Introduction to DMFT

Transcription

1 Introduction to DMFT Lecture 3 : Introduction to cluster methods 1 Toulouse, June 13th 27 O. Parcollet SPhT, CEA-Saclay 1. Cluster DMFT methods. 2. Application to high-tc superconductors.

2 General references for Cluster DMFTs 2 G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, Rev. Mod. Phys. 78, 865 (26) A. Georges, G. Kotliar, W. Krauth and M. Rozenberg, Rev. Mod. Phys. 68, 13, (1996). G. Biroli, O. Parcollet, G. Kotliar, PRB, 69,2518 (24) T. Maier et al, Rev. Mod. Phys. 77,127 (25)

3 3 DMFT is a good starting point to study Mott physics. But it has many limitations...

4 Is the Mean Field picture correct? 4 Favorable comparisons (See lecture 1), but : Stability with 1/d corrections. Shape of the transition line. Description of the insulator. Uc1,Uc2, Tc beyond mean field. 8 κ-(bedt-ttf) 2 Cu[N(CN) 2 ]Cl 7 P c 1 (T) T * Met Pc 2 (T)!!"" max 6 Semiconductor T * Ins (d#/dp) max T/W 5 Bad metal T (K) 4 3 Mott insulator 2 A.F insulator Fermi liquid P (bar) Clusters : an interpolation between mean field and d=2,3

5 The self-energy in not local! 5 In DMFT, no k-dependence of the self-energy. Consequences : Effective mass and Z are linked Z = m m Finite temperature lifetime, Z are constant along the FS. Not sufficient for high-tc. G latt (k, ω) = 1 ω + µ ɛ k Σ latt (k, ω) A(k, ω = + ) Σ latt (k, ω) = Σ impurity (ω) G 1 G 1 c Shen et al. Science 37, 91 (25) Clusters reintroduce some k-dependence in Σ

6 DMFT is only 1 site in a bath... 6 d-wave superconductivity? or DDW? (need at least a link) Competition AF-SC? Non trivial insulator a la RVB? (need at least a singlet?) Effect of J in the paramagnet not in DMFT e.g. cut divergence of the effective mass (See slave-bosons or large N, e.g. G. Kotliar, Les Houches 1988).!!#!"#$% strange metal ()!" $% *$ &' %&% )*+,-.+/,+ %&'( /1,-.+/,+ $ Clusters fix these problems (to some extent)

7 Cluster extensions of DMFT Principle : a finite number of sites in a self consistent bath. local quantum fluctuations short range quantum fluctuations 7 G G Interpolate between DMFT and finite dimensions Finite size systems BUT with boundary conditions G. Many choices : Type of clusters (e.g. shape, size) G (iω n ) = F lattice [G c ](iω n ) Self-consistency condition : How to approximate lattice quantities from cluster quantities? Cluster DMFT is not unique

8 How to build cluster methods? 8 3 points of view on DMFT : DMFT is 1 site in a self-consistent bath. Real space cluster : CDMFT (G. Kotliar et al. PRL ) DMFT is about neglecting the k-dependence of Σ Σ piecewise constant in the Brillouin zone: DCA M.H. Hettler, A.N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke,H.R. Krishnamurthy PRB 98 DMFT is an approximation of the Luttinger-Ward functional Φ φ AIM (G ii ) Higher approximation on phi. (A. Georges, G. Kotliar, W. Krauth and M. Rozenberg, Rev. Mod. Phys. 68, 13, (1996)) There are other methods (more later)! Equivalent for 1 site but lead to different cluster methods.

