Interaction effects in graphene
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1 Interaction effects in graphene Markus Müller collaborations with Lars Fritz (Harvard) Subir Sachdev (Harvard) Jörg Schmalian (Iowa) The Abdus Salam ICTP Trieste Sean Hartnoll (Harvard) Pavel Kovtun (Victoria) Taiwan National University, 23 rd December, 2009
2 Outline Relativistic physics in graphene, quantum critical systems and conformal field theories Strong coupling features in collision-dominated transport as inspired by AdS-CFT results Comparison with strongly coupled fluids (via AdS-CFT) Graphene: an almost perfect quantum liquid
3 Dirac fermions in graphene (Semenoff 84, Haldane 88) Honeycomb lattice of C atoms Tight binding dispersion 2 massless Dirac cones in the Brillouin zone: (Sublattice degree of freedom pseudospin)
4 Dirac fermions in graphene (Semenoff 84, Haldane 88) Honeycomb lattice of C atoms Tight binding dispersion 2 massless Dirac cones in the Brillouin zone: (Sublattice degree of freedom pseudospin) Fermi velocity (speed of light ) Coulomb interactions: Fine structure constant
5 Dirac fermions in graphene (Semenoff 84, Haldane 88) Honeycomb lattice of C atoms Tight binding dispersion 2 massless Dirac cones in the Brillouin zone: (Sublattice degree of freedom pseudospin) Fermi velocity (speed of light ) Coulomb interactions: Fine structure constant
6 Relativistic fluid at the Dirac point D. Sheehy, J. Schmalian, Phys. Rev. Lett. 99, (2007). Relativistic plasma physics of interacting particles and holes! Non-degenerate Fermi gas!
7 Relativistic fluid at the Dirac point D. Sheehy, J. Schmalian, Phys. Rev. Lett. 99, (2007). Relativistic plasma physics of interacting particles and holes! Strongly coupled, nearly quantum critical fluid at = 0 Strong coupling!
8 Relativistic fluid at the Dirac point D. Sheehy, J. Schmalian, Phys. Rev. Lett. 99, (2007). Relativistic plasma physics of interacting particles and holes! Strongly coupled, nearly quantum critical fluid at = 0 Strong coupling! Very similar as for quantum criticality (e.g. SIT) and in their associated CFT s
9 Other relativistic fluids: Bismuth (3d Dirac fermions with very small mass) Systems close to quantum criticality (with z = 1) Example: Superconductor-insulator transition (Bose-Hubbard model) Maximal possible relaxation rate! Damle, Sachdev (1996) Bhaseen, Green, Sondhi (2007). Hartnoll, Kovtun, MM, Sachdev (2007) Conformal field theories (critical points) E.g.: strongly coupled Non-Abelian gauge theories (akin to QCD): Exact treatment via AdS-CFT correspondence! C. P. Herzog, P. Kovtun, S. Sachdev, and D. T. Son (2007) Hartnoll, Kovtun, MM, Sachdev (2007)
10 Are Coulomb interactions strong? Fine structure constant (QED concept) Large!
11 Are Coulomb interactions strong? Fine structure constant (QED concept) Large! r s (Wigner crystal concept) Small!? n-independent!
12 Are Coulomb interactions strong? Fine structure constant (QED concept) Large! r s (Wigner crystal concept) Small!? Recall QED/QCD: n-independent! The coupling strength depends on the scale. Different theories have different scale behavior! is the high energy limit of the coupling. But we care about ( )!
