Quantum Cluster Methods (CPT/CDMFT)

Quantum Cluster Methods (CPT/CDMFT) David Sénéchal Département de physique Université de Sherbrooke Sherbrooke (Québec) Canada Autumn School on Correlated Electrons Forschungszentrum Jülich, Sept. 24, 2015 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 1 / 49

Outline 1 Preliminaries: The Green function 2 Cluster Perturbation Theory (CPT) 3 CPT : examples 4 Cluster Dynamical Mean Field Theory (CDMFT) 5 CDMFT : Applications David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 2 / 49

The Hubbard model Hamiltonian: hopping amplitude H = t r,r c r,r,σ creation operator (chemical potential µ = t r r ) r σ c r σ + U n number of spin electrons at r r n r r local Coulomb repulsion w(x) x David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 4 / 49

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: 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 5 / 49

Spectral representation 1 G αβ (z) = Ω c α r r c z E r + E β Ω 0 Q-matrix: Q αr = r>0 eigenstate with N + 1 particles Ω cα r (r > 0) Spectral representation: r c α Ω (r < 0) r<0 eigenstate with N 1 particles + Ω c β r 1 r c α Ω z + E r E 0 and ω r = Er E 0 (r > 0) E 0 E r (r < 0) Q αr Q β r G αβ (z) = z ω r r (partial fractions) David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 6 / 49

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 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 7 / 49

Spectral function Spectral function: Translation-invariant system: r>0 A αβ (ω) = 2 Im G αβ (ω + i0 + ) G(k, z) = Ω c k r 2 1 + z E r + E 0 r<0 Ω c k r 2 1 z + E r E 0 A(k, ω) = 2 Im G(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 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 8 / 49

non-interacting limit (U = 0) G(z) = 1 z t Ω = c k 0 ε k <µ (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 0.6 1 1.4 0.6 1.0 1.4 k y k y π π k x π π π k x π David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 9 / 49

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) -4-3 -2-1 0 1 2 3 4 ω/t Spectral function, half-filling, NN hopping N(ω) 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0-4 -3-2 -1 0 1 2 3 4 ω/t Associated density of states David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 10 / 49

Self-energy Interacting Green function: 1 G(z) = z t Σ(z) Local limit at half-filling (t = 0, µ = U/2): self-energy G(z) = 1/2 z + U/2 + 1/2 z U/2 = 1 z U2 4z (π,0) Analytic structure: Σ(z) = U2 4z + U 2 S Σ αβ (z) = Σ αβ + αr S β r z σ r (π,0) -1-0.5 0 0.5 1 Spectral function, half-filled HM (t = 0) r David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 11 / 49 (π,π) (0,0) ω/u

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) 0 0 0 Σ (2) 0 Σ =...... 0 0 Σ (n) Varieties: CPT : Cluster Perturbation Theory VCA : Variational Cluster Approximation CDMFT : Cluster Dynamical Mean Field Theory CDIA : Cluster Dynamical Impurity Approximation David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 13 / 49

Tiling the lattice into clusters Tiling of the triangular lattice with 6-site clusters David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 14 / 49

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 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 15 / 49

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) David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 16 / 49

Superlattices and reduced Brillouin zones e 2 e 1 K k k (π, π) (0, 0) ( π, π) 10-site cluster Brillouin zones David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 17 / 49

CPT for translation invariant tilings site within cluster one-body index = (R, k Green function and t ic diagonal in k. G independent of k. CPT formula written as G 1 (z, k) = G 1 (z) t ic ( k) where matrices are now in (R, σ) space. reduced wavevector, σ) spin David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 18 / 49

Interlude : Fourier transforms Unitary matrices performing Fourier transforms: U γ k,r = 1 L e ik r U Γ k = e i k r N r N 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 U c r 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 19 / 49

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) 6 3 0 ω 3 6 (0) 6 3 0 ω 3 6 Green function periodization Self-energy periodization David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 20 / 49

Periodization (2) Periodization as a change of basis: G( k + K, k + K, z) = Λ c ( k)g(z)λ c ( k) K K = 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. David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 21 / 49

One-dimensional example Spectral function of the half-filled HM with increasing U/t: David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 23 / 49