9 Reminder : DMFT equations (1 site, 1 orbital) 9 H = J ij σ i σ j H = ijσ t ij c iσ c jσ + Un i n i m = σ H eff = Jh eff σ m = tanh(βh eff ) h eff = zjm S eff = β G 1 (iω n) = G c (τ) = T c(τ)c () Seff c σ(τ)g 1 (τ τ )c σ (τ ) + ( k Σ = G 1 G 1 c β 1 iω n + µ t(k) Σ(iω n ) dτun (τ)n (τ) ) 1 + Σ(iω n ) Evaluation of lattice quantities. G latt (k, ω) = 1 ω + µ ɛ k Σ latt (k, ω) Σ latt (k, ω) = Σ impurity (ω) G 1 G 1 c

10 C-DMFT 1 4 Anderson impurities coupled to an effective bath DMFT Cluster DMFT = G S eff = β dτdτ c µ(τ)g 1,µν (τ, τ )c ν (τ ) + G β Superlattice dτu(n i n i )(τ) G 1 (iω n) = [ K R.B.Z. G cµν (τ) = T c µ (τ)c ν() Seff 1 µ, ν 4 Σ c = G 1 G 1 c ( ] 1 1 iω n + µ ˆt(K) Σ c (iω n )) + Σ c (iω n ) CDMFT equations

11 C-DMFT (2) IMPURITY PROBLEM 11 How to evaluate lattice quantities? Need to restore the translation invariance by periodization : G SELF CONSISTENCY G c, Σ A Lattice ij = with w > and α,β:α β=i j αβ;α β=x w α,β A Cluster αβ w α,β 1 Cluster to lattice conversion x as L c Which quantity should we periodize? Most irreducible (Σ rather than G!) Σ-periodisation versus M-periodization (cumulant) That choice is part of the cluster method. Simplest case : w =1/L²

12 C-DMFT : Σ-Periodization The original proposal (G. Kotliar et al. PRL ) Example : 2x2 cluster on a square lattice, w= const 12 Cluster site labeling 3 t t 4 1 t 2 Σ Lattice (k) = i=1 Σ Cluster ii + 1 [ (Σ Cluster 12 + Σ Cluster ) 34 cos(kx )+ 2 ( Σ Cluster 24 + Σ Cluster ) 13 cos(ky ) + Σ Cluster 14 cos(k x + k y ) + Σ Cluster 23 cos(k x k y ) ] Cluster quantities harmonics on the lattice Size of cluster = resolution in k space

13 C-DMFT : M-Periodization (1) 13 Σ- periodization generates spurious mid-gap states in Mott insulator (B. Kyung, A.M. Tremblay et al) Definition of the irreducible cumulant : Sum of all diagrams 1-particle irreducible in an expansion around the atomic limit (i.e. in t, not in U). For a presentation of this diagrammatics : (W. Metzner, PRB 43, ) Relation with the self-energy : The Green function is : M 1 (k, ω) = ω + µ Σ(k, ω) G(k, ω) = ( t(k) M 1 (k, ω) ) 1

14 C-DMFT : M-Periodization (2) 14 In DMFT, Σ and M are local. Hubbard, 1/2 filled,2x2 CDMFT, U/D = 2, ED solver, cluster quantities (T. Stanescu, G. Kotliar PRB 74, 12511, 26) M is more localized than Σ. M 11 M Σ 11 Im[M ij ] -.1 Im[Σ ij ] 2-2 Σ 13 Σ 11 + Σ ω n ω n

15 C-DMFT : M-Periodization (3) 15 Periodize the irreducible cumulant Same formula as for the self-energy : M Lattice (k) = 1 4 (T. Stanescu, G. Kotliar PRB 74, 12511, 26) 4 i=1 M Cluster ii [ (M Cluster 12 + M34 Cluster ) cos(kx )+ ( M Cluster 24 +M13 Cluster ) cos(ky )+M14 Cluster cos(k x +k y )+M23 Cluster cos(k x k y ) ] A non-linear relation Σ lattice (k, ) can have singularity! Some results (pockets) will rely on this periodization

16 C-DMFT : test of periodization procedure 16 Consistency check : G Cluster (ω) = k G Lattice (k, ω) Example of 1/2 filled Hubbard : cluster lattice M lattice Σ -.5 Im[G 11 ] ω n (T. Stanescu, G. Kotliar PRB 74, 12511, 26)

