Equilibrium and non-equilibrium dynamics of SYK models Strongly interacting conformal field theory in condensed matter physics, Institute for Advanced Study, Tsinghua University, Beijing, June 25-27, 207 Subir Sachdev Talk online: sachdev.physics.harvard.edu HARVARD
Quantum matter with quasiparticles: Quasiparticles are additive excitations: The low-lying excitations of the many-body system can be identified as a set {n } of quasiparticles with energy " E = P n " + P, F n n +... Note: The electron liquid in one dimension and the fractional quantum Hall state both have quasiparticles; however, the quasiparticles do not have the same quantum numbers as an electron.
Quantum matter with quasiparticles: Quasiparticles eventually collide with each other. Such collisions eventually leads to thermal equilibration in a chaotic quantum state, but the equilibration takes a long time. In a Fermi liquid, this time is of order ~E F /(k B T ) 2 as T! 0, where E F is the Fermi energy.
Quantum matter without quasiparticles: No quasiparticle decomposition of low-lying states Rapid thermalization
Local thermal equilibration or phase coherence time, ' : There is an lower bound on ' in all manybody quantum systems as T! 0, ' C ~ k B T, where C is a T -independent constant. Systems without quasiparticles have ' ~ k B T, K. Damle and S. Sachdev, PRB 56, 874 (997) S. Sachdev, Quantum Phase Transitions, Cambridge (999)
A simple model of a metal with quasiparticles NX H = (N) /2 i,j= t ij c i c j +... c i c j + c j c i =0, c i c j + c j c i = ij 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
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 (!) =(/N ) P (! " ). (!) E F!
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 = n ", Quasiparticle excitations with spacing /N where n =0,. Shown are all values of E for a single cluster of size N = 2. The " have a level spacing /N.
The Sachdev-Ye-Kitaev (SYK) model H = (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 (98)) (2N) 3/2 NX i,j,k,`= J 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 = X c i N c i i c i c i J ij;k` are independent random variables with J ij;k` = 0 and J ij;k` 2 = J 2 N!yields critical strange metal. S. Sachdev and J. Ye, PRL 70, 3339 (993) A. Kitaev, unpublished; S. Sachdev, PRX 5, 04025 (205)
The Sachdev-Ye-Kitaev (SYK) model Many-body level spacing 2 N = e N ln 2 Non-quasiparticle excitations with spacing e S GP S 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 = 2. The T! 0 state has an entropy S GP S with S GP S N = G + ln(2) 4 < ln 2 = 0.464848.. where G is Catalan s constant, for the half-filled case Q =/2. GPS: A. Georges, O. Parcollet, and S. Sachdev, PRB 63, 34406 (200) W. Fu and S. Sachdev, PRB 94, 03535 (206)
SYK and black holes Black hole horizon SS, PRL 05, 5602 (200) T 2 ~x The SYK model has dual description in which an extra spatial dimension,, emerges. The curvature of this emergent spacetime is described by Einstein s theory of general relativity
SYK and black holes GPS Bekenstein-Hawking entropy black hole entropy AdS 2 T 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt charge density Q T 2 = Z S = d 4 x p ĝ ~x ˆR +6/L 2 4 ˆF µ ˆF µ SS, PRL 05, 5602 (200) The BH entropy is proportional to the size of T 2, and hence the surface area of the black hole. Mapping to SYK applies when temperature /(size of T 2 ).
