From the SYK model, to a theory of the strange metal, and of quantum gravity in two spacetime dimensions

Transcription

1 HARVARD From the SYK model, to a theory of the strange metal, and of quantum gravity in two spacetime dimensions ARO MURI review, University of Maryland October 13, 2017 Subir Sachdev Talk online: sachdev.physics.harvard.edu

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3 Quantum matter without quasiparticles Resistivity 0 + AT Strange (or bad ) Metal T h/e 2 BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW +nematic 1.0 Superconductivity 0 S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, PRB 81, (2010)

4 The Sachdev-Ye-Kitaev (SYK) model Pick a set of random sites/orbitals

5 The SYK model Place electrons randomly on some sites/orbitals

6 The SYK model Entangle electrons pairwise randomly

7 The SYK model Entangle electrons pairwise randomly

8 The SYK model Entangle electrons pairwise randomly

9 The SYK model Entangle electrons pairwise randomly

10 The SYK model Entangle electrons pairwise randomly

11 The SYK model Entangle electrons pairwise randomly

12 The SYK model This describes both a strange metal and a black hole!

13 H = The SYK model (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)) 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` c i c j c k c` µ X i c i c j + c j c i =0, c i c j + c j c i = ij Q = 1 X c i N c i i c i c i U ij;k` are independent random variables with U ij;k` = 0 and U ij;k` 2 = U 2 N!1yields critical strange metal. S. Sachdev and J. Ye, PRL 70, 3339 (1993) A. Kitaev, unpublished; S. Sachdev, PRX 5, (2015)

14 Creating and probing the Sachdev-Ye-Kitaev model with ultracold gases: Towards experimental studies of quantum gravity Prog. Theor. Exp. Phys. 2017, 083I01 Ippei Danshita 1,,MasanoriHanada 1,2,3, and Masaki Tezuka 4 (b) E m (a) Atomic site E a,4 Molecular site E m,2 (2) E 3,4 E a,3 E a,2 E a,1 MB E m,1 E m (2) E 2,4 (2) E 2,3 (2) E 1,4 (2) E 1,3 PA lasers (2) E 1,2

15 A simple model of a metal with quasiparticles Pick a set of random positions

16 A simple model of a metal with quasiparticles Place electrons randomly on some sites

17 A simple model of a metal with quasiparticles Electrons move one-by-one randomly

18 A simple model of a metal with quasiparticles Electrons move one-by-one randomly

19 A simple model of a metal with quasiparticles Electrons move one-by-one randomly

20 A simple model of a metal with quasiparticles Electrons move one-by-one randomly

21 A simple model of a metal with quasiparticles 1 NX H = (N) 1/2 i,j=1 t ij c i c j +... c i c j + c j c i =0, c i c j + c j c i = ij 1 N X i c i c i = Q t ij are independent random variables with t ij = 0 and t ij 2 = t 2 Fermions occupying the eigenstates of a N x N random matrix

22 A simple model of a metal with quasiparticles p Let " be the eigenvalues of the matrix t ij / N. The fermions will occupy the lowest NQ eigenvalues, upto the Fermi energy E F.Thedensity of states is (!) =(1/N ) P (! " ). (!) E F!

23 A simple model of a metal with quasiparticles There are 2 N many body levels with energy Many-body level spacing 2 N E = NX =1 n ", Quasiparticle excitations with spacing 1/N where n =0, 1. Shown are all values of E for a single cluster of size N = 12. The " have a level spacing 1/N.

24 A simple model of a metal with quasiparticles p Let " be the eigenvalues of the matrix t ij / N. The fermions will occupy the lowest NQ eigenvalues, upto the Fermi energy E F.Thedensity of states is (!) =(1/N ) P (! " ). (!) " level spacing 1/N E F!

25 A simple model of a metal with quasiparticles There are 2 N many body levels with energy Many-body level spacing 2 N E = NX =1 n ", Quasiparticle excitations with spacing 1/N where n =0, 1. Shown are all values of E for a single cluster of size N = 12. The " have a level spacing 1/N.

26 H = The SYK model (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)) 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` c i c j c k c` µ X i c i c j + c j c i =0, c i c j + c j c i = ij Q = 1 X c i N c i i c i c i U ij;k` are independent random variables with U ij;k` = 0 and U ij;k` 2 = U 2 N!1yields critical strange metal. S. Sachdev and J. Ye, PRL 70, 3339 (1993) A. Kitaev, unpublished; S. Sachdev, PRX 5, (2015)

27 The SYK model Many-body level spacing 2 N = e N ln 2 Non-quasiparticle excitations with spacing e Ns 0 There are 2 N many body levels with energy E, which do not admit a quasiparticle decomposition. Shown are all values of E for a single cluster of size N = 12. The T! 0 state has an entropy S GP S = Ns 0 with s 0 = G + ln(2) 4 < ln 2 = where G is Catalan s constant, for the half-filled case Q =1/2. GPS: A. Georges, O. Parcollet, and S. Sachdev, PRB 63, (2001) W. Fu and S. Sachdev, PRB 94, (2016)

