new quantum “disordering” magnetic order in insulators, metals, and...
TRANSCRIPT
![Page 1: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/1.jpg)
Quantum “disordering” magnetic order in
insulators, metals, and superconductors
HARVARDTalk online: sachdev.physics.harvard.edu
Perimeter Institute, Waterloo, May 29, 2010
Saturday, May 29, 2010
![Page 2: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/2.jpg)
HARVARD
Max Metlitski, HarvardarXiv:1005.1288Cenke Xu, Harvard
arXiv:1004.5431
Eun Gook Moon, HarvardarXiv:1005.3312
Saturday, May 29, 2010
![Page 3: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/3.jpg)
1. Quantum “disordering” magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases
2. Unified theory of spin liquids Majorana liquids
3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order
Outline
Saturday, May 29, 2010
![Page 4: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/4.jpg)
1. Quantum “disordering” magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases
2. Unified theory of spin liquids Majorana liquids
3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order
Outline
Saturday, May 29, 2010
![Page 5: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/5.jpg)
Ground state has long-range Néel order
Square lattice antiferromagnet
H =�
�ij�
Jij�Si · �Sj
Order parameter is a single vector field �ϕ = ηi�Si
ηi = ±1 on two sublattices
��ϕ� �= 0 in Neel state.
Saturday, May 29, 2010
![Page 6: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/6.jpg)
H =�
�ij�
Jij�Si · �Sj + · · ·+ · · ·
Add perturbations so ground state no longer has long-range Néel order
Square lattice antiferromagnet
Saturday, May 29, 2010
![Page 7: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/7.jpg)
H =�
�ij�
Jij�Si · �Sj + · · ·+ · · ·
Add perturbations so ground state no longer has long-range Néel order
Square lattice antiferromagnet
Describe the resulting state by an effective theory of fluctuationsof the Neel order:
Rz(x, τ)��Neel
�
where R is a SU(2) spin rotation matrix related to the Neel order
Rz ≡�
z↑ −z∗↓z↓ z∗↑
�; �ϕ = z∗α�σαβzβ
Ginzburg-Landau paradigm: Effective action for Rz(x, τ), de-duced by symmetries, describes quantum transitions and phases“near” the Neel state.
Saturday, May 29, 2010
![Page 8: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/8.jpg)
Underlying electrons cannot be ignored even though charged excitations are fully gapped.
Order parameter description is incomplete
They endow topological defects in the order parameter (hedgehogs, vortices...) with Berry phases:
the defects acquire additional degeneracies and transform non-trivially under lattice space group
e.g. with non-zero crystal momentum
Saturday, May 29, 2010
![Page 9: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/9.jpg)
Metals (in the cuprates)
H0 = −�
i<j
tijc†iαciα ≡
�
k
εkc†kαckα
• Begin with free electrons.
Γ
Hole states
occupied
Electron states
occupied
Γ
Saturday, May 29, 2010
![Page 10: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/10.jpg)
Fermi surface+antiferromagnetism
Γ
Hole states
occupied
Electron states
occupied
Γ
The electron spin polarization obeys�
�S(r, τ)�
= �ϕ(r, τ)eiK·r
where K is the ordering wavevector.
