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AdS/CFT and the Fermions of condensed matter Jan Zaanen QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 1 The Gross list: the 14 Big Questions 1. The origin of the universe? 2. What is dark matter? 2004 11. What is space-time? 14. New states of matter: are there generic non-Fermi liquid states of interacting condensed matter? Solution of the Fermion sign problem?? ? QuickTime™ and a decompressor are needed to see this picture. Photoemission spectrum ‘Avatars of metallic phases’ Kachru et al., arXiv:0909.2639 4 The ‘fermionic’ excitement (a) Hydrodynamics: ‘Planckian dissipation’ (Hartnoll, Sachdev, Harvard) (b) The emergent (heavy) Fermi liquid (Schalm, Leiden) (c) The ‘marginal-’, ‘critical’ non-Fermi-liquids (Liu, MIT) (d) Holographic superconductivity (Hartnoll, Harvard) 5 Plan 1. Quantum critical electron systems and the great questions of condensed matter physics 2. Hairy black holes versus zero temperature entropy in quantum critical heavy fermion systems. 3. Fermion signs versus the AdS/CFT emergent (non) Fermi-liquids. 6 Quantum critical matter Quark gluon plasma High Tc superconductors Heavy fermions Metamagnetic ruthenate T Tc Quantum critical 4 (P 2 nd ) nt poi l a ic crit Tri Li ne a i ti c cr of FM 2 o in lp ts 1s t 4 pq 0 3 1 2 2 3 1 P 7 H The quantum in the kitchen: Landau’s miracle Kinetic energy Electrons are waves Fermi energy Pauli exclusion principle: every state occupied by one electron Fermi momenta k=1/wavelength Fermi surface of copper Unreasonable: electrons strongly interact !! Landau’s Fermi-liquid: the highly collective low energy quantum excitations are like electrons that do not interact. 8 BCS theory: fermions turning into bosons Fermi-liquid + attractive interaction Bardeen Cooper Schrieffer Quasiparticles pair and Bose condense: Ground state D-wave SC: Dirac spectrum BCS k uk v kck ck vac. 9 ‘Shankar/Polchinski’ functional renormalization group UV: weakly interacting Fermi gas Integrate momentum shells: functions of running coupling constants interaction Fermi sphere All interactions (except marginal Hartree) irrelevant => Scaling limit might be perfectly ideal Fermi-gas 10 The end of weak coupling interaction Strong interactings: Fermi gas as UV starting point does not make sense! => ‘emergent’ Fermi liquid fixed point remarkably resilient (e.g. 3He) Fermi sphere => Non Fermi-liquid/non ‘HartreeFock’ (BCS etc) states of fermion matter? 11 Fermion sign problem Imaginary time path-integral formulation Boltzmannons or Bosons: Fermions: integrand non-negative negative Boltzmann weights probability of equivalent classical system: (crosslinked) ringpolymers non probablistic: NP-hard problem (Troyer, Wiese)!!! Fractal Cauliflower (romanesco) Quantum critical cauliflower Quantum critical cauliflower Quantum critical cauliflower Quantum critical cauliflower Quantum criticality or ‘conformal fields’ 18 Quantum Phase transitions Quantum scale invariance emerges naturally at a zero temperature continuous phase transition driven by quantum fluctuations: JZ, Science 319, 1205 (2008) 19 Fermionic renormalization group The Magic of AdS/CFT! QuickTime™ and a decompressor are needed to see this picture. boundary: d-dim space-time strings Black holes Wilson-Fisher RG: based on Boltzmannian statistical physics gluons quarks Hawking radiation 20 The ‘fermionic’ excitement (a) Hydrodynamics: ‘Planckian dissipation’ (b) The emergent (heavy) Fermi liquid (c) The ‘marginal-’, ‘critical’ non-Fermi-liquids (d) Holographic superconductivity 21 Quantum critical matter Quark gluon plasma High Tc superconductors Heavy fermions Metamagnetic ruthenate T Tc Quantum critical 4 (P 2 nd ) nt poi l a ic crit Tri Li ne a i ti c cr of