28 The SYK model Many-body level spacing 2 N = e N ln 2 Non-quasiparticle excitations with spacing e Ns 0 There are 2 N many body levels with energy E, which do not admit a quasiparticle decomposition. Shown are all values of E for a single cluster of size N = 12. The T! 0 state has an entropy S GP S = Ns 0 with s 0 = G + ln(2) 4 < ln 2 = No quasiparticles! where G is Catalan s constant, for the half-filled case Q =1/2. E 6= P n " + P F, n n +... GPS: A. Georges, O. Parcollet, and S. Sachdev, PRB 63, (2001) W. Fu and S. Sachdev, PRB 94, (2016)

29 Low energy, many-body density of states (E) e Ns 0 sinh( p 2(E E 0 )N ) Low temperature entropy S = Ns 0 + N T +... T = 0 fermion Green s function G( ) 1/2 at large. (Fermi liquids with quasiparticles have G( ) 1/ ) The SYK model A. Georges, O. Parcollet, and S. Sachdev, PRB 63, (2001) D. Stanford and E. Witten, A. M. Garica-Garcia, J.J.M. Verbaarschot, D. Bagrets, A. Altland, and A. Kamenev, A. Kitaev, unpublished J. Maldacena and D. Stanford, S. Sachdev and J. Ye, PRL 70, 3339 (1993) T>0Green sfunction has conformal invariance G (T/sin( k B T /~)) 1/2 A. Georges and O. Parcollet PRB 59, 5341 (1999)

30 Low energy, many-body density of states (E) e Ns 0 sinh( p 2(E E 0 )N ) Low temperature entropy S = Ns 0 + N T +... T = 0 fermion Green s function G( ) 1/2 at large. (Fermi liquids with quasiparticles have G( ) 1/ ) The SYK model A. Georges, O. Parcollet, and S. Sachdev, PRB 63, (2001) D. Stanford and E. Witten, A. M. Garica-Garcia, J.J.M. Verbaarschot, D. Bagrets, A. Altland, and A. Kamenev, A. Kitaev, unpublished J. Maldacena and D. Stanford, S. Sachdev and J. Ye, PRL 70, 3339 (1993) T>0Green sfunction has conformal invariance G (T/sin( k B T /~)) 1/2 Study of non-equilibrium quench dynamics shows eq ~/(k B T ). A. Georges and O. Parcollet PRB 59, 5341 (1999) A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, arxiv:

31 t... See also A. Georges and O. Parcollet PRB 59, 5341 (1999) arxiv: [pdf, other] Title: A strongly Building correlated metal built a from metal Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures U X X H = X x X... U ijkl,x c ix c jx c kx c lx + X X t ij,xx 0c i,x c j,x 0 i<j,k<l hxx 0 i i,j U ijkl 2 = 2U 2 N 3 Gaussian distrib d t ij,x,x 0 2 = t 2 0 /N. sm, one studies

32 See also A. Georges and O. Parcollet PRB 59, 5341 (1999) arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures Low coherence scale E c t2 0 U (a)

33 See also A. Georges and O. Parcollet PRB 59, 5341 (1999) arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures Low coherence scale E c t2 0 U For E c <T <U,the resistivity,, and entropy density, s, are (a) h e 2 T E c, s = s 0

34 See also A. Georges and O. Parcollet PRB 59, 5341 (1999) arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures Low coherence scale E c t2 0 U For T<E c,the resistivity,, and entropy density, s, are = h e 2 " c 1 + c 2 T E c 2 # (a) s s 0 T E c

35 Quantum matter without quasiparticles: No quasiparticle decomposition of low-lying states: E 6= P n " + P, F n n +... Thermalization and many-body chaos in the shortest possible time of order ~/(k B T ). These are also characteristics of black holes in quantum gravity.

36 Black holes have an entropy and a temperature, T H. Black holes The entropy is proportional to their surface area. They relax to thermal equilibrium in a time ~/(k B T H ).

37 LIGO September 14, 2015 The Hawking temperature, T H influences the radiation from the black hole at the very last stages of the ring-down (not observed so far). The ring-down (approach to thermal equilibrium) happens very rapidly in a time ~ = 8 GM k B T H c 3 8 milliseconds.

38 Black hole horizon SYK and black holes AdS 2 T 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt charge density Q T 2 = 1 ~x Quantum gravity on the 1+1 dimensional spacetime AdS2 (when embedded in AdS4) is holographically matched to the 0+1 dimensional SYK model S. Sachdev, PRL 105, (2010); A. Kitaev (unpublished); J. Maldacena, D. Stanford, and Zhenbin Yang, arxiv: