Quantum matter without quasiparticles: SYK models, strange metals, and black holes University of Maryland, College Park, December 13, 2016 Subir Sachdev Talk online: sachdev.physics.harvard.edu HARVARD Aavishkar Patel, Harvard Wenbo Fu, Harvard Antoine Georges, Paris Yingfei Gu, Stanford Richard Davison, Harvard Kristan Jensen, SFSU Quantum matter with quasiparticles: • Landau quasi-particles & holes • Phonon • Magnon • Roton • Plasmon • Polaron • Exciton • Laughlin quasiparticle • Bogoliubovon • Anderson-Higgs mode • Massless Dirac Fermions • Weyl fermions • …. Quantum matter with quasiparticles: Most generally, a quasiparticle is an “additive” excitation: Quasiparticles can be combined to yield additional excitations, with energy determined by the energies and densities of the constituents. Such a procedure yields all the low-lying excitations. Then we can apply the Boltzmann-Landau theory to make predictions for dynamics. Resistivity ⇢ + AT ↵ ⇠ 0 T Strange BaFe2(As1-xPx)2 0 TSDW Metal Tc α" 2.0 AF 1.0 +nematicSDW 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, Physical Review B 81, 184519 (2010) Strange metals 1 k T = ↵ B ⌧ ~ J. A. N. Bruin, H. Sakai, R. S. Perry, A. P. Mackenzie, Science. 339, 804 (2013) Quantum matter without quasiparticles: No quasiparticle structure to excitations. But how can we be sure that no quasiparticles exist in a given system? Perhaps there are some exotic quasiparticles inaccessible to current experiments…….. Consider how rapidly the system loses “phase coherence”, reaches local thermal equilibrium, or becomes “chaotic” Quantum matter without quasiparticles: No quasiparticle structure to excitations. But how can we be sure that no quasiparticles exist in a given system? Perhaps there are some exotic quasiparticles inaccessible to current experiments…….. Consider how rapidly the system loses “phase coherence”, reaches local thermal equilibrium, or becomes “chaotic” Local thermal equilibration or phase coherence time, ⌧': There is an lower bound on ⌧' in all many-body quantum • systems as T 0, ! ~ ⌧' >C , kBT where C is a T -independent constant. Systems without quasiparticles have ⌧' ~/(kBT ). ⇠ In systems with quasiparticles, ⌧' is parametrically larger • at low T ; 2 e.g. in Fermi liquids ⌧' 1/T , ⇠ ∆/(kB T ) and in gapped insulators ⌧' e where ∆ is the energy gap. ⇠ K. Damle and S. Sachdev, PRB 56, 8714 (1997) S. Sachdev, Quantum Phase Transitions, Cambridge (1999) A bound on quantum chaos: In classical chaos, we measure the sensitivity of the position at time • t, q(t), to variations in the initial position, q(0), i.e. we measure @q(t) 2 =( q(t),p(0) )2 @q(0) { }P.B. ✓ ◆ By analogy, we define ⌧L as the Lyapunov time over which the wave- • function of a quantum system is scrambled by an initial perturbation. This scrambling can be measured by 2 [Aˆ(t), Bˆ(0)] et/⌧L ⇠ ⌧ This quantum time was argued to obey lower bound 1 ~ ⌧L . ≥ 2⇡ kBT There is no analogous bound in classical mechanics. A. I. Larkin and Y. N. Ovchinnikov, JETP 28, 6 (1969) J. Maldacena, S. H. Shenker and D. Stanford, arXiv:1503.01409 A bound on quantum chaos: In classical chaos, we measure the sensitivity of the position at time • t, q(t), to variations in the initial position, q(0), i.e. we measure @q(t) 2 =( q(t),p(0) )2 @q(0) { }P.B. ✓ ◆ By analogy, we define ⌧L as the Lyapunov time over which the wave- • function of a quantum system is scrambled by an initial perturbation. This scrambling can be measured by 2 [Aˆ(t), Bˆ(0)] et/⌧L ⇠ ⌧ This quantum time was argued to obey lower bound 1 ~ ⌧L . ≥ 2⇡ kBT ThereQuantum is no analogous matter bound without in classical quasiparticles mechanics. fastest possible many-body quantum chaos ⇡ Quantum matter without quasiparticles: The Sachdev-Ye-Kitaev Black holes with (SYK) models AdS2 horizons Fermi surface coupled to a gauge field 2 ( i~a) 1 2 [ ,a]= † @ ia r µ + ( ~a) L ⌧ − ⌧ − 2m − 2g2 r⇥ ✓ ◆ Quantum matter without quasiparticles: The Sachdev-Ye-Kitaev Black holes with (SYK) models AdS2 horizons Same low energy theory Fermi surface coupled to a gauge field 2 ( i~a) 1 2 [ ,a]= † @ ia r µ + ( ~a) L ⌧ − ⌧ − 2m − 2g2 r⇥ ✓ ◆ Quantum matter without quasiparticles: The Sachdev-Ye-Kitaev Black holes with (SYK) models AdS2 horizons ~ ~ ⌧L = ⌧L = 2⇡kBT 2⇡kBT v T 1/2 v T 1/2 B ⇠ B ⇠ ~ ⌧L = Fermi surface coupled 2.48 kBT 5/3 to a gauge field v