DMFT and beyond : IPAM, Los Angeles, Jan. 26th 2009 O. Parcollet Institut de Physique Théorique CEA-Saclay, France

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1 DMFT and beyond : From quantum impurities to high temperature superconductors 1 IPAM, Los Angeles, Jan. 26th 29 O. Parcollet Institut de Physique Théorique CEA-Saclay, France Coll : M. Ferrero (IPhT), L. de Leo, A. Georges (CPHT, Polytechnique) M. Civelli (ILL, Grenoble), P. Cornaglia (Bariloche), G. Kotliar (Rutgers)

2 Outline 2 1. Physical motivations 2. DMFT and clusters 3. Selective Mott transition in k space 4. Two gaps in the SC phase

3 !Mott insulator Mott physics 3 Mott transition = Metal-Insulator transition due to interactions Hubbard model, a minimal model for theorists. H = t ij c iσ c jσ + Un i n i, n iσ c iσ c iσ ij,σ=, δ = 1 n + n How is a metal(superconductor) destroyed close to a Mott transition? Example : Cuprates!"#$% U/t Mott insulator!# ()!" $%? ω metal *$ &' %&% )*+,-.+/,+ %&'( /1,-.+/,+ Doped Mott insulators Anderson (1987),... $ ρ(ω) metal δ

4 Nodes and Antinodes 4 d-wave gap and order parameter (with nodes) (k) ( cos(kx ) cos(k y ) ) +...! Antinodal direction!"#$%!# () Nodal direction!" $% *$ &' %&% )*+,-.+/,+ %&'( /1,-.+/,+ $ ϕ ARPES Ding et al,1996 ky kx π

5 Fermi Arcs Spectral intensity map at Fermi level (ARPES) Bi2212 : Kanigel et al. Nature 2,447 (26) A(k, ω = ) 5 Ca2-xNaxCuO2Cl2 Quasi-Particle Shen et al. Science 37, 91 (25) No Quasi-Particle Good Fermi liquid Y Y Y M M M! M! M! Arcs shrinks for δ δ M

6 Nature of the Fermi Arcs? 6 What is the low temperature normal state? Vanishing arcs at T=? Nodal liquid? Kanigel et al. Nature 2,447 (26) M Y M Y M? Quantum oscillations in strong magnetic field (Shubnikov-de Haas) N. Doiron-Leyraud at al, Nature, 27 Low temperature, suppress SC with field Pocket Fermi surface in the normal state?! M! M! Fermi liquid with strong variations of Z, Tcoh, m* along the Fermi surface? Tcoh Antinode < T < Tcoh Node?

7 Mott insulator SC phase 7 Evolution of the superconducting gap with doping δ Non-BCS behaviour : (δ) T c (δ) PHYSICAL REVIEW B 67, !!"#$% T*!# () *$!" &' $% Tc %&% )*+,-.+/,+ %&'( /1,-.+/,+ $ δ Sutherland,23 But recent experiments suggest this is too simple...

8 Electronic Raman spectroscopy χ (ω) Ov. 92 K 4 cm 1 Ov. 92 K 417 cm 1 8 M. LeTacon, A. Sacuto, A. Georges, G. Kotliar, Y Gallais, D. Colson and A. Forget, Nature Physics, 2, 537,26 Opt. 95 K T << T c T > T c 515 cm 1 Opt. 95 K T << T c T > T c 55 cm 1 HgBa2CuO4+δ 2 set of measures, probing Nodal Region (NR) and Antinodal Region (ANR) Und. 89 K Und. 86 K 38 cm 1 34 cm 1 Und. 89 K Und. 86 K 66 cm 1 72 cm 1 Hole doping ( π, ) ( π, π) ( π, ) ( π, π) δ Γ NR B 2g ANR B 1g Γ Und. 78 K 36 cm 1 Und. 78 K No BCS fit Two energy scales in the SC phase Und. 63 K 23 cm 1 Und. 63 K Raman shift ω (cm 1 )

9 Gap at antinode (maximum) increases for δ. Two gaps in SC phase? Energy (mev) Gap close to node (slope) decreases for δ, like Tc. 8 4 E pg Pseudogap Optimal Doping Normal Metal E Superconductor sc Hole doping (x) δ T c (K) picture from arxiv: Bi ARPES 48 Bi RS-B1g 45,46 Bi SIS 34,35 Bi STM 38 Bi SIN 34 Tl ARPES 56 Tl HC 68 Y HC 67 Bi ARPES 49 Bi INS 64 Bi RS-B2g 45,46 Bi SIS 35 Tl INS 66 Y AR 69 Y INS 64 Raman experiments. M. LeTacon et al., Nature Physics, 2, 537,26 See also ARPES experiments. Tanaka et al,science 314, 191, (26) v = d (φ) dφ Still debated... for δ Node

10 Nodal-Antinodal dichotomy : summary 1 Normal phase : Fermi Arcs Quasi-particle in the nodes, pseudogap in the antinode Superconducting phase : Two behaviours of the gap of 1-particle Green function(?) Phenomenological idea : Node like an ordinary Fermi liquid/sc Antinode more like an insulator Can we understand this from a systematic microscopic calculation e.g. of Hubbard model? What is the mechanism? In this talk : DMFT approach

11 Outline Physical motivations 2. DMFT and clusters 3. Selective Mott transition in k space 4. Two gaps in the SC phase

12 Dynamical Mean Field Theory A. Georges, G. Kotliar, W. Krauth and M. Rozenberg, Rev. Mod. Phys. 68, 13, (1996) G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, OP, C. Marianetti, Rev. Mod. Phys. 78, 865 (26) Ising model (Weiss) : A single spin in an effective field. Fermionic Hubbard model (Kotliar-Georges, 92) Anderson impurity with an effective band determined self-consistently 12 H = ɛ c σc σ + Un n σ=, }{{} Local site + V kσ ξ kσ c σ + h.c. + ɛ kσ ξ kσ ξ kσ k,σ=, k,σ=, }{{} Coupled to an effective electronic bath S = Bath β Weiss field c σ(τ)g 1 σ (τ τ )c σ (τ )+ G 1 σ (iω n) iω n + ɛ k β dτun (τ)n (τ) V kσ 2 iω ɛ kσ G

13 G c (τ) = Tc(τ)c () Seff Gc DMFT Self-consistency Impurity Solver G 13 S eff = β c σ(τ)g 1 (τ τ )c σ (τ )+ Continuous Time QMC : Sum diagrams with Monte-Carlo Expansion in U : A.N. Rubtsov et al., Phys. Rev. B (25) Expansion around atomic limit : [P. Werner's talk] P. Werner, A. Comanac, L. de Medici, M. Troyer, A. J. Millis, PRL (26) β CT-AUX [P. Werner's talk]: E.Gull et al., EPL (28) No sign problem (1 site DMFT). Exact diagonalization M.Caffarel and W. Krauth PRL (1994) dτun (τ)n (τ)

14 Ising H = J ij DMFT equations (simplest case) σ i σ j H = ij σ Hubbard t ij c iσ c jσ + Un i n i i 14 m = σ G c (τ) = Tc(τ)c () Seff H eff = Jh eff σ S eff = β c σ(τ)g 1 (τ τ )c σ (τ )+ β dτun (τ)n (τ) h eff = zjm Σ(iω n ) G 1 (iω n) G 1 c (iω n ) ( ) 1 G 1 (iω 1 n)= + Σ(iω n ) iω n + µ ɛ k Σ(iω n ) k }{{} G lattice (k, iω n ) The self-energy on the lattice is local : in metals, Z, m*, coherence temperature, finite temperature lifetime are constant along the Fermi surface.

15 A diagrammatic point of view Σ ij = δφ δg ji De Dominicis, Martin (1964) 15 Φ Hubbard [G ij ]= 2 particle-irreducible (2PI) diagrams = φ 1 (G ii ) + φ 2 (G i,j )+ φ 3 (G i,j,g i,k,g j,k )+... i i,j i,j,k }{{}}{{} Local Non local DMFT approximation (exact in d limit) Metzner, Vollhardt (1989) Φ Hubbard (G ij ) i φ 1 (G ii ) Local 2PI diagrams of Hubbard = 2PI diagrams of Anderson impurity φ 1 (G ii )=Φ Anderson (G ii ) Kotliar, Georges (1992) DMFT sums local 2 PI diagrams, with a sign-free QMC.

16 From Mean Field to a controllable method : clusters Principle : systematic interpolation between Φ Hubbard and Φ DMFT with a finite number of sites in a self consistent bath. One way to bring control to DMFT [See also A. Lichtenstein's talk] local quantum fluctuations T. Maier et al, Rev. Mod. Phys. 77,127 (25) short range quantum fluctuations 16 G G Real space cluster (CDMFT) DMFT on a superlattice A. Lichtenstein and M. Katsnelson, PRB 62, R9283 (2) G. Kotliar et al. PRL , Reciprocal space cluster method (DCA) Cluster method in k-space : Σ piecewise constant on B.Z. M.H. Hettler, A.N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke,H.R. Krishnamurthy PRB (1998)

17 C-DMFT 17 DMFT on a superlattice of clusters. Cluster site labeling 3 t t 4 Superlattice 1 t 2 1 µ, ν 4 R,R : position of the cluster. μ,ν= cluster site labels. Φ CDMF T (G) = R Φ 4sites (G µ,r;ν,r G ρ,r;λr = ) Same equations as multiorbital DMFT (4 sites as orbitals)

18 DCA Cluster method in k-space : Σ piecewise constant on B.Z. M.H. Hettler, A.N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke,H.R. Krishnamurthy PRB (1998) 18 Example for 2x2 cluster on square lattice. Σ(k, iω n ) Σ c (k c (k), iω n ) (,!) (!,!) Cluster momenta k c (,) K (!,) k k ~ Φ DCA (G) = N sites Φ(G(k)) U(k1,k 2,k 3,k 4 )=U DCA (k 1,k 2,k 3,k 4 ) U DCA (k 1, k 2, k 3, k 4 ) = δ Kc (k 1 )+K c (k 2 ),K c (k 3 )+K c (k 4 )/N sites Cluster size = momentum resolution T. Maier et al, Rev. Mod. Phys. 77,127 (25)

19 Is DMFT a good starting point? 19 Compare to experiments... Cluster corrections small at small U, high T, large doping e.g. good for ultra-cold fermions. Why?...

20 DMFT : a good starting point for Mott physics 2 A(k, ω) Lattice Quasi-particle-peak Anderson impurity 1 π ImG c(ω) Free electrons ω Fermi liquid Mott physics : Hubbard band (localized) vs Q.P. peak (delocalized) ω Hubbard bands Abrikosov-Suhl resonance Local Fermi liquid with coherence temperature TK Nozières, 1974 DMFT : from an analogy to a mean field formalism

21 Outline Physical motivations 2. DMFT and clusters 3. Selective Mott transition in k space 4. Two gaps in the SC phase

22 Minimal cluster DMFT approach to normal phase 22 Two complementary points of view on cluster DMFT : A systematic numerical method (largest cluster) e.g. 16 sites : A. Macridin et al., PRL 97, 3641 (26) A mean field method (smallest cluster) M. Ferrero, P. S. Cornaglia, L. De Leo, O. Parcollet, G. Kotliar, A. Georges, arxiv: Ca2-xNaxCuO2Cl2 Shen et al. Science 37, 91 (25) Goal : Describe the normal phase (Fermi arcs, pseudogap) with a minimal cluster of 2 sites. Search for a simple (analytical) picture of the mechanism.