SYK and black holes Black hole quasi-normal modes relax to thermal equilibrium in a time ~/(k B T H ), where T H is the Hawking temperature. T 2 ' ~/(k B T ) ~x The SYK model has dual description in which an extra spatial dimension,, emerges. The curvature of this emergent spacetime is described by Einstein s theory of general relativity SS, PRL 05, 5602 (200)
Quantum Quench of the SYK model Andreas Eberlein Julia Steinberg Valentin Kasper A. Eberlein, V. Kasper, S. Sachdev and J. Steinberg, arxiv:706.xxxxx
Quench from pq fermion + q fermion interactions for t<0, to only q fermion interactions for t>0. H = f(t)(i) pq 2 X applei <i 2 <...<i pq applen X j i i 2...i pq i i2... ipq + g(t)(i) q 2 j 0 i i 2...i q i i2... iq applei <i 2 <...<i q applen for t<0, f(t) = and g(t) =; for t>0, f(t) = 0 and g(t) =. hj 2 i...i pq i = J 2 p (pq )! N pq, J p (t) =J p f(t) hj 02 i...i q i = J 2 (q )! N q, J(t) =Jg(t)
i @ @t G > (t,t 2 )= Kadanoff-Baym equations ig > (t,t 2 )=ht C (t ) (t + 2 )i, G< (t,t 2 )= G > (t 2,t ) i q Z t + i q Z t2 dt 3 J(t )J(t 3 ) (G > ) q (t,t 3 ) (G < ) q (t,t 3 ) G > (t 3,t 2 ) dt 3 J(t )J(t 3 )(G > ) q (t,t 3 ) G > (t 3,t 2 ) G < (t 3,t 2 ) i pq Z t dt 3 J p (t )J p (t 3 ) (G > ) pq (t,t 3 ) (G < ) pq (t,t 3 ) G > (t 3,t 2 ) i @ @t 2 G > (t,t 2 )= + i pq Z t2 dt 3 J p (t )J p (t 3 )(G > ) pq (t,t 3 ) G > (t 3,t 2 ) G < (t 3,t 2 ), i q Z t + i q Z t2 dt 3 J(t 3 )J(t 2 ) G > (t,t 3 ) G < (t,t 3 ) (G > ) q (t 3,t 2 ) dt 3 J(t 3 )J(t 2 )G > (t,t 3 ) (G > ) q (t 3,t 2 ) (G < ) q (t 3,t 2 ) i pq Z t dt 3 J p (t 3 )J p (t 2 ) G > (t,t 3 ) G < (t,t 3 ) (G > ) pq (t 3,t 2 ) + i pq Z t2 dt 3 J p (t 3 )J p (t 2 )G > (t,t 3 ) (G > ) pq (t 3,t 2 ) (G < ) pq (t 3,t 2 ).
Numerical solutions of Kadanoff-Baym equations p=/2, q=4 J 2,i =0.5, J 2,f = 0, J 4,i = J 4,f =, T i =0.04J 4 ig K (T,ω)/A(T,ω) 0 T J 4 = 50 T J 4 =0 T J 4 = 50 2.5 0 2.5 ω/j 4,f Determine an e ective inverse temperature from such data, and then fit to e (T )= f + exp ( T ), where T = (t + t 2 )/2, and obtain the relaxation rate,.
Numerical solutions of Kadanoff-Baym equations p=/2, q=4 As T f! 0, the relaxation rate T f
q!limit of Kadano -Baym equations G > (t,t 2 )= i 2 apple + q g(t,t 2 )+... Z @ t2 g(t,t 2 )=2 @t +2 @ @t 2 g(t,t 2 )=2 +2 Z t2 Z t Z t dt 3 J (t )J (t 3 )e g(t,t 3 ) dt 3 J p (t )J p (t 3 )e pg(t,t 3 ) dt 3 J (t 3 )J (t 2 )e g(t 3,t 2 ) dt 3 J p (t 3 )J p (t 2 )e pg(t 3,t 2 ) Z t Z t2 Z t dt 3 J (t )J (t 3 ) he i g(t,t 3 ) + e g(t 3,t ) Z t2 J 2 (t) =qj 2 (t)2 q, J 2 p (t) =qj 2 p (t)2 pq dt 3 J p (t )J p (t 3 ) he i pg(t,t 3 ) + e pg(t 3,t ) dt 3 J (t 3 )J (t 2 ) he i g(t 3,t 2 ) + e g(t 2,t 3 ) dt 3 J p (t 3 )J p (t 2 ) he i pg(t 3,t 2 ) + e pg(t 2,t 3 ) These non-linear, partial, integro-differential equations are exactly solvable!
q!limit of Kadano -Baym equations From a derivative of the Kadano -Baym equations, we obtain @ 2 @t @t 2 g(t,t 2 )=2J (t )J (t 2 )e g(t,t 2 ) +2J p (t )J p (t 2 )e pg(t,t 2 ). For t > 0 and t 2 > 0, this is the two-dimensional Liouville equation. The most general solution is of the form apple h 0 g(t,t 2 )=ln (t )h 0 2(t 2 ) J 2 (h (t ) h 2 (t 2 )) 2. A remarkable and significant feature of this expression is that it is exactly invariant under SL(2,C) transformations h (t)! ah (t)+b ch (t)+d, where =, 2 and a, b, c, d are arbitrary complex numbers.
q!limit of Kadano -Baym equations Substituting back into the Kadano -Baym equations, we find that for generic initial conditions in the regime t < 0 and t 2 < 0, the solutions for h (t ) and h 2 (t 2 ) at t > 0 and t 2 > 0 can be written as h (t) = aei e t + b ce i e t + d, h 2 (t) = ae i e t + b ce i e t + d. The complex constants a, b, c, d, and the real constants, are determined by the initial conditions in the t < 0 and t 2 < 0 quadrant of the t -t 2 plane. Substituting back into the SL(2,C) invariant Green s function, we obtain g(t,t 2 )=ln apple 2 4J 2 sinh 2 ( (t t 2 )/2+i ), =2J sin( ), which is (as expected) independent of a, b, c, d. The surprising feature is that depends only upon t = t t 2, and is independent of T =(t + t 2 )/2.
