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