids 6 69, 1958)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 10 / 32
R − T MeasurementsFor SmB6
(Allen et al., PRB 20, 12(1979).)
Metal?at ≈ 300KAg ≈ 10−6Ωcm,Hg ≈ 10−4Ωcm
(Ziman, Theory ofTransport in Metals, 1960)
Insulator?at ≈ 20KSi ≈ 104Ωcm,glass ≈ 1012Ωcm
(Fritzsche, J. Phys. Chem.Solids 6 69, 1958)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 10 / 32
R − T MeasurementsFor SmB6
(Allen et al., PRB 20, 12(1979).)
Metal?at ≈ 300KAg ≈ 10−6Ωcm,Hg ≈ 10−4Ωcm
(Ziman, Theory ofTransport in Metals, 1960)
Insulator?
at ≈ 20KSi ≈ 104Ωcm,glass ≈ 1012Ωcm
(Fritzsche, J. Phys. Chem.Solids 6 69, 1958)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 10 / 32
R − T MeasurementsFor SmB6
(Allen et al., PRB 20, 12(1979).)
Metal?at ≈ 300KAg ≈ 10−6Ωcm,Hg ≈ 10−4Ωcm
(Ziman, Theory ofTransport in Metals, 1960)
Insulator?at ≈ 20KSi ≈ 104Ωcm,glass ≈ 1012Ωcm
(Fritzsche, J. Phys. Chem.Solids 6 69, 1958)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 10 / 32
R − T MeasurementsFor SmB6
(Allen et al., PRB 20, 12(1979).)
Metal?at ≈ 300KAg ≈ 10−6Ωcm,Hg ≈ 10−4Ωcm
(Ziman, Theory ofTransport in Metals, 1960)
Insulator?at ≈ 20KSi ≈ 104Ωcm,glass ≈ 1012Ωcm
(Fritzsche, J. Phys. Chem.Solids 6 69, 1958)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 10 / 32
Outline
The StageI Electronic Configuration and Structure.I The early R vs T Measurement on SmB6
Act II Anderson Impurity ModelI Kondo EffectI Kondo Lattice
Act III 1D Topological Insulator
Act IIII 1D Topological Kondo InsulatorI Some understanding of the experiments.I Comments on 3D TKI.
Conclusions
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 11 / 32
Towards Kondo Effect..
Now SmB6 has f levels, so how about a single f level in a metal?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 12 / 32
Towards Kondo Effect..
Now SmB6 has f levels, so how about a single f level in a metal?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 12 / 32
Towards Kondo Effect..
Now SmB6 has f levels, so how about a single f level in a metal?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 12 / 32
Towards Kondo Effect..
Now SmB6 has f levels, so how about a single f level in a metal?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 12 / 32
Towards Kondo Effect..
Now SmB6 has f levels, so how about a single f level in a metal?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 12 / 32
Towards Kondo Effect..
Now SmB6 has f levels, so how about a single f level in a metal?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 12 / 32
Anderson Impurity Model
Hamiltonian(Bath)
H =∑kσ
εk C†kσCkσ (1)
Impurity Site∑σ
(εf C†fσCfσ) + Un↑n↓ (2)
Hybridization
∑k ,σ
(V√Ω
C†fσCkσ +V√Ω
∗C†kσCfσ) (3)
Anderson(1961)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 13 / 32
Anderson Impurity Model
Hamiltonian(Bath)
H =∑kσ
εk C†kσCkσ (1)
Impurity Site∑σ
(εf C†fσCfσ) + Un↑n↓ (2)
Hybridization
∑k ,σ
(V√Ω
C†fσCkσ +V√Ω
∗C†kσCfσ) (3)
Anderson(1961)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 13 / 32
Anderson Impurity Model
Hamiltonian(Bath)
H =∑kσ
εk C†kσCkσ (1)
Impurity Site∑σ
(εf C†fσCfσ) + Un↑n↓ (2)
Hybridization
∑k ,σ
(V√Ω
C†fσCkσ +V√Ω
∗C†kσCfσ) (3)
Anderson(1961)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 13 / 32
Kondo Effect
T TK
low T Hamiltonian
H =∑
kσ ε(k)C†kσCkσ +Js · S
Hewson, The Kondo problem to heavy fermions (CUP), 1997.Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 14 / 32
Kondo Effect
T TK
low T Hamiltonian
H =∑
kσ ε(k)C†kσCkσ +Js · S
Hewson, The Kondo problem to heavy fermions (CUP), 1997.Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 14 / 32
Kondo Effect
T TK
low T Hamiltonian
H =∑
kσ ε(k)C†kσCkσ +Js · S
Hewson, The Kondo problem to heavy fermions (CUP), 1997.Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 14 / 32
Kondo Effect
T TK
low T Hamiltonian
H =∑
kσ ε(k)C†kσCkσ +Js · S
Hewson, The Kondo problem to heavy fermions (CUP), 1997.Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 14 / 32
Kondo Effect
T TK
low T Hamiltonian
H =∑
kσ ε(k)C†kσCkσ +Js · S
Hewson, The Kondo problem to heavy fermions (CUP), 1997.Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 14 / 32
The Kondo Lattice
Single Impurity(or f level)
at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level...okay! So, put a f level at each site!and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local moment
at low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level...okay! So, put a f level at each site!and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level...okay! So, put a f level at each site!and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?
There is no impurity there, eachSm atom has a f level...okay! So, put a f level at each site!and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level.
..okay! So, put a f level at each site!and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level...
okay! So, put a f level at each site!and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level...okay! So, put a f level at each site!
and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level...okay! So, put a f level at each site!and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level...okay! So, put a f level at each site!and at low T
Kondo Lattice Hamiltonian
H =∑
k εk c†k ck +∑
i JK Si · si +JH∑
i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level...okay! So, put a f level at each site!and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck
+∑
i JK Si · si +JH∑
i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level...okay! So, put a f level at each site!and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck +
∑i JK Si · si
+JH∑
i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
The Kondo Lattice
Single Impurity(or f level)at high T → local momentat low T ,→ Js · S.
