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![Page 1: Quantum criticality of Fermi surfaces in two dimensions HARVARD Talk online: sachdev.physics.harvard.edu.](https://reader035.fdocuments.net/reader035/viewer/2022062806/56649f535503460f94c78834/html5/thumbnails/1.jpg)
Quantum criticality of Fermi surfaces in two
dimensions
HARVARDTalk online: sachdev.physics.harvard.edu
![Page 2: Quantum criticality of Fermi surfaces in two dimensions HARVARD Talk online: sachdev.physics.harvard.edu.](https://reader035.fdocuments.net/reader035/viewer/2022062806/56649f535503460f94c78834/html5/thumbnails/2.jpg)
Yejin Huh, Harvard
Max Metlitski, Harvard
Yejin Huh, Harvard
Max Metlitski, Harvard
HARVARD
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1. Quantum criticality of Fermi points: Dirac fermions in d-wave superconductors
2. Quantum criticality of Fermi surfaces: Onset of spin density wave order in the cuprates
Outline
![Page 4: Quantum criticality of Fermi surfaces in two dimensions HARVARD Talk online: sachdev.physics.harvard.edu.](https://reader035.fdocuments.net/reader035/viewer/2022062806/56649f535503460f94c78834/html5/thumbnails/4.jpg)
1. Quantum criticality of Fermi points: Dirac fermions in d-wave superconductors
2. Quantum criticality of Fermi surfaces: Onset of spin density wave order in the cuprates
Outline
![Page 5: Quantum criticality of Fermi surfaces in two dimensions HARVARD Talk online: sachdev.physics.harvard.edu.](https://reader035.fdocuments.net/reader035/viewer/2022062806/56649f535503460f94c78834/html5/thumbnails/5.jpg)
The cuprate superconductors
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Ground state has long-range Néel order
Square lattice antiferromagnet
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Central ingredients in cuprate phase diagram: antiferromagnetism,
superconductivity, and change in Fermi surface
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d-wave superconductivity in cuprates
Hole states
occupied
Electron states
occupied
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d-wave superconductivity in cuprates
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d-wave superconductivity in cuprates
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d-wave superconductivity in cuprates
4 two-component Dirac fermions
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Nematic order in YBCO
V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer , Science 319, 597 (2008)
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Broken rotational symmetry in the pseudogap phase of a high-Tc superconductor
R. Daou, J. Chang, David LeBoeuf, Olivier Cyr-Choiniere, Francis Laliberte, Nicolas Doiron-Leyraud, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis TailleferarXiv: 0909.4430
S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature 393, 550 (1998).
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d-wave superconductivity in cuprates
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d-wave superconductivity in cuprates
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Lattice rotation symmetry breaking
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Time-reversal symmetry breaking
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M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (2000)E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson,
Phys. Rev. B 77, 184514 (2008).
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Discrete symmetry breaking in d-wave superconductors
4 two-component Dirac fermions
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Discrete symmetry breaking in d-wave superconductors
4 two-component Dirac fermions
Ising field theory
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M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)
Ising order and Dirac fermions couple via a “Yukawa” term.
Nematic ordering
Time reversal symmetry breaking
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M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)
Ising order and Dirac fermions couple via a “Yukawa” term.
Nematic ordering
Time reversal symmetry breaking
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Integrating out the fermions yields an effective action for the scalar order
parameter
Expansion in number of fermion spin
components Nf
Y. Huh and S. Sachdev, Physical Review B 78, 064512 (2008).
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Integrating out the fermions yields an effective action for the nematic order
parameter
Expansion in number of fermion spin
components Nf
E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson, arXiv:0705.4099
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Integrating out the fermions yields an effective action for the T-breaking order
parameter
Expansion in number of fermion spin
components Nf
E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson, arXiv:0705.4099
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Integrating out the fermions yields an effective action for the scalar order
parameter
Expansion in number of fermion spin
components Nf
Y. Huh and S. Sachdev, Physical Review B 78, 064512 (2008).
![Page 27: Quantum criticality of Fermi surfaces in two dimensions HARVARD Talk online: sachdev.physics.harvard.edu.](https://reader035.fdocuments.net/reader035/viewer/2022062806/56649f535503460f94c78834/html5/thumbnails/27.jpg)
1. Quantum criticality of Fermi points: Dirac fermions in d-wave superconductors
2. Quantum criticality of Fermi surfaces: Onset of spin density wave order in the cuprates
Outline
![Page 28: Quantum criticality of Fermi surfaces in two dimensions HARVARD Talk online: sachdev.physics.harvard.edu.](https://reader035.fdocuments.net/reader035/viewer/2022062806/56649f535503460f94c78834/html5/thumbnails/28.jpg)
1. Quantum criticality of Fermi points: Dirac fermions in d-wave superconductors
2. Quantum criticality of Fermi surfaces: Onset of spin density wave order in the cuprates
Outline
![Page 29: Quantum criticality of Fermi surfaces in two dimensions HARVARD Talk online: sachdev.physics.harvard.edu.](https://reader035.fdocuments.net/reader035/viewer/2022062806/56649f535503460f94c78834/html5/thumbnails/29.jpg)
“Large” Fermi surfaces in cupratesHole
states occupied
Electron states
occupied
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Spin density wave theory
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Spin density wave theory
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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).
