Dynamical Mean Field Theory. Quantum Cluster Methods

Transcription

1 Dynamical Mean Field Theory and Quantum Cluster Methods David Sénéchal Département de physique & Institut Quantique Université de Sherbrooke International Summer School on Computational Quantum Materials June 1, 2016 Institut Quantique 1 / 119

2 Outline 1 Introduction 2 Dynamical Mean Field Theory 3 Cluster Perturbation Theory 4 Cluster Dynamical Mean Field Theory 5 The self-energy functional approach 6 Exact Diagonalizations 2 / 119

3 Outline 1 Introduction 2 Dynamical Mean Field Theory 3 Cluster Perturbation Theory 4 Cluster Dynamical Mean Field Theory 5 The self-energy functional approach 6 Exact Diagonalizations 3 / 119

4 The Hubbard model Hamiltonian: hopping amplitude t H = r,r c r,r,σ creation operator (chemical potential µ = t rr ) rσ c r σ + U nr n r number of spin electrons at r r local Coulomb repulsion w(x) x 4 / 119

5 The Green function Hilbert space of dimension 4 L (L: # of sites) The many-body ground state Ω contains too much information A lot of useful information is contained in the one-particle Green function: (here at T = 0) matrix G 1 Gαβ(z) = Ω c α c z H + E β Ω + 1 Ω c c β α Ω 0 z + H E 0 Retarded Green function: G.S. energy G R (t) = iθ(t) Ω {c α (t), c β (0)} Ω = GR (ω) = G(ω + i0 + ) Approximation schemes for G are easier to implement 5 / 119

6 Spectral representation 1 G αβ (z) = Ω c α r r c z E r + E β Ω + Ω c β r 1 r c α Ω 0 z + E r E 0 Q-matrix: r>0 eigenstate with N + 1 particles r<0 eigenstate with N 1 particles Q αr = Ω cα r (r > 0) r c α Ω (r < 0) and ω r = Er E 0 (r > 0) E 0 E r (r < 0) Spectral representation: Q αr Q β r G αβ (z) = z ω r r (partial fractions) 6 / 119

7 Spectral representation (cont.) Completeness relations: Q αr Q β r c = Ω α c β + c β c α Ω r = δ αβ Asymptotic behavior: lim G αβ(z) = δ αβ z z 7 / 119

8 Spectral function Spectral function: Translation-invariant system: r>0 A αβ (ω) = 2 ImG αβ (ω + i0 + ) G(k, z) = Ω c k r z E r + E 0 r<0 Ω c k r 2 1 z + E r E 0 A(k, ω) = 2 ImG(k, ω + i0 + ) = Ω c k r 2 2πδ(ω E r + E 0 ) r>0 prob. of electron with ɛ = E r E 0 + Ω c k r 2 2πδ(ω + E r E 0 ) r<0 prob. of hole with ɛ = E r E 0 8 / 119

9 non-interacting limit (U = 0) G(z) = 1 z t Ω = c k 0 ε k <0 (Fermi sea) G(z, k) = 1 z ɛ k ɛ k = t 0,r e ik r r π t = 0, t = 0 π t = 0.3t, t = 0.2t k y k y π π k x π π π k x π 9 / 119

10 non-interacting limit (cont.) Spectral function & density of states: A(k, ω) = 2πδ(ω ɛ k ) ρ(ω) = A(k, ω) k (π,0) (π,π) (0,0) (π,0) ω/t Spectral function, half-filling, NN hopping N(ω) ω/t Associated density of states 10 / 119

11 Self-energy Interacting Green function: 1 G(z) = z t Σ(z) Local limit at half-filling (t = 1 2 U1): self-energy G(z) = 1/2 z + U/2 + 1/2 z U/2 = 1 Analytic structure: Σ(z) = U2 4z + U 2 Σ αβ (z) = Σ αβ Hartree-Fock S αr S β r + z σ r r z U2 4z (π,0) (π,π) (0,0) (π,0) ω/u Spectral function, half-filled HM (t = 0) 11 / 119

12 Outline 1 Introduction 2 Dynamical Mean Field Theory Approximation schemes The cavity method The hybridization function The self-consistency condition Impurity solvers The Mott transition 3 Cluster Perturbation Theory 4 Cluster Dynamical Mean Field Theory 5 The self-energy functional approach 12 / 119

13 Approximation schemes Hartree-Fock Σ(ω, k) Σ(, k) is frequency-independent Can be absorbed in new dispersion relation ɛ (k) Approximation equivalent to new one-body Hamiltonian DMFT Σ(ω, k) Σ(ω) is momentum-independent System still fundamentally interacting Approximated by single site with effective medium 13 / 119

14 The cavity method Action of the Hubbard model: β S[c rσ, c rσ ] = dτ c rσ τc rσ + t rr c rσ c r σ + U n r n r One-site effective action: 1 e S eff. [c Z 0σ, c 0σ ] eff. 0 = 1 Z r,σ r 0,σ r,r,σ c rσ c rσ e S 0 r 14 / 119

15 The cavity method (cont.) Effective action (exact form, spin indices suppressed): S eff. = S 0 + dτ η r η 1 r η n r... η 1 r G env n r 1 r n(τ 1 τ rn, τ 1 τ n ) n=1 r 1,...,r n where η r = t r0 c r0 acts like a source field and S 0 = β 0 dτ c 0 τ c 0 µn 0 + Un 0 n 0 One can show that in the d limit, if t t/ 2d, only n = 1 survives. DMFT approximation: S eff. = S 0 + t 0r t 0r dτ dτ c 0 (τ)c 0 (τ )G env r,r (τ, τ ) r,r 15 / 119

16 The cavity method (cont.) Effective action: S eff. = dτ dτ c 1 0 (τ) 0 (τ τ )c 0 (τ ) + U β 1 0 (iω n) = iω n + µ t 0r t 0r G env r,r (iω n ) r,r G env r,r (iω n ) unknown. Rather, treat 1 0 as an adjustable dynamical mean field Only single particles hop on and off the site The environment is uncorrelated Nonlocal in time: no Hamiltonian involving c 0 only 0 dτ n 0 n 0 16 / 119