17 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 Example for 2x2 cluster on square lattice. G(k c, iω n ) = k 1 iω n + µ t( k) Σ(k c, iω n ) Σ(k, iω n ) Σ c (k c (k), iω n ) G 1 (k c, iω n ) = G 1 (k c, iω n ) + Σ c (k c, iω n ) Impurity model : same as for CDMFT. G, Σ cyclic on the cluster (,!) (!,!) Cluster momenta k c (,) K (!,) k k ~

18 DCA (2) 18 A real space formulation : CDMFT self-consistency condition with a modified hopping Also valid in the AF phase. (G. Biroli, O. Parcollet, G. Kotliar, PRB, 69,2518 (24)) S eff = β dτdτ c µ(τ)g 1,µν (τ, τ )c ν (τ ) + G cµν (τ) = T c µ (τ)c ν() S eff β dτun µ n µ (τ) G 1 (iω n) = [ K R.B.Z. Σ c = G 1 G 1 c ( ] 1 1 iω n + µ ˆt DCA (K) Σ c (iω n )) + Σ c (iω n ) t DCA αβ (K) = k c e ik c(α β) t(k + k c )

19 DCA (3) 19 Lattice quantities : In DCA, translation invariance is preserved by construction But there is still a need for cluster to lattice conversion Use spline interpolation to get a smooth Σ(k,ω) on the lattice (M. Jarrell et al.).

20 Functional point of view 2 Functional formulation of DMFT Γ BK [G ij ] = Tr ln G ij Tr(g ij 1 G ij) + Φ BKLW [G ij ] G ij (t) T c i (t)c j () Σ ij = δφ BKLW δg ij DMFT as an approximation of the Baym-Kadanoff functional. Φ φ AIM (G ii ) Exact in large dimension (Metzner-Vollhardt,1989) Anderson impurity model = machinery to solve this approximation (Kotliar-Georges 1992)

21 Φ-derivability CDMFT : the impurity problem is Φ-derivable : 21 Φ CDMF T (G) = R Φ(G µ,r;ν,r G ρ,r;λr = ) but the lattice conversion breaks the Φ-derivability. DCA is also Φ-derivable (but not the lattice conversion!). 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 Coarse-graining of the momentum conservation at the vertex. T. Maier et al, Rev. Mod. Phys. 77,127 (25) Φ-derivability conservative approximation (Baym-Kadanoff) In particular, Luttinger Theorem...

22 Luttinger Theorem 22 In Fermi liquid, volume of the Fermi surface is conserved. Derivation is based on the existence of the Φ functional (See e.g. Abrikosov-Gorkov-Dzyaloshinky, sect 19.4) When Σ becomes singular : problem! Fermi surface : location of poles of G(k,) Luttinger surface : location of zeros of G(k,) BUT when Σ has a singularity, volume enclosed by both surfaces not necessarily conserved! (A. Georges, O. Parcollet and S.Sachdev, PRB , (21)) G ω Σ

23 Nested Schemes 23 Take a higher approximation of the Luttinger Ward functional. Φ = φ 1 (G ii ) + φ 2 (G ii, G jj, G ij ) + i <ij> Apply it to a 1 site and a 2 site problem : φ 1site (G ii ) = φ 1 (G ii ) φ 2sites (G ii, G jj, G ij ) = φ 2 (G ii, G jj, G ij ) + φ 1 (G ii ) + φ 1 (G jj ) Introducing z, the connectivity of the lattice : Φ Nested (1 z) i φ 1site (G ii ) + <ij> φ 2sites (G ii, G jj, G ij )

24 Nested Schemes (2) Complete equations : (A. Georges, G. Kotliar, W. Krauth and M. Rozenberg, Rev. Mod. Phys. 68, 13, (1996)) Σ loc = δφ δg ii = δφ 1site δg ii + z ( δφ2sites δg ii δφ 1site δg ii ) 24 Σ nn = δφ 2sites δg ij Σ latt = Σ loc (iω n ) + t(k)σ nn (iω n ) G loc = k 1 iω n t(k) Σ loc (iω n ) t(k)σ nn (iω n ) G nn = k G = e ik. δ iω n t(k) Σ loc (iω n ) t(k)σ nn (iω n ) ( iωn t(k) Σ latt (K) ) 1 K R.B.Z.