13 Are Coulomb interactions strong? Coulomb interactions: Unexpectedly strong! nearly quantum critical!
14 Are Coulomb interactions strong? Coulomb interactions: Unexpectedly strong! nearly quantum critical! RG: (μ = 0) Strong coupling! Cb marginal! Gonzalez et al., PRL 77, 3589 (1996)
15 Are Coulomb interactions strong? Coulomb interactions: Unexpectedly strong! nearly quantum critical! RG: (μ = 0) Strong coupling! Cb marginal! RG flow of v F RG flow of Coulomb only marginally irrelevant for μ = 0! Gonzalez et al., PRL 77, 3589 (1996)
16 Are Coulomb interactions strong? Coulomb interactions: Unexpectedly strong! nearly quantum critical! RG: (μ = 0) Strong coupling! Cb marginal! RG flow of v F RG flow of Coulomb only marginally irrelevant for μ = 0! But: (μ > 0) For Gonzalez et al., PRL 77, 3589 (1996) screening kicks in, short ranged Cb irrelevant
17 Consequences for transport 1. Collision-limited conductivity in clean undoped graphene 2. Emergent relativistic invariance at low frequencies! 3. Graphene is a perfect quantum liquid: very small viscosity!
18 Hydrodynamic approach to transport
19 Strong coupling in undoped graphene MM, L. Fritz, and S. Sachdev, PRB Inelastic scattering rate (Electron-electron interactions) μ >> T: standard 2d Fermi liquid C: Independent of the Coulomb coupling strength!
20 Strong coupling in undoped graphene MM, L. Fritz, and S. Sachdev, PRB Inelastic scattering rate (Electron-electron interactions) Relaxation rate ~ T, like in quantum critical systems! Fastest possible rate! μ >> T: standard 2d Fermi liquid μ < T: strongly coupled relativistic liquid
21 Strong coupling in undoped graphene MM, L. Fritz, and S. Sachdev, PRB Inelastic scattering rate (Electron-electron interactions) Relaxation rate ~ T, like in quantum critical systems! Fastest possible rate! μ >> T: standard 2d Fermi liquid μ < T: strongly coupled relativistic liquid Heisenberg uncertainty principle for well-defined quasiparticles
22 Strong coupling in undoped graphene MM, L. Fritz, and S. Sachdev, PRB Inelastic scattering rate (Electron-electron interactions) Relaxation rate ~ T, like in quantum critical systems! Fastest possible rate! μ >> T: standard 2d Fermi liquid μ < T: strongly coupled relativistic liquid Heisenberg uncertainty principle for well-defined quasiparticles As long as (T) ~ 1, energy uncertainty is saturated, scattering is maximal Nearly universal strong coupling features in transport, similarly as at the 2d superfluid-insulator transition [Damle, Sachdev (1996, 1997)]
23 Strong coupling in undoped graphene MM, L. Fritz, and S. Sachdev, PRB Inelastic scattering rate (Electron-electron interactions) Relaxation rate ~ T, like in quantum critical systems! Fastest possible rate! μ >> T: standard 2d Fermi liquid μ < T: strongly coupled relativistic liquid 2. Elastic scattering rate (Scattering from charged impurities) Subdominant at high T, low disorder
24 Strong coupling in undoped graphene MM, L. Fritz, and S. Sachdev, PRB Inelastic scattering rate (Electron-electron interactions) Relaxation rate ~ T, like in quantum critical systems! Fastest possible rate! μ >> T: standard 2d Fermi liquid μ < T: strongly coupled relativistic liquid 2. Elastic scattering rate (Scattering from charged impurities) Subdominant at high T, low disorder 3. Deflection rate due to magnetic field (Cyclotron frequency of non-interacting particles with thermal energy )
25 Strong coupling in undoped graphene MM, L. Fritz, and S. Sachdev, PRB Inelastic scattering rate (Electron-electron interactions) Relaxation rate ~ T, like in quantum critical systems! Fastest possible rate! μ >> T: standard 2d Fermi liquid μ < T: strongly coupled relativistic liquid 2. Elastic scattering rate (Scattering from charged impurities) Subdominant at high T, low disorder 3. Deflection rate due to magnetic field (Cyclotron frequency of non-interacting particles with thermal energy ) Hydrodynamic regime: (collision-dominated)
26 Hydrodynamics Hydrodynamic collision-dominated regime Long times, Large scales
27 Hydrodynamics Hydrodynamic collision-dominated regime Long times, Large scales Local equilibrium established: Study relaxation towards global equilibrium Slow modes: Diffusion of the density of conserved quantities: Charge Momentum Energy
28 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Energy-momentum tensor Current 3-vector 3-velocity: Dissipative current No energy current Viscous stress tensor (Reynold s tensor)
29 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Energy-momentum tensor Current 3-vector 3-velocity: Dissipative current No energy current Viscous stress tensor (Reynold s tensor) + Thermodynamic relations Gibbs-Duheme 1 st law of thermodynamics
30 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Conservation laws (equations of motion): Charge conservation
31 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Conservation laws (equations of motion): Charge conservation Energy/momentum conservation Coulomb interaction
32 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Conservation laws (equations of motion): Charge conservation Energy/momentum conservation Weak disorder momentum relaxation Coulomb interaction
33 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Conservation laws (equations of motion): Charge conservation Energy/momentum conservation Coulomb interaction Dissipative current and viscous tensor?
