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The SYK model HARVARD Subir Sachdev Aspen Center for Physics August 13, 2018

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  • The SYK model

    HARVARD

    Subir Sachdev

    Aspen Center for PhysicsAugust 13, 2018

  • • Note: The electron liquid in one dimension and the fractionalquantum Hall state both have quasiparticles; however, the quasi-

    particles do not have the same quantum numbers as an electron.

    What are quasiparticles ?

    • Quasiparticles are additive excitations:The low-lying excitations of the many-body systemcan be identified as a set {n↵} of quasiparticles withenergy "↵

    E =P

    ↵ n↵"↵ +P

    ↵,� F↵�n↵n� + . . .

    In a lattice system ofN sites, this parameterizes the energyof ⇠ e↵N states in terms of poly(N) numbers.

  • • Quasiparticles eventually collide with each other. Suchcollisions eventually leads to thermal equilibration in achaotic quantum state, but the equilibration takes a longtime. In a Fermi liquid, this time diverges as

    ⌧eq ⇠~EF

    (kBT )2, as T ! 0,

    where EF is the Fermi energy.

    What are quasiparticles ?

  • • Quasiparticles eventually collide with each other. Suchcollisions eventually leads to thermal equilibration in achaotic quantum state, but the equilibration takes a longtime. In a Fermi liquid, this time diverges as

    ⌧eq ⇠~EF

    (kBT )2, as T ! 0,

    where EF is the Fermi energy.

    What are quasiparticles ?

    • This time is much longer than the ‘Planckian time’ ~/(kBT ),which we will find in systems without quasiparticle ex-citations.

    ⌧eq �~

    kBT, as T ! 0.

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  • 1. Random matrix quasiparticle model q=2, complex SYK

    2. Matter without quasiparticles q=4, complex SYK

    3. The Schwarzian theory

    4.Connections to black holes with AdS2 horizons

  • 1. Random matrix quasiparticle model q=2, complex SYK

    2. Matter without quasiparticles q=4, complex SYK

    3. The Schwarzian theory

    4.Connections to black holes with AdS2 horizons

  • A simple model of a metal with quasiparticles

    Pick a set of random positions

  • Place electrons randomly on some sites

    A simple model of a metal with quasiparticles

  • Place electrons randomly on some sites

    A simple model of a metal with quasiparticles

  • Electrons move one-by-one randomly

    A simple model of a metal with quasiparticles

  • Electrons move one-by-one randomly

    A simple model of a metal with quasiparticles

  • Electrons move one-by-one randomly

    A simple model of a metal with quasiparticles

  • Electrons move one-by-one randomly

    A simple model of a metal with quasiparticles

  • Electrons move one-by-one randomly

    A simple model of a metal with quasiparticles

  • Electrons move one-by-one randomly

    A simple model of a metal with quasiparticles

  • Electrons move one-by-one randomly

    A simple model of a metal with quasiparticles

  • Fermions occupying the eigenstates of a N x N random matrix

    tij are independent random variables with tij = 0 and |tij |2 = t2

    A simple model of a metal with quasiparticles

    H =1

    (N)1/2

    NX

    i,j=1

    tijc†i cj � µ

    X

    i

    c†i ci

    cicj + cjci = 0 , cic†j + c

    †jci = �ij

    1

    N

    X

    i

    c†i ci = Q

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  • A simple model of a metal with quasiparticles

    ⇢(!) =� 1⇡ ImG(!)