q!limit of Kadano -Baym equations Indeed, from the KMS condition, we find g describes a state in thermal equilibrium at an inverse temperature f = 2( 2 ). As T f! 0,! 0, and =2 T f, the maximal chaos rate. The system is thermal at t =0 + Actually, this may be a pre-thermal state (pointed out by Aavishkar Patel). The next correction G > (t,t 2 )= i apple + 2 q g(t,t 2 )+ q 2 g 2(t,t 2 )... may lead to a g 2 which relaxes at a rate T f.
q!limit of Kadano -Baym equations This result has a remarkable connection to the Schwarzian action. Consider the Euler-Lagrange equation of motion of a Lagrangian, L, which is the Schwarzian of h(t) The equation of motion is L[h(t)] = h000 (t) h 0 (t) 3 2 h 00 (t) h 0 (t) 2. This is exactly solved by [h 0 (t)] 2 h 0000 (t)+3[h 00 (t)] 3 4h 0 (t)h 00 (t)h 000 (t) =0. h(t) = ae t + b ce t + d.
SYK and AdS2 AdS 2 T 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt charge density Q T 2 = ~x Einstein-Maxwell theory + cosmological constant Mapping to SYK applies when temperature /(size of T 2 ) SS, PRL 05, 5602 (200)
J. Maldacena, D. Stanford, and Zhenbin Yang, arxiv:606.0857; K. Jensen, arxiv:605.06098; J. Engelsoy, T.G. Mertens, and H. Verlinde, arxiv:606.03438 SYK and AdS2 Same long-time e ective action involving Schwarzian derivatives of a time reparameterization! h( ). AdS 2 T 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt charge density Q T 2 = ~x Einstein-Maxwell theory + cosmological constant Mapping to SYK applies when temperature /(size of T 2 )
Thermal diffusivity and chaos in metals without quasiparticles Mike Blake Aavishkar Patel Wenbo Fu Yingfei Gu Richard Davison
SYK model H = (2N) 3/2 NX i,j,k,`= J ij;k` c i c j c k c` µ X i c i c i c i c j + c j c i =0, c i c j + c j c i = ij Q = X c N i c i 3 4 i 2 5 6 J 3,5,7,3 J 4,5,6, 7 0 8 9 2 3 J 8,9,2,4 J ij;k` are independent random variables with J ij;k` = 0 and J ij;k` 2 = J 2 N!yields critical strange metal. 4 A. Kitaev, unpublished; S. Sachdev, PRX 5, 04025 (205) 2 S. Sachdev and J. Ye, PRL 70, 3339 (993)
SYK and AdS2 Reparametrization and phase zero modes We can write the path integral for the SYK model as Z Z = DG(, )D (, 2 )e NS[G, ] for a known action S[G, ]. We find the saddle point, G s, s, and only focus on the Nambu-Goldstone modes associated with breaking reparameterization and U() gauge symmetries by writing G(, 2 )=[f 0 ( )f 0 ( 2 )] /4 G s (f( ) f( 2 ))e i ( ) i ( 2 ) (and similarly for ). Then the path integral is approximated by Z Z = Df( )D ( )e NS eff [f, ]. J. Maldacena and D. Stanford, arxiv:604.0788; R. Davison, Wenbo Fu, A. Georges, Yingfei Gu, K. Jensen, S. Sachdev, arxiv.62.00849; S. Sachdev, PRX 5, 04025 (205); J. Maldacena, D. Stanford, and Zhenbin Yang, arxiv:606.0857; K. Jensen, arxiv:605.06098; J. Engelsoy, T.G. Mertens, and H. Verlinde, arxiv:606.03438
SYK and AdS2 Z Z = Df( )D ( )e NS eff [f, ]. Symmetry arguments, and explicit computations, show that the e ective action is S e [f, ]= K 2 Z /T 0 d (@ + i(2 ET )@ ) 2 4 2 Z /T 0 d {tan( T ( + ( ), }, where f( ) + ( ), and we have used the Schwarzian: {g, } g000 3 g 0 2 g 00 g 0 2. K and are related to the compressibility and the linear-t specific heat, and is a derivative of the entropy: @Q K =, S(T )=S 0 + T +..., 2 E = ds 0 @µ dq T All these results apply unchanged to the gravitational theory on AdS 2. J. Maldacena and D. Stanford, arxiv:604.0788; R. Davison, Wenbo Fu, A. Georges, Yingfei Gu, K. Jensen, S. Sachdev, arxiv.62.00849; S. Sachdev, PRX 5, 04025 (205); J. Maldacena, D. Stanford, and Zhenbin Yang, arxiv:606.0857; K. Jensen, arxiv:605.06098; J. Engelsoy, T.G. Mertens, and H. Verlinde, arxiv:606.03438