What has this to do with SmB6?There is no impurity there, eachSm atom has a f level...okay! So, put a f level at each site!and at low T
Kondo Lattice HamiltonianH =
∑k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 15 / 32
Outline
The StageI Electronic Configuration and Structure.I The early R vs T Measurement on SmB6
Act II Anderson Impurity ModelI Kondo EffectI Kondo Lattice
Act III 1D Topological Insulator
Act IIII 1D Topological Kondo InsulatorI Some understanding of the experiments.I Comments on 3D TKI.
Conclusions
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 16 / 32
1− D Topological Insulator!
t1 t1 t1 t1t2 t2 t2 t2
Tight binding dispersion(0 h(k)
h∗(k) 0
)h(k) = t1 + t2eik
h(k) = (t1 + t2 cos(k)) + it2 sin(k)
x
iy
hx
hy
Semi Infinite Chain?
t1 t1t2 t2t2
An edge state exists if t2 > t1!And we can see this from the bulkdispersion,
x
iy
hyt2 > t1
t2 < t1
hx
Shockley(1939), Pershoguba et al.(PRB 86, 075304)(2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 17 / 32
1− D Topological Insulator!
t1 t1 t1 t1t2 t2 t2 t2
Tight binding dispersion
(0 h(k)
h∗(k) 0
)h(k) = t1 + t2eik
h(k) = (t1 + t2 cos(k)) + it2 sin(k)
x
iy
hx
hy
Semi Infinite Chain?
t1 t1t2 t2t2
An edge state exists if t2 > t1!And we can see this from the bulkdispersion,
x
iy
hyt2 > t1
t2 < t1
hx
Shockley(1939), Pershoguba et al.(PRB 86, 075304)(2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 17 / 32
1− D Topological Insulator!
t1 t1 t1 t1t2 t2 t2 t2
Tight binding dispersion(0 h(k)
h∗(k) 0
)h(k) = t1 + t2eik
h(k) = (t1 + t2 cos(k)) + it2 sin(k)
x
iy
hx
hy
Semi Infinite Chain?
t1 t1t2 t2t2
An edge state exists if t2 > t1!And we can see this from the bulkdispersion,
x
iy
hyt2 > t1
t2 < t1
hx
Shockley(1939), Pershoguba et al.(PRB 86, 075304)(2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 17 / 32
1− D Topological Insulator!
t1 t1 t1 t1t2 t2 t2 t2
Tight binding dispersion(0 h(k)
h∗(k) 0
)h(k) = t1 + t2eik
h(k) = (t1 + t2 cos(k)) + it2 sin(k)
x
iy
hx
hy
Semi Infinite Chain?
t1 t1t2 t2t2
An edge state exists if t2 > t1!And we can see this from the bulkdispersion,
x
iy
hyt2 > t1
t2 < t1
hx
Shockley(1939), Pershoguba et al.(PRB 86, 075304)(2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 17 / 32
1− D Topological Insulator!
t1 t1 t1 t1t2 t2 t2 t2
Tight binding dispersion(0 h(k)
h∗(k) 0
)h(k) = t1 + t2eik
h(k) = (t1 + t2 cos(k)) + it2 sin(k)
x
iy
hx
hy
Semi Infinite Chain?
t1 t1t2 t2t2
An edge state exists if t2 > t1!And we can see this from the bulkdispersion,
x
iy
hyt2 > t1
t2 < t1
hx
Shockley(1939), Pershoguba et al.(PRB 86, 075304)(2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 17 / 32
1− D Topological Insulator!
t1 t1 t1 t1t2 t2 t2 t2
Tight binding dispersion(0 h(k)
h∗(k) 0
)h(k) = t1 + t2eik
h(k) = (t1 + t2 cos(k)) + it2 sin(k)
x
iy
hx
hy
Semi Infinite Chain?
t1 t1t2 t2t2
An edge state exists if t2 > t1!And we can see this from the bulkdispersion,
x
iy
hyt2 > t1
t2 < t1
hx
Shockley(1939), Pershoguba et al.(PRB 86, 075304)(2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 17 / 32
1− D Topological Insulator!
t1 t1 t1 t1t2 t2 t2 t2
Tight binding dispersion(0 h(k)
h∗(k) 0
)h(k) = t1 + t2eik
h(k) = (t1 + t2 cos(k)) + it2 sin(k)
x
iy
hx
hy
Semi Infinite Chain?
t1 t1t2 t2t2
An edge state exists if t2 > t1!
And we can see this from the bulkdispersion,
x
iy
hyt2 > t1
t2 < t1
hx
Shockley(1939), Pershoguba et al.(PRB 86, 075304)(2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 17 / 32
1− D Topological Insulator!
t1 t1 t1 t1t2 t2 t2 t2
Tight binding dispersion(0 h(k)
h∗(k) 0
)h(k) = t1 + t2eik
h(k) = (t1 + t2 cos(k)) + it2 sin(k)
x
iy
hx
hy
Semi Infinite Chain?
t1 t1t2 t2t2
An edge state exists if t2 > t1!And we can see this from the bulkdispersion,
x
iy
hyt2 > t1
t2 < t1
hx
Shockley(1939), Pershoguba et al.(PRB 86, 075304)(2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 17 / 32
And the Story Ahead..
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉h(k) = (t1 + t2 cos(k)) + it2 sin(k)
Lower eigenvector, u(k) =
1√2
(ei tan−1(hy/hx )
−1
)hx = t1 + t2 cos(k)
hy = t2 sin(k)
take derivative and the innerproductand integrate,
0.5 1.0 1.5 2.0t2
0.1
0.2
0.3
0.4
0.5
P
t1=1
P=0/1, trivial/non-trivial
t1 t1 t1 t1t2 t2 t2 t2
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 18 / 32
And the Story Ahead..
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉
h(k) = (t1 + t2 cos(k)) + it2 sin(k)
Lower eigenvector, u(k) =
1√2
(ei tan−1(hy/hx )
−1
)hx = t1 + t2 cos(k)
hy = t2 sin(k)
take derivative and the innerproductand integrate,
0.5 1.0 1.5 2.0t2
0.1
0.2
0.3
0.4
0.5
P
t1=1
P=0/1, trivial/non-trivial
t1 t1 t1 t1t2 t2 t2 t2
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 18 / 32
And the Story Ahead..
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉h(k) = (t1 + t2 cos(k)) + it2 sin(k)
Lower eigenvector, u(k) =
1√2
(ei tan−1(hy/hx )
−1
)hx = t1 + t2 cos(k)
hy = t2 sin(k)
take derivative and the innerproductand integrate,
0.5 1.0 1.5 2.0t2
0.1
0.2
0.3
0.4
0.5
P
t1=1
P=0/1, trivial/non-trivial
t1 t1 t1 t1t2 t2 t2 t2
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 18 / 32
And the Story Ahead..
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉h(k) = (t1 + t2 cos(k)) + it2 sin(k)
Lower eigenvector, u(k) =
1√2
(ei tan−1(hy/hx )
−1
)hx = t1 + t2 cos(k)
hy = t2 sin(k)
take derivative and the innerproductand integrate,
0.5 1.0 1.5 2.0t2
0.1
0.2
0.3
0.4
0.5
P
t1=1
P=0/1, trivial/non-trivial
t1 t1 t1 t1t2 t2 t2 t2
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 18 / 32
And the Story Ahead..
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉h(k) = (t1 + t2 cos(k)) + it2 sin(k)
Lower eigenvector, u(k) =
1√2
(ei tan−1(hy/hx )
−1
)hx = t1 + t2 cos(k)
hy = t2 sin(k)
take derivative and the innerproduct
and integrate,
0.5 1.0 1.5 2.0t2
0.1
0.2
0.3
0.4
0.5
P
t1=1
P=0/1, trivial/non-trivial
t1 t1 t1 t1t2 t2 t2 t2
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 18 / 32
And the Story Ahead..
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉h(k) = (t1 + t2 cos(k)) + it2 sin(k)
Lower eigenvector, u(k) =
1√2
(ei tan−1(hy/hx )
−1
)hx = t1 + t2 cos(k)
hy = t2 sin(k)
take derivative and the innerproductand integrate,
0.5 1.0 1.5 2.0t2
0.1
0.2
0.3
0.4
0.5
P
t1=1
P=0/1, trivial/non-trivial
t1 t1 t1 t1t2 t2 t2 t2
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 18 / 32
And the Story Ahead..
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉h(k) = (t1 + t2 cos(k)) + it2 sin(k)
Lower eigenvector, u(k) =
1√2
(ei tan−1(hy/hx )
−1
)hx = t1 + t2 cos(k)
hy = t2 sin(k)
take derivative and the innerproductand integrate,
0.5 1.0 1.5 2.0t2
0.1
0.2
0.3
0.4
0.5
P
t1=1
P=0/1, trivial/non-trivial
t1 t1 t1 t1t2 t2 t2 t2
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 18 / 32
And the Story Ahead..
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉h(k) = (t1 + t2 cos(k)) + it2 sin(k)
Lower eigenvector, u(k) =
1√2
(ei tan−1(hy/hx )
−1
)hx = t1 + t2 cos(k)
hy = t2 sin(k)
take derivative and the innerproductand integrate,
0.5 1.0 1.5 2.0t2
0.1
0.2
0.3
0.4
0.5
P
t1=1
P=0/1, trivial/non-trivial
t1 t1 t1 t1t2 t2 t2 t2
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 18 / 32
Reality Check!
You told about a single impurity in a metal
I and some low energy system of JS.sI also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.s
I also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.sI also something about Kondo Lattice..
I HamiltonianH =
∑k εk c†
k ck +∑
i JK Si · si +JH∑
i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.sI also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.sI also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.sI also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.sI also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.sI also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.sI also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.sI also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization P
I and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.sI also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Reality Check!
You told about a single impurity in a metalI and some low energy system of JS.sI also something about Kondo Lattice..I Hamiltonian
H =∑
k εk c†k ck +
∑i JK Si · si +JH
∑i,j Si · Sj
and then 1D topological insulator,
t1 t1 t1 t1t2 t2 t2 t2
I and about a quantity polarization PI and edge states? etc.
..And the title of your talk is SmB6?
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 19 / 32
Outline
The StageI Electronic Configuration and Structure.I The early R vs T Measurement on SmB6
Act II Anderson Impurity ModelI Kondo EffectI Kondo Lattice
Act III 1D Topological Insulator
Act IIII 1D Topological Kondo InsulatorI Some understanding of the experiments.I Comments on 3D TKI.
Conclusions
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 21 / 32
1D Topological Kondo Insulator
HamiltonianH = Hc + HH + HK
Hc = −t∑
j,σ(c†j+1σcjσ + H.c)
HH = JH∑
i Sj · Sj+1
HK =∑
j,αβJK (j)
2 Sj ·p†j,ασαβpj,β
pj,σ ≡ cj+1,σ − cj−1σ.
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 22 / 32
1D Topological Kondo Insulator
HamiltonianH = Hc + HH + HK
Hc = −t∑
j,σ(c†j+1σcjσ + H.c)
HH = JH∑
i Sj · Sj+1
HK =∑
j,αβJK (j)
2 Sj ·p†j,ασαβpj,β
pj,σ ≡ cj+1,σ − cj−1σ.
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 22 / 32
1D Topological Kondo Insulator
HamiltonianH = Hc + HH + HK
Hc = −t∑
j,σ(c†j+1σcjσ + H.c)
HH = JH∑
i Sj · Sj+1
HK =∑
j,αβJK (j)
2 Sj ·p†j,ασαβpj,β
pj,σ ≡ cj+1,σ − cj−1σ.
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 22 / 32
1D Topological Kondo Insulator
HamiltonianH = Hc + HH + HK
Hc = −t∑
j,σ(c†j+1σcjσ + H.c)
HH = JH∑
i Sj · Sj+1
HK =∑
j,αβJK (j)
2 Sj ·p†j,ασαβpj,β
pj,σ ≡ cj+1,σ − cj−1σ.
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 22 / 32
1D Topological Kondo Insulator
HamiltonianH = Hc + HH + HK
Hc = −t∑
j,σ(c†j+1σcjσ + H.c)
HH = JH∑
i Sj · Sj+1
HK =∑
j,αβJK (j)
2 Sj ·p†j,ασαβpj,β
pj,σ ≡ cj+1,σ − cj−1σ.
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 22 / 32
1D Topological Kondo Insulator
HamiltonianH = Hc + HH + HK
Hc = −t∑
j,σ(c†j+1σcjσ + H.c)
HH = JH∑
i Sj · Sj+1
HK =∑
j,αβJK (j)
2 Sj ·p†j,ασαβpj,β
pj,σ ≡ cj+1,σ − cj−1σ.