+
Saturday, May 29, 2010
![Page 11: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/11.jpg)
Spin density wave theory
In the presence of spin density wave order, �ϕ at wavevector K =(π,π), we have an additional term which mixes electron states withmomentum separated by K
Hsdw = �ϕ ·�
k,α,β
ck,α�σαβck+K,β
where �σ are the Pauli matrices. The electron dispersions obtainedby diagonalizing H0 +Hsdw for �ϕ ∝ (0, 0, 1) are
Ek± =εk + εk+K
2±
��εk − εk+K
2
�+ ϕ2
This leads to the Fermi surfaces shown in the following slides forhalf-filling
Saturday, May 29, 2010
![Page 12: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/12.jpg)
Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Half-filled band
Saturday, May 29, 2010
![Page 13: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/13.jpg)
Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Half-filled band
Saturday, May 29, 2010
![Page 14: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/14.jpg)
Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Hot spots
Half-filled band
Saturday, May 29, 2010
![Page 15: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/15.jpg)
Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Fermi surface breaks up at hot spotsinto electron and hole “pockets”
Hole pockets
Hot spots
Half-filled band
Saturday, May 29, 2010
![Page 16: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/16.jpg)
Increasing SDW order
ΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Insulator with Neel order has electronsfilling a band, and no Fermi surface
Hot spots
Insulator
Half-filled band
Saturday, May 29, 2010
![Page 17: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/17.jpg)
Increasing SDW order
ΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Insulator with Neel order has electronsfilling a band, and no Fermi surface
Hot spots
Insulator
Half-filled band
Saturday, May 29, 2010
![Page 18: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/18.jpg)
Square lattice antiferromagnet
Rz(x, τ)��Neel
�
Perform SU(2) rotation Rz on filled band of electrons:
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
��ψ+
ψ−
�
ψ± states filled
ψ± states empty
Saturday, May 29, 2010
![Page 19: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/19.jpg)
Square lattice antiferromagnet
Rz(x, τ)��Neel
�
Perform SU(2) rotation Rz on filled band of electrons:
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
��ψ+
ψ−
�
This is invariant under
zα → eiθzα ; ψ+ → e−iθψ+ ; ψ− → eiθψ.−
We obtain a U(1) gauge theory of
• bosonic neutral spinons zα;
• spinless, charged fermions ψ± occupying filled bands;
• an emergent U(1) gauge field Aµ.Saturday, May 29, 2010
![Page 20: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/20.jpg)
The Neel phase is the Higgs state with �zα� �= 0.
In the quantum “disordered” phase, with �zα� = 0 and
zα excitations gapped, let us examine the theory for the
ψ± fermions. For simplicity, we focus on the honeycomb
lattice, where this can be written in Dirac notation:
Lψ = iψγµ(∂µ − iAµσ
z)ψ +mψρyψ
where �σ/�ρ are Pauli matrices in spin/valley space.
Saturday, May 29, 2010
![Page 21: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/21.jpg)
The Neel phase is the Higgs state with �zα� �= 0.
In the quantum “disordered” phase, with �zα� = 0 and
zα excitations gapped, let us examine the theory for the
ψ± fermions. For simplicity, we focus on the honeycomb
lattice, where this can be written in Dirac notation:
Lψ = iψγµ(∂µ − iAµσ
z)ψ +mψρyσzψ
where �σ/�ρ are Pauli matrices in spin/valley space.
Nature of quantum “disordered” phase
Saturday, May 29, 2010
![Page 22: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/22.jpg)
The Neel phase is the Higgs state with �zα� �= 0.
In the quantum “disordered” phase, with �zα� = 0 and
zα excitations gapped, let us examine the theory for the
ψ± fermions. For simplicity, we focus on the honeycomb
lattice, where this can be written in Dirac notation:
Lψ = iψγµ(∂µ − iAµσ
z − iBµρy)ψ +mψρyσzψ
where �σ/�ρ are Pauli matrices in spin/valley space.
Introduce an external gauge field Bµ toprobe the structure of the gapped ψ± phase
Nature of quantum “disordered” phase
Saturday, May 29, 2010
![Page 23: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/23.jpg)
Nature of quantum “disordered” phase
After integrating out the fermions, the quantum spin
Hall physics implies a mutual Chern-Simons term be-
tween Aµ and Bµ
Leff =i
2π�µνλAµ∂νBλ
Changing theAµ flux (analog of electric field in QSHE),
induces a Bµ charge (analog of spin in QSHE).
Saturday, May 29, 2010
![Page 24: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/24.jpg)
Monopoles in Aµ carry Bµ charge.
This endows Aµ monopoles withnon-zero crystal momentum.
Nature of quantum “disordered” phase
After integrating out the fermions, the quantum spin
Hall physics implies a mutual Chern-Simons term be-
tween Aµ and Bµ
Leff =i
2π�µνλAµ∂νBλ
Changing theAµ flux (analog of electric field in QSHE),
induces a Bµ charge (analog of spin in QSHE).