v F T 1/3 B ⇠ e4/3γ1/3 2 ( i~a) 1 2 [ ,a]= † @ ia r µ + ( ~a) L ⌧ − ⌧ − 2m − 2g2 r⇥ ✓ ◆ ⌧L: the Lyapunov time to reach quantum chaos The Sachdev-Ye-Kitaev Figure credit: L. Balents (SYK) model: A theory of a • strange metal Dual theory of • gravity on AdS2 Fastest possible • quantum chaos ~ with ⌧L = 2⇡kBT Infinite-range model with quasiparticles 1 N H = t c†c + ... (N)1/2 ij i j i,j=1 X cicj + cjci =0 ,cicj† + cj†ci = δij 1 c†c = N i i Q i X t are independent random variables with t = 0 and t 2 = t2 ij ij | ij| Fermions occupying the eigenstates of a N x N random matrix Infinite-range model with quasiparticles Feynman graph expansion in tij.., and graph-by-graph average, yields exact equations in the large N limit: 1 G(i!)= , ⌃(⌧)=t2G(⌧) i! + µ ⌃(i!) − G(⌧ =0−)= . Q G(!) can be determined by solving a quadratic equation. Im G(!) − µ ! Infinite-range model with quasiparticles Now add weak interactions 1 N 1 N H = t c†c + J c†c†c c (N)1/2 ij i j (2N)3/2 ij;k` i j k ` i,j=1 X i,j,k,X`=1 J are independent random variables with J = 0 and J 2 = J 2.We ij;k` ij;k` | ij;k`| compute the lifetime of a quasiparticle, ⌧↵, in an exact eigenstate ↵(i) of the free particle Hamitonian with energy E↵. By Fermi’s Golden rule, for E↵ at the Fermi energy 1 = ⇡J 2⇢2 dE dE dE f(E )(1 f(E ))(1 f(E ))δ(E + E E E ) ⌧ 0 β γ δ β − γ − δ ↵ β − γ − δ ↵ Z ⇡3J 2⇢2 = 0 T 2 4 where ⇢0 is the density of states at the Fermi energy. Fermi liquid state: Two-body interactions lead to a scattering time of quasiparticle excitations from in (random) single-particle eigen- 2 states which diverges as T − at the Fermi level. ⇠ SYK model 1 N H = J c†c†c c µ c†c (2N)3/2 ij;k` i j k ` − i i i i,j,k,X`=1 X cicj + cjci =0 ,cicj† + cj†ci = δij 1 = c†c Q N i i i 1 X 2 3 4 5 J4,5,6,11 6 J3,5,7,13 7 10 8 9 12 11 J8,9,12,14 14 12 13 2 2 Jij;k` are independent random variables with Jij;k` = 0 and Jij;k` = J N yields critical strange metal. | | !1 S. Sachdev and J. Ye, PRL 70, 3339 (1993) A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015) SYK model Feynman graph expansion in Jij.., and graph-by-graph average, yields exact equations in the large N limit: 1 G(i!)= , ⌃(⌧)= J 2G2(⌧)G( ⌧) i! + µ ⌃(i!) − − − G(⌧ =0−)= . Q Low frequency analysis shows that the solutions must be gapless and obey 1 A ⌃(z)=µ pz + ... , G(z)= − A pz for some complex A. The ground state is a non-Fermi liquid, with a continuously variable density . Q S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993) SYK and AdS2 Non-zero GPS entropy as T 0, S(T 0) = NS0 + ... • Not a ground state degeneracy! : due! to an exponen- tially small (in N) many-body level spacing at all energies down to the ground state energy. A. Georges, O. Parcollet, and S. Sachdev, PRB 63, 134406 (2001) This entropy, and other dynamic correlators of the SYK • models, imply that the SYK model is holographically dual to black holes with an AdS2 horizon. The Bekenstein- Hawking entropy of the black hole equals NS0: GPS = BH. S. Sachdev, PRL 105, 151602 (2010) SYK and AdS2 Non-zero GPS entropy as T 0, S(T 0) = NS0 + ... • Not a ground state degeneracy! : due! to an exponen- tially small (in N) many-body level spacing at all energies down to the ground state energy. A. Georges, O. Parcollet, and S. Sachdev, PRB 63, 134406 (2001) This entropy, and other dynamic correlators of the SYK • models, imply that the SYK model is holographically dual to black holes with an AdS2 horizon. The Bekenstein- Hawking entropy of the black hole equals NS0: GPS = BH. S. Sachdev, PRL 105, 151602 (2010) SYK and AdS2 GPS BH entropy entropy 2 charge T 2 density AdS2 T Q ds2 =(d⇣2 dt⇥2)/⇣2 + d~x2 Gauge field:− A =( /⇣)dt E ⇣ = 1 ⇣ ~x Einstein-Maxwell theory S. Sachdev, PRL , 151602 (2010) + cosmological constant 105 Mapping to SYK applies when temperature 1/(size of T2) ⌧ SYK and AdS2 1 G(i!)= , ⌃(⌧)= J 2G2(⌧)G( ⌧) i! + µ ⌃(i!) − − − 1 A ⌃(z)=µ pz + ... , G(z)= − A pz S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993) SYK and AdS2 1 G(i!)= , ⌃(⌧)= J 2G2(⌧)G( ⌧) i! + µ ⌃(i!) − − X X− 1 A ⌃(z)=µ pz + ... , G(z)= X− A pz At frequencies J,thei! + µ can be dropped, and without it⌧ equations are invariant under the reparametrization and gauge transformations ⌧ = f(σ) 1/4 g(σ1) G(⌧1, ⌧2)=[f 0(σ1)f 0(σ2)]− G(σ1, σ2) g(σ2) 3/4 g(σ1) ⌃(⌧1, ⌧2)=[f 0(σ1)f 0(σ2)]− ⌃(σ1, σ2) g(σ2) where f(σ) and g(σ) are arbitrary functions.