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 22 / 32
Equation Overdose
H = Hc + HH + HK
H = −t∑
j,σ(c†j+1σcjσ + H.c) + JH∑
i Sj · Sj+1 +∑
j,αβJK(j)
2 Sj ·p†j,ασαβpj,β
Sj =∑σσ′ f
†jσσσσ′ fjσ′ ; nf ,j = 1
HH =−JH
∑j,αβ(f †j+1,αfjα)(f †j,β fj+1β)
HK =
−JK∑
j,αβ
(f †j,αpjα
)(p†j,β fjβ
)
H →Hc +
∑j,σ
(∆j f†j+1,σfj,σ + H.c +
|∆j |2JH
)+∑
j,σ
(Vj (c† j+1σ − c† j−1σ)fjσ + H.c +
|Vj |2JK (j)
)+∑
j λj (nf ,j − 1)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 23 / 32
Equation Overdose
H = Hc + HH + HK
H = −t∑
j,σ(c†j+1σcjσ + H.c) + JH∑
i Sj · Sj+1 +∑
j,αβJK(j)
2 Sj ·p†j,ασαβpj,β
Sj =∑σσ′ f
†jσσσσ′ fjσ′ ; nf ,j = 1
HH =−JH
∑j,αβ(f †j+1,αfjα)(f †j,β fj+1β)
HK =
−JK∑
j,αβ
(f †j,αpjα
)(p†j,β fjβ
)
H →Hc +
∑j,σ
(∆j f†j+1,σfj,σ + H.c +
|∆j |2JH
)+∑
j,σ
(Vj (c† j+1σ − c† j−1σ)fjσ + H.c +
|Vj |2JK (j)
)+∑
j λj (nf ,j − 1)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 23 / 32
Equation Overdose
H = Hc + HH + HK
H = −t∑
j,σ(c†j+1σcjσ + H.c) + JH∑
i Sj · Sj+1 +∑
j,αβJK(j)
2 Sj ·p†j,ασαβpj,β
Sj =∑σσ′ f
†jσσσσ′ fjσ′ ;
nf ,j = 1
HH =−JH
∑j,αβ(f †j+1,αfjα)(f †j,β fj+1β)
HK =
−JK∑
j,αβ
(f †j,αpjα
)(p†j,β fjβ
)
H →Hc +
∑j,σ
(∆j f†j+1,σfj,σ + H.c +
|∆j |2JH
)+∑
j,σ
(Vj (c† j+1σ − c† j−1σ)fjσ + H.c +
|Vj |2JK (j)
)+∑
j λj (nf ,j − 1)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 23 / 32
Equation Overdose
H = Hc + HH + HK
H = −t∑
j,σ(c†j+1σcjσ + H.c) + JH∑
i Sj · Sj+1 +∑
j,αβJK(j)
2 Sj ·p†j,ασαβpj,β
Sj =∑σσ′ f
†jσσσσ′ fjσ′ ; nf ,j = 1
HH =−JH
∑j,αβ(f †j+1,αfjα)(f †j,β fj+1β)
HK =
−JK∑
j,αβ
(f †j,αpjα
)(p†j,β fjβ
)
H →Hc +
∑j,σ
(∆j f†j+1,σfj,σ + H.c +
|∆j |2JH
)+∑
j,σ
(Vj (c† j+1σ − c† j−1σ)fjσ + H.c +
|Vj |2JK (j)
)+∑
j λj (nf ,j − 1)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 23 / 32
Equation Overdose
H = Hc + HH + HK
H = −t∑
j,σ(c†j+1σcjσ + H.c) + JH∑
i Sj · Sj+1 +∑
j,αβJK(j)
2 Sj ·p†j,ασαβpj,β
Sj =∑σσ′ f
†jσσσσ′ fjσ′ ; nf ,j = 1
HH =−JH
∑j,αβ(f †j+1,αfjα)(f †j,β fj+1β)
HK =
−JK∑
j,αβ
(f †j,αpjα
)(p†j,β fjβ
)
H →Hc +
∑j,σ
(∆j f†j+1,σfj,σ + H.c +
|∆j |2JH
)+∑
j,σ
(Vj (c† j+1σ − c† j−1σ)fjσ + H.c +
|Vj |2JK (j)
)+∑
j λj (nf ,j − 1)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 23 / 32
Equation Overdose
H = Hc + HH + HK
H = −t∑
j,σ(c†j+1σcjσ + H.c) + JH∑
i Sj · Sj+1 +∑
j,αβJK(j)
2 Sj ·p†j,ασαβpj,β
Sj =∑σσ′ f
†jσσσσ′ fjσ′ ; nf ,j = 1
HH =−JH
∑j,αβ(f †j+1,αfjα)(f †j,β fj+1β)
HK =
−JK∑
j,αβ
(f †j,αpjα
)(p†j,β fjβ
)
H →Hc +
∑j,σ
(∆j f†j+1,σfj,σ + H.c +
|∆j |2JH
)+∑
j,σ
(Vj (c† j+1σ − c† j−1σ)fjσ + H.c +
|Vj |2JK (j)
)+∑
j λj (nf ,j − 1)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 23 / 32
Equation Overdose
H = Hc + HH + HK
H = −t∑
j,σ(c†j+1σcjσ + H.c) + JH∑
i Sj · Sj+1 +∑
j,αβJK(j)
2 Sj ·p†j,ασαβpj,β
Sj =∑σσ′ f
†jσσσσ′ fjσ′ ; nf ,j = 1
HH =−JH
∑j,αβ(f †j+1,αfjα)(f †j,β fj+1β)
HK =
−JK∑
j,αβ
(f †j,αpjα
)(p†j,β fjβ
)
H →Hc +
∑j,σ
(∆j f†j+1,σfj,σ + H.c +
|∆j |2JH
)+∑
j,σ
(Vj (c† j+1σ − c† j−1σ)fjσ + H.c +
|Vj |2JK (j)
)+∑
j λj (nf ,j − 1)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 23 / 32
Equation Overdose
H = Hc + HH + HK
H = −t∑
j,σ(c†j+1σcjσ + H.c) + JH∑
i Sj · Sj+1 +∑
j,αβJK(j)
2 Sj ·p†j,ασαβpj,β
Sj =∑σσ′ f
†jσσσσ′ fjσ′ ; nf ,j = 1