17 The hybridization function 0 has the analytic properties of a Green function: poles on the real axis and positive residues. Define the hybridization function Γ (z): 1 0 (z) = z + µ Γ (z) Γ (z) has the analytic properties of a self-energy and can represented by a (quasi-infinite) set of poles: θ 2 r Γ (z) = z ε r r Interacting Green function of the effective theory for the single site: 1 G s (z) = z + µ Γ (z) Σ(z) 17 / 119

18 Hamiltonian representation G s can be obtained from the following AIM Hamiltonian: N b H AIM = θ r r=1 hyb. amplitude c 0 a r + H.c N b + ε r r=1 bath orbital bath energy a r a r µc 0 c 0 + Un 0 n 0 ε 5 ε 4 ε 3 ε 7 ε 6 θ θ 4 5 θ3 θ 6 θ 2 θ 7 θ 1 ε 1 site θ 8 ε 2 One-body matrix: µ θ[1 Nb ] T = θ [N b 1] ɛ [Nb N b ] ε 8 18 / 119

19 Proof (U = 0) G full (z) = Gs G bs G sb G b Need to invert a block matrix: z + µ θ block equations: G 1 θ Gs z ɛ z + µ full (z) = z T = θ G bs G sb G b = 1 θ z ɛ therefore (z + µ)g s θ G bs = 1 and θ G s + (z ɛ)g bs = 0 G bs = (z ɛ) 1 θ G s = (z + µ) θ 1 z ɛ θ G s = 1 G 1 s = z + µ θ 1 z ɛ θ = z + µ Γ (z) 19 / 119

20 Proof (cont.) where Γ (z) = θ 1 z ɛ θ = If U 0, simply add the self-energy: G s (z) = N b r=1 θ 2 r z ε r 1 z + µ Γ (z) Σ(z) 20 / 119

21 The self-consistency condition Lattice model Green function in the DMFT approximation: G(iω n, k) = 1 iω n ɛ(k) Σ(iω n ) The local Green function Ḡ(iω n ) = 1 G(iω n, k) N must coincide with G s (iω n ): Ḡ(iω n ) 1 = iω n + µ Γ (iω n ) Σ(iω n ) k = 1 0 (iω n) Σ(iω n ) 21 / 119

22 The DMFT self-consistency loop Initial guess for Γ (iω n ) Impurity solver: Compute G s (iω n ) Ḡ(iω n ) = 1 N k G 1 0 (iω n, k) Σ(iω n ) 1 Γ (iω n ) iω n + µ Ḡ(iω n ) Σ(iω n ) No Γ converged? Yes exit 22 / 119

23 The impurity solver Methods for solving the impurity Hamiltonian: Perturbation theory (2nd order, NCA) Numerical Renormalization Group (NRG) Quantum Monte Carlo (QMC) Infinite bath: only Γ (iω n ) is needed. Finite temperature Hirsch-Fye (time grid) or Continuous-time (no discretization error) But: sign problem Exact diagonalizations restricted to small, discrete baths (explicit form of H AIM ) Hence self-consistency relation only approximately satisfied real-frequency information zero temperature Other real frequency methods: CI, natural basis, etc. 23 / 119

24 The DMFT self-consistency loop (discrete bath version) 1 Start with a guess value of (θ r, ɛ r ) 2 Compute the impurity Green function G s (iω n ) (ED) 3 Compute the lattice-averaged (or local) Green function Ḡ(iω n ) = 1 N k 1 G 1 0 (k) Σ(iω n) and 4 Minimize the following distance function: d(θ, ɛ) = (iω n ) tr ω n W weights G (iω n) = Ḡ 1 + Σ(iω n ) s (iω n ) Ḡ 1 (iω n ) over the set of bath parameters. Thus obtain a new set (θ r, ɛ r ). 5 Go back to step (2) until convergence / 119

25 Application: The Mott transition 64 A. Georges et al.: Dynamical mean-field theory of... Density of states N(ω) for the half-filled Hubbard FIG. 30. Local spectral density pdr(v) at T=0, for several model on the Bethe lattice with interactions U/D = values of U, obtained by the iterated perturbation theory approximation. The first four curves (from top to bottom, U/D 1, 2, 2.5, 3, 4. Iterated perturbation theory. Zhang et al., Phys. Rev. Lett. 70, 1666 (1993). =1,2,2.5,3) correspond to an increasingly correlated metal, while the bottom one (U/D=4) is an insulator. disappears continuously (at T=0) at a critical value v, and the substitution into the self-consistency condition implies that G 1 0 ;iv, which 1 site is another way of saying that the effective bath in the Anderson model picture has a gap. We know from the theory of an U=t U=12t Anderson impurity embedded in an insulating medium that the Kondo effect U=8t does not take place. The U=14t impurity model ground state is a doubly degenerate local moment. Thus, the superposition of two magnetic Hartree- Fock solutions is qualitatively a self-consistent ansatz. If this ansatz is placed into Eq. (221), we are led to a closed (approximate) equation for G(iv n ): D 4 G 3 28D 2 vg 2 14~4v 2 1D 2 2U 2!G216v50. (234) This approximation corresponds to the first-order approximation in the equation of motion decoupling schemes reviewed in Sec. VI.B.4. It is similar in spirit to the Hubbard III approximation Eq. (173) (Hubbard, 1964), which would correspond to pushing this scheme one step further. 6These approximations /t are valid 6 for very large U but become quantitatively worse as U is N(ω) for the half-filled 3D Hubbard model. Exact reduced. They would predict a closure of the gap at diagonalization solver. Zhang and Imada, Phys. Rev. U c 5D B for 76, (234) (U c (2007). 5)D for Hubbard III). The failure of these approximations, when continued into the metallic phase, is due to their inability to capture the Kondo effect which builds up the Fermi-liquid quasiparticles. They are qualitatively valid in the Mott insulating 25 / 119

