Bekenstein-Hawking entropy and strange metals

Transcription

1 HARVARD Bekenstein-Hawking entropy and strange metals CMSA Colloquium Harvard University September 16, 2015 Subir Sachdev Talk online: sachdev.physics.harvard.edu

2 Quantum matter without quasiparticles 1. A solvable model of an ordinary metal 2. A solvable model of a strange metal 3. Holography and charged black holes 4. The (slightly less) strange metal in graphene

3 Quantum matter without quasiparticles 1. A solvable model of an ordinary metal 2. A solvable model of a strange metal 3. Holography and charged black holes 4. The (slightly less) strange metal in graphene

4 Infinite-range model of a Fermi liquid 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

5 Infinite-range model of a Fermi liquid Feynman graph expansion in t ij.., and graph-by-graph average, yields exact equations in the large N limit: G(i!) = 1 i! + µ (i!) G( =0 )=Q., ( ) =t 2 G( ) G(!) can be determined by solving a quadratic equation. Im G(!) µ! Fermions occupying eigenstates with a semi-circular density of states

6 Ordinary quantum matter: the Fermi liquid (FL) Fermi surface Fermi surface separates empty and occupied states in momentum space. Area enclosed by Fermi surface = Q. Momenta of low energy excitations fixed by density of all electrons. k y k x Long-lived electron-like quasiparticle excitations near the Fermi surface: lifetime of quasiparticles 1/T 2.

7 Ordinary quantum matter: the Fermi liquid (FL) Fermi surface Fermi surface separates empty and occupied states in momentum space. Area enclosed by Fermi surface = Q. Momenta of low energy excitations fixed by density of all electrons. k y k x Long-lived electron-like quasiparticle excitations near the Fermi surface: lifetime of quasiparticles 1/T 2. (Thermal conductivity) T (Electrical conductivity) = 2 kb 2 3e 2

8 Fluid in Graphene iedemann-franz Law I Wiedemann-Franz law in a Fermi liquid: apple T 2 kb 2 3e W K ~'v 3"0 t ~ 2.5 O I ~ ' I Bi Carrier concentration (cm 3) ~ I 1 ~ t I t ; I SiGe 2 Cs AI Au Mg SiBe3 /. / Nb~b.,~o/Cu F i Sb As Ag Ga 9 i e c N 2.5 c- O " 2.0 SiGe z ~oe \ SiG% i \ SiGe 1 Se H~ Y W1 Cs2 Fe Ir Co z Li2 AUz Au3 AI2 All \." \\\!!,,L T, oytw, p, t Feq z,~/ Cr '' \ X~., K Pt,\ Ni2 u A ~" Yb ~o 1 Er / / 2 As Sb ~ Rh g t31~ Bi 2 Bi I Cq! I I r I I t f I [ I I [ I i I t I I I I I I e I 101o Electrical conductivity (~,~-1 cm-1) G. S. Kumar, G. Prasad, and R.O. Pohl, J. Mat. Sci. 28, 4261 (1993)

9 Quantum matter without quasiparticles 1. A solvable model of an ordinary metal 2. A solvable model of a strange metal 3. Holography and charged black holes 4. The (slightly less) strange metal in graphene

10 Quantum matter without quasiparticles 1. A solvable model of an ordinary metal 2. A solvable model of a strange metal 3. Holography and charged black holes 4. The (slightly less) strange metal in graphene

11 Infinite-range model of a strange metal H = 1 (NM) 1/2 NX i,j=1 MX, =1 J ij c i c i c j c j c i c j + c j c i =0, c i c j + c j c i = ij 1 M X c i c i = Q J ij are independent random variables with J ij = 0 and J 2 ij = J 2 N!1at M = 2 yields spin-glass ground state. N!1and then M!1yields critical strange metal S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

12 Infinite-range model of a strange metal H = 1 (2N) 3/2 NX i,j,k,`=1 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 = 1 X c N i c i 3 4 i J 3,5,7,13 J 4,5,6, J 8,9,12, J ij;k` are independent random variables with J ij;k` = 0 and J ij;k` 2 = J 2 N!1yields same critical strange metal; simpler to study numerically A. Kitaev, unpublished; S. Sachdev, arxiv:

13 Infinite-range strange metals Feynman graph expansion in J ij.., and graph-by-graph average, yields exact equations in the large N limit: G(i!) = 1 i! + µ (i!) G( =0 )=Q., ( ) = J 2 G 2 ( )G( ) Low frequency analysis shows that the solutions must be gapless and obey (z) =µ 1 Ap z +..., G(z) = A pz for some complex A. Let us also define e (z) = (z) µ. S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

14 Infinite-range strange metals At frequencies J, the equations for G and can be written as Z d 2 G( 1, 2 ) e ( 2, 3 )= ( 1 3 ) e ( 1, 2 )= J 2 [G( 1, 2 )] 2 G( 2, 1 ) These equations are invariant under the reparametrization and gauge transformations = f( ) G( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] e ( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] where f( ) and g( ) are arbitrary functions. 1/4 g( 1 ) g( 2 ) G( 1, 2) 3/4 g( 1 ) g( 2 ) e ( 1, 2) A. Georges and O. Parcollet PRB 59, 5341 (1999) A. Kitaev, unpublished S. Sachdev, arxiv: These equations and invariances have similarities to those of the large N limit of quantum spins at the spatial boundary of a CFT2 (multi-channel Kondo problems) O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta PRB 58, 3794 (1998)

15 Infinite-range strange metals S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993) A. Georges, O. Parcollet, and S. Sachdev Phys. Rev. B 63, (2001) Local fermion density of states! 1/2,! > 0 (!) = Im G(!) e 2 E! 1/2,! < 0. E encodes the particle-hole asymmetry While E determines the low energy spectrum, it is determined by the total fermion density Q: Q = 1 4 (3 tanh(2 E)) 1 tan 1 e 2 E.

16 Infinite-range strange metals At non-zero temperature, T, the Green s function also fully determined by E. G R (!) = ice i (2 T ) where =1/4 and e 2 E = i! 2 T + ie i! 2 T + ie sin( + ) sin( ). G R (!)G A (!) ReG R (!) ImG R (!) S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993) A. Georges and O. Parcollet PRB 59, 5341 (1999) A. Georges, O. Parcollet, and S. Sachdev Phys. Rev. B 63, (2001)!

17 Infinite-range strange metals The entropy per site, S, has a non-zero limit as T! 0, and can be viewed as each site acquiring the universal boundary entropy of the multichannel Kondo problem. This Q =2 E O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta Phys. Rev. B 58, 3794 (1998) A. Georges, O. Parcollet, and S. Sachdev Phys. Rev. B 63, (2001)

18 Free spin has entropy ln(2s + 1)

19 Kondo-screened spin has no entropy Metal

20 Critically-screened spin has irrational entropy N. Andrei and C. Destri, PRL 52, 364 (1984). A. M. Tsvelick, J. Phys. C 18, 159 (1985). I. A eck and A. W. W. Ludwig, PRL 67, 161 (1991). S. Sachdev, C. Buragohain, and M. Vojta, Science 286, 2479 (1999). CFT

21 Infinite-range strange metals The entropy per site, S, has a non-zero limit as T! 0, and can be viewed as each site acquiring the universal boundary entropy of the multichannel Kondo problem. Critically-screened This entropy obeys spin has irrational N. Andrei and C. Destri, PRL 52, 364 (1984). A. M. Tsvelick, J. Phys. C 18, 159 (1985). I. A eck and A. W. W. Ludwig, PRL 67, 161 (1991). S. Sachdev, C. Buragohain, and M. Vojta, Science 286, 2479 (1999). Q =2 E O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta Phys. Rev. B 58, 3794 (1998) A. Georges, O. Parcollet, and S. Sachdev Phys. Rev. B 63, (2001)