25 Many other possible schemes! 25 For a review of some methods : T. Maier et al, Rev. Mod. Phys. 77,127 (25) Chain DMFT : a chain in a self-consistent bath (see below) (S. Biermann et al) Extended DMFT and Extended cluster DMFT. (Q. Si et al., see e.g. K. Haule) Cluster perturbation theory. Not self-consistent clusters A.M. Tremblay, D. Sénéchal et al., T. Maier et al, Rev. Mod. Phys. 77,127 (25) Self-energy functional (M. Potthof et al.) PCDMFT : use Σ(k,ω) in the self-consistency (G. Biroli, O. Parcollet, G. Kotliar, PRB, 69,2518 (24))...

26 So what is the best cluster method? 26

27 Classical limit A large body of work in classical statistical mechanics (Bethe, Kikuchi, Cf Domb-Green series) 27 How do our cluster methods connect to this? For example large U limit of Falikov-Kimball model = Ising model (G. Biroli, O. Parcollet, G. Kotliar, PRB, 69,2518 (24)) H = ijσ t ijσ c iσ c jσ + Un i n i,with t = CDMFT reduces to Ising cluster with a bath at the boundary. H = H Ising + Jσ 1 σ 3 3 J 1 3 Exercise : show that at large U, one finds the Ising Weiss theory. Example of 2x2 cluster

28 Classical limit (2) 28 DCA : Cluster is cyclic, no boundary. An effective J on ALL links. (G. Biroli, O. Parcollet, G. Kotliar, PRB, 69,2518 (24)) Nested scheme (2 sites) leads to Bethe-Kikuchi method. Solve 1 site and 2 sites problems and fix the field h so that local magnetization is the same in the 2 problems : H (1) = zhs (1) H (2) = (z 1)h(S (2) 1 S (2) 2 ) + JS(2) 1 S(2) 2 < S (1) > =< S (2) 1 >= < S (2) 2 > Convergence of Tc vs size much faster than CDMFT or DCA. BUT...

29 Causality issue 29 Causality = Im Σ < (definite negative matrix) Strong Causality property : guarantee that Im Σ < for any bath G Hence there will not be any causality violation at any step in the DMFT iterative loop. Quantum impurity problem is causal by construction : the problem lies in the self-consistency. It is not obvious to have a causal scheme : Nested schemes show causality violations CDMFT, DCA are proven to be causal (See original papers) What is the origin of this problem?

30 Origin of the causality problem 3 Using Cutkovsky-t Hooft-Veltmann cutting technique. (G. Biroli, O. Parcollet, G. Kotliar, PRB, 69,2518 (24)) Diagrammatic expansion of Use Keldysh technique ImΣ R (ω)!! "!!!- #!!! $$$," """"#!"$##!$ $%&%' ( $$( """"#!"$##!$ $%&%' $%&%' ) $$) """"#!"$##$!$ $%&%' * $$$$$$* """"#!"$##$!$ + $$$$$$+ ImΣ R (ω) is a quadratic form in half-diagrams R and L Can we form a square? Nested scheme is not causal (was known empirically before). Roughly speaking, problem arises when self-energy not local enough See also A. Fuhrmann, S. Okamoto, H. Monien, and A. J. Millis PRB 75, (27)

31 Convergence properties 31 Debate in the literature about CDMFT and DCA convergence (See controversy by Biroli et al. vs Jarrell et al., 22) DCA (cluster of linear size L) : CDMFT : A priori : But : δm m m L= 1/L 2 δm 1/L δm center e L/ξ Cavity construction is not exact : error of order 1 at the boundary. Spurious transition in one dimension. Large sizes : Clusters DMFT are not better than finite systems!