26 The Mott transition (cont.) U= 14 U= 13 ε 3 ε 2 U= 12 U= 11 ε 4 U= 10 U= 8 U= 6 U= 4 U= ω N(ω) in the 2D, half-filled Hubbard model. Exact diagonalization solver with N b = 7. ε 5 θ 4 θ 3 θ 2 θ 1 ε 1 θ 5 θ θ 7 6 ε 6 ε 7 26 / 119

27 Breaking the exponential barrier (a) b 3 (b) (c) b 2 i b 1 b 0 i b 1 b 2 b 3 b 4 b 5 b 6 b 7 i c 1 c 2 c 3 b c 4 c 5 b -1 b -2 v 1 v 2 v 3 v 4 v 5 b -3 Lu et al., Phys. Rev. B90, (2014). 27 / 119

28 Breaking the exponential barrier (cont.) Lu et al., Phys. Rev. B90, (2014). 28 / 119

29 Real-space DMFT : Ultra-cold atoms Antiferromagnetic order in cold atom systems with harmonic trap H = t r,r c rσ c r σ r,r,σ + U n r n r r (V r µ)n r,σ r,σ 29 / 119

30 Outline 1 Introduction 2 Dynamical Mean Field Theory 3 Cluster Perturbation Theory kinematics CPT : examples 4 Cluster Dynamical Mean Field Theory 5 The self-energy functional approach 6 Exact Diagonalizations 30 / 119

31 Real-space cluster methods: General idea Tile the lattice with small units (clusters) Solve an approximate, effective problem on each cluster Use the self-energies Σ (j) (z) to approximate the full self-energy: Σ (1) Σ (2) 0 Σ = Σ (n) Varieties: CPT : Cluster Perturbation Theory VCA : Variational Cluster Approximation CDMFT : Cluster Dynamical Mean Field Theory CDIA : Cluster Dynamical Impurity Approximation 31 / 119

32 Tiling the lattice into clusters Tiling of the triangular lattice with 6-site clusters 32 / 119

33 Cluster Perturbation Theory Cluster decomposition of the one-body matrix: t (1,1) t (1,2) t (1,n) t (2,1) t (2,2) t (2,n) t = t (n,1) t (n,2) t (n,n) t = t diagonal blocks + t ic Cluster Green function: G (j) 1 (z) = z t (j,j) Σ (j) (z) CPT Green function for the full system: G 1 cpt (z) = G (j) 1 (z) t j inter-cluster blocks ic 33 / 119

34 Cluster Perturbation Theory (cont.) lattice Hamiltonian H = H cluster Hamiltonian + H ic H ic = (t ic ) αβ c α c β α,β Treat H ic at lowest order in Perturbation theory At this order, the Green function is G 1 (z) = G 1 (z) t ic cluster Green function matrix C. Gros and R. Valenti, Phys. Rev. B 48, 418 (1993) D. Sénéchal, D. Perez, and M. Pioro-Ladrière. Phys. Rev. Lett. 84, 522 (2000) 34 / 119

35 Superlattices and reduced Brillouin zones e 2 e 1 K k k (π, π) (0, 0) 10-site cluster ( π, π) Brillouin zones 35 / 119

36 CPT for translation invariant tilings site within cluster one-body index = (R, k Green function and t ic are diagonal in k. G is independent of k. The CPT formula may be written as G 1 (z, k) = G 1 (z) t ic ( k) where matrices are now in (R, σ) space. reduced wavevector, σ) spin (or band) 36 / 119

37 Interlude : Fourier transforms Unitary matrices performing Fourier transforms: U γ k,r = 1 N e ik r U Γ k r = L N e i k r U c K,R = 1 L e ik R complete superlattice cluster Various representations of the annihilation operator c(k) = U γ kr c r c K ( k) = U Γ k r U c KR c r+r r r,r c R ( k) = U Γ k r c r+r c r,k = r Caveat: U γ U Γ U c The matrix Λ = U γ (U Γ U c ) 1 relates (K, k) to k: c( k + K) = Λ K,K ( k)c K ( k) R U c KR c r+r 37 / 119

38 Periodization CPT breaks translation invariance, which needs to be restored: G per. (k, z) = 1 e ik (R R ) G RR ( k, z) L R,R Periodizing (1D half-filled HM, 12-site cluster): (π) (π) (π/2) (π/2) (0) ω 3 6 Green function periodization (0) ω 3 6 Self-energy periodization 38 / 119

39 Periodization (2) Periodization as a change of basis: G( k + K, k + K, z) = Λ c ( k)g(z)λ c ( k) KK = 1 e i( k+k K 1 ) R e i( k+k K L 2 1 ) R G K1 K (z) 1 R,R,K 1,K 1 = 1 e i( k+k) R e i( k+k ) R GRR ( k, z) L R,R Set K = K : the spectral function is a partial trace and thus involves diagonal elements only Replace k by k = k + K in G RR ( k, z), which leaves t ic ( k) unchanged Since this is a change of basis, analytic properties are still OK. 39 / 119

40 One-dimensional example Spectral function of the half-filled HM with increasing U/t: 40 / 119

41 One-dimensional example (cont.) Spectral function of the half-filled HM with increasing L at U = 4t: π L = 2 L = 4 L = 6 π/2 0 π L = 8 L = 10 L = 12 π/ ω ω ω 41 / 119

42 One-dimensional example (cont.) Spectral gap at half-filling: periodic cluster vs CPT 5 Mott gap U L =12(cluster) L (cluster) L =12 L exact 42 / 119

43 Application: Pseudogap in h-doped cuprates 2D Hubbard model, t = 0.3t, t = 0.2t, 3 4 cluster, n = 5/6 Sénéchal and Tremblay, PRL 92, (2004) 43 / 119

