Mott transition : beyond Dynamical Mean Field Theory O. Parcollet 1. Cluster methods. 2. CDMFT 3. Mott transition in frustrated systems : hot-cold spots. Coll: G. Biroli (SPhT), G. Kotliar (Rutgers) Ref: cond-mat/38577, cond-mat/37587 Grenoble-Atelier Théorique-12/11/23 p.1
Beyond DMFT : Motivations for cluster methods Stability : Is DMFT a good starting point? Overcome some limitations of DMFT : k dependence of Σ. Variations of Z, m, τ on the Fermi surface. Describe more complex orders : e.g. d-sc,ddw,cdw,af. Independent variation of Z and m Effect of J in the paramagnetic phase. Better description of insulator. Experimental motivations 2D organics High T c superconductors : Phase diagram. Competition AF / d-sc Pseudo-gap (AF fluctuations, DDW, stripes, RVB?). Hot and Cold spots Grenoble-Atelier Théorique-12/11/23 p.2
Cluster DMFT DMFT Cluster DMFT G G Interpolate between DMFT and finite dimensions. Finite size systems but G = Boundary Conditions. Choices : Type of cluster (e.g. size L c ). Self-consistency condition : G = F [G imp ]. Cluster quantities = Lattice quantities. Cluster DMFT is not unique Grenoble-Atelier Théorique-12/11/23 p.3
Various Cluster DMFT Schemes CDMFT (Cellular DMFT, or free cluster ) (G. Kotliar et al.) Real space cluster. multiorbital DMFT. DCA (M.H. Hettler, M. Jarrell, H.R. Krishnamurthy et al.) Reciprocal-space cluster. Σ(k) piecewise constant in B.Z. NCS : Nested Cluster Scheme, pair scheme. (G. Kotliar, A. Georges) Φ P S = (1 z) i φ 1site (G ii ) + <ij> φ pair (G ii, G jj, G ij ) φ 1site : One impurity problem; φ pair : Two impurities problem PCDMFT : Periodised CDMFT. (A. Lichtenstein et al.; G. Biroli et al.) Chain DMFT (S. Biermann et al.) 1d chain in a self-consistent bath. Quasi-1d materials. Cluster perturbation theory (A.M. Tremblay et al.) Grenoble-Atelier Théorique-12/11/23 p.4
How to choose? Requirements Convergence to the exact solution for L c (infinite cluster). Describes all phases including the symmetry broken phases AF,SC, CDW, DDW, stripes... Causality : ImΣ(ω), ImG(ω) Pair scheme is not causal, while CDMFT, DCA are causal. Proof using cutting rules. (G. Biroli, OP, G. Kotliar) Compare the methods : Simplest insulator : Ising model. Classical limit of Falikov-Kimball model (G. Biroli, OP, G. Kotliar) Test in 1d models (C.J. Bolech et al. PRB 67, 7511 (23) ), large-n models. CDMFT and DCA pass the tests. Grenoble-Atelier Théorique-12/11/23 p.5
Dynamical Mean Field Theory Spatial Mean Field approximation (exact for d ) Self consistent quantum impurity model Hubbard model Classical Ising model H = ijσ t ij c iσ c jσ + Un i n i H = J ij σ i σ j S eff = β G 1 (iω n) = G(τ) = T c(τ)c () Seff c σ(τ)g 1 (τ τ )c σ (τ ) + ( k Σ = G 1 G 1 c β dτun (τ)n (τ) m = σ H eff = Jh eff σ m = tanh(βh eff ) ) 1 1 + Σ(iω n ) h iω n + µ ɛ k Σ(iω n ) eff = zjm Grenoble-Atelier Théorique-12/11/23 p.6
CDMFT H = R m µr n ν ˆt µν (R m R n )c + R m µ c R n ν+ R 1 µr 2 νr 3 ρr 4 ς U µνρς ({R i })c + R 1 µ c+ R 2 ν c R 4 ςc R3 ρ. β S eff = dτdτ c µ(τ)g 1,µν (τ, τ )c ν (τ ) + β dτu αβγδ (R = )(c αc β c γc δ )(τ G cµν (τ) = T c µ (τ)c ν() S eff G 1 (iω n) = Σ c = G 1 G 1 c [ K R.B.Z. ( iω n + µ ˆt(K) Σ c (iω n )) 1 ] 1 + Σ c (iω n ) Grenoble-Atelier Théorique-12/11/23 p.7