HH =−JH
∑j,αβ(f †j+1,αfjα)(f †j,β fj+1β)
HK =
−JK∑
j,αβ
(f †j,αpjα
)(p†j,β fjβ
)
H →Hc +
∑j,σ
(∆j f†j+1,σfj,σ + H.c +
|∆j |2JH
)
+∑j,σ
(Vj (c† j+1σ − c† j−1σ)fjσ + H.c +
|Vj |2JK (j)
)+∑
j λj (nf ,j − 1)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 23 / 32
Equation Overdose
H = Hc + HH + HK
H = −t∑
j,σ(c†j+1σcjσ + H.c) + JH∑
i Sj · Sj+1 +∑
j,αβJK(j)
2 Sj ·p†j,ασαβpj,β
Sj =∑σσ′ f
†jσσσσ′ fjσ′ ; nf ,j = 1
HH =−JH
∑j,αβ(f †j+1,αfjα)(f †j,β fj+1β)
HK =
−JK∑
j,αβ
(f †j,αpjα
)(p†j,β fjβ
)
H →Hc +
∑j,σ
(∆j f†j+1,σfj,σ + H.c +
|∆j |2JH
)+∑
j,σ
(Vj (c† j+1σ − c† j−1σ)fjσ + H.c +
|Vj |2JK (j)
)+
∑j λj (nf ,j − 1)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 23 / 32
Equation Overdose
H = Hc + HH + HK
H = −t∑
j,σ(c†j+1σcjσ + H.c) + JH∑
i Sj · Sj+1 +∑
j,αβJK(j)
2 Sj ·p†j,ασαβpj,β
Sj =∑σσ′ f
†jσσσσ′ fjσ′ ; nf ,j = 1
HH =−JH
∑j,αβ(f †j+1,αfjα)(f †j,β fj+1β)
HK =
−JK∑
j,αβ
(f †j,αpjα
)(p†j,β fjβ
)
H →Hc +
∑j,σ
(∆j f†j+1,σfj,σ + H.c +
|∆j |2JH
)+∑
j,σ
(Vj (c† j+1σ − c† j−1σ)fjσ + H.c +
|Vj |2JK (j)
)+∑
j λj (nf ,j − 1)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 23 / 32
Insulator in Kondo Lattice?
HamiltonianH = −t
∑jσ(C† j+1,σCj,σ + h.c.) +∑
jσ V (C† j+1,σ −C† j−1,σ)fjσ + h.c +∑jσ ∆(f † j+1,σfj,σ + h.c) +
∑j λnf ,j
New fermions
Sj · σαβpj,β ≡(
2VJK
)fj,α
p† j,βσβα · Sj ≡(
2VJK
)f † j,α
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 24 / 32
Insulator in Kondo Lattice?
HamiltonianH = −t
∑jσ(C† j+1,σCj,σ + h.c.)
+∑jσ V (C† j+1,σ −C† j−1,σ)fjσ + h.c +∑jσ ∆(f † j+1,σfj,σ + h.c) +
∑j λnf ,j
New fermions
Sj · σαβpj,β ≡(
2VJK
)fj,α
p† j,βσβα · Sj ≡(
2VJK
)f † j,α
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 24 / 32
Insulator in Kondo Lattice?
HamiltonianH = −t
∑jσ(C† j+1,σCj,σ + h.c.) +∑
jσ V (C† j+1,σ −C† j−1,σ)fjσ + h.c
+∑jσ ∆(f † j+1,σfj,σ + h.c) +
∑j λnf ,j
New fermions
Sj · σαβpj,β ≡(
2VJK
)fj,α
p† j,βσβα · Sj ≡(
2VJK
)f † j,α
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 24 / 32
Insulator in Kondo Lattice?
HamiltonianH = −t
∑jσ(C† j+1,σCj,σ + h.c.) +∑
jσ V (C† j+1,σ −C† j−1,σ)fjσ + h.c +∑jσ ∆(f † j+1,σfj,σ + h.c) +
∑j λnf ,j
New fermions
Sj · σαβpj,β ≡(
2VJK
)fj,α
p† j,βσβα · Sj ≡(
2VJK
)f † j,α
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 24 / 32
Insulator in Kondo Lattice?
HamiltonianH = −t
∑jσ(C† j+1,σCj,σ + h.c.) +∑
jσ V (C† j+1,σ −C† j−1,σ)fjσ + h.c +∑jσ ∆(f † j+1,σfj,σ + h.c) +
∑j λnf ,j
New fermions
Sj · σαβpj,β ≡(
2VJK
)fj,α
p† j,βσβα · Sj ≡(
2VJK
)f † j,α
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 24 / 32
Insulator in Kondo Lattice?
HamiltonianH = −t
∑jσ(C† j+1,σCj,σ + h.c.) +∑
jσ V (C† j+1,σ −C† j−1,σ)fjσ + h.c +∑jσ ∆(f † j+1,σfj,σ + h.c) +
∑j λnf ,j
New fermions
Sj · σαβpj,β ≡(
2VJK
)fj,α
p† j,βσβα · Sj ≡(
2VJK
)f † j,α
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 24 / 32
Insulator in Kondo Lattice?
HamiltonianH = −t
∑jσ(C† j+1,σCj,σ + h.c.) +∑
jσ V (C† j+1,σ −C† j−1,σ)fjσ + h.c +∑jσ ∆(f † j+1,σfj,σ + h.c) +
∑j λnf ,j
New fermions
Sj · σαβpj,β ≡(
2VJK
)fj,α
p† j,βσβα · Sj ≡(
2VJK
)f † j,α
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 24 / 32
Insulator in Kondo Lattice?