    AAACUnicdZJNbxMxEIad8FVCgVCOXCxSpCCVyBtlaS9IFT0AtyKRtlIcRV7vbGLVHyt7tm202h/Er+HCgf4TxAknDRIgGMnSq3lnNOPHzkqtAjJ23Wrfun3n7r2t+50H2w8fPe4+2TkJrvISxtJp588yEUArC2NUqOGs9CBMpuE0Oz9a+acX4INy9hMuS5gaMbeqUFJgTM26R9w6ZXOw2NnlfuFon3JnYC5e0jec01e88ELWSV PzUjWUm8xd1R9Mw/fou/6mcHfW7bEB209HrxPKBgljB2kaRTpK03RIkwFbR49s4njW/c5zJysTp0otQpgkrMRpLTwqqaHp8CpAKeS5mMMkSisMhL38QpVhLaf1+t4NfRHNnBbOx2ORrrO/N9fChLA0Waw0Ahfhb2+V/Jc3qbA4mNbKlhWClTeDikpTdHQFkebKg0S9jEJIr+LaVC5EJIURdYcHiO9g57ioOcIVXqo8zqlHyjYR1S8e9P/iZBgpDpKPw97h2w20LfKMPCd9kpB9ckjek2MyJpJ8Jl/IN3Ld+tr60Y6/5Ka03dr0PCV/RHv7JzrOtBs=AAACUnicdZJNbxMxEIad8FVCgVCOXCxSpCCVyBtlaS9IFT0AtyKRtlIcRV7vbGLVHyt7tm202h/Er+HCgf4TxAknDRIgGMnSq3lnNOPHzkqtAjJ23Wrfun3n7r2t+50H2w8fPe4+2TkJrvISxtJp588yEUArC2NUqOGs9CBMpuE0Oz9a+acX4INy9hMuS5gaMbeqUFJgTM26R9w6ZXOw2NnlfuFon3JnYC5e0jec01e88ELWSV PzUjWUm8xd1R9Mw/fou/6mcHfW7bEB209HrxPKBgljB2kaRTpK03RIkwFbR49s4njW/c5zJysTp0otQpgkrMRpLTwqqaHp8CpAKeS5mMMkSisMhL38QpVhLaf1+t4NfRHNnBbOx2ORrrO/N9fChLA0Waw0Ahfhb2+V/Jc3qbA4mNbKlhWClTeDikpTdHQFkebKg0S9jEJIr+LaVC5EJIURdYcHiO9g57ioOcIVXqo8zqlHyjYR1S8e9P/iZBgpDpKPw97h2w20LfKMPCd9kpB9ckjek2MyJpJ8Jl/IN3Ld+tr60Y6/5Ka03dr0PCV/RHv7JzrOtBs=AAACUnicdZJNbxMxEIad8FVCgVCOXCxSpCCVyBtlaS9IFT0AtyKRtlIcRV7vbGLVHyt7tm202h/Er+HCgf4TxAknDRIgGMnSq3lnNOPHzkqtAjJ23Wrfun3n7r2t+50H2w8fPe4+2TkJrvISxtJp588yEUArC2NUqOGs9CBMpuE0Oz9a+acX4INy9hMuS5gaMbeqUFJgTM26R9w6ZXOw2NnlfuFon3JnYC5e0jec01e88ELWSV PzUjWUm8xd1R9Mw/fou/6mcHfW7bEB209HrxPKBgljB2kaRTpK03RIkwFbR49s4njW/c5zJysTp0otQpgkrMRpLTwqqaHp8CpAKeS5mMMkSisMhL38QpVhLaf1+t4NfRHNnBbOx2ORrrO/N9fChLA0Waw0Ahfhb2+V/Jc3qbA4mNbKlhWClTeDikpTdHQFkebKg0S9jEJIr+LaVC5EJIURdYcHiO9g57ioOcIVXqo8zqlHyjYR1S8e9P/iZBgpDpKPw97h2w20LfKMPCd9kpB9ckjek2MyJpJ8Jl/IN3Ld+tr60Y6/5Ka03dr0PCV/RHv7JzrOtBs=AAACUnicdZJNbxMxEIad8FVCgVCOXCxSpCCVyBtlaS9IFT0AtyKRtlIcRV7vbGLVHyt7tm202h/Er+HCgf4TxAknDRIgGMnSq3lnNOPHzkqtAjJ23Wrfun3n7r2t+50H2w8fPe4+2TkJrvISxtJp588yEUArC2NUqOGs9CBMpuE0Oz9a+acX4INy9hMuS5gaMbeqUFJgTM26R9w6ZXOw2NnlfuFon3JnYC5e0jec01e88ELWSV PzUjWUm8xd1R9Mw/fou/6mcHfW7bEB209HrxPKBgljB2kaRTpK03RIkwFbR49s4njW/c5zJysTp0otQpgkrMRpLTwqqaHp8CpAKeS5mMMkSisMhL38QpVhLaf1+t4NfRHNnBbOx2ORrrO/N9fChLA0Waw0Ahfhb2+V/Jc3qbA4mNbKlhWClTeDikpTdHQFkebKg0S9jEJIr+LaVC5EJIURdYcHiO9g57ioOcIVXqo8zqlHyjYR1S8e9P/iZBgpDpKPw97h2w20LfKMPCd9kpB9ckjek2MyJpJ8Jl/IN3Ld+tr60Y6/5Ka03dr0PCV/RHv7JzrOtBs=

    �2t� µ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

    2t� µ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0

    AAACGnicdZDPattAEMZXSZq4bps4ybGXJU6hh2JWxop9NO2lRwdqO2CJsFqN7SWrlbI7cmKEn6OXHtpX6S302kveJMes/xSS0n4w8DHfDDP84lxJi4zde1vbOy929yovq69ev9k/qB0eDWxWGAF9kanMXMTcgpIa+ihRwUVugKexgmF89WmZD2dgrMz0F5znEKV8ouVYCo6uFYU6kzoBjdVTdnpZq7MGawetM5+yhs9YJwicCVpBEDSp32Ar1clGvcvaQ5hkokjdvlDc2pHPcoxKblAKBYtqWFjIubjiExg5q3kK9kMyk7ld2ahc/b+g71yY0HFmXGmkq+7T5ZKn1s7T2E2mHKf272zZ/Fc2KnDciUqp8wJBi/WhcaEoZnQJgybSgEA1d4YLI93bVEy54QIdsmpowfHUE5yWIcIt3sjE3SlbUi8cqj886P/NoOkoNvzzZr37cQOtQt6SE/Ke+KRNuuQz6ZE+EeSafCXfyQ/vm/fTu/N+rUe3vM3OMXkm7/cjQoqiKw==AAACGnicdZDPattAEMZXSZq4bps4ybGXJU6hh2JWxop9NO2lRwdqO2CJsFqN7SWrlbI7cmKEn6OXHtpX6S302kveJMes/xSS0n4w8DHfDDP84lxJi4zde1vbOy929yovq69ev9k/qB0eDWxWGAF9kanMXMTcgpIa+ihRwUVugKexgmF89WmZD2dgrMz0F5znEKV8ouVYCo6uFYU6kzoBjdVTdnpZq7MGawetM5+yhs9YJwicCVpBEDSp32Ar1clGvcvaQ5hkokjdvlDc2pHPcoxKblAKBYtqWFjIubjiExg5q3kK9kMyk7ld2ahc/b+g71yY0HFmXGmkq+7T5ZKn1s7T2E2mHKf272zZ/Fc2KnDciUqp8wJBi/WhcaEoZnQJgybSgEA1d4YLI93bVEy54QIdsmpowfHUE5yWIcIt3sjE3SlbUi8cqj886P/NoOkoNvzzZr37cQOtQt6SE/Ke+KRNuuQz6ZE+EeSafCXfyQ/vm/fTu/N+rUe3vM3OMXkm7/cjQoqiKw==AAACGnicdZDPattAEMZXSZq4bps4ybGXJU6hh2JWxop9NO2lRwdqO2CJsFqN7SWrlbI7cmKEn6OXHtpX6S302kveJMes/xSS0n4w8DHfDDP84lxJi4zde1vbOy929yovq69ev9k/qB0eDWxWGAF9kanMXMTcgpIa+ihRwUVugKexgmF89WmZD2dgrMz0F5znEKV8ouVYCo6uFYU6kzoBjdVTdnpZq7MGawetM5+yhs9YJwicCVpBEDSp32Ar1clGvcvaQ5hkokjdvlDc2pHPcoxKblAKBYtqWFjIubjiExg5q3kK9kMyk7ld2ahc/b+g71yY0HFmXGmkq+7T5ZKn1s7T2E2mHKf272zZ/Fc2KnDciUqp8wJBi/WhcaEoZnQJgybSgEA1d4YLI93bVEy54QIdsmpowfHUE5yWIcIt3sjE3SlbUi8cqj886P/NoOkoNvzzZr37cQOtQt6SE/Ke+KRNuuQz6ZE+EeSafCXfyQ/vm/fTu/N+rUe3vM3OMXkm7/cjQoqiKw==AAACGnicdZDPattAEMZXSZq4bps4ybGXJU6hh2JWxop9NO2lRwdqO2CJsFqN7SWrlbI7cmKEn6OXHtpX6S302kveJMes/xSS0n4w8DHfDDP84lxJi4zde1vbOy929yovq69ev9k/qB0eDWxWGAF9kanMXMTcgpIa+ihRwUVugKexgmF89WmZD2dgrMz0F5znEKV8ouVYCo6uFYU6kzoBjdVTdnpZq7MGawetM5+yhs9YJwicCVpBEDSp32Ar1clGvcvaQ5hkokjdvlDc2pHPcoxKblAKBYtqWFjIubjiExg5q3kK9kMyk7ld2ahc/b+g71yY0HFmXGmkq+7T5ZKn1s7T2E2mHKf272zZ/Fc2KnDciUqp8wJBi/WhcaEoZnQJgybSgEA1d4YLI93bVEy54QIdsmpowfHUE5yWIcIt3sjE3SlbUi8cqj886P/NoOkoNvzzZr37cQOtQt6SE/Ke+KRNuuQz6ZE+EeSafCXfyQ/vm/fTu/N+rUe3vM3OMXkm7/cjQoqiKw== !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

