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Spin liquids on the triangular lattice
HARVARDTalk online: sachdev.physics.harvard.edu
ICFCM, Sendai, Japan, Jan 11-14, 2011
Monday, January 10, 2011
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Outline
κ
1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea
2. Quantum-disordering magnetic order Application to -(ET)2Cu2(CN)3
3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2 ?
Monday, January 10, 2011
![Page 3: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/3.jpg)
Outline
κ
1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea
2. Quantum-disordering magnetic order Application to -(ET)2Cu2(CN)3
3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2 ?
Monday, January 10, 2011
![Page 4: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/4.jpg)
Fractionalization of the electron spin and charge
Monday, January 10, 2011
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�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
� �ψ+
ψ−
�
Fractionalization of the electron spin and charge
A) Quantum “disordering” magnetic order
Monday, January 10, 2011
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neutral bosonicspinons which transform to a
rotating reference from along the local antiferromagnetic order
Fractionalization of the electron spin and charge
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
� �ψ+
ψ−
�
A) Quantum “disordering” magnetic order
Monday, January 10, 2011
![Page 7: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/7.jpg)
spinlesscharge -efermions
Fractionalization of the electron spin and charge
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
� �ψ+
ψ−
�
A) Quantum “disordering” magnetic order
Monday, January 10, 2011
![Page 8: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/8.jpg)
SU(2)spin
Fractionalization of the electron spin and charge
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
� �ψ+
ψ−
�
A) Quantum “disordering” magnetic order
Monday, January 10, 2011
![Page 9: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/9.jpg)
U(1)charge
Fractionalization of the electron spin and charge
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
� �ψ+
ψ−
�
A) Quantum “disordering” magnetic order
Monday, January 10, 2011
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S. Sachdev, M. A. Metlitski, Y. Qi, and S. Sachdev Phys. Rev. B 80, 155129 (2009)
Fractionalization of the electron spin and charge
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
� �ψ+
ψ−
�
U × U−1
Theory has SU(2)sgauge invariance
A) Quantum “disordering” magnetic order
Monday, January 10, 2011
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�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
Fractionalization of the electron spin and charge
B) Projected Fermi seaX.-G. Wen, P. A. Lee,
O. Motrunich, M. P. A. Fisher
Monday, January 10, 2011
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charge e spinless bosons (or “rotors”) whose fluctuations
project onto the single electron states on each site
Fractionalization of the electron spin and charge
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
X.-G. Wen, P. A. Lee, O. Motrunich, M. P. A. FisherB) Projected Fermi sea
Monday, January 10, 2011
![Page 13: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/13.jpg)
neutralfermionicspinons
Fractionalization of the electron spin and charge
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
B) Projected Fermi seaX.-G. Wen, P. A. Lee,
O. Motrunich, M. P. A. Fisher
Monday, January 10, 2011
![Page 14: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/14.jpg)
Fractionalization of the electron spin and charge
SU(2)pseudospin ⊃ U(1)charge
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
B) Projected Fermi seaX.-G. Wen, P. A. Lee,
O. Motrunich, M. P. A. Fisher
Monday, January 10, 2011
![Page 15: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/15.jpg)
SU(2)spin
Fractionalization of the electron spin and charge
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
B) Projected Fermi seaX.-G. Wen, P. A. Lee,
O. Motrunich, M. P. A. Fisher
Monday, January 10, 2011
![Page 16: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/16.jpg)
Fractionalization of the electron spin and charge
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
U × U−1
Theory has SU(2)pgauge invariance
B) Projected Fermi seaX.-G. Wen, P. A. Lee,
O. Motrunich, M. P. A. Fisher
Monday, January 10, 2011
![Page 17: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/17.jpg)
On the triangular lattice, this leads to aU(1) spin liquid with a Fermi surface of spinons
which has been proposed to apply to EtMe3Sb[Pd(dmit)2]2.
This spin liquid has a thermal Hall responsewhich is not observed.
Fractionalization of the electron spin and charge
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
B) Projected Fermi seaX.-G. Wen, P. A. Lee,
O. Motrunich, M. P. A. Fisher
Monday, January 10, 2011
![Page 18: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/18.jpg)
On the triangular lattice, this leads to aU(1) spin liquid with a Fermi surface of spinons
which has been proposed to apply to EtMe3Sb[Pd(dmit)2]2.
This spin liquid has a thermal Hall responsewhich is not observed.
Fractionalization of the electron spin and charge
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
B) Projected Fermi seaX.-G. Wen, P. A. Lee,
O. Motrunich, M. P. A. Fisher
Monday, January 10, 2011
![Page 19: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/19.jpg)
Complete fractionalization: separate excitations carrying spin, charge, and Fermi statistics
Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, 057201 (2010)Monday, January 10, 2011
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Complete fractionalization: separate excitations carrying spin, charge, and Fermi statisticsDecompose electron operator into real fermions, χ:
c↑ = χ1 + iχ2 ; c↓ = χ3 + iχ4
Introduce a 4-component Majorana fermion ζi, i = 1 . . . 4and a SO(4) matrix R, and decompose:
χ = R ζ
Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, 057201 (2010)Monday, January 10, 2011
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Complete fractionalization: separate excitations carrying spin, charge, and Fermi statisticsDecompose electron operator into real fermions, χ:
c↑ = χ1 + iχ2 ; c↓ = χ3 + iχ4
Introduce a 4-component Majorana fermion ζi, i = 1 . . . 4and a SO(4) matrix R, and decompose:
χ = R ζ
Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, 057201 (2010)
SO(4) ∼= SU(2)pseudospin×SU(2)spin
Monday, January 10, 2011
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Complete fractionalization: separate excitations carrying spin, charge, and Fermi statisticsDecompose electron operator into real fermions, χ:
c↑ = χ1 + iχ2 ; c↓ = χ3 + iχ4
Introduce a 4-component Majorana fermion ζi, i = 1 . . . 4and a SO(4) matrix R, and decompose:
χ = R ζ
Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, 057201 (2010)
O ×OT
SO(4)gauge∼= SU(2)p;gauge×SU(2)s;gauge
Monday, January 10, 2011
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Complete fractionalization: separate excitations carrying spin, charge, and Fermi statisticsDecompose electron operator into real fermions, χ:
c↑ = χ1 + iχ2 ; c↓ = χ3 + iχ4
Introduce a 4-component Majorana fermion ζi, i = 1 . . . 4and a SO(4) matrix R, and decompose:
χ = R ζ
Cenke Xu and S. Sachdev, Phys. Rev. Lett. 105, 057201 (2010)
By breaking SO(4)gauge with different Higgs fields, we can reproduce essentially all earlier theories of spin liquids.
We also find many new spin liquid phases, some with Majorana fermion excitations which carry neither spin nor charge
Monday, January 10, 2011
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Outline
κ
1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea
2. Quantum-disordering magnetic order Application to -(ET)2Cu2(CN)3
3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2 ?
Monday, January 10, 2011
![Page 25: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/25.jpg)
Outline
κ
1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea
2. Quantum-disordering magnetic order Application to -(ET)2Cu2(CN)3
3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2 ?
Monday, January 10, 2011
![Page 26: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/26.jpg)
H =�
�ij�
Jij�Si · �Sj + . . .