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/2 0 4 0 4 4 0 4 4 0 4 ω ω ω David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 24 / 49

One-dimensional example (cont.) Spectral gap at half-filling: periodic cluster vs CPT 5 Mott gap 4.5 4 3.5 3 2.5 2 0.03 0.02 0.01 0 0.2 0.4 0.6 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 U L =12(cluster) L (cluster) L =12 L exact David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 25 / 49

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, 126401 (2004) David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 26 / 49

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, 126401 (2004) David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 27 / 49

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% U= 8, n = 7/6 0% kx F. Ronning et al., PRB 67, 165101 (2003) 90% Sénéchal and Tremblay, PRL 92, 126401 (2004) David Sénéchal (Sherbrooke) (0, 0) (π, π) N.P. Armitage et al., PRL 88, 257001 (2002) Quantum Cluster Methods Jülich school 28 / 49

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 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 29 / 49

Averages of one-body operators General one-body operator: O = s αβ c α c β 0 p Average: O = s αβ c α c β = L N k α,β s = s dω 2π Re tr s( k)g( k, iω) trs( k) iω p David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 30 / 49

The dynamical mean field Action formalism: Z = [dc α d c α ]e is[c, c] α S[c, c] = = interaction d t c α (t)(iδ αβ t t αβ )c β (t) H1(c, c) α,β d t d t α,β c α (t)g 1 0,αβ (t t )c β (t ) H 1 (c, c)δ(t t ) intra cluster S = S (j) + S j i, j inter cluster (i, j) i Γ S (i, j) S (j) env. David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 32 / 49

The hybridization function S eff [c, c] = d t d t α,β c α (t)g 1 0,αβ (t t )c β (t ) + d t H 1 (c, c) G 1 0 (ω) = ω t c Γ (ω) The hybridization function has the analytic structure of a self-energy: N b θ αr θ β r Γ αβ (z) = z ɛ r (the constant piece Γ is dropped in CDMFT) N b is potentially infinite. r David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 33 / 49

The Anderson impurity model hybridization bath energy N b H AIM = t c,αβ c α c β + θ αr c α a r + H.c. + ɛr a r a r + H 1 α,β α,r r bath annihilation operator ɛ 3 ɛ 4 ɛ 1 θ 1 ɛ 3 θ 3 + + θ + 2 θ + 1 ɛ + 2 ɛ + 1 + + θ 7 θ 3 θ 4 ɛ 7 ɛ 8 θ 8 θ 2 θ 4 + + 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) David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 34 / 49

The Anderson impurity model (cont.) Non interacting Green function (cluster+bath): G full 0 (z) = 1 z T tc θ where T = θ ɛ G full 0 (z) 1 z tc θ A11 A = θ = 12 B11 B = 12 z ɛ A 21 A 22 B 21 B 22 Need to extract the cluster component of G full 0 (z), i.e. B 11: A 11 B 11 + A 12 B 21 = 1 B 21 = A 1 22 A 21B 11 A11 A 12 A 1 22 A 21 B11 = 1 G 1 0 c = z t c Γ (z) Γ (z) = θ 1 z ɛ θ 1 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 35 / 49

The Anderson impurity model (cont.) Interacting case: one must simply add the cluster self-energy (no self-energy on the bath). The cluster Green function is then G 1 c (z) = z t c Γ (z) Σ(z) = G 1 0 (z) Σ(z) David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 36 / 49

The self-consistency condition The local, cluster Green function G c must be consistent with the projection Ḡ to the cluster of the lattice Green function: Ḡ(z) = L 1 z t ( k) Σ(z) N k This condition may be statisfied (up to statistical errors) in QMC. It can only be satisfied approximately in ED (i.e. with a finite bath). Instead, one minimizes the distance function: d(θ, ɛ) = iω n W weight (iωn) Tr c (iω n ) Ḡ 1 (iω n ) G 1 Matsubara freqs from a fictitious temperature β 1 2 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 37 / 49

The CDMFT algorithm Initial guess for Γ (iω n ) Impurity solver: Compute G c (iω n ) Ḡ(iω n ) = L N k G 1 0 (iω n, k) Σ(iω n ) 1 Γ (iω n ) iω n t c + µ Ḡ(iω n ) Σ(iω n ) (QMC) minimize ω n W (iω n ) tr G 1 c (iω n ) Ḡ 1 (iω n ) 2 (ED) No Γ converged? Yes exit David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 38 / 49