22 Infinite-range strange metals The entropy per site, S, has a non-zero limit as T! 0, and can be viewed as each site acquiring the universal boundary entropy of the multichannel Kondo problem. This This entropy N. Andrei and C. Destri, PRL 52, 364 (1984). A. M. Tsvelick, J. Phys. C 18, 159 (1985). I. A eck and A. W. W. Ludwig, PRL 67, 161 (1991). S. Sachdev, C. Buragohain, and M. Vojta, Science 286, 2479 (1999). @µ Q =2 E = =2 Q Note that S and E involve low-lying states, while Q depends upon all states, and details of the UV structure O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta Phys. Rev. B 58, 3794 (1998) A. Georges, O. Parcollet, and S. Sachdev Phys. Rev. B 63, (2001)

23 Quantum matter without quasiparticles 1. A solvable model of an ordinary metal 2. A solvable model of a strange metal 3. Holography and charged black holes 4. The (slightly less) strange metal in graphene

24 Quantum matter without quasiparticles 1. A solvable model of an ordinary metal 2. A solvable model of a strange metal 3. Holography and charged black holes 4. The (slightly less) strange metal in graphene

25 Holography x i r J. McGreevy Key idea: ) Implement r as an extra dimension, and map to a local theory in d+2 spacetime dimensions.

26 Holography x i r For a relativistic CFT in d spatial dimensions, the proper length, ds, in the holographic space is fixed by demanding the scale transformation (i =1...d) x i! x i, t! t, ds! ds J. McGreevy

27 Holography x i r This gives the unique metric J. McGreevy ds 2 = 1 r 2 dt 2 + dr 2 + dx 2 i This is the metric of anti-de Sitter space AdS d+2.

28 AdS/CFT correspondence at zero temperature Einstein gravity S E = Z d d+2 x p g apple 1 2apple 2 R + d(d + 1) L 2 AdS d+2 R d,1 Minkowski CFTd+1 x i r ds 2 = L r 2 dr 2 dt 2 + d~x 2

29 AdS/CFT correspondence at non-zero temperature Einstein gravity S E = Z d d+2 x p g apple 1 2apple 2 R + d(d + 1) L 2 AdS-Schwarzschild R d,1 Minkowski CFT d+1 at Hawking temperature T of horizon x i r R 2 apple L dr ds 2 2 = f(r)dt 2 + d~x 2 r f(r) with f(r) =1 (r/r) d+1 and T =(d + 1)/(4 R).

30 AdS/CFT correspondence at non-zero temperature Einstein gravity S E = Z d d+2 x p g apple 1 2apple 2 R + d(d + 1) L 2 AdS-Schwarzschild R d,1 Minkowski CFT d+1 at Hawking temperature T of horizon x i r R Entropy density of CFT d+1, S T d Bekenstein-Hawking entropy density, S BH T d

31 AdS/CFT correspondence at non-zero temperature Einstein gravity S E = Z d d+2 x p g apple 1 2apple 2 R + d(d + 1) L 2 AdS-Schwarzschild R d,1 Minkowski CFT d+1 at Hawking temperature T of horizon x i r R For SU(N) SYM in d = 3, S BH =( 2 /2)N 2 T 3.Butthereis(still) no confirmation of this from a field-theory computation on SYM. Gubser, Klebanov, Peet 96

32 AdS/CFT correspondence at non-zero temperature Einstein gravity S E = Z d d+2 x p g apple 1 2apple 2 R + d(d + 1) L 2 AdS-Schwarzschild R d,1 Minkowski CFT d+1 at Hawking temperature T of horizon x i r R Correspondence in d = 1: S = S BH = 3 ct, where c = 12 L/apple 2 is the central charge of the CFT 2. Strominger 97

33 Einstein-Maxwell theory S EM = Charged black branes Z d d+2 x p g apple 1 2apple 2 R + d(d + 1) R 2 L 2 gf 2 F 2 AdS-Reissner-Nordstrom r Electric flux hqi =0 Quantum matter on the boundary with a variable charge density Q of a global U(1) symmetry. A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, 99