Saturday, May 29, 2010
![Page 25: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/25.jpg)
Neel state�zα� �= 0 �zα� = 0
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
Valence bond solid (VBS)
szszc
Nature of quantum “disordered” phase
Saturday, May 29, 2010
![Page 26: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/26.jpg)
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)
C. Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Saturday, May 29, 2010
![Page 27: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/27.jpg)
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)
C. Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Saturday, May 29, 2010
![Page 28: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/28.jpg)
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)
C. Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Quantum “disordering”
spiral order leads to a Z2 spin liquid
Saturday, May 29, 2010
![Page 29: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/29.jpg)
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)
C. Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Saturday, May 29, 2010
![Page 30: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/30.jpg)
Z2 spin liquidValence bond solid
(VBS)
Neel antiferromagnet
Spiral antiferromagnet
M
Phase diagram of frustrated antiferromagnets
Described by a deconfined Z2
gauge theory, with topolog-ical degeneracy on a torus,and gapped spinon and vi-son excitations with mutualsemionic statistics
N. Read and S. Sachdev, Phys.Rev. Lett. 63, 1773 (1991).(also X.-G. Wen, Phys. Rev. B44, 2664 (1991))
N. Read and S. SachdevPhys. Rev. Lett. 63, 1773 (1991)
C. Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Saturday, May 29, 2010
![Page 31: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/31.jpg)
Z2 spin liquidValence bond solid
(VBS)
Neel antiferromagnet
Spiral antiferromagnet
M
Phase diagram of frustrated antiferromagnets
M
N. Read and S. SachdevPhys. Rev. Lett. 63, 1773 (1991)
C. Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Saturday, May 29, 2010
![Page 32: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/32.jpg)
Z2 spin liquidValence bond solid
(VBS)
Neel antiferromagnet
Spiral antiferromagnet
M
Phase diagram of frustrated antiferromagnets
M
N. Read and S. SachdevPhys. Rev. Lett. 63, 1773 (1991)
C. Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)
Multicritical point M described by
a doubled Chern-Simons theory;
non-supersymmetric analog of the ABJM model
Saturday, May 29, 2010
![Page 33: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/33.jpg)
Phase diagram of J1-J2-J3 antiferromagnet on the square lattice
~1/S
N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)Saturday, May 29, 2010
![Page 34: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/34.jpg)
Phase diagram of J1-J2-J3 antiferromagnet on the square lattice
~1/S
Collinear magnetic
order(Neel)
N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)Saturday, May 29, 2010
![Page 35: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/35.jpg)
Phase diagram of J1-J2-J3 antiferromagnet on the square lattice
~1/S
ValenceBondSolids
N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)Saturday, May 29, 2010
![Page 36: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/36.jpg)
Phase diagram of J1-J2-J3 antiferromagnet on the square lattice
~1/S
N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)Saturday, May 29, 2010
![Page 37: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/37.jpg)
Phase diagram of J1-J2-J3 antiferromagnet on the square lattice
~1/S
Spiralmagnetic
order
N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)Saturday, May 29, 2010
![Page 38: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/38.jpg)
Phase diagram of J1-J2-J3 antiferromagnet on the square lattice
~1/S Z2spin
liquids
N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)Saturday, May 29, 2010
![Page 39: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/39.jpg)
Phase diagram of J1-J2-J3 antiferromagnet on the square lattice
~1/SM
M
N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)Saturday, May 29, 2010
![Page 40: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/40.jpg)
1. Quantum “disordering” magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases
2. Unified theory of spin liquids Majorana liquids
3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order
Outline
Saturday, May 29, 2010
![Page 41: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/41.jpg)