H = J
�
�ij�
�Si · �Sj ; �Si ⇒ spin operator with S = 1/2
Monday, January 10, 2011
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Found in -(ET)2Cu[N(CN)2]Clκ
Anisotropic triangular lattice antiferromagnet
Classical ground state for small J’/J
Monday, January 10, 2011
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Anisotropic triangular lattice antiferromagnet
Classical ground state for large J’/JFound in Cs2CuCl4
Monday, January 10, 2011
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Valence bond solid
Anisotropic triangular lattice antiferromagnet
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)Monday, January 10, 2011
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Valence bond solid
Anisotropic triangular lattice antiferromagnet
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)Monday, January 10, 2011
![Page 31: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/31.jpg)
Valence bond solid
Anisotropic triangular lattice antiferromagnet
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)Monday, January 10, 2011
![Page 32: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/32.jpg)
Valence bond solid
Anisotropic triangular lattice antiferromagnet
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)Monday, January 10, 2011
![Page 33: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/33.jpg)
M. Tamura, A. Nakao and R. Kato, J. Phys. Soc. Japan 75, 093701 (2006)Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, Phys. Rev. Lett. 99, 256403 (2007)
Observation of a valence bond solid (VBS) in ETMe3P[Pd(dmit)2]2
Spin gap ~ 40 K J ~ 250 K
X-ray scattering
Monday, January 10, 2011
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Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Triangular lattice antiferromagnet
Monday, January 10, 2011
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P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
Triangular lattice antiferromagnet
Monday, January 10, 2011
![Page 36: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/36.jpg)
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
Triangular lattice antiferromagnet
Monday, January 10, 2011
![Page 37: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/37.jpg)
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
Triangular lattice antiferromagnet
Monday, January 10, 2011
![Page 38: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/38.jpg)
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
Triangular lattice antiferromagnet
Monday, January 10, 2011
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P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
Triangular lattice antiferromagnet
Monday, January 10, 2011
![Page 40: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/40.jpg)
Excitations of the Z2 Spin liquid
=A spinon
Monday, January 10, 2011
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Excitations of the Z2 Spin liquid
=A spinon
Monday, January 10, 2011
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Excitations of the Z2 Spin liquid
=A spinon
Monday, January 10, 2011
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Excitations of the Z2 Spin liquid
=A spinon
Monday, January 10, 2011
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• A characteristic property of a Z2 spin liquidis the presence of a spinon pair condensate
• A vison is an Abrikosov vortex in the paircondensate of spinons
• Visons are are the dark matter of spin liq-uids: they likely carry most of the energy,but are very hard to detect because they donot carry charge or spin.
Excitations of the Z2 Spin liquid
A vison
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)Monday, January 10, 2011
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=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)Monday, January 10, 2011
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=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)Monday, January 10, 2011
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=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)Monday, January 10, 2011
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=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)Monday, January 10, 2011
![Page 49: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/49.jpg)
=
-1
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)Monday, January 10, 2011
![Page 50: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/50.jpg)
=
-1
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)Monday, January 10, 2011
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Mutual Chern-Simons Theory Express theory in terms of the physical excitations of the Z2 spinliquid: the spinons, zα, and the visons. After accounting for Berryphase effects, the visons can be described by complex fields va,which transforms non-trivially under the square lattice space groupoperations.
The spinons and visons have mutual semionic statistics, and thisleads to the mutual CS theory at k = 2:
L =2�
α=1
�|(∂µ − iaµ)zα|2 + sz|zα|2
�
+Nv�
a=1
�|(∂µ − ibµ)va|2 + sv|va|2
�
+ik
2π�µνλaµ∂νbλ + · · ·
Cenke Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Monday, January 10, 2011
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Z2 spin liquidValence bond solid
(VBS)
Neel antiferromagnet
Spiral antiferromagnet
M
Phase diagram of frustrated antiferromagnetsN. Read and S. SachdevPhys. Rev. Lett. 63, 1773 (1991)
Cenke Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Monday, January 10, 2011
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Z2 spin liquidValence bond solid
(VBS)
Neel antiferromagnet
Spiral antiferromagnet
M
Phase diagram of frustrated antiferromagnetsN. Read and S. SachdevPhys. Rev. Lett. 63, 1773 (1991)
Vanishing of gap to bosonic spinon
excitations
Cenke Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Monday, January 10, 2011
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Z2 spin liquidValence bond solid
(VBS)
Neel antiferromagnet
Spiral antiferromagnet
M
Phase diagram of frustrated antiferromagnetsN. Read and S. SachdevPhys. Rev. Lett. 63, 1773 (1991)
Cenke Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Vanishing of gap to bosonic vison
excitations
Monday, January 10, 2011
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Z2 spin liquidValence bond solid
(VBS)
Neel antiferromagnet
Spiral antiferromagnet
M
Phase diagram of frustrated antiferromagnetsProposal:
κ-(ET)2Cu2(CN)3is here
Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett.
102, 176401 (2009)
Monday, January 10, 2011
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• Originally motivated by NMR relation. The
quantum critical point has O(4) symmetry
and has 1/T1 ∼ T η with η = 1.374(12).