CDMFT (2) 2- Lattice quantities cluster quantities Γ µνρς ^t(k) Impurity Solver G^ G^ c Σ^ c Self Consistency Lattice self energy is computed at the end Cluster to Lattice Conversion Restore the translation invariance on the original lattice : Σ lattice ij = α,β:α β=i j w α,β (Σ c ) αβ ^ Σ lat with w > (causality) and αβ;α β=x w α,β 1 x as L c Simplest case : w αβ = 1 L 2 c Phase with broken translation invariance captured by the cluster (AF) (if the cluster is big enough). Prescription for Σ lattice is not unique. Grenoble-Atelier Théorique-12/11/23 p.8
Anisotropic frustrated Hubbard model A effective model for 2D-Organics κ-(bedt-ttf) 2 Cu[N(CN) 2 ]Cl : H = i,j,σ t i,j c i,σ c j,σ + i ( U n i 1 )( n i 1 ) 2 2 Solve CDMFT for 2 2 cluster with QMC. 3 t t 4 1 t 2 Explicit expression of the lattice self-energy : Σ lattice (k) = 1 4 Σ ii + 1 (Σ12 ) + Σ 34 cos(kx )+ 4 2[ i=1 ] ( ) Σ24 + Σ 13 cos(ky ) + Σ 14 cos(k x + k y ) + Σ 23 cos(k x k y ) Grenoble-Atelier Théorique-12/11/23 p.9
Role of frustration.1.8 T/D.6.4 METAL CROSSOVER INSULATOR.2. 1 2 3 4 5 Uc1 Uc2 U/D Unfrustrated model t Large Néel temperature T N. Frustrated model t /t 1 Frustration destroys the AF order : Frustration is essential to see the Mott transition Grenoble-Atelier Théorique-12/11/23 p.1
Paramagnetic phases : DMFT versus cluster methods Computation of AF in DMFT (two sublattices e.g. Bethe Lattice) G 1 Aσ (iω n) = iω n + µ t 2 G Bσ (iω n ) σ =, G Bσ = G A σ DMFT : paramagnetic equations = equations for a frustrated model Example of the Bethe lattice with second neighbour t 1 = t AB G 1 (iω n) = iω n + µ (t 2 1 + t 2 2)G(iω n ) t 2 = t AA = In DMFT, simply solve the paramagnetic equations. In CDMFT, there is AF fluctuations inside the cluster : = In CDMFT one must solve a frustrated model Grenoble-Atelier Théorique-12/11/23 p.11
Frustrated model (t /t = 1) : a DMFT-like Mott transition... Difficult for QMC : frustration = sign problems. No AF order (at the temperature of the QMC). DMFT-like Mott transition. Double occupation d occ = i n i n i DMFT CDMFT.8.25.1 double occupation.7.6.5.4 [T Tc] QMC.1.5..5.5..5.1 [T Tc] IPT d occ.2.15.1 d occ.8.6.4.2 2 2.2 2.4 2.6 2.8 3 U.3.5.2 2.1 2.2 2.3 2.4 2.5 U 1 2 3 4 U (Kotliar et al. condmat/316) T/D = 1/2, 1/3, 1/4, 1/44 Grenoble-Atelier Théorique-12/11/23 p.12
... with k-dependent self-energy 3 regions in U U/D 2.2 : Metal but no k-dependance of Σ lattice 2.25 U/D 2.3 : Metal with k-dependance of Σ lattice. 2.35 U/D : Finite temperature insulator. Σ (ω = ) = t more isotropic Im G 11 -.2 -.4 -.6 -.8-1 Im Σ 11 -.5-1 Im Σ 14.5.4.3.2-1.2-1.4-1.5-2 1 2 3 4 5 ω 1 2 3 ω.1.5 1 1.5 2 2.5 ω U = 2, 2.1, 2.2, 2.25, 2.29, 2.31, 2.34, 2.43, 2.5 T/D = 1/44. Grenoble-Atelier Théorique-12/11/23 p.13
Hot and Cold spots A(k, ω = ) in the metallic region : U/D = 2., 2.25 Grenoble-Atelier Théorique-12/11/23 p.14
Summary and Perspectives Mott transition in 2 2 CDMFT with QMC. Frustration is crucial but solving a very frustrated model is difficult. Close to transition : Off-diagonal cluster self-energy. k-dependance of Σ lattice. Hot and cold regions. Isotropic model. Zero temperature computations (exact diagonalisation). Effect of doping. Approximate solvers for cluster DMFT equations. Grenoble-Atelier Théorique-12/11/23 p.15