44 Application: Pseudogap in e-doped cuprates 2D Hubbard model, t = 0.3t, t = 0.2t, 3 4 cluster, n = 7/6 Sénéchal and Tremblay, PRL 92, (2004) 44 / 119

45 Application: Fermi surface maps U= 2, n = 5/6 (π,π) (0,0) U= 4, n = 7/6 U= 8, n = 5/6 0% ky 90% kx F. Ronning et al., PRB 67, (2003) U= 8, n = 7/6 0% (0, 0) (π, π) 90% Sénéchal and Tremblay, PRL 92, (2004) N.P. Armitage et al., PRL 88, (2002) 45 / 119

46 CPT : Conclusion Approximation scheme for the one-body Green function Yields approximate values for the averages of one-body operators Exact at U = 0 Exact at t i j = 0 Exact short-range correlations Allows all values of the wavevector Controlled by the size of the cluster But : No long-range order, no self-consistency Higher Green functions still confined to the cluster = A first step towards CDMFT or VCA 46 / 119

47 Outline 1 Introduction 2 Dynamical Mean Field Theory 3 Cluster Perturbation Theory 4 Cluster Dynamical Mean Field Theory The hybridization function The CDMFT self-consistency condition Applications The Dynamical Cluster Approximation (DCA) 5 The self-energy functional approach 6 Exact Diagonalizations 47 / 119

48 Generalization of DMFT to small clusters H AIM H Simple adaptation of DMFT Scalar equations matrix equations H H H H H H H H H H Dynamical mean field 0 : S eff [c, c ] = β 0 dτdτ α,β c 1 α (τ) 0,αβ (τ τ )c β (τ ) + β 0 dτ H 1 (c, c ) 48 / 119

49 The hybridization function In the frequency domain: 1 hybridization function 0 (iω n) = iω n t Γ(iω n ) where 0 (iω n ) = Spectral representation of Γ: N b Γ αβ (iω n ) = Corresponding Hamiltonian: r θ αr θ β r iω n ɛ r = θ H = t αβ c α c β + U n i n i + α,β hybridization matrix i θ rα r,α 1 iω n ɛ θ β (c α a r + H.c.) + 0 bath orbital e iω nτ 0 (τ) ɛ r r bath energies a r a r 49 / 119

50 Discrete bath systems ɛ 3 ɛ 4 ɛ 1 θ 1 ɛ 3 θ θ + 2 θ + 1 ɛ + 2 ɛ θ 7 θ 3 θ 4 ɛ 7 ɛ 8 θ 8 θ 2 θ θ 1 ɛ 1 ɛ 5 ɛ 6 θ 5 θ 6 ɛ 2 ɛ 4 θ 2 ɛ 2 θ 1 θ 2 (A) (B) ɛ 1 ɛ 2 (C) (B) : Liebsch and Ishida, Journal of Physics: Condensed Matter 24 (2012), no. 5, / 119

51 The CDMFT Procedure (discrete bath) 1 Start with a guess value of (θ αr, ɛ r ). 2 Calculate the cluster Green function G (ω) (ED). 3 Calculate the superlattice-averaged Green function Ḡ(ω) = k 1 G 1 0 ( k) Σ(ω) and 4 Minimize the following distance function: 1 0 (ω) = Ḡ 1 + Σ(ω) d(θ, ɛ) = W (iω n ) tr G 1 (iω n ) Ḡ 1 (iω n ) ω n over the set of bath parameters. Thus obtain a new set (θ αr, ɛ r ). 5 Go back to step (2) until convergence / 119

52 The CDMFT self-consistency loop Initial guess for Γ(iω n ) Impurity solver: Compute G (iω n ) Ḡ(iω n ) = L N k G 1 0 (iω n, k) Σ(iω n ) 1 Γ(iω n ) iω n t + µ Ḡ(iω n ) Σ(iω n ) (QMC) minimize ω n W (iω n ) tr G 1 (iω n ) Ḡ 1 (iω n ) 2 (ED) No Γ converged? Yes exit 52 / 119

53 Application: The Mott transition T T T DMFT CDMFT DIA Uc1(T ) Uc2(T ) Uc(T ) metal insulator metal Uc1(T ) Uc2(T ) Uc(T ) insulator Uc1(T ) Uc2(T ) metal insulator Uc(T ) U M. Balzer et al., Europhys. Lett. 85, (2009) U U U=t U=8t 1 site U=12t U=14t U=t U=5t 4 sites U=7t U=10t 6 /t 6 Y.Z. Zhang, M. Imada, Phys. Rev. B 76, (2007) /t / 119

54 Application: the Mott transition (QMC) 0.2 a) U/t=5.0 b) U/t=5.4 A(ω) c) U/t=5.6 d) U/t=5.8 A(ω) ω ω H. Park et al, PRL 101, (2008) 54 / 119

55 Application: the Mott transition (ED) 0.2 U = 5.0 U = 5.4 A(ω) U = 5.6 U = 5.8 A(ω) ω ω solutions from M. Balzer et al., Europhys. Lett. 85, (2009) 55 / 119

56 First-order character of the Mott transition (CDMFT) The Mott transition is seen in CDMFT as a hysteresis of the double occupancy This shows up nicely in a simulation of BEDT organic superconductors D t=0.7t t=t t t t U/t B. Kyung, A.M.S. Tremblay, Phys. Rev. Lett. 97, (2006) 56 / 119

57 Digression: Superconductivity Superconductivity is described by pairing fields: = rr c r c r + H.c r,r s-wave pairing: rr = δ rr d x 2 y 2 pairing: rr = 1 if r r = ±x 1 if r r = ±y + + d x y pairing: rr = 1 if r r = ±(x + y) 1 if r r = ±(x y) + + Pairing fields are introduced in the bath, and measured on the cluster 57 / 119

58 Digression: Superconductivity (cont.) Pairing fields violate particle number conservation The Hilbert space is enlarged to encompass all particle numbers with a given total spin Use the Nambu formalism: a particle-hole transformation on the spin-down sector: c α c α and a r a r Structure of the one-body matrix: c a c a t θ 0 0 θ ɛ 0 b 0 0 t θ 0 θ ɛ b 58 / 119