Coupled SYK models J jklm J 0 jklm j k l m j k l m Figure : A chain of coupled SYK sites: each site contains N fermion with SYK interaction. The coupling between nearest neighbor sites are four fermion interaction with two from each site. Yingfei Gu, Xiao-Liang Qi, and D. Stanford, arxiv:609.07832 R. Davison, Wenbo Fu, A. Georges, Yingfei Gu, K. Jensen, S. Sachdev, arxiv.62.00849
Coupled SYK and AdS4 AdS 2 R 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt charge density Q R 2 = ~x S = Z d 4 x p ĝ ˆR +6/L 2 2 2X i= (@ ˆ' i ) 2 4 ˆF µ ˆF µ Einstein-Maxwell-axion theory with saddle point ˆ' i = kx i leading to momentum disspation!,
Coupled SYK and AdS4 The response functions of the density, Q, and the energy, E exhibit di usion hq; Qi k,! he µq; Qi k,! /T = i!( i! + Dk 2 ) + s he µq; Qi k,! he µq; E µqi k,! /T where the di usivities are related to the thermoelectric conductivities by the Einstein relations D = T apple s. The Seebeck co-e cient (thermopower), /, is given exactly by a thermodynamic derivative @S 0 = @Q The coupled-syk and AdS4 models realize a disordered metal with no quasiparticle excitations. (a strange metal ) R. Davison, Wenbo Fu, A. Georges, Yingfei Gu, K. Jensen, S. Sachdev, arxiv.62.00849
Quantum chaos: The growth of chaos is characterized by D {c(x, t),c (0, 0)} 2E exp L t x v B in terms of the Lyapunov time L butterfly velocity v B. ~/(2 k B T ) and the In the SYK and AdS 2 holographic models we find L = ~/(2 k B T ) and a non-universal v B T /2. However the thermal di usivity (which controls the di usion of temperature) D T = apple c, where c is the specific heat at fixed density, is given exactly by D T = v 2 B L. There is no universal relationship between the charge di usivity, D c, and v B. R. Davison, Wenbo Fu, A. Georges, Yingfei Gu, K. Jensen, S. Sachdev, arxiv.62.00849
Coupled SYK and AdS4 Matching correlators for thermodynamics, thermoelectric diffusion, and quantum chaos AdS 2 R 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt charge density Q R 2 = ~x S = Z d 4 x p ĝ ˆR +6/L 2 2 2X i= (@ ˆ' i ) 2 4 ˆF µ ˆF µ Einstein-Maxwell-axion theory with saddle point ˆ' i = kx i leading to momentum disspation!,
Fermi surface coupled to a gauge field k y k x L[,a]= @ ia (r i~a) 2 2m µ + 2g 2 (r ~a)2
Fermi surface coupled to a gauge field Compute out-of-time-order correlator to diagnose quantum chaos Strongly-coupled theory with no quasiparticles and fast scrambling: L ~ 2.48 kb T /3 vb DT NT 4. 4/3 e 2 0.42 vb L 5/3 vf /3 N is the number of fermion flavors, vf is the Fermi velocity, the Fermi surface curvature, e is the gauge coupling constant. is A. A. Patel and S. Sachdev, arxiv:6.00003
Quantum chaos: We examined chaos and transport in a wide-class of holographic fixed points. In general, chaos and transport can be extracted from the fixed point solution, while the AdS 2 geometries (and SYK models) required consideration of the leading irrelevant operator. For the holographic fixed points, we find ~ L = 2 k B T v B T /z D T = z 2z 2 v2 B L, where z is the dynamic critical exponent. There is no general relation to chaos for other transport co-e cients. M. Blake, R. A. Davison, and S. Sachdev, arxiv.62.00849
Quantum chaos: We define the chaos parameters by D {c(x, t),c (0, 0)} 2E exp L t x v B and the thermal di usivity close to local equilibrium by Then we find the general relation @T @t = D T r 2 T. D T = Cv 2 B L where C is a numerical constant of order unity, independent of shortdistance details. Quantum chaos is intimately linked to the loss of phase coherence from electron-electron interactions. As the time derivative of the local phase is determined by the local energy, phase fluctuations and chaos are linked to interaction-induced energy fluctuations, and hence thermal di usivity. M. Blake, R. A. Davison, and S. Sachdev, arxiv.62.00849