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Itinerant Electrons in Frustrated Magnets:emerging chiral insulators,
macroscopic degeneracies and [Fractional] Quantum Hall liquids
Jeroen van den Brink
Jörn Venderbos, Stefanos Kourtis, Maria Daghofer, Sanjeev Kumar
Dutch Science Foundation
MPI-PKS 03.04.2012
Outline
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Itinerant electrons in cubic manganites
Outline
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Itinerant electrons in cubic manganites
Triangular lattice: chiral spin textures
Outline
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Itinerant electrons in cubic manganites
Triangular lattice: chiral spin textures
Checkerboard: Dirac + topological mass
Outline
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Itinerant electrons in cubic manganites
Triangular lattice: chiral spin textures
Relation to multi-orbital Hubbard system
Checkerboard: Dirac + topological mass
Outline
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Itinerant electrons in cubic manganites
Triangular lattice: chiral spin textures
Relation to multi-orbital Hubbard system
Conclusions
Checkerboard: Dirac + topological mass
Perovskite crystal structure of Pr1-xCaxMnO3
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Mn4+ / Mn3+
Oxygen2- Pr3+/Ca2+
0 < x < 1
• t2g orbitals
• eg orbitals
Local considerations
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
––
––
–– ––
––––
Mn4+ / Mn3+
• t2g orbitals
• eg orbitals
Local considerations
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
––
––
–– ––
––––
eg
t2g
5x5xeg
t2g
Mn (3+) = 3d4 Mn (4+) = 3d3
Mn4+ / Mn3+
Magnetic orderings
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
PM
FM AF
AF-CAF-AFM
AF-G
Mn4+ 3d3
Mn3+ 3d4
x
Magnetic A- and C-phase
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
x = 0.5 – 0.6
A-phase
x = 0.6 – 1
C-phase
2D ferro planescoupled antiferro
1D ferro chainscoupled antiferro
Superexchange: antiferromagnetism
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
core-core super-exchange
x=1 → CaMnO3 → Mn4+ 3d3
eg
t2g
eg
t2g JAF
€
Eij = J cos θij
Superexchange: antiferromagnetism
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
G-type antiferromagnetism
core-core super-exchange
x=1 → CaMnO3 → Mn4+ 3d3
eg
t2g
eg
t2g JAF
€
Eij = J cos θij
Double exchange: ferromagnetism
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
conduction electron double-exchange
x~0.3 → both Mn3+ 3d4 and Mn4+ 3d3 present
eg
t2g
teg
t2g
DeGennes, Zener, Anderson/Hasegawa€
tij = t cos θij 2
Double exchange: ferromagnetism
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
conduction electron double-exchange
x~0.3 → both Mn3+ 3d4 and Mn4+ 3d3 present
eg
t2g
teg
t2g
DeGennes, Zener, Anderson/Hasegawa€
tij = t cos θij 2
Ferromagnetism FRUSTRATION
Interplay between orbital, spin and charge
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
JvdB, Khomskii, PRL 82, 1016 (1999)
eg
t2g
teg
t2g JAF
orbital degrees of freedom adapt to spin orientation
vice versa, magnetic ordering adapts to orbital order
Interplay between orbital, spin and charge
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
JvdB, Khomskii, PRL 82, 1016 (1999)
eg
t2g
teg
t2g JAF
Formally: DDEX model
orbital degrees of freedom adapt to spin orientation
vice versa, magnetic ordering adapts to orbital order
�z
2,z
2
hijikz
= 1, �z
2,z
2
hijikx
=14, �x
2�y
2,x
2�y
2
hijikx
=34
H =X
hiji,�
tij�↵�ij c+
i↵,�cj�,� � JH
X
i
Si · ~�i + JAF
X
hiji
Si · Sj
Degenerate double-exchange Hamiltonian
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Strong coupling of spin-charge-orbital
Degrees of freedom: local moments, orbitals, itinerant fermions
Low energy sector of multi-orbital Hubbard model
�z
2,z
2
hijikz
= 1, �z
2,z
2
hijikx
=14, �x
2�y
2,x
2�y
2
hijikx
=34
H =X
hiji,�
tij�↵�ij c+
i↵,�cj�,� � JH
X
i
Si · ~�i + JAF
X
hiji
Si · Sj
Degenerate double-exchange Hamiltonian
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Strong coupling of spin-charge-orbital
Degrees of freedom: local moments, orbitals, itinerant fermions
Large JH limit ⇒ spinless fermions with S-dependent hopping
Si · Sj = cos(✓i � ✓j)
Low energy sector of multi-orbital Hubbard model
H =X
hiji
tij�↵�ij c+
i↵cj� + JAF
X
hiji
Si · Sj
tij = t[cos(✓i/2) cos(✓j/2) + sin(✓i/2) sin(✓j/2)e�i(�i��j)]
Degenerate double-exchange Hamiltonian
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Large JH limit ⇒ spinless fermions with S-dependent hopping
H =X
hiji
tij�↵�ij c+
i↵cj� + JAF
X
hiji
Si · Sj
tij = t[cos(✓i/2) cos(✓j/2) + sin(✓i/2) sin(✓j/2)e�i(�i��j)]
