(Cluster) Dynamical Mean Field Theory: insights into physics of cuprates and other materials

Transcription

1 (Cluster) Dynamical Mean Field Theory: insights into physics of cuprates and other materials A. J. Millis

2 Three talks presented at Emergence of Inhomogeneous Phases in Strongly Correlated Electron Systems 2009 ICAM-I2CAM Cargèse Summer Workshop: Emergent Quantum Phenomenna from the Nano to the Macro World.

3 Collaborators *P. Werner (CU->ETH) *L. De Medici (Rutgers/Paris) *E. Gull (ETH->Columbia) *X. (Sunny)Wang (Columbia) A. Comanac (CU->finance) N. Lin (Columbia) M. Troyer (ETH) M. Capone (Roma) Support: NSF-DMR

4 References: Phys. Rev. Lett. 97, (2006). Nature Physics 4, (2008). EPL (2008) arxiv: arxiv: ArXiv: (Phys. Rev. B. in press) ArXiv: (Phys. Rev. B. in press)

5 Outline Introduction: why DMFT (?Doesnt Mean a *&^ Thing?) The method in brief The numerical problem and recent breakthroughs Cuprates 1: correlation strength and reduction to 1 band model Cuprates 2: angular dependence in the doping and interaction driven Mott transition Conclusions

6 Condensed Matter Physics: Historically, two themes Long wavelength/low energy physics: --Broken symmetries --Collective excitations --Universality=>field theories Connection to reality : --Wave functions and energetics --Band structures <=>crystal structure --Particularity=>material design

7 In strongly correlated materials: these two themes intersect?universal? collective behavior: --cuprate (?pnictide?) superconductivity --quantized hall effect --heavy fermion materials Particularity: --why is Tc so high for doped CuO2 planes or Pu 115 heavy fermions --why is this manganite a high Tc ferromagnet and that one charge ordered insulator --what is hidden order e.g. URu2Si2

8 In addition: Basic questions --relate crystal structure/quantum chemistry to relevant low energy physics --Non-universal (higher T, shorter length scale) effects important --Phenomena (e.g. Mott transition) not easily or usefully describable by field theory --Possible role of short ranged, fluctuating (in time) order

9 Because these questions are not easy to address often: (heated) disputes

10 Gottfried Wilhelm von Leibniz dream... If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, and say to each other, Let us calculate.

11 Of course, calculations can only be as good as the ideas you put into them: creativity and insight are essential But: enormously important to have reliable methods of obtaining well defined answers to well posed questions --Band theory --1 dimension ( luttinger liquid ) theory --Dynamical Mean Field Theory

12 DMFT: the method

13 Work-horse of Theoretical Materials Science: Density Functional Theory Basic Theorem (Hohenberg and Kohn): functional of electron density n(r): minimized at physical density; value at minimum gives ground state energy Φ[{n(r)}] =Φ univ [{n(r)}]+ (dr)v lattice (r)n(r) Useful because: *Have uncontrolled (but apparently good) approximations to *Have efficient way to carry out minimization Φ Φ

14 Density functional band theory Good for: Total energies Crystal structures Phonon Frequencies Identification of relevant electronic states Not so good for: Dynamics Thermodynamics Phase transitions Local moment/mott physics

15 Density Functional Theory: Difficulties Φ[{n(r)}] =Φ univ [{n(r)}]+ (dr)v lattice (r)n(r) Density is not the optimal variable: phases with quite different physical properties have almost the same density Parcollet talk: electron spectral function may be more better variable

16 Dynamical Mean Field Method Metzner/Vollhardt; Mueller Hartmann KOTLIAR/GEORGES Standard many-body theory=>exists a functional of self energy F [{Σ(p, ω}] =F univ [{Σ(p, ω}]+tr [ ln ( G 1 0 (p, ω) Σ(p, ω))] extremized at correct self energy and from which ALL RESPONSE FUNCTIONS can be extracted. Compare density functional theory Φ[{n(r)}] =Φ univ [{n(r)}]+ (dr)v lattice (r)n(r) Function of one static quantity:=> only ground state props Quantity (density) not necessarily best variable

17 DMFT 2: general idea F [{Σ(p, ω}] =F univ [{Σ(p, ω}]+tr [ ln ( G 1 0 (p, ω) Σ(p, ω))] Until recently, F univ only known perturbatively=> general formalism was of limited use (Kotliar/Georges): there is an?accurate? approximation Σ p (ω) Σ approx p (ω) = a φ a (p)σ a (ω) and convenient procedure for doing minimization over restricted sub-space of approximate self energies

18 DMFT 3: broader context Σ p (ω) Σ approx p (ω) = a φ a (p)σ a (ω) Different choices of basis function => different flavors of DMFT (1-site, DCA, CDMFT...). Different flavors: different physics, different computational complexity

19 DMFT 4: theoretical structure F [{Σ(p, ω}] =F univ [{Σ(p, ω}]+tr [ ln ( G 1 0 (p, ω) Σ(p, ω))] Σ p (ω) Σ approx p (ω) = a φ a (p)σ a (ω) => F approx = F univ [{Σ a (ω)}]+tr [ ln ( G 1 0 (p, ω) a φ a (p)σ a (ω) )] Funiv: functional of small number of functions of frequency=> generating functional for quantum impurity model (0+1 dimensional field theory)

20 DMFT 4: theoretical structure, continued F approx = F univ [{Σ a (ω)}]+tr [ ln ( G 1 0 (p, ω) a φ a (p)σ a (ω) )] Stationarity w.r.t variations in self energy: δf univ δσ a (ω) GQI a (ω) = Fixes quantum impurity model φ a (p) (dp) ω ε p Σ approx p (ω) Note: each component a of self energy=> 1 orbital in impurity model

21 DMFT 5: technicalities Challenge: impurity solver <=>find local (d-d) green functions of H QI = H loc [{d a,d a } + p,a ( Vpa d ac pa + H.c ). + H bath [{c pac pa }] ex: Slater-Kanamori d-multiplet interactions

22 DMFT 5: technicalities, ii For many years, methods limited. Could only do 1 orbital, simple on-site repulsion, and this with limited precision. Recent years: 3 breakthroughs Exact diagonalization (Caffarel/Krauth, Capone) CT-QMC Weak coupling: Rubtsov; Parcollet/Gull CT-QMC Strong Coupling: Werner

23 Exact Diagonalization Exact diagonalization (Caffarel/Krauth; Capone; Liebsch) H QI = H loc [{d a,d a } + p,a ( Vpa d ac pa + H.c ). + H bath [{c pac pa }] approximate continuous bath by small number (typically 8-11) of intelligently chosen states Direct access to real frequency Now works very fast Still requires babysitting (choose states rightly) multiorbital models--too many states (but cf Liebsch)

24 CT-QMC: basic idea (illustrate w/ Werner method) H QI = H loc [{d a,d a } + p,a ( Vpa d ac pa + H.c ). + H bath [{c pac pa }] interaction representation with respect to Hloc, Hband [ Z = Tr T τ e p,a ( V I pa d a (τ)c pa(τ)+h.c.) ] formal expansion in V = 1 β dτ 1...dτ k Tr k! k 0 [ ] T τ ˆV I (τ 1 )... ˆV I (τ k ) sample series stochastically: add/remove V; accept or reject by usual importance sampling

25 Technical issues Tr[c...]--determinant=> sum all contractions at once. Essential to do this, or face serious sign problem. Tr[d...]--product of matrices in Hilbert space of Hloc. n orbitals=>4 n dimensional matrices. No fast update for trace or quick way to know if Tr of product vanishes.

26 Advantages: Continuous time=> more points where G varies fast=> Much better energies Mean perturbation order lower at strong coupling

27 All methods: must invert matrix of size k~1/temp. Realistic systems now within reach

28 DMFT: CT-QMC Impurity Solvers Expand in V Strong coupling Advantages: --Strong couplings --arbitrary interactions Disadvantage: --N orbitals=>cost ~2 N --Imaginary time Expand in U Weak coupling (CT-AUX) Advantages: --N orbitals=>cost ~N --highly accurate Disadvantage: --Only Hubbard U --not strong coupling --Imaginary time --sigm problem

29 DMFT Summary: New (improved) methods: many problems now open to attack Analogy: early days of density functional theory: we have an apparently useful structure. Task now: do things, see what works and what does not--and if possible, understand why

30 Application: Interaction Strength in Cuprates and Reduction to One Band model

31 Interaction strength in cuprates High Tc: add carriers to insulating parent compound Low freq absorption strength ~ number of dopants Uchida 91 Comanac 08 But: La2CuO4: 1 free carrier per unit cell =>? why insulating?

32 Slater: One possible answer: long ranged Neel order long ranged order doubles unit cell Amplitude of order large enough: insulator

33 But--Neel temperature only ~350K; gap ~2eV => Generally accepted (P. W. Anderson) view:

34 But--Neel temperature only ~350K; gap ~2eV => Generally accepted (P. W. Anderson) view: La2CuO4: Mott Insulator:

35 But--Neel temperature only ~350K; gap ~2eV => Generally accepted (P. W. Anderson) view: La2CuO4: Mott Insulator: strong on-site interaction suppresses double occupancy : each site has either 1 or no spins; only holes move

36 But--Neel temperature only ~350K; gap ~2eV => Generally accepted (P. W. Anderson) view: La2CuO4: Mott Insulator: strong on-site interaction suppresses double occupancy : each site has either 1 or no spins; only holes move Gap without long ranged order

37 But--Neel temperature only ~350K; gap ~2eV => Generally accepted (P. W. Anderson) view: La2CuO4: Mott Insulator: strong on-site interaction suppresses double occupancy : each site has either 1 or no spins; only holes move Gap without long ranged order In gap spectral weight ~doping

38 But--Neel temperature only ~350K; gap ~2eV => Generally accepted (P. W. Anderson) view: La2CuO4: Mott Insulator: strong on-site interaction suppresses double occupancy : each site has either 1 or no spins; only holes move Gap without long ranged order In gap spectral weight ~doping Reduced Hilbert space: gauge theory

39 But--Neel temperature only ~350K; gap ~2eV => Generally accepted (P. W. Anderson) view: La2CuO4: Mott Insulator: strong on-site interaction suppresses double occupancy : each site has either 1 or no spins; only holes move Gap without long ranged order In gap spectral weight ~doping Reduced Hilbert space: gauge theory Spatial structure not fundamental

40 Mott insulator <=>?gap with no long ranged order? Ex: 2d nested Hubbard model H = 2 (cosk x + cosk y )c kσ c kσ + U k i At half filling, T=0: arbitrarily small U>0 =>2 sublattice Neel order=>gap n i n i Heisenberg symmetry: at any T>0: no LRO < S i S j > e Ri Rj ξ Gap surely persists if ξ 1 => gap w/o LRO not a good def ξ e J T

41 Canonical definition: Mott if insulating in absence of intersite correlations (i.e. from purely local physics) Implemented by single-site DMFT Bethe lattice. bandwidth =4 density n=1 T=0 correlation-driven metal insulator transition at U=Uc2 Gives precise theoretical meaning to Mott transition. Question for cuprates, other materials: what is U wrt Uc2

42 Issue: experimental measure of correlation strength Eskes et al PRB High energy physics is complicated Need: experiment which directly couples to effective interaction relevant for low energy physics U ~ 10eV

43 Conductivity: sensitive to Mott blocking effect Optical process: move electron from one site to another. If correlation strong, most final states are blocked

44 Conductivity: sensitive to Mott blocking effect Optical process: move electron from one site to another. If correlation strong, most final states are blocked

45 Conductivity: sensitive to Mott blocking effect Optical process: move electron from one site to another. If correlation strong, most final states are blocked

46 Conductivity: sensitive to Mott blocking effect Optical process: move electron from one site to another. If correlation strong, most final states are blocked Forbidden

47 Conductivity: sensitive to Mott blocking effect Optical process: move electron from one site to another. If correlation strong, most final states are blocked Forbidden

48 Conductivity: sensitive to Mott blocking effect Optical process: move electron from one site to another. If correlation strong, most final states are blocked Allowed Forbidden

49 Qualitative conductivity of correlated states (plus interband transitions) σ(ω) =σ ÿſſ#$% ɏ#'(ω)+σ (ɏ#ſ* % ' (ω) Conductivity Frequency If interband transitions well separated, correlated conductivity can be extracted and analysed *Identify correlated contribution. *Energy scale=> U *Compare to band theory *Magnitude of renormalization=> blocking strength

50 Points of comparison Uchida 91 *magnitude of gap *Magnitude of abovegap conductivity *rate at which low freq oscillator strength is added

51 In gap oscillator strength: comp to band theory Comanac 08

52 Issue: to convert pictures to quantitative analysis Need: Optical matrix elements Theory to handle interactions

53 Matrix elements: typical correlated-electron materials: tight binding description ok H = ) t i j (c j,σ c i,σ + H.c i,j. + H int Couple to electromagnetic field: Peierl s phase (note usual assumption: Hint indep of A: could generalize...) t i j t i j e i e hc A ( R i R j) (ε p ε p e for 1 band model <=> c A ) Will compare 1-band and 3-band (Cu-O) models

54 1 site DMFT Hubbard model 3-band (pd) model tij chosen to reproduce conduction bands tpd chosen to reproduce conduction bands

55 3 band model Phase diagram, 1 hole/cell Many-body DOS = ε p ε d plays role of U Zaanen Sawatxky Allen 85

56 Conductivities: 1 band vs 3 band

57 Conductivities: 1 band vs 3 band Half filling band 1 band Hubbard 0.06 σ Frequency [ev]

58 Conductivities: 1 band vs 3 band Half filling band 1 band Hubbard 0.06 σ Frequency [ev]

59 Conductivities: 1 band vs 3 band Half filling 0.1 doping band 1 band Hubbard band 1 band Hubbard σ σ Frequency [ev] Frequency [ev]

60 Conductivities: 1 band vs 3 band Half filling 0.1 doping band 1 band Hubbard band 1 band Hubbard σ σ Frequency [ev] Conclusion: Hubbard-model OK for low-intermediate energy physics Frequency [ev]

61 Technicality: Estimating the gaps Gaps: real frequency information => *Analytical continuation: requires very good data; still has uncontrolled systematic errors *Exact diagonalization: wide level spacing

62 Estimating Gaps: Max/Ent analytic continuation broadens gap edges: simple example--optimize broadening Γ Result: cost funct Γ

63 We (==Xin Sunny Wang) found Most robust method: continue self energy Find gap from Real part, quasiparticle equation Note: no evidence for subgap excitations: Im Sigma=0 for w inside qp gap Much more info: ArXiv/ (PRB in press)

64 AF phase Hubbard Model, U=8t Continue G Continue Sigma n vs mu

65 PM phase, 2d square lattice Hubbard U=12t >Uc2 Different continuations: gap edge robust ED: pole at gap edge

66 AF vs PM: U=12t PM AF AF=> increase in gap by ~80%

67 Optical conductivity: U=12t High Tc: t~0.37ev PM phase gap about right-- AF gap too big if U~Uc2

68 Comparison to Experiment Square lattice Hubbard U=12t=Uc band Hubbard 300 σ Frequency [ev] Cond far too low at gap edge AF (not shown: gap too big)

69 Physics of small conductivity at gap edge False color intensity map of Im G Top of valence band: p=0 Bottom of cond band: p=pi,pi E =>PM Mott state is indirect gap insulator *AF folds bands back =>direct gap p,p

70 Conclusion: Intermediate correlation strength U<Uc2=>Cuprates not Mott insulators Antiferromagnetic order or correlations stabilizes Mott gap and therefore may be expected to play a crucial role in doped materials.

71 Obvious questions Gap size: too big for AF Not much T dependence to gap (as far as known) but T-dep to AF Single-site DMFT +AF not reliable picture

72 Step towards theory multisite (cluster) dmft

73 What clusters bring Σ(p, ω) Σ(ω) Single-site DMFT: momentum averaged quantities Σ(p, ω) a φ a (p)σ a (ω) Multi-site DMFT: increased momentum resolution Multi-site DMFT: new physics associated with short ranged order

74 Interlude: pseudogap in high Tc

75 Metal-insulator transition in cuprates Electron-doped: relatively conventional. Hole doped: strange metal

76 Metal Etrange: Emerging consensus on importance of emergent (k and r-space) structure Abdel-Jawad...Hussey (Magneto) transport=> increase in anisotropy of scattering rate as approach SC from higher doping Reproduced by Functional RG (Ossednik,Honerkamp, Rice)

77 Metal etrange sous-dopage normal state gap Huefner et al Rep. Prog. Phys. 71, (2008)

78 Angular dependent Mott transition Pseudo gap mainly at zone edge Raman scattering: photon in/photon out=> many geometries B2g B1g Guyard..Sacuto..Phys. Rev. B, (2008)

79 Normal and superfluid transport Uemura 91 ρ S δ Pushp, Pasupathy Yazdani 09 good part of gap ~ doping Implication: supercurrent and thus normal current carried only by near-node states

80 Momentum-space differentiation: Functional RG Lauechli, Honerkamp, Rice) theory Result: gaps near antinode but not near node

81 Suppression of charge response as approach insulating state Real space: Decrease density of mobile holes Non! k-space: Decrease size of good region of fermi surface Si!

82 Step towards theory: Multisite (Cluster) DMFT

83 Step towards theory: Multisite (Cluster) DMFT 2-site Ferrero et al PRB

84 Step towards theory: Multisite (Cluster) DMFT 2-site 4-site Ferrero et al PRB Gull et al EPL

85 Step towards theory: Multisite (Cluster) DMFT 2-site 4-site 8-site Ferrero et al PRB Gull et al EPL Gull et al PRB

86 Step towards theory: Multisite (Cluster) DMFT 2-site 4-site 8-site Ferrero et al PRB Gull et al EPL Gull et al PRB 16 and 32--Maier/Jarrell

87 Step towards theory: Multisite (Cluster) DMFT 2-site 4-site 8-site Ferrero et al PRB Gull et al EPL Gull et al PRB 16 and 32--Maier/Jarrell Tradeoff: momentum resolution vs ability to compute

88 Step towards theory: Multisite (Cluster) DMFT 2-site 4-site 8-site Ferrero et al PRB Gull et al EPL Gull et al PRB 16 and 32--Maier/Jarrell Tradeoff: momentum resolution vs ability to compute Larger clusters <=> sign problem

89 Paramagnetic insulator stabilized at lower U T Hubbard model n=1 insulator U (4) ~5t 4-DCA U (8) ~6.5t 8-DCA U (1) =12t=Uc2 1-DCA

90 Characterize Insulator

91 Characterize Insulator Choose zero of energy to be chemical potential

92 Characterize Insulator Choose zero of energy to be chemical potential Metal if quasiparticle equation

93 Characterize Insulator Choose zero of energy to be chemical potential Metal if quasiparticle equation G 1 (k, ω) =ω ε k ReΣ(k, ω) = 0

94 Characterize Insulator Choose zero of energy to be chemical potential Metal if quasiparticle equation G 1 (k, ω) =ω ε k ReΣ(k, ω) = 0 is satisfied for some k near w=0

95 Characterize Insulator Choose zero of energy to be chemical potential Metal if quasiparticle equation G 1 (k, ω) =ω ε k ReΣ(k, ω) = 0 is satisfied for some k near w=0 To kill the metal: make renormalized chemical potential ReΣ larger than max ε k

96 Single site: insulating state pole in sigma splits the band Σ(z) = ReΣ 2 z ω 0 (dk)img

97 Single-site: pre-formed gaps; interaction dep MIT via pole splitting Σ(z) 2 1 z ω z ω 2 P. Cornaglia data U=0.85 Uc2 1 U=0.85U c2 5 0 U=0.85U c2 Im G 0.5 Im Σ Energy Energy

98 Single-site: pre-formed gaps: doping dep MIT via pole splitting ImΣ(ω) Σ(z) 2 1 z ω z ω ω δ= δ= δ= δ= P. Werner, N. Lin Two poles: --large, mid-gap: gives basic structure of insulator --small at at band edge: doping induced states

99 Ferrero et al PRB 09 2 sites: sector-selective band-like transition Band associated with - patch pushed above EF No pole in self energy

100 4 sites: new insights into insulating phase

101 What to look at G(τ = β 2 )= dx 4π A sector (x) cosh x 2T βg(τ = β 2 )=β dx 4π A sector (x) cosh x 2T = dy 4π A sector (2Ty) coshy As T->0, integral is peaked at x=0 => picks out fermi surface density of states

102 4 and 1 site dmft: Density of states vs U,T Gull et al, EPL 08

103 Analytically continued density of states vs mean field theory

104 Physics of transition: compare 1 site and 4 site U=5t Weak coupling transition: decrease in E interaction = UD =>gap is consequence of short range order

105 Contribution of impurity model states to partition function Order most probably of placquette singlet type

106 Similar physics previously noted Lee-Rice-Anderson PRL 73 Civelli et al PRL (2005) Kyung et al CMDFT +ED (4-sites) Kyung et al PRB (2006)

107 When you dope: pseudogap 4 site; as fn of e-doping ArXiv: site h-doping.08 arxiv:

108 4-site DCA U=9t (high Tc band structure: t =-0.3) temperature t/5 ~800K A(!! ""# ""##$##%& ""##$#'' ""##$(() ""##$(* Slight decrease in gap at half filling (~4.5t ~1.8eV) pseudo gap reduced !

109 Implications for optical conductivity

110 Conductivity: methodology and uncertainties Current as function of vector potential: Expand to linear order in A [ ] Ĵ(A)G(t; j(t) =Tr { A}) j(t) = dt χ jj (t t )A(t ) χ jj = χ dia + χ bubble + χ vertex

111 Conductivity: methodology and uncertainties Current as function of vector potential: j(t) =Tr [ Ĵ(A)G(t; { A}) ] Expand to linear order in A χ jj = χ dia + χ bubble + χ vertex χ dia (t t ) = Tr [ ˆKG(t = 0) ] δ(t t ) K = δj δa

112 Conductivity: methodology and uncertainties Current as function of vector potential: j(t) =Tr [ Ĵ(A)G(t; { A}) ] Expand to linear order in A χ jj = χ dia + χ bubble + χ vertex [ χ bubble (t t ĴG(t ) = Tr t ) ] ĴG(t t), G 1 = i t ε k A Σ(p A; ω; A) J = ε k k

113 Conductivity: methodology and uncertainties Current as function of vector potential: j(t) =Tr [ Ĵ(A)G(t; { A}) ] Expand to linear order in A χ jj = χ dia + χ bubble + χ vertex [ ] χ vertex (t t ĴG(t ) = Tr t1 ) ˆΓ(t1 t,t t 2 )G(t 2 t), G 1 = i t ε k A Σ(p A; ω; A) Γ = δσ/δ A 4-site cluster: vertex only from explicit k-dep of Σ

114 Conductivity: methodology and uncertainties Best case: U=6t, t =0 Method: imaginary time; also sign problem limits range of numerics, quality of data Analytical continuation needed: 2 methods--continue self energy, continue conductivity--note negative cond in 2nd method

115 Conductivity: importance of vertex corrections Best case: U=6t, t =0 Vertex correction: --increases above gap spectral weight --sharpens gap edge =>captures aspects of band folding, coherence factors assoc to insulating state

116 0.4 Conductivity U=9t, t =-0.3t continue # continue " 0.3 "(!) !

117 Conductivity: preliminary results "!!" T=t/10 "!!" continue # continue " continue # continue " T=t/ ! ! Gap magnitude, cond value agree w/data

118 Doping (continue conductivity) "!!" Temp ~ 400K 1000 Ω 1 cm 1 ##$ ##$$%$$& ##$$%$'( ##$$%$&) ##$$%*+, eV! t=0.37ev

119 Spectral Weight 0.5 Partial Integral Drude Peak (up to 0.2) Mid-IR (up to 1) Total KE Doping Kband=0.81t

120 Structure in conductivity at ~0.8t due to pseudogap in many-body DOS

121 Conclusion: Hubbard model with moderate correlations contains the essential aspects of the Mott transition as observed in cuprates

122 More detailed look at the physics

123 8 sites: direct (coarse grained) view of variation around fermi surface 2 sectors containing noninteraction fermi surface requires solution of 8-site impurity model: weak coupling expansion up to U~8t, moderate T focus on directly measurable quantities Werner et al PRB in press 09

124 Half filled, vary U sector-selective transition t =0 van Hove sing in sector C=>T-dependence

125 Gap opens gradually Pole in self energy turns on gradually

126 Density (in sector) pinned to 1/2

127 Transition is particle-hole asymmetric (for t not 0)

128 Transition not controlled by van Hove physics E. Gull to be published

129 Side remark: No increase in nematic fluctuations near transition Basic picture looks like that proposed by Lauechli/ Honerkamp/Rice

130 Summary: New (improved) methods: many problems now open to attack Analogy: early days of density functional theory: we have an apparently useful structure. Task now: do things, see what works and what does not--and if possible, understand why

131 Summary Dynamical mean field theory (note--can do realistic rotationally invariant Hunds/multiorbital interactions) Cuprates: 1 band model OK, E<4eV Cuprates: moderate correlations, =>gap, associated with spatial structure of correlations/order doping dep<=>how are correlations unwound New route to metal-insulator transition: selector selective gradually wipe out fermi surface (for h-doped not e-doped) beginning with (0,Pi) point associated with short ranged correlations (but precise connection not yet established) Many physical/conceptual challenges Many open questions