32 Convergence properties (2) 32 For large clusters, one has to improve CDMFT by using more the result at the center : See G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, Rev. Mod. Phys. 78, 865 (26) Reweigh the self-energy in the self-consistency condition. Σ Cluster αβ Σ w-cdmft α β = w α β αβ ΣCluster αβ α β w αβ γδ = δ α β,γ δf c (α)f c (β) f c (α) 2 = 1 α Remove spurious transition in 1d. Tc convergence close to DCA. This debate is interesting only if one can solve large clusters at large U!

33 Test the cluster method in one dimension 33 A priori the worst case for a mean field methods Computation of short range physics, thermodynamics. DMFT can NOT capture Luttinger liquid large distance physics. e.g. : occupation vs chemical potential (M. Capone, M. Civelli, S.S. Kancharla, C. Castellani, G. Kotliar, PRB ) BA = Bethe Ansatz 1.75 n.5 CDMFT PCDMFT BA DMFT µ

34 Test the cluster method in one dimension (2) 34 What about dynamical quantities? Comparison to DMRG (in Matsubara, with Hallberg s algorithm) (M. Capone, M. Civelli, S.S. Kancharla, C. Castellani, G. Kotliar, PRB ) Im G U/t=1 DMRG CDMFT PCDMFT DMFT U/t=7 Re G U/t= ! " U/t= ! "

35 35 1. Cluster methods. 2. Application to high-tc superconductors.

36 Is the DMFT scenario for Mott transition confirmed by clusters? 36

37 U-driven Mott transition 37 Frustrated model: signature of Mott transition in double occupancy, as in 1 site DMFT. T/W Frustration is essential! (hard for QMC) DMFT C-DMFT (2x2) ("(+.25.1,-./ ("(* ("() ("(& ("(% 9:!:3; >?@ ("(#( ("((& ("(((!("((&!("((& ("((( ("((& ("(#( 9:!:3; <=:2 d occ T d occ U ("($.5 ("(!!"#!"!!"$!"%!"& ' U (Kotliar et al. condmat/316) O. Parcollet et al. PRL 23 But...

38 Cluster corrections close to Mott transition 38 Fixed T/D = 45>Tc, various U. Anisotropic Hubbard model T/D U/D? A(k, ω = + ) DMFT metal Metal. Hot-Cold spots Finite T insulator U/D U/D U/D Σ 11 c 1 + ( ) 1 1 Z iωn Σ 11 c 2 + ( ) 1 1 Z iωn Σ 11 c 3 Σ 12, Σ 14 Σ 14 k Σ lattice Modulation of the finite T lifetime DMFT metal : a generic feature at small U, large doping.

39 Hot Cold regions due to local Mott physics 39 Metal DMFT metal Anisotropic metal t =!.3t t =!.3t t =!.3t n=1 : Mott A(k, ω = + ) Hole doped Teff= D/128, U=16t t,t isotropic Hubbard model 2x2 CDMFT,ED.8 m AF order vs t /t t /t 2 2 n=.73 n=.7 2 t =+.3t 2 t =+.9t n=.89 Electron doped Highly frustrated M. Civelli, M. Capone, S. S. Kancharla, O.P and G. Kotliar Phys. Rev. Lett. 95, 1642 (25) 2 n=.9 t =+.3t 2 t =+.9t n=.96 2 n=.95 n=.69 n=.92 n=.96. x t =+.3t 2 t =+.9t 2

40 Renormalization of the Fermi surface 4 t eff (k) = t(k) ReΣ lattice (k, ) Fermi Surface can be strongly renormalized by interactions close to the Mott transition. Model dependent effect! Position of cold regions for hole/electrons doped similar to ARPES 4.5 t eff /t t eff /t (t eff /t eff )/(t /t) t /t= -.3 t /t= +.3 t eff /t t eff /t (t eff /t eff )/(t /t) n M. Civelli, M. Capone, S. S. Kancharla, O.P and G. Kotliar Phys. Rev. Lett. 95, 1642 (25)