Degenerate double-exchange Hamiltonian
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Large JH limit ⇒ spinless fermions with S-dependent hopping
tij ! ei� ! eiR
A·dl
Effective vector potential
Cooperative lifting of magnetic frustration
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
F
AF
AF
F
AFM A and C phases are due to orbital order
JvdB, Khomskii, PRL 82, 1016 (1999)
Cooperative lifting of magnetic frustration
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
spontaneous generation of very narrow bands
F
AF
AF
F
AFM A and C phases are due to orbital order
JvdB, Khomskii, PRL 82, 1016 (1999)
Phase diagram of DDEX model
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Efremov, JvdB, Khomskii, Nature Mat. (2004)
Transition from FM to A-type to CE-type to C-type as a function of x
Double- and super-exchange on a triangular lattice
J.W.F. Venderbos et al November 24th 2011Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Double- and super-exchange on a triangular lattice
J.W.F. Venderbos et al November 24th 2011Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
H =X
hiji
tijc+i↵cj� + JAF
X
hiji
Si · Sj
Double and superexchange on a triangular lattice
J.W.F. Venderbos et al November 24th 2011Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
First neglect orbital degrees of freedom 1/2 electron/site
H =X
hiji
tijc+i↵cj� + JAF
X
hiji
Si · Sj
Double and superexchange on a triangular lattice
J.W.F. Venderbos et al November 24th 2011Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
First neglect orbital degrees of freedom
Expected types of order
1/2 electron/site
Chiral spin state on triangular lattice
Kumar & JvdB, PRL 105, 216405 (2010), Martin & Batista, PRL 101, 156402 (2008), Akagi & Motome, JPSJ 79, 083711 (2010), Akagi, Udagawa & Motome, PRL 108, 096401 (2012).
J.W.F. Venderbos et al November 24th 2011Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Chiral spin state on triangular lattice
− non-coplanar state with 4-site unit cell − Si ·Sj = -1/3 for all NN pairs; finite chirality χijk =⟨Si ·Sj ×Sk⟩ − closed loops around triangle lead to a Berry phase π/2
Kumar & JvdB, PRL 105, 216405 (2010), Martin & Batista, PRL 101, 156402 (2008), Akagi & Motome, JPSJ 79, 083711 (2010), Akagi, Udagawa & Motome, PRL 108, 096401 (2012).
J.W.F. Venderbos et al November 24th 2011Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Chiral spin state on triangular lattice
Electronsin
effective magnetic
field
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Chiral spin state on triangular lattice
Electronsin
effective magnetic
field
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Chiral spin state on triangular lattice
Electronsin
effective magnetic
field
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Chiral spin state on triangular lattice
Electronsin
effective magnetic
field
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Chiral spin state on triangular lattice
Electronsin
effective magnetic
field
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Chiral spin state on triangular lattice
Electronsin
effective magnetic
field
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Chiral spin state on triangular lattice
Edge states
Electronsin
effective magnetic
field
Phase diagram double- and superexchange model
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Kumar & JvdB, PRL 105 216405 (2010)
Topological non-trivial Chern insulator
Phase diagram double- and superexchange model
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Kumar & JvdB, PRL 105 216405 (2010)
Topological non-trivial Chern insulator
1/2 electron/site: |C|=1 chiral edge states
�n
xy
=e2
hC
n
Cn =�i
2⇡
Z
BZd2kz ·rk ⇥An(k)
An(k) = �ihun(k)|rk|un(k)i
Transversal Hall conductivity
Chern number
Berry connection
Topological states of matter
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
time
2D - Quantum Anomalous Hall: QH on a lattice without external fields
1988
2D - Quantum Spin Hall phase time-reversal invariant
20053D - topological insulators time-reversal invariant single Dirac cones
2007
2D - Fractional Quantum HallTsui et al PRL 48, 1559 (1982)
1982
2D - Integer Quantum Hallvon Klitzing et al PRL 45, 494 (1980)1980
Topological states of matter
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
time
2D - Quantum Anomalous Hall: QH on a lattice without external fields
1988
2D - Quantum Spin Hall phase time-reversal invariant
20053D - topological insulators time-reversal invariant single Dirac cones
2007
2D - Fractional Quantum HallTsui et al PRL 48, 1559 (1982)
1982
2D - Integer Quantum Hallvon Klitzing et al PRL 45, 494 (1980)1980
Thouless, Kohmoto, Nightingale, den Nijs PRL 49, 405 (2007)
Topological states of matter
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
time
2D - Quantum Anomalous Hall: QH on a lattice without external fields
1988
2D - Quantum Spin Hall phase time-reversal invariant
20053D - topological insulators time-reversal invariant single Dirac cones
2007
2D - Fractional Quantum HallTsui et al PRL 48, 1559 (1982)
1982
2D - Integer Quantum Hallvon Klitzing et al PRL 45, 494 (1980)1980
Haldane PRL 61, 2015 (1988)
Thouless, Kohmoto, Nightingale, den Nijs PRL 49, 405 (2007)
Topological states of matter
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
time
2D - Quantum Anomalous Hall: QH on a lattice without external fields
1988
2D - Quantum Spin Hall phase time-reversal invariant
20053D - topological insulators time-reversal invariant single Dirac cones
2007
2D - Fractional Quantum HallTsui et al PRL 48, 1559 (1982)
1982
2D - Integer Quantum Hallvon Klitzing et al PRL 45, 494 (1980)1980
Haldane PRL 61, 2015 (1988)
Thouless, Kohmoto, Nightingale, den Nijs PRL 49, 405 (2007)
tei�
Topological states of matter
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
time
2D - Quantum Anomalous Hall: QH on a lattice without external fields
1988
2D - Quantum Spin Hall phase time-reversal invariant
2005Kane & Mele PRL 95, 146802 (2005) König et al Science 318, 766 (2007)Fu, Kane, Mele PRL 98, 106803 (2007)
3D - topological insulators time-reversal invariant single Dirac cones
2007
2D - Fractional Quantum HallTsui et al PRL 48, 1559 (1982)
1982
2D - Integer Quantum Hallvon Klitzing et al PRL 45, 494 (1980)1980
Haldane PRL 61, 2015 (1988)
Thouless, Kohmoto, Nightingale, den Nijs PRL 49, 405 (2007)
tei�
HK+ = ~v
F
(px
�x + py
�y) + m+ �z
HK� = ~v
F
(px
�x + py
�y) + m� �z
Mass generation in graphene
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
K+ K�
Dirac fermions
-
HK+ = ~v
F
(px
�x + py
�y) + m+ �z
HK� = ~v
F
(px
�x + py
�y) + m� �z
Mass generation in graphene
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
K+ K�
Dirac fermions
K+ K�
-
HK+ = ~v
F
(px
�x + py
�y) + m+ �z
HK� = ~v
F
(px
�x + py
�y) + m� �z
Mass generation in graphene
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
K+ K�
Dirac fermions
K+ K�
sublattice potential
-
HK+ = ~v
F
(px
�x + py
�y) + m+ �z
HK� = ~v
F
(px
�x + py
�y) + m� �z
m+ = m� = m
m
Mass generation in graphene
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
K+ K�
Dirac fermions
K+ K�
m
sublattice potential
-
HK+ = ~v
F
(px
�x + py
�y) + m+ �z
HK� = ~v
F
(px
�x + py
�y) + m� �z
m+ = m� = m
m
Mass generation in graphene
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
K+ K�
Dirac fermions
K+ K�
m
sublattice potential
Giovannetti, Khomyakov, Brocks, Kelly & JvdB PRB 76, 074103 (2007)
e.g. graphene on BN
-
HK+ = ~v
F
(px
�x + py
�y) + m+ �z
HK� = ~v
F
(px
�x + py
�y) + m� �z
m+ = m� = m
m
Mass generation in graphene
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
K+ K�
Dirac fermions
K+ K�
m
sublattice potential imaginary hopping
Giovannetti, Khomyakov, Brocks, Kelly & JvdB PRB 76, 074103 (2007)
e.g. graphene on BN
Haldane PRL 61, 2015 (1988)
tei�
-
HK+ = ~v
F
(px
�x + py
�y) + m+ �z
HK� = ~v
F
(px
�x + py
�y) + m� �z
+m �m
m+ = m� = m
Mass generation in graphene
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
K+ K�
Dirac fermions
K+ K�
sublattice potential imaginary hopping
m+ = +m
m� = �mGiovannetti, Khomyakov, Brocks, Kelly & JvdB PRB 76, 074103 (2007)
e.g. graphene on BN
Haldane PRL 61, 2015 (1988) QAH
tei�
-
Checkboard lattice: flux phase (“graphene”)
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Checkboard lattice: flux phase (“graphene”)
Jeroen van den Brink
exchange and hopping equal on all bonds
Workshop on Impurities and Textures in Unconventional Magnets
Checkboard lattice: flux phase (“graphene”)
Jeroen van den Brink
Venderbos, Daghofer, JvdB & Kumar, arxiv 1202.3340
exchange and hopping equal on all bonds
flux phase for large JAF/t
Workshop on Impurities and Textures in Unconventional Magnets
Checkboard lattice: flux phase (“graphene”)
Jeroen van den Brink
Venderbos, Daghofer, JvdB & Kumar, arxiv 1202.3340
exchange and hopping equal on all bonds
Spins Electrons Dirac fermions
K+ K�
flux phase for large JAF/t
Workshop on Impurities and Textures in Unconventional Magnets
Checkboard lattice: flux phase (“graphene”)
Jeroen van den Brink
Venderbos, Daghofer, JvdB & Kumar, arxiv 1202.3340
exchange and hopping equal on all bonds
Spins Electrons Dirac fermions
K+ K�
flux phase for large JAF/t
Workshop on Impurities and Textures in Unconventional Magnets
Venderbos, Daghofer, JvdB & Kumar, arxiv 1202.3340
Checkboard: umbrella ferromagnet
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
decrease JAF/t
Venderbos, Daghofer, JvdB & Kumar, arxiv 1202.3340
Checkboard: umbrella ferromagnet
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
decrease JAF/t
Venderbos, Daghofer, JvdB & Kumar, arxiv 1202.3340
Checkboard: umbrella ferromagnet
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
decrease JAF/t
HK+ = ~v
F
(px
�x + py
�y) + m �z
HK� = ~v
F
(px
�x + py
�y)�m �z
+m �m
K+ K�
Realization of Haldane’s QAH state
-
Switchable Quantum Hall state
J.W.F. Venderbos et al
Umbrella ferromagnet: switchable edge states
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Hall conductivity edge states (bands on cylinder)
Jörn Venderbos, poster 20
Switchable Quantum Hall state
J.W.F. Venderbos et al
Umbrella ferromagnet: switchable edge states
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Chirality of edge state switchable by reversing magnetization
Hall conductivity edge states (bands on cylinder)
Jörn Venderbos, poster 20
Venderbos, Daghofer & JvdB, PRL 107, 116401 (2011)
Reintroduce orbital degrees of freedom
e g orbital degrees of freedom
∆ crystal field splittingt/t’ orbital dependent hopping
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Fix a topological non-trivial spin-texture
triangular kagome
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Fix a topological non-trivial spin-texture
triangular
kagome
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Fix a topological non-trivial spin-texture
triangular
kagome
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
very narrow topologically
non-trivialbands
Degeneracy of topological bands: FQH
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
In systems with (almost) degenerate topologically non-trivial bands
additional interactions (U) can lead to Fractional Quantum Hall stateTang et al, PRL 106, 236802 (2011)Sun et al, PRL 106, 236803 (2011) Neupert et al, PRL 106, 236804 (2011)
Degeneracy of topological bands: FQH
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
In systems with (almost) degenerate topologically non-trivial bands
Landau levels in Integer Quantum Hall systemFlat bands in Quantum Anomalous Hall system
additional interactions (U) can lead to Fractional Quantum Hall stateTang et al, PRL 106, 236802 (2011)Sun et al, PRL 106, 236803 (2011) Neupert et al, PRL 106, 236804 (2011)
Degeneracy of topological bands: FQH
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
M
W = bandwidth
In systems with (almost) degenerate topologically non-trivial bands
Landau levels in Integer Quantum Hall systemFlat bands in Quantum Anomalous Hall system
additional interactions (U) can lead to Fractional Quantum Hall state
requires hierarchy of energy scales: W < interaction U < gap M
Tang et al, PRL 106, 236802 (2011)Sun et al, PRL 106, 236803 (2011) Neupert et al, PRL 106, 236804 (2011)
Flatness figure of merit
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Venderbos, Daghofer & JvdB, PRL 107, 116401 (2011)
Orbital degrees of freedom cause topologically flat bands without invoking very long-range hopping
Flatness figure of merit
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Venderbos, Daghofer & JvdB, PRL 107, 116401 (2011)
Exact diagonalization: stabilization of 1/3 FQH stateStefanos Kourtis, poster 12
Orbital degrees of freedom cause topologically flat bands without invoking very long-range hopping
Start from multi-orbital t2g Hubbard model
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Start from multi-orbital t2g Hubbard model
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Start from multi-orbital t2g Hubbard model
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Start from multi-orbital t2g Hubbard model
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Start from multi-orbital t2g Hubbard model
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Start from multi-orbital t2g Hubbard model
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
5/2 electron/site
Interaction terms and decoupling
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Intra-orbital
Interaction terms and decoupling
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Intra-orbital
Inter-orbital
Hund’s rule
Interaction terms and decoupling
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Intra-orbital