34 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Dissipative current and viscous tensor? Heat current Entropy current
35 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Dissipative current and viscous tensor? Heat current Entropy current Positivity of entropy production (Second law):
36 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Dissipative current and viscous tensor? Heat current Entropy current Positivity of entropy production (Second law):
37 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Dissipative current and viscous tensor? Heat current Entropy current Positivity of entropy production (Second law):
38 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Dissipative current and viscous tensor? Heat current Entropy current Positivity of entropy production (Second law): Irrelevant for response at k 0
39 Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Dissipative current and viscous tensor? Heat current Entropy current Positivity of entropy production (Second law): Irrelevant for response at k 0 One single transport coefficient (instead of two)!
40 Meaning of Q? At zero doping (particle-hole symmetry): Q = ( xx imp = 0)<! Interaction-limited conductivity of the pure system! How is it possible that ( xx imp = 0) is finite??
41 Collision-limited conductivity Damle, Sachdev, (1996). Fritz et al. (2008), Kashuba (2008) Finite conductivity in a pure system at particle-hole symmetry ( = 0)!
42 Collision-limited conductivity Finite conductivity in a pure system at particle-hole symmetry ( = 0)! Key: Charge current without momentum! Damle, Sachdev, (1996). Fritz et al. (2008), Kashuba (2008) (particle) (hole) Pair creation/annihilation leads to current decay Finite collision-limited conductivity!
43 Collision-limited conductivity Damle, Sachdev, (1996). Fritz et al. (2008), Kashuba (2008) Finite conductivity in a pure system at particle-hole symmetry ( = 0)! Key: Charge current without momentum (particle) (hole) Pair creation/annihilation leads to current decay Finite collision-limited conductivity! Marginal irrelevance of Coulomb: Maximal possible relaxation rate, set only by temperature
44 Collision-limited conductivity Damle, Sachdev, (1996). Fritz et al. (2008), Kashuba (2008) Finite conductivity in a pure system at particle-hole symmetry ( = 0)! Key: Charge current without momentum (particle) (hole) Pair creation/annihilation leads to current decay Finite collision-limited conductivity! Marginal irrelevance of Coulomb: Maximal possible relaxation rate, set only by temperature Nearly universal conductivity
45 Collision-limited conductivity Damle, Sachdev, (1996). Fritz et al. (2008), Kashuba (2008) Finite conductivity in a pure system at particle-hole symmetry ( = 0)! Key: Charge current without momentum (particle) (hole) Pair creation/annihilation leads to current decay Finite collision-limited conductivity! Marginal irrelevance of Coulomb: Maximal possible relaxation rate, set only by temperature Nearly universal conductivity
46 Collision-limited conductivity Damle, Sachdev, (1996). Fritz et al. (2008), Kashuba (2008) Finite conductivity in a pure system at particle-hole symmetry ( = 0)! Key: Charge current without momentum (particle) (hole) Pair creation/annihilation leads to current decay Finite collision-limited conductivity! Marginal irrelevance of Coulomb: Maximal possible relaxation rate, set only by temperature Nearly universal conductivity Marginal irrelevance of Coulomb:
47 Back to Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Elements discussed so far: Conservation laws (equations of motion): Charge conservation Energy/momentum conservation Dissipative current (relating electrical and energy current)
48 Thermoelectric response S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Charge and heat current: Thermo-electric response etc.