41 Phase diagram of the Hubbard model? 41

42 Does the Hubbard model have d-sc?.1 T c 42 Previous works with clusters : A. Lichtenstein et al. PRB 62, R9283 (2) DCA : M. Jarrell et al, PRL 85,1524 (21)! T [ev] Pseudogap T N T * Fermi Liquid like *$!# ()!"#$% strange metal!" &' $%.2 AF d-wave superconductivity ! U/D=2 %&% )*+,-.+/,+ %&'( /1,-.+/,+ $ Large Clusters at U/D=1 (DCA), up to 26 sites : Tc.2t T. Maier et al., PRL 95, 2371 (25) 2x2 CDMFT also has d-sc phase, but at lower T (M. Civelli, K. Haule). T c t/1 T DCA 2x2 c All cluster methods consistently predicts d-sc, AF, with different Tc

43 AF, d-sc : coexistence or competition? 43 M. Capone, G. Kotliar Phys. Rev. B 74, (26) Qualitative difference between large and small U. Small U : coexistence between AF + d-sc Higher U, first order transition. U/t=4 %)$.6 AFM Pure Solution dsc Pure Solution AFM Mixed Solution dsc Mixed Solution +,- 8 '(!"#!"#$%&.4.2 %)$*'(! ED solution, 8 sites in bath.5.1!

44 Two energy scales in SC phase 44 Raman experiments. Mesure the gap around the node and at the antinode. T << T c Antinodal B 1g B 1g B 2g Raman Hg-121 (This work) Bi-2212 (ref. 4) Bi-2212 (ref. 3) Y-123 (ref. 5) Y-123 (ref. 4) LSCO (ref. 4) Bi-2212 max from other techniques Tunnelling (refs 32,33,34) ARPES (refs 23,24,28) Nodal B 2g max 4T c /T c = 4 ( (.16 p) 2 ) M. LeTacon et al., Nature Physics, 2, 537, Doping p

45 2 gaps in high-tc superconductors M. Civelli, M.Capone, A. Georges, K. Haule, O. Parcollet, T Stanescu, G. Kotliar, arxiv: Solution of Hubbard model, 2x2 cluster, ED solver, SC phase Cluster quantities : "!"#$" Im# $) /t Im# $* +t Im# $$ +t NS T/t=1/1 NS T/t=1/32 SC T/t=1/3.5 1! n /t ano Re# $* /t "!"#$% "!"#$" "!"#"& "!"#"' "!"#"( "!"#"% "!"#")( "!"#") "!"#"*( "!"#"*.5 1! n /t.6.3 Anomalous part non-monotonic in δ No FL at δ.1 : ImΣ 13 () not zero, does not scale like T²

46 2 gaps in high-tc superconductors M. Civelli, M.Capone, A. Georges, K. Haule, O. Parcollet, T Stanescu, G. Kotliar, arxiv: Analyze one particle spectrum, with/without anomalous Σ FL in the nodes Pseudo-gap at antinodes (seen previously in normal phase, see below) Asymmetric spectra close to δ= in SC $ 2 $ k anod k nod nodal A ""#$%& ""#$%' ""#$#( ""#$#& ""#$#) ""#$#* ""#$# !/t.8.16 " k x C Z nod Z anod.8.16 " antinodal # ano = B -.2.2!!t antinodal D,#-$. ky (,) nodal kx -.2.2!/t,$-$. antinodal

47 ! 2 gaps in high-tc superconductors M. Civelli, M.Capone, A. Georges, K. Haule, O. Parcollet, T Stanescu, G. Kotliar, arxiv: Low frequency analysis close to the node : QP velocity to FS Anomalous velocity // to FS Slope of the gap to the node.5 A(k, ω) Z nod δ ( ω v nod = Z nod k ξ k v = Z nod k Σ ano (k) ) vnod 2 k2 + v2 k2.25 v nod /(a o t) v "!"a o t)

48 2 gaps in high-tc superconductors M. Civelli, M.Capone, A. Georges, K. Haule, O. Parcollet, T Stanescu, G. Kotliar, arxiv: Decompose the gap in the 1 particle spectrum. sc = 2 tot 2 nor sc (k) Z anod Σ ano (k, ).1 " tot /t " nor /t " sc /t= ( " 2 tot -"2 nor ).5 /t " sc /t= Z anod # ano /t.5 Two gaps picture similar to experiments !

49 Fermi surface in normal phase? 49

50 PRL 97, (26) P H Y S I C A L R E V I E W The inverse propagator 1d-2d G 1 k;! transition 5 in Eq. (1) plays the role of a the gap an long-range, mapped time-dependent C. Berthod, T. onto Giamarchi, an effective S. Biermann,A. 1DGeorges, problem PRL 97,13641 described(26) by the action S Chain of spinless eff S fermions eff R hopping amplitude. It must be duefrequen to exc determined from the requirement with d H that the in-chain Green s int with to describ eral req function G calculated from Snext-neighbor eff coincides repulsion with thev, t=1 coupled by inter-chain hopping t. S eff X Z form ior isof no k? -summed Green s function d d c y of the r G 1 original r 2D model, r ; c r : ever, byeven fitti G k;!! rr k 1d + RPA approach "? k;! 1 [5,15]. This requirement implies : F.H.L. Essler, A.M. Tsvelik, PRB 65, , (22) much promin bett (1) count which resid Chain-DMFT G 1 : a periodic chain (32 sites) + DMFT in the transverse The inverse propagator G direction. Keep k resolution within 1 k;! k;!! k G 1 k; in! Eq. (1) R G k; plays! ; the role (2) which we of a the chain. Well controlled at long-range, time-dependent hopping amplitude. It must be int the k;!, gap with small k determined t. the in-chain momentum, due to from the Solve with Hirsch-Fye QMC. R requirement k 2t cosk the using the that p the in-chain Green s to des bare dispersion, function Gandcalculated R z sign Re z from S eff 1=z coincides 2 2t? with 2 resultingk? the k;! form is from -summed the integration Green s function over the of the transverse original energy order in p 2D model, G k; S! eff X! Z ever, ev "? 2t? cos k?. Equations d d k " c? y (1) k; and r G 1 r! (2) 1 can [5,15]. be readily r ; This c r requirement from theimplies assumption that the lattice self-energy see that th data and m : much b derived rr count r does not depend G 1 k;! on transverse! momentum k k G 1?, and is given results in k;! R G k;! ; (2) which by the effective in-chain self-energy: region we int k; with k k; the! in-chain G 1 momentum, k 2t cosk the using k;! model fits G 1 k;! : p (3) and t bare dispersion, and R z sign Re z 1=z 2 2t? 2 resulting will be from ourthe mainintegration concern here. overinthe order transverse to evaluate energy order i? val k;! fore use th k;! formation

51 1d-2d transition C. Berthod, T. Giamarchi, S. Biermann,A. Georges, PRL 97,13641 (26) 51 3 regimes in t : insulator, metal with pocket FS, metal with simple FS. 2 1 t =.2 t =.91 t t =.45 t =.33 t Fit of the self-energy Energy / t Re[Σ(k, ω)] (a) ξ k 2t cos(k ) π/2 π k t =.6 t = (b) π/2 π k + + π Σ(k, ω) = ( /2)2 ω + λξ k + (1 λ)ξ k + Σ int Red term : simple harmonics expansion in cumulant, not in Σ Energy / t 1 2 (c) π/2 π k (c') (b') π π k Finite temperature fit Zero temperature extrapolation + k π Divergence of Σ, line of Shaded area : domain covered by the free dispersion ξ k 2t cos(k )

52 1d-2d transition C. Berthod, T. Giamarchi, S. Biermann,A. Georges, PRL 97,13641 (26) 52 Z along the Fermi surface ARPES curves.7 π A(k, ) I(k) A(k, ) I(k).6.5 t = Z k.2.1. (a) k π/2 π/2 π k (a) π π k π (b) π k π.5 1 Large variation of Z along FS Hot spot remains in the metal I(k) : with some k resolution. The rear part of the FS can not be seen in experiments.