34 Einstein-Maxwell theory S EM = Charged black branes Z d d+2 x p g apple 1 2apple 2 R + d(d + 1) R 2 L 2 gf 2 F 2 AdS-Reissner-Nordstrom r Electric flux hqi =0 Quantum matter on the boundary with a variable charge density Q of a global U(1) symmetry. Realizes a strange metal: a state with an unbroken global U(1) symmetry with a continuously variable charge density, Q, at T = 0 which does not have any quasiparticle excitations. A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, 99

35 General Relativity of charged black branes charge density Q AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 ~x Near-horizon metric is AdS 2, with near-horizon electric field E. As T! 0, there is a non-zero Bekenstein-Hawking entropy, S BH Both E and S BH are determined by Q, and both vanish as Q! 0. Near the boundary, A = µdt,whereµ is the chemical potential

36 Quantum fields on charged black branes T. Faulkner, Hong Liu, J. McGreevy, and D. Vegh, PRD 83, (2011) charge density Q AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt Dirac fermion of mass m = 1 Boundary Green s function of at T = 0! (1 2 ),! > 0 ImG(!) e 2 E (1 2! ),! < 0. where the fermion scaling dimension ~x is a function of m E encodes the particle-hole asymmetry

37 Quantum fields on charged black branes Conformal mapping to T>0 = 0 charge density Q = 1 ds 2 = d 2 /(1 2 / 2 0) (1 2 / 2 0)dt 2 / 2 + d~x 2 Gauge field: A = E(1/ 1/ 0 )dt with 0 =1/(2 T ) Boundary Green s function of at T>0 is fully determined by E G R (!) = where e 2 E = ice i (2 T ) sin( + ) sin( ). i! 2 T + ie i! 2 T + ie Dirac fermion of mass m T. Faulkner, Hong Liu, J. McGreevy, and D. Vegh, PRD 83, (2011) ~x

38 General Relativity of charged black branes Conformal mapping to T>0 = 0 charge density Q ds 2 = d 2 /(1 2 / 2 0) (1 2 / 2 0)dt 2 / 2 + d~x 2 Gauge field: A = E(1/ 1/ 0 )dt with 0 =1/(2 T ) = 1 ~x As T! 0, there is a non-zero Bekenstein-Hawking entropy, S BH. Using Gauss s Law, it can be shown that µ(t )= as T! 0. 2 ET + constant Using a thermodynamic Maxwell relation (also obeyed = T Q

39 General Relativity of charged black branes Conformal mapping to T>0 = 0 charge density Q ds 2 = d 2 /(1 2 / 2 0) (1 2 / 2 0)dt 2 / 2 + d~x 2 Gauge field: A = E(1/ 1/ 0 )dt with 0 =1/(2 T ) = 1 ~x As T! 0, there is a non-zero Bekenstein-Hawking entropy, S BH. Using Gauss s Law, it can be shown that µ(t )= as T! 0. 2 ET + constant Using a thermodynamic Maxwell relation (also obeyed = A. Sen hep-th/ S. Sachdev T Q

40 General Relativity of charged black branes Conformal mapping to T>0 = 0 charge density Q ds 2 = d 2 /(1 2 / 2 0) (1 2 / 2 0)dt 2 / 2 + d~x 2 Gauge field: A = E(1/ 1/ 0 )dt with 0 =1/(2 T ) = 1 ~x As T! 0, there is a non-zero Bekenstein-Hawking entropy, S BH. Using Gauss s Law, it can be shown that µ(t )= as T! 0. 2 ET + constant Using a thermodynamic Maxwell relation (also obeyed = A. Sen hep-th/ S. Sachdev T Q Also obeyed by Wald entropy in higher-derivative gravity.