23 Orbital selective transition in k-space M. Ferrero, P. S. Cornaglia, L. De Leo, O. Parcollet, G. Kotliar, A. Georges, arxiv: DCA method with two patches of equal volume β β S eff = dτdτ c µ(τ)g 1,µν (τ, τ )c ν (τ )+ dτun µ n µ (τ) G 1 ± (iω n)= 1 + Σ ± (iω n ) iω n + µ ɛ(k) Σ ± (iω n ) Using even/odd basis µ =1, 2 c ± =(c 1 ± c 2 )/ 2 Two cluster momenta : (,) and ( CTQMC solution using Werner s et al algorithm k P ± 1 π ky (,) π ( ) kx Antinodes Nodes Fermi surface Inner patch P+ Outer patch P-

24 Orbital selective transition in k-space 24 At high doping/temperature, DMFT not corrected by cluster terms. Around 16%, orbital corresponding to outer patch P- becomes insulating : μ - Σ_() reaches the band edge of P- patch Quasi-particles only exists in the inner patch.8.6 " ! Gap of the outer Green function G_ Re "() " + " ! band edge of P- patch µ#" - β=2 Statistical weight FIG. 1: (Color online) Left: real part of Σ ± () doping level, computed with RISB (lines) and S

25 Orbital selective transition in k-space Ordinary slave boson mean field : RVB theory of cuprates c σ = bf σ, Anderson, Science (1987), G. Kotliar, J. Liu Phys. Rev. B 38, 5142 (1988) f σf σ + b b = 1 σ 25 Low energy solution of 1 site DMFT (Brinkman-Rice) No k dependence of Σ! Here, low energy is given by Rotationnally Invariant Slave Bosons F.Lechermann, A. Georges, G. Kotliar, OP, PRB (27) Re "() " + " - µ#" - β=2 Line = RISB Dots = QMC Statistical weight Cluster : k dependency as a multiorbital problem + RISB : slave bosons for multiorbital problem Cluster + RISB = a slave boson method with k dependency!

26 Orbital selective transition in k-space 26 With CTQMC and Rotationnally Invariant Slave Bosons methods, it is possible to compute the relative weight of various cluster states. Singlet state dominates at low doping Some similarities with RVB approach Anderson, Science (1987), G. Kotliar, J. Liu Phys. Rev. B 38, 5142 (1988) but here self-energy depends on k (impossible to get with ordinary slave bosons). Re "() " + " - µ#" - Statistical weight S β=2 E T ! !

27 Minimal cluster approach to Fermi Arcs M. Ferrero, P. S. Cornaglia, L. De Leo, O. Parcollet, G. Kotliar, A. Georges, arxiv: Spectral function at Fermi level (DCA + interpolation...) Maximum contrast around 1 % Doping A(k, ω = ) DMFT metal " ky 6%!"!" " kx " ky 14%!"!" " kx A(#,) / A(,) " ky 1%!"!" " kx # [degrees] as a minimal clu space differentiati of its simplicity, moderate numeric derstanding can such as rotationa culated STM and the phenomenolo The low-doping r tion. Mott phys of coherent quas tative agreement the weak/interme VB-DMFT, this s selective transitio We thank F. Le useful discussions

28 Comparison with larger clusters π ky (,) π ( ) kx Antinodes Nodes Fermi surface Inner patch P+ Outer patch P- Only two cluster momenta : (,) and ( far from Fermi surface!