53 Pocket Fermi surface in 2d? (T. Stanescu, G. Kotliar PRB 74, 12511, 26) 53 CDMFT, Hubbard model, U/D = 2, vs δ, ED solver. At low δ, a line of zeros of G appears and the topology of the FS changes. At finite temperature/resolution, ARPES does not see the second part of the FS. 1d : Luttinger surface appears at low δ and evolve with δ r(k) A(k) (π,) (π,) n =.78 (π,π) (π,π) (π,) (,) (,π) (,) (π,) n =.92 C A B (π,π) (,π) (π,π) Max Cumulant periodization is necessary here. Discussion : resolution? Experiments on YBCO in high field : pocket Fermi surface. N. Doiron-Leyraud at al, Nature, 27 (,) (,π) (,) (,π) Self-energy plot/ ARPES r(k, ω) = t(k) µ Σ(k, ω)

54 Hidden quantum critical point? K. Haule, G. Kotliar, condmat/65149 Signatures of a critical point in the normal phase at δ.1 (t-j model, NCA solver) Im!( " ) Large scattering rate in the (,π) component of the cluster Σ coherence scale Tc # #=.2 #=.1 #=.18 #=.3 #(") Power law in optics at optimal doping.1.1 " 1!=.2 $=.2!=.2 $=.36!=.18 $=.2!=.18 $=.36 " %2/3 a) T/t NCA solution : hint towards a RKKY/Kondo QCP. Singlet (RKKY) QCP? Kondo δ

55 Will CDMFT unify high-tc theory?? 55 Doped Mott insulator strange metal Quantum phase transition hidden below the SC. T spin gap Fermi liquid!!"#$% quantum critical superconductor x RVB (Anderson 87) Kotliar-Liu (88) Prediction of d-sc! *$!# ()!" &' %&% )*+,-.+/,+ %&'( /1,-.+/,+ $% $ T ordered x Fermi liquid T c

56 Low energy theory? 56

57 RVB in slave bosons picture G. Kotliar, J. Liu Phys. Rev. B 38, 5142 (1988) 57 t-j model in slave boson, no AF order, d-wave superconductivity Schematic phase diagram strange metal Pseudo-gap T spin gap Fermi liquid Boson condenses superconductor x

58 Low energy solution of Cluster DMFT? 58 Slave boson = 1 low energy theory of 1 site DMFT. Σ is independant of k ( Σ(k, ω) = const + ω 1 1 ), Z = δ Z Generalization : rotationally-invariant slave-boson F. Lechermann, A. Georges, G. Kotliar, O.Parcollet, arxiv: Describe multiplets (for realistic systems) Describe Z(k) (variation along the Fermi surface) Tested against CDMFT at low energy (Hot/cold region e.g.)

59 Heavy fermions 59 Another class of strongly correlated materials Quantum critical points : scenario under debate... Theoretical model : Periodic Anderson Model. H = t c iσ c jσ µ c iσ c iσ + V ) (f iσ c iσ + h.c. ij i i +(E f µ) i f iσ f iσ + U i n f i nf i (1)

60 Cluster for Anderson lattice Cluster DMFT solution of the Anderson lattice model (2 sites, ED solvers) L. De Leo, M. Civelli, G. Kotliar, condmat/72559; see also work by Q. Si (extended DMFT), P. Sun et al... 6 Work in progress : test various scenarios T K /T (CDMFT) T K /T (DMFT) Magnetization Z (AF maj. spin) Z (AF min. spin) Z (PM) V

61 Summary 61 CDMFT : a dynamical RVB? Not only! Various phases (AF, PG, d-sc) SC phase : 2 gaps Normal phase : strong dichotomy node/antinodes. Pocket FS. Hidden quantum critical point? Towards a unified theory with RVB and QCP?! *$!# ()!"#$%!" &' %&% )*+,-.+/,+ %&'( /1,-.+/,+ $% $ Work still in progress : Low energy solution : build a simple picture out of DMFT results. Vertex calculation/ Real ω exact solution? Improve k resolution (patch basis)

62 62 Tomorrow S. Biermann : DMFT and realistic calculations!