Hole pockets
Electron pockets
Hole-doped cuprates
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Hole pockets
Electron pockets
Hole-doped cuprates
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).
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Spin density wave theory in hole-doped cuprates
A. J. Millis and M. R. Norman, Physical Review B 76, 220503 (2007). N. Harrison, Physical Review Letters 102, 206405 (2009).
Incommensurate order in YBa2Cu3O6+x
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Electron pockets
Hole pockets
Electron-doped cuprates
D. Senechal and A.-M. S. Tremblay, Physical Review Letters 92, 126401 (2004)J. Lin, and A. J. Millis, Physical Review B 72, 214506 (2005).
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T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J.
Wosnitza, and R. Gross,
Phys. Rev. Lett. 103, 157002 (2009).
Quantum oscillations
Quantum oscillations
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Nature 450, 533 (2007)
Quantum oscillations
Quantum oscillations
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Nature 450, 533 (2007)
Quantum oscillations
Quantum oscillations
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Theory of quantum criticality in the cuprates
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Evidence for connection between linear resistivity and
stripe-ordering in a cuprate with a low Tc
Linear temperature dependence of resistivity and change in the Fermi surface at the pseudogap critical point of a high-Tc superconductorR. Daou, Nicolas Doiron-Leyraud, David LeBoeuf, S. Y. Li, Francis Laliberté, Olivier Cyr-Choinière, Y. J. Jo, L. Balicas, J.-Q. Yan, J.-S. Zhou, J. B. Goodenough & Louis Taillefer, Nature Physics 5, 31 - 34 (2009)
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Theory of quantum criticality in the cuprates
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Theory of quantum criticality in the cuprates
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Theory of quantum criticality in the cuprates
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Theory of quantum criticality in the cuprates
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Theory of quantum criticality in the cuprates
Criticality of the coupled dimer antiferromagnet at
x=xs
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Theory of quantum criticality in the cuprates
Criticality of the topological change in Fermi surface at x=xm
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Hc2
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Hc2
Quantum oscillations
Quantum oscillations
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Hsdw
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Hsdw
Neutron scattering & muon resonance
Neutron scattering & muon resonance
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V. Galitski and S. Sachdev, Physical Review B 79, 134512 (2009).
Eun Gook Moon and S. Sachdev, Physical Review B 80, 035117 (2009).
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G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223
Similar phase diagram for CeRhIn5
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Hertz-Moriya-Millis (HMM) theory
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Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).
Hertz-Moriya-Millis (HMM) theory
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Max Metlitski
M. Metlitski and S. Sachdev, to appear
Ar. Abanov, A.V. Chubukov, and J. Schmalian, Advances in Physics 52, 119 (2003)
Sung-Sik Lee, arXiv:0905.4532.
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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).
Hole pockets
Electron pockets
Hole-doped cuprates
![Page 70: Quantum criticality of Fermi surfaces in two dimensions HARVARD Talk online: sachdev.physics.harvard.edu.](https://reader035.fdocuments.net/reader035/viewer/2022062806/56649f535503460f94c78834/html5/thumbnails/70.jpg)
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).
Hole pockets
Electron pockets
Hole-doped cuprates
![Page 71: Quantum criticality of Fermi surfaces in two dimensions HARVARD Talk online: sachdev.physics.harvard.edu.](https://reader035.fdocuments.net/reader035/viewer/2022062806/56649f535503460f94c78834/html5/thumbnails/71.jpg)
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Y. Huh and S. Sachdev, Phys. Rev. B 78, 064512 (2008).
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RG-improved Migdal-Eliashberg theory
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RG-improved Migdal-Eliashberg theory
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RG-improved Migdal-Eliashberg theory
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Dynamical Nesting
RG-improved Migdal-Eliashberg theory
Bare Fermi surface
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Dynamical Nesting
RG-improved Migdal-Eliashberg theory
Dressed Fermi surface
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Dynamical Nesting
RG-improved Migdal-Eliashberg theory
Bare Fermi surface
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Dynamical Nesting
RG-improved Migdal-Eliashberg theory
Dressed Fermi surface
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RG-improved Migdal-Eliashberg theory
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Dangerous
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Double line representation• A way to compute the order of a diagram.
• Extra powers of N come from the Fermi-surface
• What are the conditions for all propagators to be on the Fermi surface?
• Concentrate on diagrams involving a single pair of hot-spots
• Any bosonic momentum may be (uniquely) written as
R. Shankar, Rev. Mod. Phys. 66, 129 (1994).S. W. Tsai, A. H. Castro Neto, R. Shankar, and D. K. Campbell, Phys. Rev. B 72, 054531 (2005).
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=
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Graph is planar after turning fermion propagators also into double lines by drawing additional dotted single line loops
for each fermion loop
=
Sung-Sik Lee, arXiv:0905.4532
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A consistent analysis requires resummation of all planar
graphs
=
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Theory for the onset of spin density wave
order in metals is strongly coupled
in two dimensions
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Naturally formulated in
route B: theory of
fluctuating Fermi pockets
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R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, 045110 (2008).
VBSand/or
nematicOnset of superconductiv
ity induces confinement