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Quantum “disordering” magnetic order in insulators, metals, and superconductors HARVARD Talk online: sachdev.physics.harvard.edu Perimeter Institute, Waterloo, May 29, 2010

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HARVARD Max Metlitski, Harvard arXiv:1005.1288 Cenke Xu, Harvard arXiv:1004.5431 Eun Gook Moon, Harvard arXiv:1005.3312

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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

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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

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Ground state has long-range Néel order Square lattice antiferromagnet

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Add perturbations so ground state no longer has long-range Néel order Square lattice antiferromagnet

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Add perturbations so ground state no longer has long-range Néel order Square lattice antiferromagnet

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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

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Metals (in the cuprates) Hole states occupied Electron states occupied

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Fermi surface+antiferromagnetism Hole states occupied Electron states occupied +

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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 Half-filled band

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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 Half-filled band

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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 Hot spots Half-filled band

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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 Fermi surface breaks up at hot spots into electron and hole “pockets” Hole pockets Hot spots Half-filled band

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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). Insulator with Neel order has electrons filling a band, and no Fermi surface Hot spots Insulator Half-filled band

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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). Insulator with Neel order has electrons filling a band, and no Fermi surface Hot spots Insulator Half-filled band

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Square lattice antiferromagnet

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Nature of quantum “disordered” phase

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N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989) Nature of quantum “disordered” phase

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Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) C. Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)

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Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) C. Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009) Quantum “disordering” spiral order leads to a Z 2 spin liquid

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Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) C. Xu and S. Sachdev, Phys. Rev. B 79, 064405 (2009)

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Phase diagram of frustrated antiferromagnets M N. Read and S. Sachdev Phys. 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

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Phase diagram of J 1 -J 2 -J 3 antiferromagnet on the square lattice ~1/S N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

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Phase diagram of J 1 -J 2 -J 3 antiferromagnet on the square lattice ~1/S Collinear magnetic order (Neel) N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

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Phase diagram of J 1 -J 2 -J 3 antiferromagnet on the square lattice ~1/S Valence Bond Solids N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

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Phase diagram of J 1 -J 2 -J 3 antiferromagnet on the square lattice ~1/S N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

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Phase diagram of J 1 -J 2 -J 3 antiferromagnet on the square lattice ~1/S Spiral magnetic order N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

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Phase diagram of J 1 -J 2 -J 3 antiferromagnet on the square lattice ~1/S Z 2 spin liquids N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

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Phase diagram of J 1 -J 2 -J 3 antiferromagnet on the square lattice ~1/S M M N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

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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

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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

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Quantum “disordering” magnetic order

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S. Sachdev, M. A. Metlitski, Y. Qi, and S. Sachdev Phys. Rev. B 80, 155129 (2009)

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Projected fermion wavefunctions (Fisher, Wen, Lee, Kim)

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Projected fermion wavefunctions neutral fermionic spinons (Fisher, Wen, Lee, Kim)

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Projected fermion wavefunctions (Fisher, Wen, Lee, Kim) charged slave boson/rotor

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Projected fermion wavefunctions (Fisher, Wen, Lee, Kim)

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Projected fermion wavefunctions (Fisher, Wen, Lee, Kim)

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Projected fermion wavefunctions (Fisher, Wen, Lee, Kim)

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Unified spin liquid theory

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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

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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

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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

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Fermi surface+antiferromagnetism Hole states occupied Electron states occupied +

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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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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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“Hot spot” “Cold” Fermi surfaces

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Hertz theory

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Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004). Hertz theory

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d -wave Cooper pairing instability in particle-particle channel + + - + -

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Emergent Pseudospin symmetry

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d -wave Cooper pairing instability in particle-particle channel + + - + -

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Bond density wave (with local Ising-nematic order) instability in particle-hole channel + + - + -

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78 “Bond density” measures amplitude for electrons to be in spin-singlet valence bond: VBS order

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79 “Bond density” measures amplitude for electrons to be in spin-singlet valence bond: VBS order

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BA C D

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BA C D Strong anisotropy of electronic states between x and y directions: Electronic “Ising-nematic” order

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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

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Conclusions Quantum “disordering” magnetic order leads to valence bond solids and Z 2 spin liquids Unified theory of spin liquids using Majorana fermions: also includes states obtained by projecting free fermion determinants

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