41 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6,11 Q = 1 X D E c i N c i. i Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < J 8,9,12,14 14 Known equation of state determines E as a function of Q Microscopic zero temperature entropy density, =2 E 12 Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 L = D + m Boundary area A b ; charge density Q ~x Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < 0. Equation of state relating E and Q depends upon the geometry of spacetime far from the AdS 2 Black hole thermodynamics (classical general relativity) =2 E A. Sen, arxiv:hep-th/ ; S. Sachdev, arxiv:

42 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6,11 Q = 1 X D E c i N c i. i Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < J 8,9,12,14 14 Known equation of state determines E as a function of Q Microscopic zero temperature entropy density, =2 E 12 Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 L = D + m Boundary area A b ; charge density Q ~x Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < 0. Equation of state relating E and Q depends upon the geometry of spacetime far from the AdS 2 Black hole thermodynamics (classical general relativity) =2 E A. Sen, arxiv:hep-th/ ; S. Sachdev, arxiv:

43 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6,11 Q = 1 X D E c i N c i. i Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < J 8,9,12,14 14 Known equation of state determines E as a function of Q 12 Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 Microscopic zero temperature Evidence for entropy density, S, AdS 2 =2 E dual of H L = D + m S. Sachdev, arxiv: Boundary area A b ; charge density Q ~x Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < 0. Equation of state relating E and Q depends upon the geometry of spacetime far from the AdS 2 Black hole thermodynamics (classical general relativity) =2 E

44 Quantum matter without quasiparticles 1. A solvable model of an ordinary metal 2. A solvable model of a strange metal 3. Holography and charged black holes 4. The (slightly less) strange metal in graphene

45 Quantum matter without quasiparticles 1. A solvable model of an ordinary metal 2. A solvable model of a strange metal 3. Holography and charged black holes 4. The (slightly less) strange metal in graphene

46 Philip Kim Jesse Crossno Kin Chung Fong Andrew Lucas

47

48 Electron Fermi surface

49 Hole Fermi surface Electron Fermi surface

50 T (K) Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2

51 (slightly less) strange metal T (K) Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2

52 Summary Transport in Strange Metals universal constraints on transport [Forster 70s] [Hartnoll, others] hydrodynamics [Lucas, Sachdev PRB] long time dynamics; renormalized IR fluid emerges few conserved quantities memory matrix appropriate microscopics for cuprates [Lucas 1506] [Donos, Gauntlett 1506] perturbative limit [Lucas JHEP] holography matrix large N theory; non-perturbative computations figure from [Lucas, Sachdev, Physical Review B (2015)] S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007)

53 Transport in Strange Metals Recall that in a Fermi liquid, the Lorenz ratio L = apple/(t ), where apple is the thermal conductivity, and is the conductivity, is given by L = 2 k 2 B 3e 2 In contrast, for a strange metal with a relativistic Hamiltonian, L diverges as the charge density Q! 0 and the impurity momentum relaxation time imp!1: L = H imp T 2 Q 1 (1 + Q 2 imp /(H Q )) 2, where H is the enthalpy density. This divergence happens because in the clean limit, apple diverges as Q! 0, while = Q, an intrinsic, finite, quantum critical conductivity.

54 Transport in Strange Metals Recall that in a Fermi liquid, the Lorenz ratio L = apple/(t ), where apple is the thermal conductivity, and is the conductivity, is given by L = 2 k 2 B 3e 2 In contrast, for a strange metal with a relativistic Hamiltonian, L diverges as the charge density Q! 0, and then the impurity momentum relaxation time imp!1: L = H imp T 2 Q 1 (1 + Q 2 imp /(H Q )) 2, where H is the enthalpy density. This divergence happens because at Q = 0, apple diverges as imp!1,while = Q, an intrinsic, finite, quantum critical conductivity. S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007)

55 Dirac Fluid in Graphene 28 Wiedemann-Franz Law Violations in Experiment Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 [Crossno et al, submitted]

56 Dirac Fluid in Graphene 28 Wiedemann-Franz Law Violations in Experiment Tbath (K) n (10 9 cm -2 ) [Crossno et al, submitted] phonon-limited disorder-limited L / L0 Wiedemann-Franz obeyed

57 Dirac Fluid in Graphene 28 Wiedemann-Franz Law Violations in Experiment Tbath (K) n (10 9 cm -2 ) [Crossno et al, submitted] phonon-limited disorder-limited L / L0 Wiedemann-Franz violated!