1. Quantum “disordering” magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases
2. Unified theory of spin liquids Majorana liquids
3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order
Outline
Saturday, May 29, 2010
![Page 42: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/42.jpg)
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
Saturday, May 29, 2010
![Page 43: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/43.jpg)
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
� �ψ+
ψ−
�Quantum “disordering” magnetic order
Saturday, May 29, 2010
![Page 44: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/44.jpg)
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
� �ψ+
ψ−
�Quantum “disordering” magnetic order
SU(2)spin
Saturday, May 29, 2010
![Page 45: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/45.jpg)
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
� �ψ+
ψ−
�Quantum “disordering” magnetic order
U(1)charge
Saturday, May 29, 2010
![Page 46: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/46.jpg)
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
�c↑c↓
�=
�z↑ −z∗↓z↓ z∗↑
� �ψ+
ψ−
�Quantum “disordering” magnetic order
U × U−1
SU(2)s;gauge
S. Sachdev, M. A. Metlitski, Y. Qi, and S. Sachdev Phys. Rev. B 80, 155129 (2009)Saturday, May 29, 2010
![Page 47: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/47.jpg)
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
Projected fermion wavefunctions (Fisher, Wen, Lee, Kim)
Saturday, May 29, 2010
![Page 48: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/48.jpg)
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
Projected fermion wavefunctions
neutralfermionicspinons
(Fisher, Wen, Lee, Kim)
Saturday, May 29, 2010
![Page 49: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/49.jpg)
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
Projected fermion wavefunctions (Fisher, Wen, Lee, Kim)
charged slaveboson/rotor
Saturday, May 29, 2010
![Page 50: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/50.jpg)
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
Projected fermion wavefunctions
SU(2)pseudospin
(Fisher, Wen, Lee, Kim)
Saturday, May 29, 2010
![Page 51: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/51.jpg)
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
Projected fermion wavefunctions
SU(2)spin
(Fisher, Wen, Lee, Kim)
Saturday, May 29, 2010
![Page 52: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/52.jpg)
�c↑c†↓
�=
�b∗1 b∗2−b2 b1
� �f1f†2
�
Global symmetry operations:
• Spin rotations, SU(2)spin
• Combine electromagnetic charge (electron number)U(1)charge with particle-hole transformations to ob-tain SU(2)pseudospin.
Projected fermion wavefunctions
U × U−1
SU(2)p;gauge
(Fisher, Wen, Lee, Kim)
Saturday, May 29, 2010
![Page 53: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/53.jpg)
Unified spin liquid theory
Decompose 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
Saturday, May 29, 2010
![Page 54: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/54.jpg)
Unified spin liquid theory
Decompose 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
SO(4) ∼= SU(2)pseudospin×SU(2)spin
Saturday, May 29, 2010
![Page 55: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/55.jpg)
Unified spin liquid theory
Decompose 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
O ×OT
SO(4)gauge∼= SU(2)p;gauge×SU(2)s;gauge
Saturday, May 29, 2010
![Page 56: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/56.jpg)
Unified spin liquid theory
Decompose 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
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
Saturday, May 29, 2010
![Page 57: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/57.jpg)
1. Quantum “disordering” magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases
2. Unified theory of spin liquids Majorana liquids
3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order
Outline
Saturday, May 29, 2010
![Page 58: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/58.jpg)
1. Quantum “disordering” magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases
2. Unified theory of spin liquids Majorana liquids
3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order
Outline
Saturday, May 29, 2010
![Page 59: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/59.jpg)
Fermi surface+antiferromagnetism
Γ
Hole states
occupied
Electron states
occupied
Γ
The electron spin polarization obeys�
�S(r, τ)�
= �ϕ(r, τ)eiK·r
where K is the ordering wavevector.