HamiltonianH = −t
∑jσ(C† j+1,σCj,σ + h.c.) +∑
jσ V (C† j+1,σ −C† j−1,σ)fjσ + h.c +∑jσ ∆(f † j+1,σfj,σ + h.c) +
∑j λnf ,j
New fermions
Sj · σαβpj,β ≡(
2VJK
)fj,α
p† j,βσβα · Sj ≡(
2VJK
)f † j,α
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 24 / 32
Understanding R vs TT TK
Js · STTK
(Allen et al., PRB 20, 12(1979).)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 25 / 32
Understanding R vs TT TK
Js · STTK
(Allen et al., PRB 20, 12(1979).)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 25 / 32
Understanding R vs TT TK
Js · STTK
(Allen et al., PRB 20, 12(1979).)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 25 / 32
Understanding R vs TT TK
Js · S
TTK
(Allen et al., PRB 20, 12(1979).)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 25 / 32
Understanding R vs TT TK
Js · S
TTK
(Allen et al., PRB 20, 12(1979).)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 25 / 32
Understanding R vs TT TK
Js · S
TTK
(Allen et al., PRB 20, 12(1979).)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 25 / 32
Understanding R vs TT TK
Js · S
TTK
(Allen et al., PRB 20, 12(1979).)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 25 / 32
Understanding R vs TT TK
Js · STTK
(Allen et al., PRB 20, 12(1979).)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 25 / 32
Topology?
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉t = ∆ = Vu(k) =
1√2
(cos(k/2)eik/2
−isin(k/2)eik/2
)
P = e/2! non − trivial
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 26 / 32
Topology?
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉t = ∆ = Vu(k) =
1√2
(cos(k/2)eik/2
−isin(k/2)eik/2
)
P = e/2! non − trivial
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 26 / 32
Topology?
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉
t = ∆ = Vu(k) =
1√2
(cos(k/2)eik/2
−isin(k/2)eik/2
)
P = e/2! non − trivial
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 26 / 32
Topology?
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉t = ∆ = Vu(k) =
1√2
(cos(k/2)eik/2
−isin(k/2)eik/2
)
P = e/2! non − trivial
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 26 / 32
Topology?
HTB =∑
k (c†kσ, f†kσ)(
−2t cos k −2i V sin k2iV sin k 2∆ cos k + λ
)(ckσfkσ
)
P = e2π
∫BZ A(k)dk
..A(k) = −i〈u(k)|∇k |u(k)〉t = ∆ = Vu(k) =
1√2
(cos(k/2)eik/2
−isin(k/2)eik/2
)
P = e/2! non − trivial
Alexandrov et al. arXiv:1403.6819v1, PRB 90, 115147(2014)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 26 / 32
But SmB6 is not a 1D system!
3D topological insulators have a different story..
and to characterize them we have four integers νo, ν1, ν2, ν3
which tells us about the kind of surface states etc.
SmB6 has a BCC lattice..In a series of papers, in which spin orbit coupling is included in thehopping elements..It was predicted that SmB6 may be a Topological Kondo Insulator whichhas robust surface states.These conclusions are drawn from a mean field calculation of the Kondolattice hamiltonian similar in spirit to what we saw for 1D problem.Look at experiments now..
Dzero et al. PRL 104,106408(2010), PRB 85, 045130 (2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 27 / 32
But SmB6 is not a 1D system!
3D topological insulators have a different story..and to characterize them we have four integers νo, ν1, ν2, ν3
which tells us about the kind of surface states etc.
SmB6 has a BCC lattice..In a series of papers, in which spin orbit coupling is included in thehopping elements..It was predicted that SmB6 may be a Topological Kondo Insulator whichhas robust surface states.These conclusions are drawn from a mean field calculation of the Kondolattice hamiltonian similar in spirit to what we saw for 1D problem.Look at experiments now..
Dzero et al. PRL 104,106408(2010), PRB 85, 045130 (2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 27 / 32
But SmB6 is not a 1D system!
3D topological insulators have a different story..and to characterize them we have four integers νo, ν1, ν2, ν3
which tells us about the kind of surface states etc.
SmB6 has a BCC lattice..In a series of papers, in which spin orbit coupling is included in thehopping elements..It was predicted that SmB6 may be a Topological Kondo Insulator whichhas robust surface states.These conclusions are drawn from a mean field calculation of the Kondolattice hamiltonian similar in spirit to what we saw for 1D problem.Look at experiments now..
Dzero et al. PRL 104,106408(2010), PRB 85, 045130 (2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 27 / 32
But SmB6 is not a 1D system!
3D topological insulators have a different story..and to characterize them we have four integers νo, ν1, ν2, ν3
which tells us about the kind of surface states etc.
SmB6 has a BCC lattice..
In a series of papers, in which spin orbit coupling is included in thehopping elements..It was predicted that SmB6 may be a Topological Kondo Insulator whichhas robust surface states.These conclusions are drawn from a mean field calculation of the Kondolattice hamiltonian similar in spirit to what we saw for 1D problem.Look at experiments now..
Dzero et al. PRL 104,106408(2010), PRB 85, 045130 (2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 27 / 32
But SmB6 is not a 1D system!
3D topological insulators have a different story..and to characterize them we have four integers νo, ν1, ν2, ν3
which tells us about the kind of surface states etc.
SmB6 has a BCC lattice..In a series of papers, in which spin orbit coupling is included in thehopping elements..
It was predicted that SmB6 may be a Topological Kondo Insulator whichhas robust surface states.These conclusions are drawn from a mean field calculation of the Kondolattice hamiltonian similar in spirit to what we saw for 1D problem.Look at experiments now..
Dzero et al. PRL 104,106408(2010), PRB 85, 045130 (2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 27 / 32
But SmB6 is not a 1D system!
3D topological insulators have a different story..and to characterize them we have four integers νo, ν1, ν2, ν3
which tells us about the kind of surface states etc.
SmB6 has a BCC lattice..In a series of papers, in which spin orbit coupling is included in thehopping elements..It was predicted that SmB6 may be a Topological Kondo Insulator whichhas robust surface states.
These conclusions are drawn from a mean field calculation of the Kondolattice hamiltonian similar in spirit to what we saw for 1D problem.Look at experiments now..
Dzero et al. PRL 104,106408(2010), PRB 85, 045130 (2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 27 / 32
But SmB6 is not a 1D system!
3D topological insulators have a different story..and to characterize them we have four integers νo, ν1, ν2, ν3
which tells us about the kind of surface states etc.
SmB6 has a BCC lattice..In a series of papers, in which spin orbit coupling is included in thehopping elements..It was predicted that SmB6 may be a Topological Kondo Insulator whichhas robust surface states.These conclusions are drawn from a mean field calculation of the Kondolattice hamiltonian similar in spirit to what we saw for 1D problem.