• Recent µSR experiments show field-induced
magnetic order at very small fields.
• Thermal conductivity is dominated by con-
tribution of visons, and activated by the vi-
son gap.
Proposal: κ-(ET)2Cu2(CN)3 is a Z2 spin liquidnear a quantum phase transition to magnetic order
Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, 176401 (2009)Monday, January 10, 2011
![Page 57: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/57.jpg)
Spin excitation in κ-(ET)2Cu2(CN)3
1/T1
Inhomogeneous relaxation
α in stretched exp
Low-lying spin excitation at low-T
Shimizu et al., PRB 70 (2006) 060510 13C NMR relaxation rate
Anomaly at 5-6 K
1/T1 ~ power law of T
Monday, January 10, 2011
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• Originally motivated by NMR relation. The
quantum critical point has O(4) symmetry
and has 1/T1 ∼ T η with η = 1.374(12).
• Recent µSR experiments show field-induced
magnetic order at very small fields.
• Thermal conductivity is dominated by con-
tribution of visons, and activated by the vi-
son gap.
Proposal: κ-(ET)2Cu2(CN)3 is a Z2 spin liquidnear a quantum phase transition to magnetic order
Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, 176401 (2009)Monday, January 10, 2011
![Page 59: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/59.jpg)
• Originally motivated by NMR relation. The
quantum critical point has O(4) symmetry
and has 1/T1 ∼ T η with η = 1.374(12).
• Recent µSR experiments show field-induced
magnetic order at very small fields.
• Thermal conductivity is dominated by con-
tribution of visons, and activated by the vi-
son gap.
Proposal: κ-(ET)2Cu2(CN)3 is a Z2 spin liquidnear a quantum phase transition to magnetic order
Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, 176401 (2009)Monday, January 10, 2011
![Page 60: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/60.jpg)
Field-induced QPT from SR Line Width
!"#$%$&'#&()%&*(*%+*(,%'-.(/0!(123(45"6.'#%#+(",(7*%8%9:('&(6-;3<(=0>/?@?@=(AB@@CD
F. Pratt et al. (ISIS, UK)preprint
Monday, January 10, 2011
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Measured Phase Boundary of the QPT!"#$%&'
H-H0
Tiny H0 implies that spin gap per spin1/2 is z ~ 3.5 mK !
H0 = 5.2(2) mT
F. Pratt et al. (ISIS, UK)preprint
Monday, January 10, 2011
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• Originally motivated by NMR relation. The
quantum critical point has O(4) symmetry
and has 1/T1 ∼ T η with η = 1.374(12).
• Recent µSR experiments show field-induced
magnetic order at very small fields.
• Thermal conductivity is dominated by con-
tribution of visons, and activated by the vi-
son gap.
Proposal: κ-(ET)2Cu2(CN)3 is a Z2 spin liquidnear a quantum phase transition to magnetic order
Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, 176401 (2009)Monday, January 10, 2011
![Page 63: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/63.jpg)
Proposal: κ-(ET)2Cu2(CN)3 is a Z2 spin liquidnear a quantum phase transition to magnetic order
• Originally motivated by NMR relation. The
quantum critical point has O(4) symmetry
and has 1/T1 ∼ T η with η = 1.374(12).
• Recent µSR experiments show field-induced
magnetic order at very small fields.
• Thermal conductivity is dominated by con-
tribution of visons, and activated by the vi-
son gap.
Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, 176401 (2009)Monday, January 10, 2011
![Page 64: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/64.jpg)
H = 0 Tesla
•Arrhenius behavior for T < Δ !
•Tiny gapΔ = 0.46 K ~ J/500
M. Yamashita et al., Nature Physics 5, 44 (2009)
Thermal conductivity of κ-(ET)2Cu2(CN)3
Monday, January 10, 2011
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Proposal: κ-(ET)2Cu2(CN)3 is a Z2 spin liquidnear a quantum phase transition to magnetic order
• Originally motivated by NMR relation. The
quantum critical point has O(4) symmetry
and has 1/T1 ∼ T η with η = 1.374(12).
• Recent µSR experiments show field-induced
magnetic order at very small fields.
• Thermal conductivity is dominated by con-
tribution of visons, and activated by the vi-
son gap.