59 Application: dsc and AF in the 2D Hubbard model Nine bath parameters Homogeneous coexistence of d x 2 y2 SC and Néel AF 0.04 ψ U = 4t U = 8t U = 12t U = 16t U = 24t M, ψ U = 8t t = 0.3t t = 0.2t M 0.08 ψ/j ψ (10 ) n n Kancharla et al., Phys. Rev. B (2008). 59 / 119

60 Mott transition and superconductivity T δc1 δc2 T MI (U MIT,T MIT ) 0.10 Tc d TWL Tχ0 400 (µ p,t p ) µ T d c µ c1 µ c2 U c2 Tc d U c1 T 0.05 Mott insulator (δ p,t p ) T Tσc Tρc,min TA,max Tχ0,max Tc d 200 T(K) U δ Sordi et al., Phys. Rev. Lett. 108 (2012), P. Semon et al., Phys. Rev. B89, (2014) 60 / 119

61 Mott transition and superconductivity (cont.) First-order, finite-doping transition with finite-t critical point: correlated metal vs pseudogap phase. The pseudogap phenomenon is related to the Widom line in first-order transitions Even though the SC order parameter is suppressed by the Mott transition, T c isn t Results obtained with an efficient CT-QCM-HYB solver. 61 / 119

62 Application: Resilience of dsc to extended interactions H = t r,r c rσ c r σ + U n r n r + V rr n r n r µ r,r,σ r r r r,σ n r,σ Question: effect of NN repulsion V on dsc in the 2D Hubbard model? V is a priori detrimental to dsc (pair breaking effect), and larger than J. But: V increases J. Exact treatment of V within the cluster; Hartree approximation between clusters. Result: a moderate V has no effect on dsc at low doping. The retarded nature of the effective pairing interaction is important. 62 / 119

63 Resilience of dsc to extended interactions (cont.) ψ (A) U = 4 V = 0 V = 0.5 V = 1 V = 1.5 (B) U = 8 IF ωf U = 8,x = 0.05 V = 0 V = 1 U = 8,x = 0.20 V = 2 V = 3 (A) (B) ψ ψ V = 0 V = 1 V = 2 V = 3 (C) U = V = 0 V = V = 2 V = 4 V = Sénéchal et al., Phys. Rev. B 87, (2013). x IF IF V = 0 V = 1 U = 16,x = 0.05 V = 2 V = 3 (C) 0.02 V = 0 V = 4 0 V = 2 V = ω / 119

64 Non-magnetic impurity in graphene G 1 ( k, z) = z t( k) (z) = z t 11 ( k) Σ 1 (z) t 12 ( k) t 13 ( k)... t 1M ( k) t 21 ( k) z t 22 ( k) Σ 2 (z) t 23 ( k)... t 2M ( k) t 31 ( k) t 32 ( k) z t 33 ( k) Σ 3 (z)... t 3M ( k) t M1 ( k) t M2 ( k) t M3 ( k)... z t M M ( k) Σ M (z) M. Charlebois et al., Phys. Rev. B91, (2015). 64 / 119

65 The Dynamical Cluster Approximation Based on periodic clusters self-consistency condition: where 1 iω n t K Γ K (ω) Σ K (ω) = L N k t K = L ɛ( k + K) N Not derivable from the Self-energy functional approach For large clusters: DCA converges better for k = 0 (average) quantities CDMFT converges better for r = 0 (local) quantities Maier et al., Rev. Mod. Phys. 77, 1027 (2005). k 1 iω n ɛ( k + K) Σ K (ω) 65 / 119

66 DCA on superconductivity 1/P d A 8A 12A 16B 16A 20A 24A 26A T/t Maier et al., Phys. Rev. Lett. 95, (2005). 66 / 119

67 Outline 1 Introduction 2 Dynamical Mean Field Theory 3 Cluster Perturbation Theory 4 Cluster Dynamical Mean Field Theory 5 The self-energy functional approach The variational principle Calculating the Potthoff functional The Variational Cluster Approximation Applications Optimization The (Cluster) Dynamical Impurity Approximation 67 / 119

68 Motivation CPT cannot describe broken symmetry states, because of the finite cluster size Idea : add a Weiss field term to the cluster Hamiltonian H, e.g., for antiferromagnetism: H M = M a e iq (π, π) r a (n a n a ) This term favors AF order, but does not appear in H, and must be subtracted from V (H = H + V ) Need a principle to set the value of M : energy minimization? Better : Potthoff s self-energy functional approach 68 / 119

69 The Luttinger-Ward functional Luttinger-Ward (or Baym-Kadanoff) functional: Φ[G] = Relation with self-energy: Legendre transform: δf[σ] δσ = δφ[g] δg M. Potthoff, Eur. Phys. J. B 32, 429 (2003) δφ[g] δg = Σ F[Σ] = Φ[G] Tr(ΣG) δg[σ] δσ functional trace ΣδG[Σ] δσ G = G 69 / 119

70 The variational principle Free energy functional: Ω t [Σ] = F[Σ] Tr ln( G 1 0t + Σ) Stationary at the physical self-energy (Euler equation): δω t [Σ] δσ = G + (G 1 0t Σ) 1 = 0 At the physical self-energy Σ, Ω t [Σ ] = grand potential Approximation strategies with variational principles: Type I : Simplify the Euler equation Type II : Approximate the functional (Hartree-Fock, FLEX) Type III : Restrict the variational space, but keep the functional exact 70 / 119

71 The Reference System To evaluate F, use its universal character : its functional form depends only on the interaction. Introduce a reference system H, which differs from H by one-body terms only (example : the cluster Hamiltonian) Suppose H can be solved exactly. Then, at the physical self-energy Σ of H, by eliminating F: Ω = F[Σ] + Tr ln( G ) Ω t [Σ] = Ω Tr ln( G ) Tr ln( G 1 0t + Σ) = Ω Tr ln( G ) + Tr ln( G) = Ω Tr ln( G ) Tr ln( G 1 + V) = Ω Tr ln(1 VG ) 71 / 119

72 The Potthoff functional Making the trace explicit, one finds Ω t [Σ] = Ω T tr ln 1 V( k)g ( k, ω) ω k = Ω T ln det 1 V( k)g ( k, ω) ω k The sum over frequencies is to be performed over Matsubara frequencies (or an integral along the imaginary axis at T = 0). The variation is done over one-body parameters of the cluster Hamiltonian H In the above example, the solution is found when Ω/ M = / 119

73 Calculating the functional I : exact form It can be shown that Tr ln( G) = T ln(1 + e βω m) + T ln(1 + e βζ m) Use the Lehmann representation of the GF: m G αβ (ω) = M. Potthoff, Eur. Phys. J. B, 36:335 (2003) Lehmann representation r poles of G m zeros of G Q αr Q β r G (ω) = Q 1 Q ω ω r ω Λ diagonal(ω r ) 73 / 119

74 Calculating the functional I : exact form (2) A similar representation holds for the CPT Green function 1 G( k, ω) = G 1 V( k) = 1 Q 1 ω Λ Q 1 V( k) 1 = Q ω L( k) Q L( k) = Λ + Q V( k)q Let ω r ( k) be the eigenvalues of L( k), i.e., the poles of G( k, ω). Then Ω(x variational parameters ) = Ω (x) ω r <0 ω r + L N k ω r ( k)<0 ω r ( k) Note: the zeros of G and G are the same, since they have the same self-energy. M. Aichhorn et al., Phys. Rev. B 74 : (2006) 74 / 119

75 Calculating the functional II : numerical integral Except for very small clusters (L 4), it is much faster to perform a numerical integration over frequencies: Ω(x) = Ω dω L (x) ln det(1 V( k)g (iω)) L(µ µ ) π N 0 k ky/(2π) The integral must be done using an adaptive method that refines a mesh where necessary. For instance: The CUBA library k x /(2π) Grid of 17,095 points used in an adaptive integration over wavevectors 75 / 119

76 The variational Cluster Approximation: procedure 1 Set up a superlattice of clusters 2 Choose a set of variational parameters, e.g. Weiss fields for broken symmetries 3 Set up the calculation of the Potthoff functional: 4 Use an optimization method to find the stationary points E.g. the Newton-Raphson method, or a quasi-newton method 5 Adopt the cluster self-energy associated with the stationary point with the lowest Ω and use it as in CPT or CDMFT. 6 Adopt the value of Ω as the best estimate of the grand potential 76 / 119

77 VCA vs Mean-Field Theory Differs from Mean-Field Theory: Interaction is left intact, it is not factorized Retains exact short-range correlations Weiss field order parameter More stringent that MFT Controlled by the cluster size Similarities with MFT: No long-range fluctuations (no disorder from Goldstone modes) Yet : no LRO for Néel AF in one dimension Need to compare different orders yet : they may be placed in competition / coexistence 77 / 119

78 Application: Néel Antiferromagnetism Used the Weiss field H M = M r e iq (π, π) r (n r n r ) Profile of Ω for the half-filled, square lattice Hubbard model: Ω U = 16 U = 8 U = 4 U = M Ω L = L = M 78 / 119

79 Néel Antiferromagnetism (cont.) order parameter M (4 ) ordering energy U Best scaling factor : q = number of links 2 number of sites M Order Parameter x1 2x1 links 2L 2x1 2x2 2x1 2x2 1 1 L 2x3 2x4 B10 2x3 2x4 B10 2x2 3x4 4x4 2x2 3x4 4x4 2x3 2x4 B10 3x4 4 2x3 2x4 B10 3x / 119

80 Application: Superconductivity Need to add a pairing field = rr c r c r + H.c r,r extended s-wave s-wave pairing: rr = δ rr d x 2 y 2 pairing: rr = d x y pairing: rr = 1 if r r = ±x 1 if r r = ±y 1 if r r = ±(x + y) 1 if r r = ±(x y) Ω d x 2 y 2 d xy cluster s-wave U = 8, µ = / 119

81 Competing SC and AF orders One-band Hubbard model for the cuprates: t = 0.3, t = 0.2, U = 8: order parameter pure Néel AF pure d x 2 y 2 coex. d x 2 y 2 coex. Néel AF n 81 / 119

82 Example: Homogeneous coexistence of dsc and AF orders 82 / 119

83 Optimization procedure Need to find the stationary points of Ω(x) with as few evaluations as possible Example: the Newton-Raphson method: Evaluate Ω at a number of points at and around x 0 that just fits a quadratic form Move to the stationary point x 1 of that quadratic form and repeat Stop when x i x i 1, or the numerical gradient Ω, converges Not robust : it converges fast when started close enough to the solution, but it can err... Proceed adiabatically through external parameter space (e.g. as function of U or µ) 83 / 119

84 The (Cluster) Dynamical Impurity Approximation (CDIA) The bath parameters (ɛ µ, θ αµ, etc) are variational parameters The bath makes a contribution to the Potthoff functional: Ω bath = ɛ α <0 One can in principle use the same procedure as in VCA but in practice it is more difficult. The presence of the bath increases the resolution of the approach in the time domain, at the cost of spatial resolution, for a fixed total number of orbitals (cluster + bath). Euler equations for the stationary point: tr G (iω n ) Ḡ(iω n ) Σ (iω n ) = 0. θ ω n ɛ α Potthoff functional 84 / 119

85 The 1D Hubbard model n θ1 & θ2 ε1 & ε ωc =1 ωc =2 ωc =5 ωc =10 SFA (A) (C) (E) 1/ω tr Σ 2 sharp SFA exact 1/ω tr Σ 2 sharp SFA µ SFA gradient ε1 & ε2 SFA gradient /ω tr Σ 2 sharp 1/ω tr Σ 2 sharp SFA ωc =1 ωc =2 ωc =5 ωc =10 (B) (D) (F) µ 85 / 119