+
Saturday, May 29, 2010
![Page 60: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/60.jpg)
Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Large Fermi surface breaks up intoelectron and hole pockets
Hole-doped cuprates
Saturday, May 29, 2010
![Page 61: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/61.jpg)
Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Hole-doped cuprates
�ϕ
�ϕ fluctuations act on thelarge Fermi surface
Saturday, May 29, 2010
![Page 62: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/62.jpg)
Start from the “spin-fermion” model
Z =�DcαD�ϕ exp (−S)
S =�
dτ�
k
c†kα
�∂
∂τ− εk
�ckα
− λ
�dτ
�
i
c†iα�ϕi · �σαβciβeiK·ri
+�
dτd2r
�12
(∇r �ϕ)2 +�ζ2
(∂τ �ϕ)2 +s
2�ϕ2 +
u
4�ϕ4
�
Saturday, May 29, 2010
![Page 63: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/63.jpg)
� = 1
� = 2
� = 4
� = 3
Low energy fermionsψ�
1α, ψ�2α
� = 1, . . . , 4
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
v�=11 = (vx, vy), v�=1
2 = (−vx, vy)Saturday, May 29, 2010
![Page 64: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/64.jpg)
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
v1 v2
ψ2 fermionsoccupied
ψ1 fermionsoccupied
Saturday, May 29, 2010
![Page 65: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/65.jpg)
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
v1 v2
“Hot spot”
“Cold” Fermi surfaces
Saturday, May 29, 2010
![Page 66: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/66.jpg)
Order parameter: Lϕ =1
2(∇r �ϕ)
2+
�ζ2
(∂τ �ϕ)2
+s
2�ϕ2
+u
4�ϕ4
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
Saturday, May 29, 2010
![Page 67: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/67.jpg)
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
Order parameter: Lϕ =1
2(∇r �ϕ)
2+
�ζ2
(∂τ �ϕ)2
+s
2�ϕ2
+u
4�ϕ4
“Yukawa” coupling: Lc = −λ�ϕ ·�ψ�†
1α�σαβψ�2β + ψ�†
2α�σαβψ�1β
�
Saturday, May 29, 2010
![Page 68: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/68.jpg)
Hertz theoryIntegrate out fermions and obtain non-local corrections to Lϕ
Lϕ =12
�ϕ2�q2 + γ|ω|
�/2 ; γ =
2πvxvy
Exponent z = 2 and mean-field criticality (upto logarithms)
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
Order parameter: Lϕ =1
2(∇r �ϕ)
2+
�ζ2
(∂τ �ϕ)2
+s
2�ϕ2
+u
4�ϕ4
“Yukawa” coupling: Lc = −λ�ϕ ·�ψ�†
1α�σαβψ�2β + ψ�†
2α�σαβψ�1β
�
Saturday, May 29, 2010
![Page 69: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/69.jpg)
Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).
Integrate out fermions and obtain non-local corrections to Lϕ
Lϕ =12
�ϕ2�q2 + γ|ω|
�/2 ; γ =
2πvxvy
Exponent z = 2 and mean-field criticality (upto logarithms)
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
Order parameter: Lϕ =1
2(∇r �ϕ)
2+
�ζ2
(∂τ �ϕ)2
+s
2�ϕ2
+u
4�ϕ4
“Yukawa” coupling: Lc = −λ�ϕ ·�ψ�†
1α�σαβψ�2β + ψ�†
2α�σαβψ�1β
�
OK in d = 3, but higher order terms contain an
infinite number of marginal couplings in d = 2
Hertz theory
Saturday, May 29, 2010
![Page 70: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/70.jpg)
Perform RG on both fermions and �ϕ,using a local field theory.
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
Order parameter: Lϕ =1
2(∇r �ϕ)
2+
�ζ2
(∂τ �ϕ)2
+s
2�ϕ2
+u
4�ϕ4
“Yukawa” coupling: Lc = −λ�ϕ ·�ψ�†
1α�σαβψ�2β + ψ�†
2α�σαβψ�1β
�
Saturday, May 29, 2010
![Page 71: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/71.jpg)
� = 1
� = 2
� = 4
� = 3
Low energy fermionsψ�
1α, ψ�2α
� = 1, . . . , 4
Saturday, May 29, 2010
![Page 72: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/72.jpg)
Hot spots have
strong instability to
d-wave pairing near
SDW critical point.
This instability is
stronger than the
BCS instability of a
Fermi liquid.
Pairing order parameter:
εαβ�ψ31αψ
11β − ψ3
2αψ12β
�
ψ31
ψ32
Saturday, May 29, 2010
![Page 73: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/73.jpg)
ψ31 ψ3
2 ψ31 ψ3
2 ψ31
d-wave Cooper pairing instability in particle-particle channel
+�ϕ
+- +-
Saturday, May 29, 2010
![Page 74: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/74.jpg)
Similar theory applies to the pnictides, and leads to s± pairing.
kx
ky
1
23
4 3
4
Q = ( !, 0)
!
2
1
Saturday, May 29, 2010
![Page 75: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/75.jpg)
Emergent Pseudospin symmetry
Continuum theory of hotspots in invariant under:
�ψ�↑
ψ�†↓
�→ U �
�ψ�↑
ψ�†↓
�
where U � are arbitrary SU(2) matrices which can bedifferent on different hotspots �.