Yang Qi, Cenke Xu and S. Sachdev, Phys. Rev. Lett. 102, 176401 (2009)Monday, January 10, 2011
![Page 66: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/66.jpg)
Outline
κ
1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea
2. Quantum-disordering magnetic order Application to -(ET)2Cu2(CN)3
3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2 ?
Monday, January 10, 2011
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Outline
κ
1. Classification of spin liquids Quantum-disordering magnetic order vs. projected Fermi sea
2. Quantum-disordering magnetic order Application to -(ET)2Cu2(CN)3
3. Fermi surfaces of spinful Majorana fermions Candidate for EtMe3Sb[Pd(dmit)2]2 ?
Monday, January 10, 2011
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Shastry-Kitaev Majorana representation
On each site, we introduce neutral Majorana fermions
γi0, γix, γiy, γiz which obey
γiαγjβ + γjβγiα = 2δijδαβ
We write the S = 1/2 spin operators as
Sjx =i
2γjyγjz
Sjy =i
2γjzγjx
Sjz =i
2γjxγjy
along with the constraint
γj0γjxγjyγjz = 1 for all j
Monday, January 10, 2011
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SU(2)-invariant spin liquids with neutral, spinful Majorana excitations
• Postulate an SU(2)-invariant effective Hamiltonian
for physical S = 1 excitations created by Majorana
fermions γix, γiy, γiz.
• Ensure there is a Projective Symmetry Group (PSG)
under which physical observables have the full trans-
lational symmetry of the triangular lattice.
• Examine stability of the effective Hamiltonian to gauge
fluctuations.
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appearMonday, January 10, 2011
![Page 70: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/70.jpg)
SU(2)-invariant spin liquids with neutral, spinful Majorana excitations
• Postulate an SU(2)-invariant effective Hamiltonian
for physical S = 1 excitations created by Majorana
fermions γix, γiy, γiz.
• Ensure there is a Projective Symmetry Group (PSG)
under which physical observables have the full trans-
lational symmetry of the triangular lattice.
• Examine stability of the effective Hamiltonian to gauge
fluctuations.
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appearMonday, January 10, 2011
![Page 71: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/71.jpg)
SU(2)-invariant spin liquids with neutral, spinful Majorana excitations
• Postulate an SU(2)-invariant effective Hamiltonian
for physical S = 1 excitations created by Majorana
fermions γix, γiy, γiz.
• Ensure there is a Projective Symmetry Group (PSG)
under which physical observables have the full trans-
lational symmetry of the triangular lattice.
• Examine stability of the effective Hamiltonian to gauge
fluctuations.
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appearMonday, January 10, 2011
![Page 72: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/72.jpg)
Majorana fermions on the triangular lattice
H = −i
�
α=x,y,z
�
i<j
tijγiαγjα
where tij is an anti-symmetric matrix withthe following symmetry
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appearMonday, January 10, 2011
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Fermi surfaces
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
1BZ
The Majorana fermionsgenerically have Fermi sur-faces with the structureshown. 6 Fermi surfacelines intersect at k = 0,and the excitation energy∼ k3 as k → 0.
Monday, January 10, 2011
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Fermi surfaces
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
1BZ
Excitation ∼ k − kF
The Majorana fermionsgenerically have Fermi sur-faces with the structureshown. 6 Fermi surfacelines intersect at k = 0,and the excitation energy∼ k3 as k → 0.
Monday, January 10, 2011
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Fermi surfaces
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appear
1BZ
Excitation ∼ k − kF
Excitation ∼ k3
The Majorana fermionsgenerically have Fermi sur-faces with the structureshown. 6 Fermi surfacelines intersect at k = 0,and the excitation energy∼ k3 as k → 0.
Monday, January 10, 2011
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• Time-reversal symetry (T ) and rotation by 60 de-
grees (Rπ/3) are broken, but T Rπ/3 is unbroken.
• The specific heat and spin susceptibility are domi-
nated by excitations near k = 0, with CV ∼ T 2/3
and χ ∼ T−1/3.
• These power-laws are quenched by impurities: de-
tails under investigation.