FM 2 o in lp ts 1s t 4 pq 0 3 1 2 2 3 1 P 22 H Quantum criticality and Planckian dissipation Quark-gluon plasma: ‘minimal viscosity’ => absorption cross section of gravitons by the black hole (AdS/CFT) s 4 k B 1 T Quantum criticality => ‘Quantum limit of dissipation’: relaxation (= entropy production) time kB T Linear resistivity in cuprates governed by ‘Planckian’ relaxation Van der Marel et al., Nature 425, 271 (2003); JZ, Nature 430, (2004), Cooper et al., Science 323, 603 (2009) 512 23 Dissipation = absorption of classical waves by Black hole! Hartnoll-Son-Starinets (2002): Viscosity: absorption cross section of abs0 gravitons by black hole 16G = area of horizon (GR theorems) Entropy densitys: Bekenstein-Hawking BH entropy = area of horizon Universal viscosity-entropy ratio for CFT’s with gravitational dual limited in large N by: s 4 k B 1 24 The quark-gluon plasma Relativistic Heavy Ion Collider Quark-gluon ‘fireball’ 25 The tiny viscosity of the QuarkGluon plasma 4kB s QG plasma: within 20% of the AdS/CFT viscosity! Quantum critical hydrodynamics: Planckian relaxation time Relaxation time : time it takes to convert work in entropy. p p s T Viscosity: Entropy density: “Planckianviscosity” s T kB T kB ?? 1 kB T Planckian relaxation time = the shortest possible conditions that can relaxation time under equilibrium only be reached when the quantum dynamics is scale invariant !! 27 Twenty three years ago … Mueller Bednorz Ceramic CuO’s, likeYBa2Cu3O7 Superconductivity jumps to ‘high’ temperatures 28 Graveyard of Theories Mott De Gennes Mueller Laughlin Bednorz Anderson Wilczek Ginzburg Abrikosov Schrieffer Leggett QuickTime™ and a decompressor are needed to see this picture. Lee Yang 29 Phase diagram high Tc superconductors ‘Stripy stuff’, spontaneous currents, phase fluctuations .. Mystery quantum critical metal The return of normalcy BCS k uk v kck ck vac. JZ, Science 315, 1372 (2007) 30 Divine resistivity 31 Critical Cuprates are Planckian Dissipators van der Marel, JZ, … Nature 2003: Optical conductivity QC cuprates Frequency less than temperature: 2 1 pr r 1(,T) , r A 2 2 4 1 r kB T [ ] const.(1 A [ 2 kB T1 kB T ]2 ) A= 0.7: the normal state of optimallly doped cuprates is a Planckian dissipator! 32 Divine resistivity ?! 33 Planckian dissipation Quark gluon plasma 4 k B T 1 s Viscosity bound Planckian relaxation time: High Tc superconductors kB T Linear resistivity normal state (JZ, Nature 430,512) : T Quantum critical 1 1 2 p 0.7 kB T Tc is set by competition between superconductor and critical normal state (e.g. arXiv:0905.1225) 34 Why the Wilsonian renormalization group fails … Superconductorinsulator QCP PRL 95, 107002 (2005) Chamon Phillips (a) Charge conservation (‘hydrodynamics’) imposes engineering scaling dimensions on current (b) Scale invariance: assume one diverging length scale. d 2 z e kB T ,T c 2 1 DC ?? T d 2 z kB T e DC T 0 c kB T 2 d =2 or 3 implies that z < 0 !? 35 The ‘fermionic’ excitement (a) Hydrodynamics: ‘Planckian dissipation’ (b) The emergent (heavy) Fermi liquid (c) The ‘marginal-’, ‘critical’ non-Fermi-liquids (d) Holographic superconductivity 36 Watching electrons: photoemission Kinetic energy Fermi energy QuickTime™ and a decompressor are needed to see this picture. Electron spectral function: probability to create or annihilate an electron at a given momentum and energy. Fermi momenta k=1/wavelength Fermi energy energy Fermi surface of copper k=1/wavelength 37 Fermi-liquid phenomenology Bare single fermion propagator ‘enumerates the fixed point’: G , k Spectral function: 1 Z 0 k 2 2m i EF vR k k F ImG(,k) A,k ,k 2 k k F 2m ,k ,k 2 2 The Fermi liquid ‘lawyer list’: - At T= 0 thespectral weight is zero at the Fermi-energy except for the quasiparticle peak at the Fermi surface: AE F ,k Z k kF - Analytical structure of the self-energy: ,k E F 2 , k EF , k F - Temperature dependence: EF k k F EF k k k F E F ,kF ,T T 2 38 ARPES: Observing Fermi liquids ‘MDC’ at EF in conventional 2D metal (NbSe2) Fermi-liquids: sharp Quasiparticle ‘poles’ 39 Cuprates: “Marginal” or “Critical” Fermi liquids Fermi ‘arcs’ (underdoped) closing to Fermi-surfaces (optimally-, overdoped). EDC lineshape: ‘branch cut’ (conformal), width propotional to energy 40 Varma’s Marginal Fermi liquid phenomenology. Fermi-gas interacting by second order perturbation theory with ‘singular heat bath’: ImP(q, ) N(0) , for | | T T N(0)sign , for | | T QuickTime™ and a decompressor are needed to see this picture. Directly observed in e.g. Raman ?? Single electron response (photoemission): G(k, ) 1 v F k kF (k, ) 2 g (k, ) ln max | |,T / c i max | |,T 2 c 1 Single particle life time max | |,T is coincident (?!) with the transport life time => linear resistivity. 41 Senthil’s critical Fermi-surface arXiv:0803.4092 Single particle spectral function: c1 k// A K,,T,g k// / zk// F zk// , ,k g gc T k c0 Time like: truly conformal (branch cuts) Space like: still a ‘critical’ Fermi surface (non conformal) Fermi-surface: higher order singularity in n(k) 42 Critical fermions at zero density: branchcut propagators ,r ,r (0,0) Two point Euclidean correlators: Analytically continue to Minkowski time => susceptibilities t,r i ,r At criticality, conformal invariance: Lorentz invariance: ,k 1 1 2 c 2 k 2 1 n ( ) 1 i Scaling dimension set by mass in AdS Dirac equation. 43 Quantum critical matter Quark gluon plasma High Tc superconductors Heavy fermions Metamagnetic ruthenate T Tc Quantum critical 4 (P 2 nd ) nt poi l a ic crit Tri Li ne a i ti c cr of FM 2 o in lp ts 1s t 4 pq 0 3 1 2 2 3 1 P 44 H Quantum critical transport in heavy fermion systems Blue = Fermi liquid: T2 Yellow= quantum critical regime: T Antiferromagnetic order Custers et al., Nature (2003) Fermions and Hertz-MillisMoriya-Lonzarich Ideologically like marginal FL: Bosonic (magnetic, etc.) order parameter drives the phase transition Supercon ductivity Fermi gas Electrons: fermion gas = heat bath damping bosonic critical fluctuations Bosonic critical fluctuations ‘back react’ as pairing glue on the electrons 46 Fermionic quantum phase transitions in the heavy fermion metals JZ, Science 319, 1205 (2008) QP effective mass 1 m EF * E F 0 m* ‘bad actors’ Paschen et al., Nature (2004) Coleman Rutgers Critical Fermi surfaces in heavy fermion systems Blue = Fermi liquid Yellow= quantum critical regime Antiferromagnetic order FL Fermi surface Coexisting critical Fermi surfaces ? FL Fermi surface Plan 1. Quantum critical electron systems and the great questions of condensed matter physics 2. Hairy black holes versus zero temperature entropy in quantum critical heavy fermion systems. 3. Fermion signs versus the AdS/CFT emergent (non) Fermi-liquids. 49 QuickTime™ and a d eco mpres sor are nee ded to s ee this picture . Hartnoll Herzog Spielberg Thorne Nature Nov 5 2009 Horowitz Fisk Grigeria MacKenzie ThomsonRonning St Andrews Los Alamos 50 The zero temperature extensive entropy ‘disaster’ The ‘extremal’ charged black hole in AdS has zero Hawking temperature but a finite horizon area. AdS-CFT The ‘seriously entangled’ quantum critical matter at zero temperature should have an extensive ground state entropy (?*##!!) 51 The holographic superconductor Hartnoll, Herzog, Horowitz, arXiv:0803.3295 Condensate (superconductor, … ) on the boundary! AdS-CFT (Scalar) matter ‘atmosphere’ QuickTime™ and a YUV420 codec decompressor are needed to see this picture. ‘Super radiance’: in the presence of matter the extremal BH is unstable => zero T entropy always avoided by low T order!!! 52 Hair as vacuum polarization 53 Fermionic quantum phase transitions in the heavy fermion metals JZ, Science 319, 1205 (2008) QP effective mass 1 m EF * E F 0 m* ‘bad actors’ Paschen et al., Nature (2004) Coleman Rutgers The metamagnetic quantum critical end point T Tc 4 (P 2n d ) int o p l tica i r c Tri Ends at critical end point a i ti c cr of FM Mackenzie Metamagnetism: first order jump from low to high magnetization in field = transition between water and steam Li ne 2 Grigera o in lp ts 1s t 4 pq 0 3 1 2 2 3 1 H ‘St. Andrews’ ruthenate (Sr3Ru2O7): critical end point can be tuned to zero temperature P 55 The ‘entropic singularity’ (Rost et al., Science 325, 1360, 2009) lim T 0 S 1 1 T H H c Zero T entropy ?? QPT 56 The naked singularity always gets covered up … Exotic quantum nematic ‘hiding’ the critical point. 1.4 2 1.0 T[K] ‘Naked’ quantum critical point dR/dT 1.2 Perspective Z. Fisk 0.8 0 0.6 0.4 QuickTime™ and a d eco mpres sor are nee ded to s ee this picture . Science 325, 1348 (2009) -2 0.2 0.0 7 8 h[T] 9 ‘The unusual new phase can be thought of as the material’s solution to the problem of lowering its entropy in accord with the third law of thermodynamics’ 57 Experimentalists: back to the entropic drawing board .. Ruthenates: St. Andrews T Tc 4 (P 2n d ) Lanthanides, actinides: Los Alamos int l po ica crit i r T Li ne a i ti c cr of FM 2 o in lp ts 1s t 4 pq 0 H 3 1 2 2 3 1 P Nailing down T=0 entropy hidden by last minute order: high precision entropy balance needed. Sorder Tc 0 C dT T Grigeria MacKenzie Thomson Ronning 58 Plan 1. Quantum critical electron systems and the great questions of condensed matter physics 2. Hairy black holes versus zero temperature entropy in quantum critical heavy fermion systems. 3. Fermion signs versus the AdS/CFT emergent (non) Fermiliquids. 59 “AdS-to-ARPES”: Fermi-liquid emerging from a quantum critical state. Cubrovic Schalm QuickTime™ and a decompressor are needed to see this picture. 60 Fermionic quantum phase transitions in the heavy fermion metals JZ, Science 319, 1205 (2008) QP effective mass 1 m EF * E F 0 m* ‘bad actors’ Paschen et al., Nature (2004) Coleman Rutgers The fermionic criticality conundrum Kinetic energy Fermi energy Pauli exclusion principle generates the Fermi-energy, Fermi surface. How to reconcile the quantum statistical scales with scale invariance? Fermi momenta k=1/wavelength Fermi surface of copper How can a (heavy) Fermi-liquid emerge from a ‘microscopic’ quantum critical state? AdS/CFT gives an answer! 62 Breaking fermionic criticality with a chemical potential Classical ‘Dirac waves’ E Fermi-surface?? Electrical monopole k 63 Fermion spectral function from RN Black hole (e.g. McGreevy, arXiv:0909.0518) Bulk Theory: Einstein-Maxwell + charged fermions S G R ,k : 1 gR 6 F 2 eAM A DM igAM m Sbnd 4 Solve Dirac equations of motion, infalling boundary condition at ReissnerNordstrom black hole horizon (horizon at z=1, boundary at z=0) Reissner-Nordstrom BH: 4T 3 q2 , 0 2q 2 2 2 2 2 ds z2 f zdt dx1 dx 2 1 dz f z z 2 A0 2q z 1 f z 1 zz 2 z 1 q 2 z 3 1. Universal: only depends on gq and m for any theory 2. Caveat: spectral function is a probe, possible dynamical instability ignored. 