29 Compare with 8 sites calculations E. Gull, P. Werner and A.J. Millis, OP 29 Solution of 8 sites DCA with CTAUX algorithm Doping driven transition 2 steps transition : sector C before sector B n! (sector) "t=4, B "t=4, C "t=2, B "t=2, C !/t B δ C Free Fermi surface A B D C Patches for 8 sites clusters

30 Compare with 8 sites calculations (II) E. Gull, P. Werner and A.J. Millis, OP Similar mechanism : Re Σ() getting large, out of bands 3 Re! (sector B) δ!/t=.35!/t=.4!/t=.5!/t=.625!/t=.75!/t=.875!/t=1!/t=1.1!/t=1.25 Re! (sector C) δ!/t=.35!/t=.4!/t=.5!/t=.625!/t=.75!/t=.875!/t=1!/t=1.1!/t= " n B A B D C " n C Confirmation of the picture found in 2 sites

31 Test interpolation M. Ferrero, P. S. Cornaglia, L. De Leo, O. Parcollet, G. Kotliar, A. Georges, arxiv: Interpolate the self-energy and the cumulant at (π,) and compare with 4 sites DCA at (π,) (black curve) Σ(k, ω) = Σ + (ω)α + (k)+σ (ω)α (k) M(k, ω) = M + (ω)α + (k)+m (ω)α (k) 31 1! α ± (k) = 1 2 {1 ± 1 2 [cos(k x) + cos(k y )]} #(i$ n ) Re# Im#!=.8 k=(%,) $ n Cumulant interpolation is a lot better

32 Outline Physical motivations 2. DMFT and clusters 3. Selective Mott transition in k space 4. Two gaps in the SC phase

33 d-sc phase in cluster DMFT CDMFT and DCA have a phase diagram with AF and d-sc A. Lichtenstein and M. Katsnelson, PRB 62, R9283 (2); M. Jarrell et al, PRL 85,1524 (21) 33 Large Clusters at U/D=1 (DCA), up to 26 sites : Tc.23t T. Maier et al., PRL 95, 2371 (25) Smallest cluster : 2x2 plaquette p= c 1 c 2 -p Plaquette cluster 3 t p t 4 -p 1 t 2 Here : a minimal cluster approach : Two gaps picture? Doping dependence of the gap? p

34 Gap at antinode (maximum) increases for δ. Two gaps in SC phase : reminder Energy (mev) Gap close to node (slope) decreases for δ, like Tc. 8 4 E pg Pseudogap Optimal Doping Normal Metal E Superconductor sc Hole doping (x) δ T c (K) picture from arxiv: Bi ARPES 48 Bi RS-B1g 45,46 Bi SIS 34,35 Bi STM 38 Bi SIN 34 Tl ARPES 56 Tl HC 68 Y HC 67 Bi ARPES 49 Bi INS 64 Bi RS-B2g 45,46 Bi SIS 35 Tl INS 66 Y AR 69 Y INS 64 Raman experiments. M. LeTacon et al., Nature Physics, 2, 537,26 See also ARPES experiments. Tanaka et al,science 314, 191, (26) v = d (φ) dφ Still debated... for δ Node

35 CDMFT result (4 sites) 35 M. Civelli, M.Capone, A. Georges, K. Haule, O. Parcollet, T Stanescu, G. Kotliar, PRL (28) A(k, ω) Z nod δ Nodes ( ω v = Z nod k Σ ano (k) t'=-.3t; U = 12t; ED solver full gap ) vnod 2 k2 + v2 k2 sc = Antinodes 2 tot 2 nor sc (k) Z anod Σ ano (k) Σano= v = d (φ) dφ Node.2.1 tot /t nor /t sc /t= ( 2 tot - 2 nor ).5 /t sc /t= Z anod Σ ano /t ( δ ) ω kσ (ω) = εk Σ nor σ (k, ω) Σ ano (k, ω) Σ ano (k, ω) ω + ε k + Σ nor σ (k, ω) G 1

36 Conclusion 36 Clusters : a systematic expansion around DMFT Useful for Mott physics A lot of recent progress on methods to solve DMFT equations. Strong differences between Nodes and Antinodes : Normal phase : destruction of quasi-particles at antinodes in a transition selective in k-space. SC phase : two components SC gap.