• The thermal conductivity is dominated by fermions
along the Fermi line. The excitations near k = 0
move too slowly to contribute to transport.
• There is no thermal Hall transport.
Physical properties
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appearMonday, January 10, 2011
![Page 77: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/77.jpg)
• Time-reversal symetry (T ) and rotation by 60 de-
grees (Rπ/3) are broken, but T Rπ/3 is unbroken.
• The specific heat and spin susceptibility are domi-
nated by excitations near k = 0, with CV ∼ T 2/3
and χ ∼ T−1/3.
• These power-laws are quenched by impurities: de-
tails under investigation.
• The thermal conductivity is dominated by fermions
along the Fermi line. The excitations near k = 0
move too slowly to contribute to transport.
• There is no thermal Hall transport.
Physical properties
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appearMonday, January 10, 2011
![Page 78: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/78.jpg)
• Time-reversal symetry (T ) and rotation by 60 de-
grees (Rπ/3) are broken, but T Rπ/3 is unbroken.
• The specific heat and spin susceptibility are domi-
nated by excitations near k = 0, with CV ∼ T 2/3
and χ ∼ T−1/3.
• These power-laws are quenched by impurities: de-
tails under investigation.
• The thermal conductivity is dominated by fermions
along the Fermi line. The excitations near k = 0
move too slowly to contribute to transport.
• There is no thermal Hall transport.
Physical properties
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appearMonday, January 10, 2011
![Page 79: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/79.jpg)
• Time-reversal symetry (T ) and rotation by 60 de-
grees (Rπ/3) are broken, but T Rπ/3 is unbroken.
• The specific heat and spin susceptibility are domi-
nated by excitations near k = 0, with CV ∼ T 2/3
and χ ∼ T−1/3.
• These power-laws are quenched by impurities: de-
tails under investigation.
• The thermal conductivity is dominated by fermions
along the Fermi line. The excitations near k = 0
move too slowly to contribute to transport.
• There is no thermal Hall transport.
Physical properties
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appearMonday, January 10, 2011
![Page 80: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/80.jpg)
• Time-reversal symetry (T ) and rotation by 60 de-
grees (Rπ/3) are broken, but T Rπ/3 is unbroken.
• The specific heat and spin susceptibility are domi-
nated by excitations near k = 0, with CV ∼ T 2/3
and χ ∼ T−1/3.
• These power-laws are quenched by impurities: de-
tails under investigation.
• The thermal conductivity is dominated by fermions
along the Fermi line. The excitations near k = 0
move too slowly to contribute to transport.
• There is no thermal Hall transport.
Physical properties
Rudro Biswas, Liang Fu, Chris Laumann, and S. Sachdev, to appearMonday, January 10, 2011
![Page 81: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/81.jpg)
Conclusions
Unified perspective on spin liquids obtained by projection of a Fermi sea, and by “quantum-disordering” magnetic order.
Z2 spin liquid proximate to magnetic order a viable candidate for -(ET)2Cu2(CN)3
Proposed novel spin liquid with Fermi surfaces of spinful Majorana fermions
κ
Monday, January 10, 2011
![Page 82: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/82.jpg)
Conclusions
Unified perspective on spin liquids obtained by projection of a Fermi sea, and by “quantum-disordering” magnetic order.
Z2 spin liquid proximate to magnetic order a viable candidate for -(ET)2Cu2(CN)3
Proposed novel spin liquid with Fermi surfaces of spinful Majorana fermions
κ
Monday, January 10, 2011
![Page 83: Spin liquids on the triangular lattice - Harvard …qpt.physics.harvard.edu/talks/icfcm11.pdf · Spin liquids on the triangular lattice HARVARD Talk online: sachdev.physics.harvard.edu](https://reader031.vdocuments.pub/reader031/viewer/2022022017/5b8452a77f8b9aef498c13b3/html5/thumbnails/83.jpg)
Conclusions
Unified perspective on spin liquids obtained by projection of a Fermi sea, and by “quantum-disordering” magnetic order.
Z2 spin liquid proximate to magnetic order a viable candidate for -(ET)2Cu2(CN)3
Proposed novel spin liquid with Fermi surfaces of spinful Majorana fermions
κ
Monday, January 10, 2011