86 The 1D Hubbard model (cont.) Ground state energy as a function of doping (U = 4t) E sharp, K + V sharp, Ω + µn SFA exact n 86 / 119

87 First-order character of the Mott transition (CDIA) SFT functional E 0 [ (V)] / L Mott insulator metal U= U c U c -0.7 hcp 4.6 U=4.2 metal variational parameter V U c M. Balzer et al., Europhys. Lett. 85, (2009) 87 / 119

88 Mott transition (CDIA vs CDMFT) The Mott transition may not show up as first-order in CDMFT, even though it is in the CDIA The choice of distance function matters D. Sénéchal, Phys. Rev. B 81 (2010), θ ε SFA gradient (A) U c1 U c2 (B) (C) 1/ω tr Σ 2 sharp VCA U 88 / 119

89 Outline 1 Introduction 2 Dynamical Mean Field Theory 3 Cluster Perturbation Theory 4 Cluster Dynamical Mean Field Theory 5 The self-energy functional approach 6 Exact Diagonalizations The Hilbert space Coding the states The Lanczos method Calculating the Green function 89 / 119

90 Exact diagonalizations vs Quantum Monte Carlo ED QMC temperature T = 0 T > 0 frequencies real/complex complex sign problem no yes system size small moderate CDMFT bath small infinite interaction strength any depends on expansion scheme 90 / 119

91 The exact diagonalization procedure 1 Build a basis 2 Construct the Hamiltonian matrix (stored or not) 3 Find the ground state (e.g. by the Lanczos method) Calculate ground state properties (expectation values, etc.) 4 Calculate a representation of the one-body Green function: Continuous-fraction representation Lehmann representation 5 Calculate dynamical properties from the Green function 91 / 119

92 The Hubbard model on a cluster of size L N and N separately conserved in the simple Hubbard model Dimension of the Hilbert space (half-filling): L! 2 d = 2 4L [(L/2)!] 2 πl L = 16 : One double-precision vector requires 1.23 GB of memory L dimension / 119

93 Two-site cluster: Hamiltonian matrix Half-filled, two-site Hubbard model: 4 states States and Hamiltonian matrix: 01, 01 01, 10 10, 01 10, 10 spin occupation U 2µ t t 0 t 2µ 0 t t 0 2µ t 0 t t U 2µ spin occupation 93 / 119

94 Six-site cluster: Hamiltonian matrix Sparse matrix structure / 119

95 Coding the states Tensor product structure of the Hilbert space: V = V N V N dimension: Example (6 sites): d = d(n )d(n ) d(n σ ) = L! N σ!(l N σ )! / 119

96 Coding the states (2) Basis of occupation number eigenstates: (c 1 )n 1 (c L )n L (c 1 )n 1 (c L )n L 0 n iσ = 0 or 1 Correspondence with binary representation of integers: b σ = (n 1σ n 2σ n Lσ ) 2 For a given (N, N ), we need a direct table: b = B (i ) b = B (i )... and a reverse table: consecutive label mod int. division i = I (b ) + d N I (b ) i = i% d N i = i/ d N 96 / 119

97 Constructing the Hamiltonian matrix Form of Hamiltonian: H = K K + V int. K = t ab c a c b K is stored in sparse form. V int. is diagonal and is stored. Matrix elements of V int. : bit_count(b & b ) Two basis states b σ and b σ are connected with the matrix K if their binary representations differ at two positions a and b. b K b = ( 1) M ab t ab M ab = a,b b 1 c=a+1 We find it practical to construct and store all terms of the Hamiltonian separately. n c 97 / 119

98 The Lanczos method Finds the lowest eigenpair by an iterative application of H Start with random vector φ 0 An iterative procedure builds the Krylov subspace: = span φ 0, H φ 0, H 2 φ 0,, H M φ 0 Lanczos three-way recursion for an orthogonal basis { φ n }: φ n+1 = H φ n a n φ n b 2 n φ n 1 a n = φ n H φ n φ n φ n b 2 n = φ n φ n φ n 1 φ n 1 b 0 = 0 98 / 119

99 The Lanczos method (2) In the basis of normalized states n = φ n / φ n φ n, the projected Hamiltonian has the tridiagonal form projector onto a 0 b b 1 a 1 b PH P = T = 0 b 2 a 2 b a N At each step n, find the lowest eigenvalue of that matrix Stop when the estimated Ritz residual T ψ E0 ψ is small enough Run again to find eigenvector ψ = n ψ n n as the φ n s are not kept in memory. 99 / 119

100 The Lanczos method: features Required number of iterations: typically from 50 to 200 Extreme eigenvalues converge first Rate of convergence increases with separation between ground state and first excited state Cannot resolve degenerate ground states : only one state per ground state manifold is picked up If one is interested in low lying states, re-orthogonalization may be required, as orthogonality leaks will occur. But then Lanczos intermediate states need to be stored. For degenerate ground states and low lying states (e.g. in DMRG), the Davidson method is generally preferable 100 / 119

101 The Lanczos method: illustration of the convergence 149 iterations on a matrix of dimension 213,840: eigenvalues of the tridiagonal projection as a function of iteration step eigenvalues iteration 101 / 119

102 Lanczos method for the Green function Zero temperature Green function: Consider the diagonal element Use the expansion G αβ (z) = G (e) (z) + G(h) αβ αβ (z) G (e) αβ (z) = Ω c 1 α c z H + E β Ω 0 G (h) αβ (z) = 1 Ω c c β α Ω z + H E 0 φ α = c α Ω = G(e) αα = φ 1 α φ α z H + E 0 1 z H = 1 z + 1 z 2 H + 1 z 3 H / 119