64 “AdS-to-ARPES”: emergent Fermi-liquid. Lorentz + Scale invariance (branchcut): 1 Schalm Cubrovic 1 c k 2 2 2 2 QP peak! Emergent heavy Fermi liquid Scale invariance (branchcut) 65 Fermi-liquid phenomenology Bare single fermion propagator ‘enumerates the fixed point’: G , k Spectral function: 1 Z 0 k 2 2m i EF vR k k F ImG(,k) A,k ,k 2 k k F 2m ,k ,k 2 2 The Fermi liquid ‘lawyer list’: - At T= 0 thespectral weight is zero at the Fermi-energy except for the quasiparticle peak at the Fermi surface: AE F ,k Z k kF - Analytical structure of the self-energy: ,k E F 2 , k EF , k F - Temperature dependence: EF k k F EF k k k F E F ,kF ,T T 2 66 “AdS-to-ARPES”: the quasiparticles. AE F ,k Z k kF ?? Schalm Cubrovic Dip in the spectrum at E F ,kF 67 “AdS-to-ARPES”: quasiparticle life times. Schalm Cubrovic Temperature dependence: Energy dependence self-energy: No constant term in "(EF ,k) "E F ,kF ,T T 2 T T R F 68 The AdS/CFT heavy Fermi liquid Landau Fermi-liquid Quasiparticle mass grows indefinitely when allowed by broken Lorentz invariance 69 “AdS-to-ARPES”: summary. Schalm Cubrovic QP mass increases indefinitely in the emergent Fermi-liquid AdS density Fermionic quantum phase transition at finite density where the Fermi surface disappears Scaling dimension Fermion fields 70 Spectral function: overview QuickTime™ and a decompressor are needed to see this picture. Non-Fermi-liquid 0.1 Landau Fermi-liquid 1 0 71 The emergent QuickTime™ and a decompressor are needed to see this picture. 2 horizon AdS Hong Liu (MIT) Horizon geometry of the extremal black hole: ‘emergent’ AdS2 => IR of boundary theory controlled by emergent CFT1(Hong Liu) Gravitational ‘mechanism’ for marginal (critical) Fermi-liquids: G1 vF k kF k, Spatial: Fermi-surface implied by matching UV-IR " 2 kF Temporal: damping controlled by AdS2, relevancy implies non Fermi-liquid QuickTime™ and a decompressor are needed to see this picture. 72 Gravitationally coding the fermion propagators (Faulkner et al. arXiv:0907.2694) T=0 extremal black hole, near horizon geometry ‘emergent scale invariant’: AdS2 R2 gk ck 2 k Matching with the UV infalling Dirac waves: GR ,k F0 k F1k F2 (k)gk Special momentum shell: | k | kF h1 i 2 GR (,k) ; ,k hgkF h2e kF kF k kF /v F ,k Space-like: IR-UV matching ‘organizes’ Fermi-surface. Time-like: IR scale invariance picked up via AdS2 self energy Miracle, this is like critical/marginal Fermi-liquids!! 73 The emergent horizon AdS2 QuickTime™ and a decompressor are needed to see this picture. arXiv:0907.2694 CFT Fermion propagator: k kF /v F ,k kF h2e GR (,k) ; ,k hg kF i kF 2 h1 k F m 2 1 kF2 1 2 6 2 3 Fermi-liquids: AdS2 QuickTime™ and a decompressor are needed to see this picture. k F 1/2 : 2 kF k kF , " k k F ‘Critical Fermi liquids with Fermi surface: k F 1/2 : (k kF ) 1 2 kF , " k kF 74 1 2 kF Two families of emergent fermion infrareds Fermi-liquid Emergent “CFT1” critical non-Fermi-liquids see Liu et al., arXiv:0907.2694 QuickTime™ and a decompressor are needed to see this picture. ",kF 2 (reproduced in Leiden) “Marginal” Fermi-liquid " ,kF , 1 “Pseudogap” phase Z 0,k 0, k 75 Thermodynamics: where are the fermions? Hartnoll et al.: arXiv:0908.2657,0912.0008 Large N limit: thermodynamics entirely determined by AdS geometry. Fermi surface dependent thermodynamics, e.g. Haas van Alphen oscillations? Leading 1/N corrections: “Fermionic one-loop dark energy” Quantum corrections: one loop using Dirac quasinormal modes: ‘generalized Lifshitz-Kosevich formula’ for HvA oscillations. 2osc. ATckF4 ck F2 osc. cos 2 3 B eB eB e cTk F2 eb T 2 1 Fn n 0 76 Collective transport: fermion currents QuickTime™ and a decompressor are needed to see this picture. Hong Liu (MIT) QuickTime™ and a decompressor are needed to see this picture. Tedious one loop calculation, ‘accidental’ cancellations: FS "1 fermion T 2 ‘Strange coincidence’ of one electron and transport lifetime of marginal fermi liquid finds gravitational explanation! 