Relativistic magnetotransport in graphene

Transcription

1 Relativistic magnetotransport in graphene Markus Müller in collaboration with Lars Fritz (Harvard) Subir Sachdev (Harvard) Jörg Schmalian (Iowa) Landau Memorial Conference June 6, 008

2 Outline Relativistic physics in graphene, quantum critical systems and conformal field theories Relativistic signatures in magnetotransport: el.+th. conductivity, Peltier, Nernst effect etc. Hydrodynamic description Collective, collision-broadened cyclotron resonance Boltzmann equation Recover and refine hydrodynamics with Boltzmann Describe relativistic-to-fermi liquid crossover Go beyond hydrodynamics

3 Dirac fermions in graphene Honeycomb lattice of C atoms (Semenoff 84, Haldane 88)

4 Dirac fermions in graphene (Semenoff 84, Haldane 88) Honeycomb lattice of C atoms Tight binding dispersion massless Dirac cones in the Brillouin zone: (Sublattice degree of freedom pseudospin) Close to the two Fermi points K, K : H v F ( p K) σ sublattice E k = v F k K

5 Dirac fermions in graphene (Semenoff 84, Haldane 88) Honeycomb lattice of C atoms Tight binding dispersion massless Dirac cones in the Brillouin zone: (Sublattice degree of freedom pseudospin) Fermi velocity (speed of light ) Close to the two Fermi points K, K : v F m s 6 c 300 H v F ( p K) σ sublattice E k = v F k K

6 Dirac fermions in graphene (Semenoff 84, Haldane 88) Honeycomb lattice of C atoms Tight binding dispersion massless Dirac cones in the Brillouin zone: (Sublattice degree of freedom pseudospin) Fermi velocity (speed of light ) Close to the two Fermi points K, K : Coulomb interactions: Fine structure constant v F m s 6 c 300 H v F ( p K) σ sublattice E k = v F k K e α ε h v F = O ( 1)

7 Relativistic fluid at the Dirac point Expect relativistic plasma physics of interacting particles and holes! Quantum critical n ( 1+ α ln[ Λ n] ) D. Sheehy, J. Schmalian, Phys. Rev. Lett. 99, 6803 (007).

8 Transport and phase diagram Expect relativistic plasma physics of interacting particles and holes!

9 Transport and phase diagram Expect relativistic plasma physics of interacting particles and holes! T = T imp ~ ρ imp

10 Transport and phase diagram Expect relativistic plasma physics of interacting particles and holes! T = Interaction limited transport Disorder limited transport T imp ~ ρ imp

11 Transport and phase diagram Expect relativistic plasma physics of interacting particles and holes! Hydrodynamics? ω << T Interaction limited transport T = Disorder limited transport T imp ~ ρ imp

12 Conductivity in and across the relativistic regime? J.-H. Chen et al. Nat. Phys. 4, 377 (008).

13 Conductivity in and across the relativistic regime? J.-H. Chen et al. Nat. Phys. 4, 377 (008). + Magnetotransport? e.g., Hall, Nernst effect?

14 Other relativistic fluids: Bismuth (3d Dirac fermions with very small mass) Effective theories close to quantum phase transitions Conformal field theories E.g.: strongly coupled Non-Abelian gauge theories (QCD): tretament via AdS-CFT

15 Low energy effective theory at quantum phase transitions Relativistic effective field theories z = 1; arise often due to particle-hole symmetry Example: Superconductor-insulator transition (SIT) Bhaseen, Green, Sondhi (PRL 07). Hartnoll, Kovtun, MM, Sachdev (PRB 07)

16 SI-transition: Bose Hubbard model Bose-Hubbard model + H = t b j bi + U ni ij i Coupling t g tunes the SI-transition U i n i QCP

17 SI-transition: Bose Hubbard model Bose-Hubbard model + H = t b j bi + U ni ij i i n i QCP Coupling t g U tunes the SI-transition ψ = 0 ψ 0 Effective action around g c ( = 0): e.g.

