Cluster Dynamical Mean Field Analysis of the Mott transition O. Parcollet CEA-Saclay FRANCE Dynamical Breakup of the Fermi Surface in a doped Mott Insulator M. Civelli, M. Capone, S. S. Kancharla, O.P., G. Kotliar. cond-mat/411696 Cluster Dynamical Mean Field Analysis of the Mott Transition O.P., G. Biroli, and G. Kotliar, Phys. Rev. Lett. 9, 64 (4) Cluster dynamical mean-field theories: Causality and classical limit G. Biroli, O.P., and G. Kotliar, Phys. Rev. B 69, 518 (4) March Meeting 5 p.1/7
Mott transition Metal-Insulator transition due to interactions. Relevant for e.g. : V O 3. D organics : κ (BEDT-TTF) Cu[N (CN) ]Cl). P. Limelette et al. PRL (3) High-Tc Cuprates are doped Mott insulators. Explanation of pseudo-gap, d-sc mechanism? DMFT : a successful method for Mott transition. Basics : A. Georges, G. Kotliar, W. Krauth, M. Rozenberg Rev. Mod. Phys. (1996) V O 3 P. Limelette et al. Science 3, 89 (3) Realistic calculations (in combination with band theory) : e.g. Pu : S. Savrasov, G. Kotliar, E. Abrahams, Nature 41, 793 (1) Correlation-assisted Peierls transition in VO S. Biermann, A. Poteryaev, A. I. Lichtenstein, A. Georges, cond-mat/415 In this talk : Beyond DMFT with cluster methods March Meeting 5 p./7
DMFT : a self-consistent impurity problem Local Quantity Effective problem Weiss Field Ising m = S i Ising spin in field h eff m = tanh(βh eff ) h eff = Jzm DMFT Local Green function G Quantum impurity Bath G (t) G(t) T c i (t)c i () G = F [G] G A. Georges, G. Kotliar (199) H Ising J i,j S is j H Hubbard = ijσ t ijc iσ c jσ + Un i n i March Meeting 5 p.3/7
Why a dynamical Mean Field? DMFT : a mean field on the local Green function G(ω) DMFT describes : ρ(ω) 1 π ImG(ω),δ = Fermi liquid with low coherence scale ɛ F = ZD. (Hubbard model, DMFT,IPT,T=) U/D=1 Coherent and incoherent part Transfer of spectral weight ImG U/D= U/D=.5 from low to high ω U/D=3 U/D=4 4 4 ω /D DMFT goes beyond a low energy quasi-particle description. March Meeting 5 p.4/7
Mott transition in DMFT : Phase diagram Frustrated Hubbard model at half filling : δ =, paramagnet A. Georges, G. Kotliar, W. Krauth, M. Rozenberg Rev. Mod. Phys. (1996).1 Coexistence region.8 T/D.6.4 METAL CROSSOVER INSULATOR First Order transition.. 1 3 4 5 Uc1 Uc U/D March Meeting 5 p.5/7
M DMFT, a starting point for some SCES? T T D organics : κ (BEDT-TTF) Cu[N (CN) ]Cl) 8 7 6 Semiconductor P c 1 ( T ) P c ( T ) * ρ(τ) M e t m a x * I n s ( d σ/ d P) m a x.3 U=4 K 3 bar W=3543 K 6 bar W=3657 K 7 bar W=3691 K 15 bar W=3943 K 1 kbar W= 5 K 5 B a d meta l. T (K) 4 3 ott ins ul a tor ρ (Ω.cm).1 1 A.F i n s u l a t o r F ermi l iq uid 1 3 4 5 6 7 8 P (bar). 5 1 15 5 3 T (K) P.Limelette et al, PRL(3). March Meeting 5 p.6/7
Cluster DMFT (1) Missing in DMFT... Various orders : e.g. d-sc,ddw,stripes k dependence of Σ(k, ω) = Z m m. Variations of Z, m, τ(t ), T coh along the Fermi surface. Non magnetic insulators (frustrated magnets?) Non-local interactions. Short range AF correlations.... but present in cluster methods short range spatial quantum fluctuations DMFT Cluster DMFT = G G March Meeting 5 p.7/7