Saturday, May 29, 2010
![Page 76: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/76.jpg)
ψ31 ψ3
2 ψ31 ψ3
2 ψ31
d-wave Cooper pairing instability in particle-particle channel
+�ϕ
+- +-
Saturday, May 29, 2010
![Page 77: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/77.jpg)
ψ31 ψ3
2 ψ31 ψ3
2 ψ31
Bond density wave (with local Ising-nematic order) instability in particle-hole channel
+�ϕ
+- +-
Saturday, May 29, 2010
![Page 78: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/78.jpg)
d-wave pairing has apartner instabilityin the particle-hole
channel
Density-wave order parameter:
�ψ3†1αψ
11α − ψ3†
2αψ12α
�
ψ31
ψ32
Saturday, May 29, 2010
![Page 79: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/79.jpg)
ψ31
ψ32
Saturday, May 29, 2010
![Page 80: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/80.jpg)
Single ordering wavevector Q:�c†k−Q/2,αck+Q/2,α
�=
Φ(cos kx − cos ky)
Q
Q
ψ31
ψ32
ψ31
ψ32
Saturday, May 29, 2010
![Page 81: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/81.jpg)
!1
"1
No modulations on sites. Modulated bond-densitywave with local Ising-nematic ordering:�
c†k−Q/2,αck+Q/2,α
�= Φ(cos kx − cos ky)
“Bond density” measures amplitude for electrons to be
in spin-singlet valence bond:
VBS order
Saturday, May 29, 2010
![Page 82: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/82.jpg)
!1
"1
No modulations on sites. Modulated bond-densitywave with local Ising-nematic ordering:�
c†k−Q/2,αck+Q/2,α
�= Φ(cos kx − cos ky)
“Bond density” measures amplitude for electrons to be
in spin-singlet valence bond:
VBS order
Saturday, May 29, 2010
![Page 83: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/83.jpg)
M. J. Lawler, K. Fujita,
Jhinhwan Lee,
A. R. Schmidt,
Y. Kohsaka, Chung Koo
Kim, H. Eisaki,
S. Uchida, J. C. Davis,
J. P. Sethna, and
Eun-Ah Kim, preprint
STM measurements of Z(r), the energy asymmetry
in density of states in Bi2Sr2CaCu2O8+δ.
Saturday, May 29, 2010
![Page 84: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/84.jpg)
B
M. J. Lawler, K. Fujita,
Jhinhwan Lee,
A. R. Schmidt,
Y. Kohsaka, Chung Koo
Kim, H. Eisaki,
S. Uchida, J. C. Davis,
J. P. Sethna, and
Eun-Ah Kim, preprint
AC
D
STM measurements of Z(r), the energy asymmetry
in density of states in Bi2Sr2CaCu2O8+δ.
Saturday, May 29, 2010
![Page 85: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/85.jpg)
B
M. J. Lawler, K. Fujita,
Jhinhwan Lee,
A. R. Schmidt,
Y. Kohsaka, Chung Koo
Kim, H. Eisaki,
S. Uchida, J. C. Davis,
J. P. Sethna, and
Eun-Ah Kim, preprint
AC
D
Strong anisotropy of electronic states between
x and y directions:Electronic
“Ising-nematic” order
STM measurements of Z(r), the energy asymmetry
in density of states in Bi2Sr2CaCu2O8+δ.
Saturday, May 29, 2010
![Page 86: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/86.jpg)
Conclusions
Theory for the onset of spin density wave in metals is strongly coupled in two dimensions
For the cuprate Fermi surface, there are strong instabilities near the quantum critical point to
d-wave pairing and
bond density waves with local Ising-nematic ordering
Saturday, May 29, 2010
![Page 87: New Quantum “disordering” magnetic order in insulators, metals, and …qpt.physics.harvard.edu/talks/perimeter10.pdf · 2010. 5. 29. · Quantum “disordering” magnetic order](https://reader033.vdocuments.site/reader033/viewer/2022060602/60562c960631f16a2725cd16/html5/thumbnails/87.jpg)
Conclusions
Quantum “disordering” magnetic order leads to valence bond solids and Z2 spin liquids
Unified theory of spin liquids using Majorana fermions: also includes states obtained by
projecting free fermion determinants
Saturday, May 29, 2010