18 SI-transition: Bose Hubbard model Bose-Hubbard model + H = t b j bi + U ni ij i i n i QCP Coupling t g U tunes the SI-transition ψ = 0 ψ 0 Effective action around g c ( = 0): e.g. Relativistic field theory in d=+1

19 Questions Transport characteristics of the relativistic plasma in lightly doped graphene and close to quantum criticality? Relativistic, hydrodynamic regime T = T imp ~ ρ imp

20 Questions Transport characteristics of the relativistic plasma in lightly doped graphene and close to quantum criticality? How does the relativistic regime connect to Fermi liquid behavior at large doping? Relativistic, hydrodynamic regime T = T imp ~ ρ imp

21 Questions Transport characteristics of the relativistic plasma in lightly doped graphene and close to quantum criticality? How does the relativistic regime connect to Fermi liquid behavior at large doping? What is the range of validity of relativistic magneto-hydrodynamics? Beyond hydrodynamics? Relativistic, hydrodynamic regime T = T imp ~ ρ imp

22 Model of graphene Graphene with Coulomb interactions and disorder H = H H1 H dis Tight binding kinetic energy

23 Model of graphene Graphene with Coulomb interactions and disorder H = H H1 H dis B-field: r r r i i ea/ c Tight binding kinetic energy

24 Model of graphene Graphene with Coulomb interactions and disorder H = H H1 H dis B-field: r r r i i ea/ c Tight binding kinetic energy Coulomb interactions

25 Model of graphene Graphene with Coulomb interactions and disorder H = H H1 H dis B-field: r r r i i ea/ c Tight binding kinetic energy Coulomb interactions Coulomb marginally irrelevant! e α ε hv r F = O ( 1) RG:

26 Model of graphene Graphene with Coulomb interactions and disorder H = H H1 H dis B-field: r r r i i ea/ c Tight binding kinetic energy Coulomb interactions Coulomb marginally irrelevant! Screening neglected (down by factor α)

27 Model of graphene Graphene with Coulomb interactions and disorder H = H H1 H dis B-field: r r r i i ea/ c Tight binding kinetic energy Coulomb interactions Coulomb marginally irrelevant! Screening neglected (down by factor α) Disorder: charged impurities

28 Time scales MM, L. Fritz, and S. Sachdev, cond-mat Inelastic scattering rate (Electron-electron interactions) τ 1 ee ~ α k B T 1 h max 1, [ T ] Relativistic regime ( < T): Relaxation rate set by temperature, like in quantum critical systems! τ 1 ~ α ee k B T h

29 Time scales MM, L. Fritz, and S. Sachdev, cond-mat Inelastic scattering rate (Electron-electron interactions) τ 1 ee ~ α k B T 1 h max 1, [ T ] Relativistic regime ( < T): Relaxation rate set by temperature, like in quantum critical systems! τ 1 ~ α ee k B T h. Elastic scattering rate (Scattering from charged impurities) Subdominant at high T τ 1 imp ~ ( Ze ε ) h ρ imp 1 max [ T, ]

30 Time scales MM, L. Fritz, and S. Sachdev, cond-mat Inelastic scattering rate (Electron-electron interactions) τ 1 ee ~ α k B T 1 h max 1, [ T ] Relativistic regime ( < T): Relaxation rate set by temperature, like in quantum critical systems! τ 1 ~ α ee k B T h. Elastic scattering rate (Scattering from charged impurities) Subdominant at high T τ 1 imp ~ ( Ze ε ) h ρ imp 1 max [ T, ] 3. Deflection rate due to magnetic field (Cyclotron frequency of non-interacting particles with typical thermal energy ) τ 1 B ~ ω typ c ~ ebv F max [ T, ]

31 Regimes MM, L. Fritz, and S. Sachdev, cond-mat Hydrodynamic regime: (collision-dominated) τ >> τ, τ, ω ee imp B

32 Regimes MM, L. Fritz, and S. Sachdev, cond-mat Hydrodynamic regime: (collision-dominated) τ >> τ, τ, ω ee imp B. Ballistic magnetotransport (large field limit) τ > τ >> τ, ω B ee imp

33 Regimes MM, L. Fritz, and S. Sachdev, cond-mat Hydrodynamic regime: (collision-dominated) τ >> τ, τ, ω ee imp B. Ballistic magnetotransport (large field limit) τ > τ >> τ, ω B ee imp 3. Disorder limited transport (inelastic scattering ineffective due to nearly conserved momentum) >> T τ < > τ 1 ee 1 imp

34 Hydrodynamic Approach

35 Hydrodynamics Hydrodynamic collision-dominated regime Long times, Large scales t >> τ ee τ >> τ, τ, ω ee imp B

36 Hydrodynamics Hydrodynamic collision-dominated regime Long times, Large scales t >> τ ee τ >> τ, τ, ω ee imp B Local equilibrium established: T loc ( r), loc ( r) ; u r loc ( r) Study relaxation towards global equilibrium Slow modes: Diffusion of the density of conserved quantities: Charge Momentum Energy