Look at experiments now..
Dzero et al. PRL 104,106408(2010), PRB 85, 045130 (2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 27 / 32
But SmB6 is not a 1D system!
3D topological insulators have a different story..and to characterize them we have four integers νo, ν1, ν2, ν3
which tells us about the kind of surface states etc.
SmB6 has a BCC lattice..In a series of papers, in which spin orbit coupling is included in thehopping elements..It was predicted that SmB6 may be a Topological Kondo Insulator whichhas robust surface states.These conclusions are drawn from a mean field calculation of the Kondolattice hamiltonian similar in spirit to what we saw for 1D problem.Look at experiments now..
Dzero et al. PRL 104,106408(2010), PRB 85, 045130 (2012)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 27 / 32
Understanding R vs T again
Why does this saturate?Wolgast(2013)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 28 / 32
Understanding R vs T again
Why does this saturate?
Wolgast(2013)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 28 / 32
Understanding R vs T again
Why does this saturate?
Wolgast(2013)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 28 / 32
Understanding R vs T again
Why does this saturate?Wolgast(2013)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 28 / 32
Another Experiment
(Kim et al. 2013)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 29 / 32
Outline
The StageI Electronic Configuration and Structure.I The early R vs T Measurement on SmB6
Act II Anderson Impurity ModelI Kondo EffectI Kondo Lattice
Act III 1D Topological Insulator
Act IIII 1D Topological Kondo InsulatorI Some understanding of the experiments.I Comments on 3D TKI.
Conclusions
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 30 / 32
Conclusions and Open questions?
Evidence of surface transport in SmB6 is there. Attempted a qualitativeunderstanding of experiments using a toy 1D model. Deeperunderstading needs us to go to 3D TKI.
To prove conclusively that SmB6 is indeed a TKI requires many morechecks!These include robustness against scattering, disorder etc. (Kim et al.Paglione Group)and ARPES studies to check the actual dispersion etc. Some of thesehave been done. (Hasan, Review)There are still some unsettled issues.(Coleman, APS Talk 2014)
SmB6 belongs to the field of heavy fermions, which has variety of kinds ofsuperconductivity, magnetic phases, fermi and non fermi liquids etc.Many of the theories regarding these involve reaching a mean fieldhamiltonian.Can these material systems be test beds for variety of models innon-interacting systems involving topology etc.
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 31 / 32
Conclusions and Open questions?
Evidence of surface transport in SmB6 is there. Attempted a qualitativeunderstanding of experiments using a toy 1D model. Deeperunderstading needs us to go to 3D TKI.To prove conclusively that SmB6 is indeed a TKI requires many morechecks!
These include robustness against scattering, disorder etc. (Kim et al.Paglione Group)and ARPES studies to check the actual dispersion etc. Some of thesehave been done. (Hasan, Review)There are still some unsettled issues.(Coleman, APS Talk 2014)
SmB6 belongs to the field of heavy fermions, which has variety of kinds ofsuperconductivity, magnetic phases, fermi and non fermi liquids etc.Many of the theories regarding these involve reaching a mean fieldhamiltonian.Can these material systems be test beds for variety of models innon-interacting systems involving topology etc.
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 31 / 32
Conclusions and Open questions?
Evidence of surface transport in SmB6 is there. Attempted a qualitativeunderstanding of experiments using a toy 1D model. Deeperunderstading needs us to go to 3D TKI.To prove conclusively that SmB6 is indeed a TKI requires many morechecks!These include robustness against scattering, disorder etc. (Kim et al.Paglione Group)
and ARPES studies to check the actual dispersion etc. Some of thesehave been done. (Hasan, Review)There are still some unsettled issues.(Coleman, APS Talk 2014)
SmB6 belongs to the field of heavy fermions, which has variety of kinds ofsuperconductivity, magnetic phases, fermi and non fermi liquids etc.Many of the theories regarding these involve reaching a mean fieldhamiltonian.Can these material systems be test beds for variety of models innon-interacting systems involving topology etc.
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 31 / 32
Conclusions and Open questions?
Evidence of surface transport in SmB6 is there. Attempted a qualitativeunderstanding of experiments using a toy 1D model. Deeperunderstading needs us to go to 3D TKI.To prove conclusively that SmB6 is indeed a TKI requires many morechecks!These include robustness against scattering, disorder etc. (Kim et al.Paglione Group)and ARPES studies to check the actual dispersion etc. Some of thesehave been done. (Hasan, Review)
There are still some unsettled issues.(Coleman, APS Talk 2014)
SmB6 belongs to the field of heavy fermions, which has variety of kinds ofsuperconductivity, magnetic phases, fermi and non fermi liquids etc.Many of the theories regarding these involve reaching a mean fieldhamiltonian.Can these material systems be test beds for variety of models innon-interacting systems involving topology etc.
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 31 / 32
Conclusions and Open questions?
Evidence of surface transport in SmB6 is there. Attempted a qualitativeunderstanding of experiments using a toy 1D model. Deeperunderstading needs us to go to 3D TKI.To prove conclusively that SmB6 is indeed a TKI requires many morechecks!These include robustness against scattering, disorder etc. (Kim et al.Paglione Group)and ARPES studies to check the actual dispersion etc. Some of thesehave been done. (Hasan, Review)There are still some unsettled issues.(Coleman, APS Talk 2014)
SmB6 belongs to the field of heavy fermions, which has variety of kinds ofsuperconductivity, magnetic phases, fermi and non fermi liquids etc.Many of the theories regarding these involve reaching a mean fieldhamiltonian.Can these material systems be test beds for variety of models innon-interacting systems involving topology etc.
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 31 / 32
Conclusions and Open questions?
Evidence of surface transport in SmB6 is there. Attempted a qualitativeunderstanding of experiments using a toy 1D model. Deeperunderstading needs us to go to 3D TKI.To prove conclusively that SmB6 is indeed a TKI requires many morechecks!These include robustness against scattering, disorder etc. (Kim et al.Paglione Group)and ARPES studies to check the actual dispersion etc. Some of thesehave been done. (Hasan, Review)There are still some unsettled issues.(Coleman, APS Talk 2014)
SmB6 belongs to the field of heavy fermions, which has variety of kinds ofsuperconductivity, magnetic phases, fermi and non fermi liquids etc.