Cluster DMFT () Interpolate between DMFT and d/3d when cluster size. Finite size systems BUT : in a bath G (like Boundary Conditions ) in thermodynamic limit (phase transitions) Static (Hartree-Fock) approximation possible at longer distance. Choices : Type, shape of clusters (e.g. (linear) size L c ). Self-consistency condition : G = F [G]. How to compute lattice quantities from cluster quantities. Various cluster methods : parametrisations of Σ(k) CDMFT (G. Kotliar, S. Savrasov, G. Pálsson, G. Biroli PRL (1)) DCA (M.H. Hettler, A.N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke,H.R. Krishnamurthy PRB (1998)) Nested Cluster Schemes, PCDMFT, Fictive impurity models,... March Meeting 5 p.8/7
In this talk Questions : Is the DMFT picture of the Mott transition stable? (Cluster as 1/d corrections). Existence of anisotropic FL (due to interactions)? Z(k), T coh (k), (m /m band )(k) Renormalization of the Fermi Surface? Orders : AF, d-sc,...? Study CDMFT, cluster, Hubbard model on square lattice for various U, t /t, δ using QMC (Hirsch-Fye) and ED (exact diagonalization) 3 4 3 4 Isotropic model : t t Anisotropic model: t t 1 t 1 t March Meeting 5 p.9/7
CDMFT H = R m µr n ν ˆt µν (R m R n )c + R m µ c R n ν+ R 1 µr ν R 3 ρr 4 ς U µνρς ({R i })c + R 1 µ c+ R ν c R 4 ςc R3 β β S eff = dτdτ c µ(τ)g 1,µν (τ, τ )c ν (τ ) + dτu αβγδ ()(c αc β c γc δ )(τ) G cµν (τ) = T c µ (τ)c ν() G 1 (iω n) = [ K R.B.Z. S eff Σ c = G 1 G 1 c ( ] 1 1 iω n + µ ˆt(K) Σ c (iω n )) + Σ c (iω n ) March Meeting 5 p.1/7
CDMFT () Γ µνρς ^t(k) Lattice quantities cluster quantities Lattice self energy is computed at the end Impurity Solver G^ G^ c Σ^ c Self Consistency Cluster to Lattice Conversion Restore translation invariance on lattice ( clusters ): Σ ^ lat 3 4 Σ lattice (k) = 1 4 4 Σ ii + [ 1 (Σ1 ) + Σ 34 cos(kx )+ i=1 1 ] ( ) Σ4 + Σ 13 cos(ky ) + Σ 14 cos(k x + k y ) + Σ 3 cos(k x k y ) Cluster self-energies parameterize the k dependence. Prescription not unique : cumulants (T. Stanescu et al. To appear), periodize the Green function (B. Kyung et al. condmat/5565, S1-7). March Meeting 5 p.11/7
Mott transition in DMFT : Phase diagram Frustrated Hubbard model at half filling : δ =, paramagnet A. Georges, G. Kotliar, W. Krauth, M. Rozenberg Rev. Mod. Phys. (1996).1 Coexistence region.8 T/D.6.4 METAL CROSSOVER INSULATOR First Order transition.. 1 3 4 5 Uc1 Uc U/D Without frustration, Néel temperature T N is large. Frustration is essential to see this Mott transition March Meeting 5 p.1/7
Paramagnetic solutions of CDMFT In DMFT : Paramagnetic sol. for t = Paramagnetic solution for t /t 1 In Cluster DMFT : different type of paramagnets Paramagnet for t = Strong AF fluctuations inside the cluster. Paramagnet with short range AF fluctuations at low resolution? Paramagnet for t /t 1 Similar to DMFT solution Can only be reached by solving a frustrated model. QMC solution, but sign problem at low temperature. Anisotropic model, t /t =.9 March Meeting 5 p.13/7
A DMFT-like Mott transition... Double occupation d occ = i n i n i. No AF order DMFT CDMFT.8.5.1 double occupation.7.6.5.4 [T Tc] QMC.1.5..5.5..5.1 [T Tc] IPT d occ..15.1 T d occ.8.6.4...4.6.8 3 U.3.5..1..3.4.5 U 1 3 4 U G. Kotliar, E. Lange, M. J. Rozenberg et al. PRL () T/D = 1/, 1/3, 1/4, 1/44 Application : Interactions-induced adiabatic cooling and antiferromagnetism of cold fermions in optical lattices F. Werner, OP, A. Georges, S. Hassan, to appear March Meeting 5 p.14/7