    Feynman graph expansion in tij.., and graph-by-graph average,yields exact equations in the large N limit:

    G(⌧) ⌘ �T⌧Dci(⌧)c

    †i (0)

    E

    G(i!) =1

    i! + µ� ⌃(i!) , ⌃(⌧) = t2G(⌧)

    G(⌧ = 0�) = Q.

    G(!) can be determined by solving a quadratic equation.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

  • A simple model of a metal with quasiparticles

    "↵ levelspacing ⇠ 1/N

    ⇢(!) =� 1⇡ ImG(!)

    AAACUnicdZJNbxMxEIad8FVCgVCOXCxSpCCVyBtlaS9IFT0AtyKRtlIcRV7vbGLVHyt7tm202h/Er+HCgf4TxAknDRIgGMnSq3lnNOPHzkqtAjJ23Wrfun3n7r2t+50H2w8fPe4+2TkJrvISxtJp588yEUArC2NUqOGs9CBMpuE0Oz9a+acX4INy9hMuS5gaMbeqUFJgTM26R9w6ZXOw2NnlfuFon3JnYC5e0jec01e88ELWSVPzUjWUm8xd1R9Mw/fou/6mcHfW7bEB209HrxPKBgljB2kaRTpK03RIkwFbR49s4njW/c5zJysTp0otQpgkrMRpLTwqqaHp8CpAKeS5mMMkSisMhL38QpVhLaf1+t4NfRHNnBbOx2ORrrO/N9fChLA0Waw0Ahfhb2+V/Jc3qbA4mNbKlhWClTeDikpTdHQFkebKg0S9jEJIr+LaVC5EJIURdYcHiO9g57ioOcIVXqo8zqlHyjYR1S8e9P/iZBgpDpKPw97h2w20LfKMPCd9kpB9ckjek2MyJpJ8Jl/IN3Ld+tr60Y6/5Ka03dr0PCV/RHv7JzrOtBs=AAACUnicdZJNbxMxEIad8FVCgVCOXCxSpCCVyBtlaS9IFT0AtyKRtlIcRV7vbGLVHyt7tm202h/Er+HCgf4TxAknDRIgGMnSq3lnNOPHzkqtAjJ23Wrfun3n7r2t+50H2w8fPe4+2TkJrvISxtJp588yEUArC2NUqOGs9CBMpuE0Oz9a+acX4INy9hMuS5gaMbeqUFJgTM26R9w6ZXOw2NnlfuFon3JnYC5e0jec01e88ELWSVPzUjWUm8xd1R9Mw/fou/6mcHfW7bEB209HrxPKBgljB2kaRTpK03RIkwFbR49s4njW/c5zJysTp0otQpgkrMRpLTwqqaHp8CpAKeS5mMMkSisMhL38QpVhLaf1+t4NfRHNnBbOx2ORrrO/N9fChLA0Waw0Ahfhb2+V/Jc3qbA4mNbKlhWClTeDikpTdHQFkebKg0S9jEJIr+LaVC5EJIURdYcHiO9g57ioOcIVXqo8zqlHyjYR1S8e9P/iZBgpDpKPw97h2w20LfKMPCd9kpB9ckjek2MyJpJ8Jl/IN3Ld+tr60Y6/5Ka03dr0PCV/RHv7JzrOtBs=AAACUnicdZJNbxMxEIad8FVCgVCOXCxSpCCVyBtlaS9IFT0AtyKRtlIcRV7vbGLVHyt7tm202h/Er+HCgf4TxAknDRIgGMnSq3lnNOPHzkqtAjJ23Wrfun3n7r2t+50H2w8fPe4+2TkJrvISxtJp588yEUArC2NUqOGs9CBMpuE0Oz9a+acX4INy9hMuS5gaMbeqUFJgTM26R9w6ZXOw2NnlfuFon3JnYC5e0jec01e88ELWSVPzUjWUm8xd1R9Mw/fou/6mcHfW7bEB209HrxPKBgljB2kaRTpK03RIkwFbR49s4njW/c5zJysTp0otQpgkrMRpLTwqqaHp8CpAKeS5mMMkSisMhL38QpVhLaf1+t4NfRHNnBbOx2ORrrO/N9fChLA0Waw0Ahfhb2+V/Jc3qbA4mNbKlhWClTeDikpTdHQFkebKg0S9jEJIr+LaVC5EJIURdYcHiO9g57ioOcIVXqo8zqlHyjYR1S8e9P/iZBgpDpKPw97h2w20LfKMPCd9kpB9ckjek2MyJpJ8Jl/IN3Ld+tr60Y6/5Ka03dr0PCV/RHv7JzrOtBs=AAACUnicdZJNbxMxEIad8FVCgVCOXCxSpCCVyBtlaS9IFT0AtyKRtlIcRV7vbGLVHyt7tm202h/Er+HCgf4TxAknDRIgGMnSq3lnNOPHzkqtAjJ23Wrfun3n7r2t+50H2w8fPe4+2TkJrvISxtJp588yEUArC2NUqOGs9CBMpuE0Oz9a+acX4INy9hMuS5gaMbeqUFJgTM26R9w6ZXOw2NnlfuFon3JnYC5e0jec01e88ELWSVPzUjWUm8xd1R9Mw/fou/6mcHfW7bEB209HrxPKBgljB2kaRTpK03RIkwFbR49s4njW/c5zJysTp0otQpgkrMRpLTwqqaHp8CpAKeS5mMMkSisMhL38QpVhLaf1+t4NfRHNnBbOx2ORrrO/N9fChLA0Waw0Ahfhb2+V/Jc3qbA4mNbKlhWClTeDikpTdHQFkebKg0S9jEJIr+LaVC5EJIURdYcHiO9g57ioOcIVXqo8zqlHyjYR1S8e9P/iZBgpDpKPw97h2w20LfKMPCd9kpB9ckjek2MyJpJ8Jl/IN3Ld+tr60Y6/5Ka03dr0PCV/RHv7JzrOtBs=

    Let "↵ be the eigenvalues of the matrix tij/pN .

    The fermions will occupy the lowest NQ eigen-values, upto the Fermi energy EF . The single-particle density of states is⇢(!) = (1/N)

    P↵ �(! � "↵), and ⇢0 ⌘ ⇢(! = 0).