Many of the theories regarding these involve reaching a mean fieldhamiltonian.Can these material systems be test beds for variety of models innon-interacting systems involving topology etc.
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 31 / 32
Conclusions and Open questions?
Evidence of surface transport in SmB6 is there. Attempted a qualitativeunderstanding of experiments using a toy 1D model. Deeperunderstading needs us to go to 3D TKI.To prove conclusively that SmB6 is indeed a TKI requires many morechecks!These include robustness against scattering, disorder etc. (Kim et al.Paglione Group)and ARPES studies to check the actual dispersion etc. Some of thesehave been done. (Hasan, Review)There are still some unsettled issues.(Coleman, APS Talk 2014)
SmB6 belongs to the field of heavy fermions, which has variety of kinds ofsuperconductivity, magnetic phases, fermi and non fermi liquids etc.Many of the theories regarding these involve reaching a mean fieldhamiltonian.
Can these material systems be test beds for variety of models innon-interacting systems involving topology etc.
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 31 / 32
Conclusions and Open questions?
Evidence of surface transport in SmB6 is there. Attempted a qualitativeunderstanding of experiments using a toy 1D model. Deeperunderstading needs us to go to 3D TKI.To prove conclusively that SmB6 is indeed a TKI requires many morechecks!These include robustness against scattering, disorder etc. (Kim et al.Paglione Group)and ARPES studies to check the actual dispersion etc. Some of thesehave been done. (Hasan, Review)There are still some unsettled issues.(Coleman, APS Talk 2014)
SmB6 belongs to the field of heavy fermions, which has variety of kinds ofsuperconductivity, magnetic phases, fermi and non fermi liquids etc.Many of the theories regarding these involve reaching a mean fieldhamiltonian.Can these material systems be test beds for variety of models innon-interacting systems involving topology etc.
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 31 / 32
Conclusions and Open questions?
Evidence of surface transport in SmB6 is there. Attempted a qualitativeunderstanding of experiments using a toy 1D model. Deeperunderstading needs us to go to 3D TKI.To prove conclusively that SmB6 is indeed a TKI requires many morechecks!These include robustness against scattering, disorder etc. (Kim et al.Paglione Group)and ARPES studies to check the actual dispersion etc. Some of thesehave been done. (Hasan, Review)There are still some unsettled issues.(Coleman, APS Talk 2014)
SmB6 belongs to the field of heavy fermions, which has variety of kinds ofsuperconductivity, magnetic phases, fermi and non fermi liquids etc.Many of the theories regarding these involve reaching a mean fieldhamiltonian.Can these material systems be test beds for variety of models innon-interacting systems involving topology etc.
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 31 / 32
The End
ReferencesEnd States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014 PRB 90,115147(2014)
Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)Hewson, Alexander Cyril. The Kondo Problem to Heavy Fermions. No. 2.Cambridge university press, 1997.
AcknowledgmentsThanks to Vijay and Aveek for many discussions. Special thanks to Sangramfor references on the experiments. Thanks to many friends for discussions.
Thank you :)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 32 / 32
The End
ReferencesEnd States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014 PRB 90,115147(2014)Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014
Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)Hewson, Alexander Cyril. The Kondo Problem to Heavy Fermions. No. 2.Cambridge university press, 1997.
AcknowledgmentsThanks to Vijay and Aveek for many discussions. Special thanks to Sangramfor references on the experiments. Thanks to many friends for discussions.
Thank you :)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 32 / 32
The End
ReferencesEnd States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014 PRB 90,115147(2014)Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)
Hewson, Alexander Cyril. The Kondo Problem to Heavy Fermions. No. 2.Cambridge university press, 1997.
AcknowledgmentsThanks to Vijay and Aveek for many discussions. Special thanks to Sangramfor references on the experiments. Thanks to many friends for discussions.
Thank you :)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 32 / 32
The End
ReferencesEnd States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014 PRB 90,115147(2014)Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)Hewson, Alexander Cyril. The Kondo Problem to Heavy Fermions. No. 2.Cambridge university press, 1997.
AcknowledgmentsThanks to Vijay and Aveek for many discussions. Special thanks to Sangramfor references on the experiments. Thanks to many friends for discussions.
Thank you :)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 32 / 32
The End
ReferencesEnd States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014 PRB 90,115147(2014)Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)Hewson, Alexander Cyril. The Kondo Problem to Heavy Fermions. No. 2.Cambridge university press, 1997.
AcknowledgmentsThanks to Vijay and Aveek for many discussions.
Special thanks to Sangramfor references on the experiments. Thanks to many friends for discussions.
Thank you :)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 32 / 32
The End
ReferencesEnd States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014 PRB 90,115147(2014)Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)Hewson, Alexander Cyril. The Kondo Problem to Heavy Fermions. No. 2.Cambridge university press, 1997.
AcknowledgmentsThanks to Vijay and Aveek for many discussions. Special thanks to Sangramfor references on the experiments.
Thanks to many friends for discussions.
Thank you :)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 32 / 32
The End
ReferencesEnd States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014 PRB 90,115147(2014)Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)Hewson, Alexander Cyril. The Kondo Problem to Heavy Fermions. No. 2.Cambridge university press, 1997.
AcknowledgmentsThanks to Vijay and Aveek for many discussions. Special thanks to Sangramfor references on the experiments. Thanks to many friends for discussions.
Thank you :)
Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 32 / 32
The End
ReferencesEnd States in a 1D topological Kondo insulator.Alexandrov et al. arXiv:1403.6819v1 26th March,2014 PRB 90,115147(2014)Topological Insulators, Topological Crystalline Insulators, and TopologicalKondo Insulators (Review Article)M. Zahid Hasan et al. arXiv:1406.1040 2nd June,2014Chapter 17 and 18, Introduction to Many Body Physics.Piers Coleman, unpublished preprint on his webpage, available onrequest (2014)Hewson, Alexander Cyril. The Kondo Problem to Heavy Fermions. No. 2.Cambridge university press, 1997.
AcknowledgmentsThanks to Vijay and Aveek for many discussions. Special thanks to Sangramfor references on the experiments. Thanks to many friends for discussions.
Thank you :)Adhip Agarwala (IISc) The Story of SmB6 October 9, 2014 32 / 32