37 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Energy-momentum tensor Current 3-vector u : Energy velocity: u = ( 1,0, 0) ν : Dissipative current ( heat current ) ρ ρux ρu y No energy current ε P P τ ν : Viscous stress tensor (Reynold s tensor)

38 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Energy-momentum tensor Current 3-vector u : Energy velocity: u = ( 1,0, 0) ν : Dissipative current ( heat current ) ρ ρux ρu y No energy current ε P P τ ν : Viscous stress tensor (Reynold s tensor) + Thermodynamic relations

39 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Conservation laws (equations of motion): Charge conservation

40 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Conservation laws (equations of motion): Charge conservation Energy/momentum conservation 1 τ imp 0 T ν δ 0 r E r π = ik ρ r Coulomb interaction k k

41 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Conservation laws (equations of motion): Charge conservation Energy/momentum conservation 1 τ imp 0 T ν δ 0 r E r π = ik ρ r Coulomb interaction k k Heat current and viscous tensor?

42 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Heat current and viscous tensor? Landau-Lifschitz, Relat. plasma physics Heat current Q Entropy current = ( ε + P) u J S = Q T

43 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current S τ ν ν A α Q Entropy current ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = = const. B ( ε + P) u J S ν = Q T + const. δ ν B α α τ αβ 0

44 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current S τ ν ν A α Q Entropy current ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = = const. B ( ε + P) u J S ν = Q T + const. δ ν B α α τ αβ 0

45 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current Q Entropy current S τ ν ν A α = ( ε + P) u J S = Q ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = const. B ν T + const. δ ν B α α τ αβ 0

46 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current Q Entropy current S τ ν ν A α = ( ε + P) u J S = Q ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = const. B ν T + const. δ ν B α α τ αβ 0 B small!

47 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current Q Entropy current S τ ν ν A α = ( ε + P) u J S = Q ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = const. B ν T + const. δ ν B α α τ αβ 0 Irrelevant for response at k 0

48 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current Q Entropy current S τ ν ν A α = ( ε + P) u J S = Q ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = const. B ν T + const. δ ν B α α τ αβ 0 Irrelevant for response at k 0 One single transport coefficient (instead of two)!

49 Meaning of σ Q? Dimension of electrical conductivity At zero doping (particle-hole symmetry): σ Q ( ) = σ ρ = 0 imp xx = Universal d.c. conductivity of the pure system σ xx ( ρ = 0 ) imp Why is finite??

50 Universal conductivity σ Q Particle-hole symmetry (ρ = 0) K. Damle, S. Sachdev, (1996). Key: Charge current without momentum (energy current)! (particle) (hole) J r 0, P r = 0 Finite quantum critical conductivity! Pair creation/annihilation leads to current decay

51 Universal conductivity σ Q Quantum critical situation: Particle-hole symmetry (ρ = 0) Key: Charge current without momentum (energy current) (particle) (hole) J r 0, Finite quantum critical conductivity! As in quantum criticality: Relaxation time set by temperature alone P r = 0 Pair creation/annihilation leads to current decay τ K. Damle, S. Sachdev, (1996). ee h α k B T

52 Universal conductivity σ Q Quantum critical situation: Particle-hole symmetry (ρ = 0) Key: Charge current without momentum (energy current) (particle) (hole) J r 0, = 0 Finite quantum critical conductivity! As in quantum criticality: Relaxation time set by temperature alone P r Pair creation/annihilation leads to current decay τ K. Damle, S. Sachdev, (1996). ee h α k B T Universal conductivity σ Drude e = ρτ m σ Q ~ k B e T v e ( kbt ) h 1 e ~ ( hv) α k T α h B

53 Universal conductivity σ Q Quantum critical situation: Particle-hole symmetry (ρ = 0) Key: Charge current without momentum (energy current) (particle) (hole) J r 0, Finite quantum critical conductivity! As in quantum criticality: Relaxation time set by temperature alone P r = 0 Pair creation/annihilation leads to current decay τ K. Damle, S. Sachdev, (1996). ee h α k B T σ Universal conductivity Drude e = ρτ σ m Q ~ k e T v B e ( kbt ) h 1 e ~ ( hv) α k T α h B Exact (Boltzmann) 0.76 e σ Q ( = 0) = α h

54 Universal conductivity σ Q Quantum critical situation: Particle-hole symmetry (ρ = 0) Key: Charge current without momentum (energy current) (particle) (hole) J r 0, Finite quantum critical conductivity! As in quantum criticality: Relaxation time set by temperature alone P r = 0 Pair creation/annihilation leads to current decay τ K. Damle, S. Sachdev, (1996). ee h α k B T σ Universal conductivity Drude e = ρτ σ m Q ~ k e T v B e ( kbt ) h 1 e ~ ( hv) α k T α h B Marginal irrelevance of Coulomb: Exact (Boltzmann) 0.76 e σ Q ( = 0) = α h 4 α log ( Λ T )

55 Thermoelectric response S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Charge and heat current: J Q = ρ ν = u ( ) ε + P u J Thermo-electric response etc.