77 Soaked in Entropy …. Entropic catastrophe! S A C Td F AT 78 The unstable RN Black hole: Fermion VEV’s! Backreaction of matter: need VEV’s/classical finite amplitude fields in AdS. Cannot be formed from single fermion fields, but bilinears: J ?? Witten: multi trace operators in AdS/CFT (hep-th/0112258) Fermion density J0 could be finite !? Back reaction on scalar potential A0, keep RN-AdS geometry fixed: , ,J J J ,I0 i B T z i 0 T gA z, k f z i 0 f z z z 2 B J 0 2T0 0 I 0 0 2 B0 I 0 T0 0 J 0 J 0 0 zz A0 gJ 0 0 3 z f z 79 The dictionary and Fermion VEV’s How do bilinears like J The fields near the boundary (z -> 0): fit in the dictionary? z z Proposal: The bilinears should go simply like the fields squared at the tree level J z 0 2 , J z 0 2 Following Witten’s proposal for multi trace operators in AdS/CFT (hepth/0112258) 80 Fermionic density back hole hair! Density switches on! First order gas-liquid transition at ‘high’ temperature: strongly interacting matter! 81 Fermionic hair and the Fermi liquid stability. Strongly renormalized EF Single Fermion spectral function: pathological temperature dependent spectral weight transfer removed. 82 The fermion probe limit as a lucky accident E z Ez z The change in BH electrical field is numerically very small But the field is diverging near the horizon A0 log 1 z 1 z The near-horizon geometry will backreact as well! 83 Luttinger volume versus bulk density Luttinger theorem: the volume in k space enclosed by the Fermi surface of a Fermi liquid does not renormalize. Given kF (from the spectral functions) the density is: q 2 k F This is within numerical accuracy equal to the CFT density following from the bulk fermion VEV’s! 84 Serious fermion hair: the Lifshitz horizon Divergence of E field at horizon: geometry has to backreact ! Solve equations of motion for Maxwell, Dirac and Einstein z f 0, z 0, zz A0 0, z 0 Metric: 2 2 2 dt exp z dx dy ds2 dz 2 f z f z z2 As for holographic superconductor this turns into a Lifshitz geometry (-> Horova): Lifshitz exponent ds 2 dt 2 exp z z 2 dx 2 dy 2 dz z z2 2 2 85 Full backreaction: the stability of the Fermi-liquid! Non Fermi-liquid Fermi surfaces disappear from the spectral functions! Perfect Fermi-liquid 86 Holographic superconductivity: stabilizing the fermions. Fermion spectrum for scalar-hair back hole (Faulkner et al., 911.340; Chen et al., 0911.282): Temperature dependence as expected for ‘quantum-critical’ superconductivity (She, JZ, 0905.1225) ‘BCS’ Gap in fermion spectrum !! Excessive temperature dependence ‘pacified’ ! 87 ‘Pseudogap’ fermions in high Tc superconductors 102 K Tc = 82 K Gap stays open above Tc 10 K But sharp quasiparticles disappear in incoherent ‘spectral smears’ in the metal Shen group, Nature 450, 81 (2007) 88 The moral of this story… The AdS/CFT correspondence: processing the fermion signs in fermionic quantum critical states The mystery of Fermi-liquid stability: in the ‘hairy’ gravitational dual it is just like a bosonic condensate!! Critical Fermi-liquid phenomenology: AdS2 horizons needed, do they occur at ‘isolated’ quantum critical points associated with bosonic order? Holographic superconductors feeds on the AdS2 T=0 entropy: employment program for condensed matter experimentalists! 89 Further reading AdS/CMT tutorials: J. Mc Greevy, arXiv:0909.0518; S. Hartnoll, arXiv:0909.3553 AdS/CMT fermions: Hong Liu et al., arXiv:0903.2477,0907.2694; K. Schalm et al. Science, and to appear any moment; T. Faulkner et al., arXiv: 0911.3402 Condensed matter: High Tc: J. Zaanen et al., Nature 430, 512, Nature Physics 2, 138; C.M. Varma et al., Phys. Rep. 361, 267417 Heavy Fermions: J. Zaanen, Science 319, 1205; von Loehneisen et al, Rev. Mod. Phys. 79, 1015 90 Thermodynamics from classical action Total action: S Sbulk Sbnd Bulk action: equation of motions Sbulk Boundary action 1 2 g p m F 4 Sbnd h F A For fermions bulk action vanishes in the classical limit Sbulk 0 But boundary action is finite => Free energy is determined by I0 and J0 Sbnd I 0 J 0 F0 91 Empty 92 Empty 93 Empty 94 Empty 95 Empty 96