49 Thermoelectric response S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, (2007). Charge and heat current: Thermo-electric response etc. i) Solve linearized conservation laws ii) Read off the response functions from the dynamic response to initial conditions! (see Kadanoff & Martin, 1960)
50 Results from Hydrodynamics
51 Response functions at B=0 Symmetry z -z : Longitudinal conductivity: Collision-limited conductivity at the quantum critical point r = 0 Drude-like conductivity, divergent for Momentum conservation (r 0)!
52 Response functions at B=0 Symmetry z -z : Longitudinal conductivity: Thermal conductivity: Relativistic Wiedemann-Franz-like relations between and in the quantum critical window!
53 B > 0 : Cyclotron resonance E.g.: Longitudinal conductivity Pole in the response Collective cyclotron frequency of the relativistic plasma
54 B > 0 : Cyclotron resonance E.g.: Longitudinal conductivity Pole in the response Collective cyclotron frequency of the relativistic plasma Intrinsic, interaction-induced broadening ( Galilean invariant systems: No broadening due to Kohn s theorem) Observable at room temperature in the GHz regime!
55 Boltzmann Approach MM, L. Fritz, and S. Sachdev, PRB 2008 Recover and refine the hydrodynamic description Describe relativistic-to-fermiliquid crossover
56 Boltzmann approach Boltzmann equation in Born approximation L. Fritz, J. Schmalian, MM, and S. Sachdev, PRB 2008 e.g.: collision-limited conductivity:
57 Relativistic hydrodynamics from microscopics Does relativistic hydro really apply to graphene even though Coulomb interactions break relativistic invariance? Yes! Within weak-coupling theory: Key point: There is a zero ( momentum ) mode of the collision integral due to translational invariance of the interactions The dynamics of the zero mode under an AC driving field reproduces relativistic hydrodynamics at low frequencies.
58 What to do beyond the weak coupling (Boltzmann) approximation [ ( ) 0]??
59 What to do beyond the weak coupling (Boltzmann) approximation [ ( ) 0]?? Answer until recently: Not much at all!
60 What to do beyond the weak coupling (Boltzmann) approximation [ ( ) 0]?? Answer until recently: Not much at all! Newest progress from string theory: 1) Look at similar theories which are very strongly coupled, but can be solved exactly 2) Try to extract the generally valid, universal part of the result and use it as a guide
61 What to do beyond the weak coupling (Boltzmann) approximation [ ( ) 0]?? Answer until recently: Not much at all! Newest progress from string theory: 1) Look at similar theories which are very strongly coupled, but can be solved exactly 2) Try to extract the generally valid, universal part of the result and use it as a guide Warning: There is no AdS-CFT mapping for graphene itself!