Inter-orbital
Hund’s rule
− density-density decoupling − allow for non-coplanar spin textures − considered all 4-site unit cells − optimize spin angles
Venderbos, Kourtis, JvdB, Daghofer, PRL 108, 126405 (2012)
Structure of topological bands
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Venderbos, Kourtis, JvdB, Daghofer, PRL 108, 126405 (2012)
Structure of topological bands
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
very narrow topologically non-trivial band
Venderbos, Kourtis, JvdB, Daghofer, PRL 108, 126405 (2012)
Structure of topological bands
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Flatness figure of merit
very narrow topologically non-trivial band
Stefanos Kourtis, poster 12ED: stabilization of FQH states
Venderbos, Kourtis, JvdB, Daghofer, PRL 108, 126405 (2012)
Structure of topological bands
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Flatness figure of merit
very narrow topologically non-trivial band
Topological states of matter
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
time
2D - Quantum Anomalous Hall: QH on a lattice without external fields
1988
2D - Quantum Spin Hall phase time-reversal invariant
2005Kane & Mele PRL 95, 146802 (2005) König et al Science 318, 766 (2007)Fu, Kane, Mele PRL 98, 106803 (2007)
3D - topological insulators time-reversal invariant single Dirac cones
2007
2D - Fractional Quantum HallTsui et al PRL 48, 1559 (1982)
1982
2D - Integer Quantum Hallvon Klitzing et al PRL 45, 494 (1980)1980
Haldane PRL 61, 2015 (1988)
Thouless, Kohmoto, Nightingale, den Nijs PRL 49, 405 (2007)
2D - Fractional Quantum Anomalous Hall: FQH on a lattice without external fields2011
Topological states of matter
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
time
2D - Quantum Anomalous Hall: QH on a lattice without external fields
1988
2D - Quantum Spin Hall phase time-reversal invariant
2005Kane & Mele PRL 95, 146802 (2005) König et al Science 318, 766 (2007)Fu, Kane, Mele PRL 98, 106803 (2007)
3D - topological insulators time-reversal invariant single Dirac cones
2007
2D - Fractional Quantum HallTsui et al PRL 48, 1559 (1982)
1982
2D - Integer Quantum Hallvon Klitzing et al PRL 45, 494 (1980)1980
Haldane PRL 61, 2015 (1988)
Thouless, Kohmoto, Nightingale, den Nijs PRL 49, 405 (2007)
2D - Fractional Quantum Anomalous Hall: FQH on a lattice without external fields2011
Venderbos, Daghofer, JvdB, Kumar, PRL 107, 076405 (2011)
Honeycomb lattice
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
− not geometrically frustrated − highly degenerate dimer covering − emergent geometric frustration
Conclusions
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Competition of super- and double-exchange + geometric frustration leads to non-trivial spin textures and topological states
Conclusions
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Competition of super- and double-exchange + geometric frustration leads to non-trivial spin textures and topological states
Checkerboard: emerging Dirac fermions + topological mass
Chirality of QAH state controlled by magnetization
Conclusions
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Competition of super- and double-exchange + geometric frustration leads to non-trivial spin textures and topological states
Orbital degrees of freedom cause band flattening, potential for FQH
Checkerboard: emerging Dirac fermions + topological mass
Chirality of QAH state controlled by magnetization
Conclusions
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Competition of super- and double-exchange + geometric frustration leads to non-trivial spin textures and topological states
Orbital degrees of freedom cause band flattening, potential for FQH
Checkerboard: emerging Dirac fermions + topological mass
Chirality of QAH state controlled by magnetization
Hubbard model on triangular lattice with t2g orbitals
Topological non-trivial groundstates are present
Potential for FQH
Conclusions
Jeroen van den Brink Workshop on Impurities and Textures in Unconventional Magnets
Competition of super- and double-exchange + geometric frustration leads to non-trivial spin textures and topological states
Orbital degrees of freedom cause band flattening, potential for FQH
Checkerboard: emerging Dirac fermions + topological mass
Chirality of QAH state controlled by magnetization
Hubbard model on triangular lattice with t2g orbitals
Topological non-trivial groundstates are present
Potential for FQH
Itinerant correlated magnets: class of magnetic materials with potentially spectacular topological properties