    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    �2t� µ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

    2t� µ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0

    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 !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

  • A simple model of a metal with quasiparticles

    Quasiparticleexcitations withspacing ⇠ 1/N

    There are 2N manybody levels with energy

    E =NX

    ↵=1

    n↵"↵,

    where n↵ = 0, 1. Shownare all values of E for asingle cluster of size

    N = 12. The "↵ have alevel spacing ⇠ 1/N .

    Many-bodylevel spacing

    ⇠ 2�N⇠ NtAAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=

  • A simple model of a metal with quasiparticles

    The grand potential ⌦(T ) at low T is (from the Sommerfeld expansion)

    ⌦(T )� E0 = N

    ✓�⇡2

    6⇢0T

    2 +O(T 4)

    ◆+ . . .

    where ⇢0 ⌘ ⇢(0) is the single particle density of states at the Fermi level.We can also define the many body density of states, D(E), via

    Z = e�⌦(T )/T =

    Z 1

    �1dED(E)e�E/T

    The inversion from ⌦(T ) to D(E) has to performed with care (it need not commutewith the 1/N expansion), and we obtain

    D(E) ⇠ exp

    r2N⇢0(E � E0)

    3

    !, E > E0 ,

    1

    N⌧ ⇢0(E � E0) ⌧ N

    andD(E) = 0 for E < E0. This is related to the asymptotic growth of the partitionsof an integer, p(n) ⇠ exp(⇡

    p2n/3). Near the lower bound, there are large sample-

    to-sample fluctuations due to variations in the lowest quasiparticle energies.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

  • Fermi liquid state: Two-body interactions lead to a scattering timeof quasiparticle excitations from in (random) single-particle eigen-states which diverges as ⇠ T�2 at the Fermi level.

    A simple model of a metal with quasiparticlesNow add weak interactions

    H =1

    (N)1/2

    NX

    i,j=1

    tijc†i cj � µ

    X

    i

    c†i ci +

    1

    (2N)3/2

    NX

    i,j,k,`=1

    Uij;k` c†i c

    †jckc`

    Uij;k` are independent random variables with Uij;k` = 0 and |Uij;k`|2 = U2. Wecompute the lifetime of a quasiparticle, ⌧↵, in an exact eigenstate ↵(i) of the

    free particle Hamitonian with energy "↵. By Fermi’s Golden rule, for "↵ at the

    Fermi energy

    1

    ⌧↵= ⇡U

    2⇢20

    Zd"�d"�d"�f("�)(1� f("�))(1� f("�))�("↵ + "� � "� � "�)

    =⇡3U

    2⇢20

    4T

    2

    where ⇢0 is the density of states at the Fermi energy, and f(✏) = 1/(e✏/T

    +1) is

    the Fermi function.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AAAB5HicbZDNSgMxFIXv1L86Vq1rN8EiuJCScaNLwY3LCvYH6lAymUwbmskMyZ1CGfoCbl25E9/Khe9iOu1CWw8EPs5JyL0nypW0SOmXV9vZ3ds/qB/6Rw3/+OS02ejZrDBcdHmmMjOImBVKatFFiUoMciNYGinRj6YPy7w/E8bKTD/jPBdhysZaJpIzdFZn1GzRNq1EtiFYQwvWGjW/X+KMF6nQyBWzdhjQHMOSGZRciYX/UliRMz5lYzF0qFkq7HU8k7mtMCyriRfk0oUxSTLjjkZSub8flyy1dp5G7mbKcGI3s6X5XzYsMLkLS6nzAoXmq4+SQhHMyHJ9EksjOKq5A8aNdGMTPmGGcXQl+a6OYHP5bejdtAPaDp4o1OEcLuAKAriFe3iEDnSBQwyv8OZZ7937WNVW89b9ncEfeZ8/wvGO1Q==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

  • 1. Random matrix quasiparticle model q=2, complex SYK

    2. Matter without quasiparticles q=4, complex SYK

    3. The Schwarzian theory

    4.Connections to black holes with AdS2 horizons

  • The Sachdev-Ye-Kitaev (SYK) model

    Pick a set of random positions

  • Place electrons randomly on some sites

    The SYK model

  • Place electrons randomly on some sites

    The SYK model

  • Place electrons randomly on some sites

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • Entangle electrons pairwise randomly

    The SYK model

  • The SYK model

    This describes both a strange metal and a black hole!