62 Compare graphene to: Strongly coupled relativistic liquids S. Hartnoll, P. Kovtun, MM, S. Sachdev (2007) Obtain exact results via string theoretical AdS CFT correspondence Response functions in particular strongly coupled relativistic fluids (for maximally supersymmetric Yang Mills theories with colors):
63 Compare graphene to: Strongly coupled relativistic liquids S. Hartnoll, P. Kovtun, MM, S. Sachdev (2007) Obtain exact results via string theoretical AdS CFT correspondence Response functions in particular strongly coupled relativistic fluids (for maximally supersymmetric Yang Mills theories with colors): Confirm the structure of the hydrodynamic response functions such as ( ). Calculate the transport coefficients for a strongly coupled theory! SUSY - SU(N): ; Interpretation: effective degrees of freedom, strongly coupled: T = O(1)
64 AdS/CFT correspondence The quantum theory of a black hole in a 3+1-dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space Maldacena, Gubser, Klebanov, Polyakov, Witten
65 AdS/CFT correspondence The quantum theory of a black hole in a 3+1-dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space A 2+1 dimensional system at its quantum critical point Maldacena, Gubser, Klebanov, Polyakov, Witten
66 AdS/CFT correspondence The quantum theory of a black hole in a 3+1-dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space A 2+1 dimensional system at its quantum critical point Maldacena, Gubser, Klebanov, Polyakov, Witten
67 Black hole temperature = temperature of quantum critical system AdS/CFT correspondence The quantum theory of a black hole in a 3+1-dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space A 2+1 dimensional system at its quantum critical point Maldacena, Gubser, Klebanov, Polyakov, Witten
68 AdS/CFT correspondence The quantum theory of a black hole in a 3+1-dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space Black hole entropy = Entropy of quantum criticality A 2+1 dimensional system at its quantum critical point Strominger, Vafa
69 Quantum critical dynamics = waves in curved space AdS/CFT correspondence The quantum theory of a black hole in a 3+1-dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space A 2+1 dimensional system at its quantum critical point Maldacena, Gubser, Klebanov, Polyakov, Witten
70 AdS/CFT correspondence The quantum theory of a black hole in a 3+1-dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space Dissipation in quantum criticality = waves falling into black hole A 2+1 dimensional system at its quantum critical point Kovtun, Policastro, Son
71 Au+Au collisions at RHIC Quark-gluon plasma can be described by QCD (nearly conformal, critical theory) Extremely low viscosity fluid!
72 Au+Au collisions at RHIC Quark-gluon plasma can be described by QCD (nearly conformal, critical theory) Extremely low viscosity fluid! SUSY - SU(N): This IS an extremely low value! Is there a lowest possible value, or a most perfect liquid?
73 Graphene a nearly perfect liquid! MM, J. Schmalian, and L. Fritz, (PRL 2009) Anomalously low viscosity (like quark-gluon plasma) Heisenberg Measure of strong coupling: Conjecture from AdS-CFT:
74 Graphene a nearly perfect liquid! MM, J. Schmalian, and L. Fritz, (PRL 2009) Anomalously low viscosity (like quark-gluon plasma) Heisenberg Measure of strong coupling: Doped Graphene & Fermi liquids: (Khalatnikov etc) Conjecture from AdS-CFT:
75 Graphene a nearly perfect liquid! MM, J. Schmalian, and L. Fritz, (PRL 2009) Anomalously low viscosity (like quark-gluon plasma) Heisenberg Measure of strong coupling: Doped Graphene & Fermi liquids: (Khalatnikov etc) Undoped Graphene: Conjecture from AdS-CFT: Boltzmann-Born Approximation:
76 Graphene T. Schäfer, Phys. Rev. A 76, (2007). A. Turlapov, J. Kinast, B. Clancy, Le Luo, J. Joseph, J. E. Thomas, J. Low Temp. Phys. 150, 567 (2008)
77 Electronic consequences of low viscosity? MM, J. Schmalian, L. Fritz, (PRL 2009) Expected viscous effects on conductance in non-uniform current flow: Decrease of conductance with length scale L
78 Electronic consequences of low viscosity? MM, J. Schmalian, L. Fritz, (PRL 2009) Electronic turbulence in clean, strongly coupled graphene? (or at quantum criticality!) Reynolds number:
79 Electronic consequences of low viscosity? MM, J. Schmalian, L. Fritz, (PRL 2009) Electronic turbulence in clean, strongly coupled graphene? (or at quantum criticality!) Reynolds number: Strongly driven mesoscopic systems: (Kim s group [Columbia]) Complex fluid dynamics! (pre-turbulent flow) New phenomenon in an electronic system!
80 Summary Strong coupling Undoped graphene is strongly coupled in a large temperature window! Nearly universal strong coupling features in transport; many similarities with strongly coupled critical fluids (described by AdS-CFT) Emergent relativistic hydrodynamics at low frequency Graphene: Nearly perfect quantum liquid! Possibility of complex (turbulent?) current flow at high bias
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