... with k-dependent self-energy Fixed T/D = 45 > T c, various U : 3 regions in U..1.8 T/D.6.4.. METAL CROSSOVER U/D? INSULATOR 1 3 4 5 Uc1 Uc DMFT metal Metal. Hot-Cold spots Finite T insulator U/D..5 U/D.3.35 U/D ) iωn Σ 11 c + ( 1 1 ) Z iωn Σ 11 c 3 Σ 11 c 1 + ( 1 1 Z Σ 1, Σ 14 Σ 14 k Σ lattice Modulation of the finite T lifetime March Meeting 5 p.15/7
Self-energies in imaginary time Im Σ 11 -.5-1 -1.5 Im Σ 14.5.4.3..1-1 3 4 5 ω.5 1 1.5.5 ω U =,.1,.,.5,.9,.31,.34,.43,.5 T/D = 1/44. March Meeting 5 p.16/7
Hot and Cold regions A(k, ω = ) in the metallic region : U/D =.,.5 March Meeting 5 p.17/7
Quasi-particle residue and effective mass Close to the Mott transition, Z m m 1 Hot Cold.95 (m * /m)z.9.85 T/D = 1/44.8 1 1.5 1.5 1.75.5.5 U March Meeting 5 p.18/7
Fermi liquid at T =? Conjecture : for T, ImΣ 14 Z (π,π) Z (,π) k-dependence in Z due to the proximity of the Mott insulator. QMC inefficient for low T and high U. = Exact Diagonalization method. March Meeting 5 p.19/7
Exact diagonalization Hamiltonian representation of the bath G with free electrons. Hoppings and energies of the Bath determined self-consistently. For clusters : 8 sites in the bath 3 t 1 t t 4 Not limited by frustration or large U. T = calculation but at finite EFFECTIVE temperature T eff. Size of the bath is small. March Meeting 5 p./7
Exact diagonalization : Comparison with QMC t =, δ =.1,U = 8t, βd = 5, D = 4t = 1, paramagnet..1 -.1 -. -.3 -.4 ImΣ 11 QMC ImΣ 1 QMC ImΣ 14 QMC ImΣ 11 ED ImΣ 1 ED ImΣ 14 ED -.5.5 5 7.5 1 ω n March Meeting 5 p.1/7
Hot and cold regions A(k, ω = + ). T eff = D/18, U = 16t, isotropic model. Condmat/411696 t =.3t n =.73,.89,.96 t = +.3t n =.7,.9,.95 t = +.9t n =.69,.9,.96. x March Meeting 5 p./7
Renormalization of the Fermi Surface t eff (k) = t(k) ReΣ lattice (k, ) Fermi Surface is renormalized by interactions close to the Mott transition. Details are model dependent. 4.5 t eff /t t eff /t (t eff /t eff )/(t /t) 3 1.5 1. t /t= -.3 t eff /t t eff /t (t eff /t eff )/(t /t) t /t= +.3.5.6.7.8.9 1 n Bare t, t can not be extracted simply from ARPES data March Meeting 5 p.3/7
Fermi liquid or Non Fermi Liquid at T =? Pseudo-gap at T =? B. Kyung et al., condmat/5565, S1-7 ω-resolution problem in ED : ω.1t = T eff D/4 T Mott!..1 -.1 -. -.3 -.4 ImΣ 11 QMC ImΣ 1 QMC ImΣ 14 QMC ImΣ 11 ED ImΣ 1 ED ImΣ 14 ED -.5.5 5 7.5 1 ω n Solution at T = is still an open question March Meeting 5 p.4/7
d-sc in CDMFT d-sc order present at T = (M. Capone and S. S. Kancharla, private communication).4.3 U = 4t U = 8t U = 1t U = 16t..1.5.1.15..5.3 δ March Meeting 5 p.5/7
Conclusion Cluster methods add k resolution to DMFT. Mott transition in frustrated systems in CDMFT : DMFT-like Mott transition...... with Hot-Cold regions due to the proximity of Mott transition. Renormalization of the Fermi surface. d-sc phase at T =. Open questions Complete solution of clusters (T = )? Fermi liquid or Non Fermi liquid? Nature of the Mott insulator? March Meeting 5 p.6/7