58 n (submitted) Observation of the Dirac fluid 5 and the breakdown of the Wiedemann-Franz Disorder Thermallylaw in graphene Limited Limited 10 Liu,1 Achim Harzheim,1 Andrew0Lucas,1 Subir Sachdev,1, 3 Jesse Crossno,1, 2 Jing K. Shi,1 Ke Wang,1 Xiaomeng Philip Kim,1, 2, Takashi Taniguchi,4 Kenji Watanabe,4 Thomas A. Ohki,5 and Kin Temperature Chung Fong5, (K) 9 Temperature (K) Thermal Conductivity (nw/k) 8 6 C 75 K e H (ev/µm2) 50 L/L0 Department of Physics, Harvard University, Cambridge, MA 02138, USA School of Engineering and Applied Sciences, John A. Paulson C Harvard University, Cambridge, MA 02138, USA 8 3 e Perimeter Institute for 16 Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 80 4 H 1-1, Tsukuba, C Ibaraki ,6 Japan 16 Namiki National Institute for Materials Science, h 70 5 Raytheon BBN Technologies, Quantum 12Information Processing Group, Cambridge, Massachusetts 02138, USA 4 60 (Dated: 12July 15, 2015) L / L0 Tbath (K) 1 2 Interactions between particles8 in quantum many-body systems-v can lead to collective behavior +V 8 is the electron-hole h described by hydrodynamics. One such system plasma in graphene 0 near the 40 plasma charge neutrality point which can 4 form a strongly coupled Dirac fluid. This charge neutral T (K) 4 of quasi-relativistic fermions is expected to exhibit a substantial enhancement of the thermal conductivity, due to decoupling of 0charge and heat currents within hydrodynamics. Employing high we report the breakdown of the Wiedemann-Franz law in 10sensitivity 5 0Johnson 5 noise 10 2 thermometry, 15 0 graphene, magnitude n (109with cm-2) a thermal conductivity an order of 6 4 larger 2than the 0 value predicted 2 4by 6 Fermi liquid theory. This result is a signature of the Dirac fluid, and constitutes 10 direct 2 evidence of n (10 cm ) Tdis Tel-ph collective motion in8 a quantum electronic fluid. D 6 FIG. 3. Disorder in and the the Dirac fluid. (A) sample. Minimum e Understanding κthe dynamics4 of many interacting partifluid (DF) FL in the same In cara FL, rier density the as relaxation a functionofofheat temperature for all three sam-recles complicated by many and charge currents is closely 4 is a formidable task in physics, 2 L 0 T ples. At temperature sample limited by disorder. σ coupled degrees of freedom. For electronic transport in lowlated as they areeach carried by theissame quasiparticles. The 0V 0 imp At high all samples 2 matter, stronge interactions can lead to a breakdown of temperature Wiedemann-Franz (WF)become law [19]limited states by thatthermal the elec2 contribution excitations. tronic Dashed lines aretoa a2 guide tothermal the eye. (B)2 The e the Fermi liquid (FL) paradigm metal s conductivity 6 of coherent quasipartiqthree samples 0 Lorentz ratio of all asimp a function ofq bathand temcles scattering o of impurities. In such situations, the is proportional to its electrical conductivity tem40 K 10 mm 1 4 perature. largest WF that violation is seen cleanest complex microscopic dynamics can be coarse-grained to The perature T, such the Lorenz ratioinlthe satisfies 0 20 K 2 momentum, energy, sample. ratio is well a1 hydrodynamic description of and (C) The gate dependence of the Lorentz V e kb 0 and time scalesfit 0 charge transport 50 on long length [1].toHyhydrodynamic theory L of Ref. = [5, 6]. Fits L0of all three(1) T samples 3 ereturn to the Fermi drodynamics has been diverse are shown at 60 K. All Tbath (K)successfully applied to a Vg (V) samples Lorentz ratio L = /(T ) H 1 = T (1 + Q /(H ))

59 Quantum matter without quasiparticles 1. A solvable model of an ordinary metal 2. A solvable model of a strange metal 3. Holography and charged black holes 4. The (slightly less) strange metal in graphene