103 Lanczos method for the Green function (2) Truncated expansion evaluated exactly in Krylov subspace generated by φ α if we perform a Lanczos procedure on φ α. Then G (e) αα is given by a Jacobi continued fraction: G (e) αα (z) = φ α φ α z a 0 z a 1 The coefficients a n and b n are stored in memory What about non diagonal elements G (e) αβ? See, e.g., E. Dagotto, Rev. Mod. Phys. 66:763 (1994) b 2 1 b 2 2 z a / 119

104 Lanczos method for the Green function (3) Trick: Define the combination G (e)+ αβ αβ (z) = Ω (c 1 α + c β ) (c α + c β ) Ω z H + E 0 G (e)+ (z) can be calculated like G(e) αα (z) Since G (e) (z) = G(e) (z), then αβ βα Likewise for G (h) αβ (z) G (e) αβ (z) = 1 2 G (e)+ αβ (z) G(e) αα (z) G(e) ββ (z) 104 / 119

105 The Lehmann representation Define the matrices Ω c α m m c β Ω Ω c β n n c α Ω G αβ (z) = + z E m m + E 0 z + E n n E 0 Q (e) αm = Ω c α m Q (h) αn = Ω c α n Then G αβ (z) = m = r Q (e) αm Q(e) βm z ω (e) m + n Q (h) αn Q(h) βn z ω (h) n Q αr Q β r z ω r QQ = / 119

106 The Band Lanczos method Define φ α = c α Ω, α = 1,..., L. Extended Krylov space : φ 1,..., φ L, H φ 1,..., H φ L,..., (H) M φ 1,..., (H) M φ L States are built iteratively and orthogonalized Possible linearly dependent states are eliminated ( deflation ) A band representation of the Hamiltonian (2L + 1 diagonals) is formed in the Krylov subspace. It is diagonalized and the eigenpairs are used to build an approximate Lehmann representation dongarra/etemplates/node131.html 106 / 119

107 Lanczos vs band Lanczos The usual Lanczos method for the Green function needs 3 vectors in memory, and L(L + 1) distinct Lanczos procedures. The band Lanczos method requires 3L + 1 vectors in memory, but requires only 2 iterative procedures ((e) et (h)). If Memory allows it, the band Lanczos is much faster. 107 / 119

108 Cluster symmetries Clusters with C 2v symmetry Clusters with C 2 symmetry 108 / 119

109 Cluster symmetries (2) Symmetry operations form a group G The most common occurences are : C 1 : The trivial group (no symmetry) C 2 : The 2-element group (e.g. left-right symmetry) C 2v : 2 reflections, 1 π-rotation C 4v : 4 reflections, 1 π-rotation, 2 π/2-rotations C 3v : 3 reflections, 3 2π/3-rotations C 6v : 6 reflections, 1 π, 2 π/3, 2 π/6 rotations States in the Hilbert space fall into a finite number of irreducible representations (irreps) of G The Hamiltonian H is block diagonal w.r.t. to irreps. Easiest to implement with Abelian (i.e. commuting) groups 109 / 119

110 Taking advantage of cluster symmetries... order of the group Reduces the dimension of the Hilbert space by G Accelerates the convergence of the Lanczos algorithm Reduces the number of Band Lanczos starting vectors by G But: complicates coding of the basis states Make use of the projection operator: dimension of irrep. See, e.g. Poilblanc & Laflorencie cond-mat/ P (α) = d α χ G g G (α) g g group character 110 / 119

111 Group characters C 2 E C 2 A 1 1 B 1 1 C 2v e c 2 σ 1 σ 2 A A B B C 4v e c 2 2c 4 2σ 1 2σ 2 A A B B E / 119

112 Taking advantage of cluster symmetries (2) Need new basis states, made of sets of binary states related by the group action: fermionic phase ψ = d α χ (α) g g b g b = φ g(b) g b G g Then matrix elements take the form ψ 2 H ψ 1 = d α χ (α) φ G h g (b) g b 2 H b 1 g When computing the Green function, one needs to use combinations of creation operators that fall into group representations. For instance (4 1): c (A) 1 = c 1 + c 4 c (A) 2 = c 2 + c 3 c (B) 1 = c 1 c 4 c (B) 2 = c 2 c / 119

113 Taking advantage of cluster symmetries (3) Example : number of matrix elements of the kinetic energy operator (Nearest neighbor) on a 3 4 cluster with C 2v symmetry: A 1 A 2 B 1 B 2 dim. 213, , , , 248 value , 640 6, 208 7, 584 5, , 983, 264 2, 936, 144 2, 884, 832 2, 911, , , 168 1, 050, 432 1, 021, , 088 2, 304 3, 232 2, / 119

114 QUESTIONS? 114 / 119

115 High-T c superconductors : structure Physics of CuO 2 planes + Cu + O + + Cu O O + Cu + O + + Cu / 119

116 Three-band model for HTSC A Hubbard model involving the three orbitals is (Emery 1987) H = (ɛ d µ) n (d) rσ + (ɛ p µ) n (p) xσ r,σ x,σ + t pd (p xσ d rσ + d rσ p xσ ) + t pp (p xσ p x σ + H.c.) r,x + U d r n (d) r n(d) r + U p x x,x n (p) x n(p) x + U pd r,x n (d) r n (p) x with the parameters (in ev, from LDA, in the hole representation) ɛ p ɛ d t pd t pp U d U p U pd / 119

117 Effect of the distance function What weights W (iω n ) to use? W 1/ω better in the underdoped region Sharp cutoff better in the overdoped region D. Sénéchal, Phys. Rev. B 81 (2010), SC order parameter SFA gradient (A) (B) 1/ω tr Σ 2 sharp tr Σ 2, ω c = n 117 / 119

118 Thermodynamic consistency The electron density n may be calculated either as n = TrG or n = Ω µ 1 The two methods give different results, except if the cluster chemical potential µ is a variational parameter 2 2 cluster U = 8 normal state n Ω µ cons. TrG TrG cons. Ω µ µ 118 / 119

119 Example clusters T9 B T / 119