Application: the Mott transition (QMC) 0.2 a) U/t=5.0 b) U/t=5.4 A(ω) 0.1 0.0 0.2 c) U/t=5.6 d) U/t=5.8 A(ω) 0.1 0.0-4 -2 0 2 4 ω -4-2 0 2 4 ω H. Park et al, PRL 101, 186403 (2008) David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 40 / 49

Application: the Mott transition (ED) 0.2 U = 5.0 U = 5.4 A(ω) 0.1 0 0.2 U = 5.6 U = 5.8 A(ω) 0.1 0-4 -2 0 2 4-4 -2 0 2 4 ω ω solutions from M. Balzer et al., Europhys. Lett. 85, 17002 (2009) David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 41 / 49

Superconductivity Pairing operator: ˆΨ = 1 N r,r g r r cr c r c r c r. In momentum space: ˆΨ = 1 g(k) c (k)c ( k) c (k)c ( k) N with the correspondence Gorkov function: k g(k) = g r,0 e ik r r 1 1 F r r (z) = Ω c r c r z H + E Ω + Ω c r c r Ω 0 z + H E 0 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 42 / 49

Trick: The Nambu formalism Particle-hole transformation on the down spin: d r σ = (c r, c r ) Anticommutation relations OK : {d r σ, d } = δ r σ r r δ σσ Pairing operator: ˆΨ = 1 dr N d r d r d r r,r g r r Gorkov function is off-diagonal block of Green function: G (z) F(z) G(z) = F (z) G (z) 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 43 / 49

Superconductivity in the square-lattice Hubbard model 0.04 ψ 0.02 0 U = 4t U = 8t U = 12t U = 16t U = 24t 1.0 0.8 0.6 M, ψ U = 8t t = 0.3t t = 0.2t M 0.08 ψ/j 0.04 0.4 0.2 ψ (10 ) 0.0 0.7 0.8 0.9 1.0 n 0.0 0.7 0.8 0.9 1.0 1.1 1.2 1.3 n Kancharla et al., Phys. Rev. B 77, 184516 (2008). David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 44 / 49

Inhomogeneous systems (many distinct clusters) G 1 ( k, z) = z t ( k) Σ(z) z t (11) ( k) Σ 1 (z) t (12) ( k)... t (1n) ( k) t (21) ( k) z t (22) ( k) Σ = 2 (z)... t (2n) ( k)...... t (n1) ( k) t (n2) ( k)... z t (nn) ( k) Σ n (z) Distance function in that case: G.F. of the j th cluster d(θ, ɛ) = W (iω n ) Tr G (j) 1 (iω n ) Ḡ (j) 1 (iω n ) j iω n 2 j th diagonal block of Ḡ David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 45 / 49

Non-magnetic impurity in graphene U = 12t, ε 0 = 11t M. Charlebois et al., Phys. Rev. B 91, 035132 (2015) David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 46 / 49

Extended interactions: Dynamical Hartree Approximation Extended interaction: H V = 1 V r 2 r n r n r = 1 2 r,r Dynamical Hartree Approximation: c V r,r intra cluster r,r n r n r + 1 2 ic r,r n r n r r,r V inter cluster 1 V ic r,r n 2 r n r ˆV ic = 1 V ic r,r ( n 2 r n r + n r n r n r n r ) r,r r,r Set n r = n r + δn r, then neglect the term quadratic in fluctuations δn r δn r David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 47 / 49

CDMFT : Conclusion Approximation scheme for the one-body Green function with exact short-range correlations and self-consistency Allows for broken symmetry phases (magnetism, superconductivity) Takes into account time-domain fluctuations better than spatial fluctuations Controlled by the size of the cluster (and size of the bath, in ED). But: Restricted to small clusters for practical reasons Clusters are not practical for incommensurate order Two-body operators and fluctuations not well described (cluster only) David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 48 / 49

THANK YOU! QUESTIONS? David Sénéchal (Sherbrooke) Quantum Cluster Methods Jülich school 49 / 49