  • A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015)S. Sachdev and J. Ye, PRL 70, 3339 (1993)

    (See also: the “2-Body Random Ensemble” in nuclear physics; did not obtain the large N limit;T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, and S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981))

    The SYK model

    H =1

    (2N)3/2

    NX

    i,j,k,`=1

    Uij;k` c†i c

    †jckc` � µ

    X

    i

    c†i ci

    cicj + cjci = 0 , cic†j + c

    †jci = �ij

    Q = 1N

    X

    i

    c†i ci

    Uij;k` are independent random variables with Uij;k` = 0 and |Uij;k`|2 = U2N ! 1 yields critical strange metal.

  • The SYK model

    U U⌃ =AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==

    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

    S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

    Feynman graph expansion in Uijk`, and graph-by-graph average, yields ex-act equations in the large N limit:

    G(i!) =1

    i! + µ� ⌃(i!) , ⌃(⌧) = �U2G2(⌧)G(�⌧)

    G(⌧ = 0�) = Q.

    Low frequency analysis shows that the solutions must be gapless and obey

    ⌃(z) = µ� ei(⇡/4+✓)

    A

    pz + . . . , G(z) =

    Ae�i(⇡/4+✓)pz

    where A = (⇡/U2 cos(2✓))1/4. The value of ✓ is universally related to Q bya Luttinger-Ward functional analysis similar to that used to establish theLuttinger theorem of Fermi liquid theory:

    Q = 12� ✓

    ⇡� sin(2✓)

    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

  • S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

    The SYK model

    X

    A. Georges, O. Parcollet, and S. Sachdev, PRB 63, 134406 (2001)

    Feynman graph expansion in Uijk`, and graph-by-graph average, yields ex-act equations in the large N limit:

    G(i!) =1

    i! + µ� ⌃(i!) , ⌃(⌧) = �U2G2(⌧)G(�⌧)

    G(⌧ = 0�) = Q.

    Low frequency analysis shows that the solutions must be gapless and obey

    ⌃(z) = µ� ei(⇡/4+✓)

    A

    pz + . . . , G(z) =

    Ae�i(⇡/4+✓)pz

    where A = (⇡/U2 cos(2✓))1/4. The value of ✓ is universally related to Q bya Luttinger-Ward functional analysis similar to that used to establish theLuttinger theorem of Fermi liquid theory:

    Q = 12� ✓

    ⇡� sin(2✓)

    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

  • The SYK model

    At T > 0, we obtain a solution with a conformal structure

    G(⌧) = �A e�2⇡ET⌧

    p1 + e�4⇡E

    ✓T

    sin(⇡T ⌧)

    ◆1/2, 0 < ⌧ < 1/T ,

    where the ‘particle-hole asymmetry’ is determined by E

    e2⇡E =sin(⇡/4 + ✓)

    sin(⇡/4� ✓) .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

    S. Sachdev, PRX 5, 041025 (2015)A. Georges and O. Parcollet PRB 59, 5341 (1999)

    Feynman graph expansion in Uijk`, and graph-by-graph average, yields ex-act equations in the large N limit:

    G(i!) =1

    i! + µ� ⌃(i!) , ⌃(⌧) = �U2G2(⌧)G(�⌧)

    G(⌧ = 0�) = Q.

    Low frequency analysis shows that the solutions must be gapless and obey

    ⌃(z) = µ� ei(⇡/4+✓)

    A

    pz + . . . , G(z) =

    Ae�i(⇡/4+✓)pz

    where A = (⇡/U2 cos(2✓))1/4. The value of ✓ is universally related to Q bya Luttinger-Ward functional analysis similar to that used to establish theLuttinger theorem of Fermi liquid theory:

    Q = 12� ✓

    ⇡� sin(2✓)

    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AAAFlHicdZTfb9s2EMfVJt4670fTDRg27OWweIWDxq5s2GsxbECzYs0eiqLF6qZAlAQn6SxzoUiFpJKqg