56 Thermoelectric response S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (007). Charge and heat current: J Q = ρ ν = u ( ) ε + P u J Thermo-electric response etc. i) Solve linearized hydrodynamic equations ii) Read off the response functions (Kadanoff & Martin 1960)

57 Results from Hydrodynamics

58 Response functions at B=0 Symmetry z -z : σ xy = α xy = κ xy = 0 Longitudinal conductivity: Universal conductivity at the quantum critical point ρ = 0 Drude-like conductivity, divergent for Momentum conservation (ρ 0)! τ, ω 0, ρ 0

59 Response functions at B=0 Symmetry z -z : σ xy = α xy = κ xy = 0 Longitudinal conductivity: Coulomb correction g = π e ( ) +O( k )

60 Response functions at B=0 Symmetry z -z : σ xy = α xy = κ xy = 0 Longitudinal conductivity: Coulomb correction g = π e ( ) +O( k ) Thermal conductivity: Relativistic Wiedemann-Franz-like relations between σ and κ in the quantum critical window!

61 Response functions at B=0 Symmetry z -z : σ xy = α xy = κ xy = 0 Longitudinal conductivity: Coulomb correction g = π e ( ) +O( k ) Thermopower: 3e π 1 k T B α dσ xx xx d Relativistic fluid! Only valid in the Fermi liquid regime, but violated in the relativistic window. T

62 B > 0 : Cyclotron resonance E.g.: Longitudinal conductivity Poles in the response ω = ± ω QC c iγ i τ Collective cyclotron frequency of the relativistic plasma ω QC c ρ B = ω FL c ( ε + P) v m F e B =

63 B > 0 : Cyclotron resonance E.g.: Longitudinal conductivity Poles in the response ω = ± ω QC c iγ i τ Collective cyclotron frequency of the relativistic plasma ω QC c ρ B = ω FL c ( ε + P) v m F e B = Intrinsic, interaction-induced broadening ( Galilean invariant systems: No broadening due to Kohn s theorem) γ = σ Q B ( ε + P) / v F

64 B > 0 : Cyclotron resonance Longitudinal conductivity Poles in the response ω = ± ω QC c iγ i τ

65 Can the resonance be observed? ω = ± ω iγ i c τ v F = c / m / s Conditions to observe collective cyclotron resonance Collison-dominated regime Small broadening Quantum critical regime } High T: no Landau quantization hω c << α k B T ρ ρ E 1 γ, τ < th = ω c ( kbt ) ( h ) v F eb hc LL = hv F << k B T Parameters: T 300K B 0.1T 11 ρ 10 cm 13 1 ω c 10 s

66 Does relativistic hydrodynamics apply? Do T and not break relativistic invariance? Validity at large chemical potential? Beyond linearization in magnetic field? Treatment of disorder?

67 Boltzmann Approach MM, L. Fritz, and S. Sachdev, cond-mat Recover and refine the hydrodynamic description Describe relativistic-to-fermiliquid crossover Go beyond hydrodynamics

68 σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± a.) +, i +, i +, i +, i +, i +, i +, i +, j +(N 1) +(N 1), i, i, i, i, j, j, j, i b.) +, i +, i +, i +, i +, i +, i 1 +(N 1) +, i +, i +, i +, i +, j +, j

69 σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± eq Linearization: f (, t) = f ( k, t) δf ( k, t) ± k ± + ±

70 σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± eq Linearization: f (, t) = f ( k, t) δf ( k, t) ± k ± + ± Great simplification: Divergence of forward scattering amplitude in d [ Amp ] Equilibration along unidimensional spatial directions At p-h symmetry: ( ) ; = C( t) eq ( k, t) f k, + δ ( t) f ± = ± eq δ E k k

71 σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± eq Linearization: f (, t) = f ( k, t) δf ( k, t) ± k ± + ± Great simplification: Divergence of forward scattering amplitude in d [ Amp ] Equilibration along unidimensional spatial directions At p-h symmetry: ( ) ; = C( t) eq ( k, t) f k, + δ ( t) f ± = ± eq δ E k k σ Q ( = 0) 0.76 α e h

72 σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± General analysis in linear response:

73 σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± General analysis in linear response: Central element of analysis: Choose appropriate basis λ=± = ( k, t) a n φ ( λ k) g, Momentum or energy-current mode Charge current mode n n

74 σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± General analysis in linear response: = Central element of analysis: Choose appropriate basis g ( k, t) a n φ ( λ, k) λ=± Momentum or energy-current mode Charge current mode n Relativistic dispersion ensures that ϕ 0 only couples to ϕ 1 for clean systems! n

75 Conductivity: σ Q L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat General doping: Clean system: Precise expression for σ Q!

76 Conductivity: σ Q L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat General doping: Clean system: Precise expression for σ Q! Gradual disappearance of relativistic physics σ Q ( = 0) σ Q 0.76 α T ( ) e h Will appear in all Boltzmann formulae below!

77 Conductivity: crossover General doping: L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Lightly disordered system: Correction to hydrodynamics

78 Conductivity: crossover General doping: L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Lightly disordered system: Correction to hydrodynamics Fermi liquid regime: Α Σ xx Ρ Ρ Ρ imp

79 Conductivity: crossover General doping: L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Lightly disordered system: Correction to hydrodynamics Fermi liquid regime: Α Σ xx Ρ 4 3 Impurity limited conductivity Universal Cb limited conductivity Ρ Ρ imp

80 Conductivity: crossover General doping: L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat Lightly disordered system: Correction to hydrodynamics Fermi liquid regime: J.-H. Chen et al. Nat. Phys. 4, 377 (008). Maryland group ( 08): Graphene Α Σ xx Ρ Universal Cb limited conductivity Ρ Ρ imp Impurity limited conductivity

81 Magnetotransport Strategy: describe the slow dynamics of the momentum mode ϕ 0 in very weak disorder and moderate magnetic field

82 Magnetotransport Strategy: describe the slow dynamics of the momentum mode ϕ 0 in very weak disorder and moderate magnetic field Result: Full thermoelectric response (for general B) obtained in terms of thermodynamic quantities + only independent transport coefficients (collision matrix elements)!

83 Magnetotransport Strategy: describe the slow dynamics of the momentum mode ϕ 0 in very weak disorder and moderate magnetic field Result: Full thermoelectric response (for general B) obtained in terms of thermodynamic quantities + only independent transport coefficients (collision matrix elements)! At small B, one transport coefficient is subdominant Relativistic hydrodynamics with only one transport coefficient σ Q is recovered! 1 1 >>τ τ ee B Corrections to hydrodynamics

84 Magnetotransport Strategy: describe the slow dynamics of the momentum mode ϕ 0 in very weak disorder and moderate magnetic field Result: Full thermoelectric response (for general B) obtained in terms of thermodynamic quantities + only independent transport coefficients (collision matrix elements)! At small B, one transport coefficient is subdominant Relativistic hydrodynamics with only one transport coefficient σ Q is recovered! τ >>τ 1 ee 1 B Cyclotron resonance: Hydrodynamics Boltzmann

85 Cyclotron resonance revisited Crossover to Fermi liquid regime: Semiclassical ω c recovered at >> T Broadening goes to zero - Kohn s theorem recovered: Non-broadening of the resonance for a single parabolic band. γ σ Q >> T ( ) 0

86 Cyclotron resonance revisited Beyond hydrodynamics: Towards ballistic magnetotransport = T Large fields τ > τ >> τ, ω B ee imp Resonance Damping

87 Strongly coupled liquids Same trends as in exact (AdS-CFT) results for strongly coupled relativistic fluids! Graphene Resonance S. Hartnoll, C. Herzog (007) Damping N = 4 SUSY SU(N) gauge theory [flows to CFT at low energy] Exact Exact MHD B MHD B

88 Summary Relativistic physics in graphene and quantum critical systems Hydrodynamic description: collective cyclotron resonance in the relativistic regime covariance: 6 frequency dependent response functions given by thermodynamics and only one parameter σ Q. Boltzmann approach Confirmed and refined hydrodynamic description Understood relativistic-to-fermi liquid crossover: From universal Coulomb-limited to disorder-limited linear conductivity in graphene From collective-broadened to semiclasscial sharp cyclotron resonance Beyond hydrodynamics: describe large fields and disorder