Quantum Cluster Methods: An introduction

Transcription

1 Quantum Cluster Methods: An introduction David Sénéchal Département de physique, Université de Sherbrooke International summer school on New trends in computational approaches for many-body systems May 31, 2012 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 1 / 87

2 Outline 1 Introduction 2 Exact Diagonalizations 3 Cluster Perturbation Theory 4 The self-energy functional approach 5 The Variational Cluster Approximation 6 Cluster Dynamical Mean Field Theory David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 2 / 87

3 Outline Introduction 1 Introduction David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 3 / 87

4 High-T c superconductors Introduction Physics of CuO 2 planes + Cu + O + + Cu O O + Cu + O + + Cu + David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 4 / 87

5 Introduction Three-band model for HTSC A Hubbard model involving the three orbitals is (Emery 1987) H = (ε d µ) + (ε p µ) i,σ j,σ + t pd (p jσ d iσ + d iσ p jσ) + t pp (p jσ p j σ + H.c.) i,j + U d i n (d) iσ n (d) i n(d) i + U p j n (p) jσ n (p) j n(p) j j,j + U pd i,j n (d) i n (p) j with the parameters (in ev, from LDA, in the hole representation) ε p ε d t pd t pp U d U p U pd David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 5 / 87

6 Introduction One-band effective model for HTSC H = t rr c rσc r σ + U i r,r,σ n r n r µ r n r Each CuO 2 unit cell is a site Hopping parameters for YBCO are (Andersen et la., 1995), in mev: layer t t t (A) (B) In practice, we take t /t = 0.3, t /t = 0.2 U is unknown really : between 4t and 12t... k y (0, 0) t t t (π, π) k x n = 1.3 n = 1.0 n = 0.7 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 6 / 87

7 Introduction Approximation strategies method Mean Field Theory MFT Σ(k) yes yes Dynamical Mean Field Theory DMFT Σ(ω) yes yes/no Cluster Perturbation Theory CPT Σ ab (ω) no no Variational Cluster Approximation VCA Σ ab (ω) yes yes Cluster Dynamical Mean Field Theory CDMFT Σ ab (ω) yes yes/no Dynamical Cluster Approximation DCA Σ(K, ω) yes yes/no acronym self-energy self-consistent variational David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 7 / 87

8 Introduction Lattice and cluster Hamiltonians One-band Hubbard model: H = r,r,σ t rr c rσc r σ + U i n r n r µ r n r H H H H H H H H H H Need for an impurity solver for H In these lectures: exact diagonalizations at zero temperature David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 8 / 87

9 Outline Exact Diagonalizations 2 Exact Diagonalizations The Hilbert space Coding the states The Lanczos method Calculating the Green function Lehmann representation David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 9 / 87

10 Exact Diagonalizations 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 quasi-continuous David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 10 / 87

11 Exact Diagonalizations 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 11 / 87

12 Exact Diagonalizations The Hilbert space 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): ( d = L! [(L/2)!] 2 ) 2 2 4L πl L = 16 : One double-precision vector requires 1.23 GB of memory L dimension David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 12 / 87

13 Exact Diagonalizations The Hilbert space Two-site cluster: Hamiltonian matrix Half-filled, two-site Hubbard model: 4 states States and Hamiltonian matrix: 01, 01 U 2µ t t 0 01, 10 t 2µ 0 t 10, 01 t 0 2µ t 10, 10 0 t t U 2µ spin occupation spin occupation David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 13 / 87

14 Exact Diagonalizations The Hilbert space Six-site cluster: Hamiltonian matrix Sparse matrix structure David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 14 / 87

15 Exact Diagonalizations Coding the states 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 )! David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 15 / 87

16 Coding the states (2) Exact Diagonalizations Coding the states 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 16 / 87

17 Exact Diagonalizations Coding the states Constructing the Hamiltonian matrix Form of Hamiltonian: H = K K + V int. K = a,b t ab c ac 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 = b 1 c=a+1 We find it practical to construct and store all terms of the Hamiltonian separately. n c David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 17 / 87

18 The Lanczos method Exact Diagonalizations 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: K = 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 18 / 87

19 Exact Diagonalizations The Lanczos method (2) The Lanczos method In the basis of normalized states n = φ n / φ n φ n, the projected Hamiltonian has the tridiagonal form projector onto K a 0 b b 1 a 1 b P HP = 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. David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 19 / 87

20 Exact Diagonalizations The Lanczos method 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, periodic re-orthogonalization may be required, as orthogonality leaks will occur For degenerate ground states and low lying states (e.g. in DMRG), the Davidson method is generally preferable David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 20 / 87

21 Exact Diagonalizations The Lanczos method 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 21 / 87

22 Exact Diagonalizations Calculating the Green function Lanczos method for the Green function Zero temperature Green function: Consider the diagonal element G αβ (ω) = G (e) αβ (ω) + G(h) αβ (ω) G (e) αβ (ω) = Ω c 1 α c β ω H + E Ω 0 G (h) αβ (ω) = 1 Ω c β c α Ω ω + H E 0 φ α = c α Ω = G (e) 1 αα = φ α φ α ω H + E 0 Use the expansion 1 z H = 1 z + 1 z 2 H + 1 z 3 H2 + David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 22 / 87

23 Exact Diagonalizations Calculating the Green function 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(ω) = ω a 0 φ α φ α ω 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 ω a 2 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 23 / 87

24 Exact Diagonalizations Calculating the Green function Lanczos method for the Green function (3) Trick: Define the combination G (e)+ αβ αβ (ω) = Ω (c 1 α + c β ) (c α + c β ) Ω ω H + E 0 G (e)+ (ω) can be calculated like G(e) αα(ω) Since G (e) αβ (ω) = G(e) βα (ω), then Likewise for G (h) αβ (ω) G (e) αβ (ω) = 1 [ 2 G (e)+ αβ ] (ω) G(e) αα(ω) G (e) ββ (ω) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 24 / 87

25 Exact Diagonalizations The Lehmann representation Lehmann representation G αβ (ω) = m Define the matrices Ω c α m m c β Ω ω E m + E 0 + n Ω c β n n c α Ω ω + E n E 0 Q (e) αm = Ω c α m Q (h) αn = Ω c α n Then G αβ (ω) = m Q (e) αmq (e) βm + ω ω m (e) n Q (h) αn Q (h) βn ω ω (h) n = r Q αr Q βr ω ω r QQ = 1 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 25 / 87

26 Exact Diagonalizations Lehmann representation 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 26 / 87

27 Exact Diagonalizations Lehmann representation 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. David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 27 / 87

28 Cluster symmetries Exact Diagonalizations Lehmann representation Clusters with C 2v symmetry Clusters with C 2 symmetry David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 28 / 87

29 Exact Diagonalizations Lehmann representation 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 29 / 87

30 Exact Diagonalizations Lehmann representation 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: See, e.g. Poilblanc & Laflorencie cond-mat/ dimension of irrep. P (α) = d α χ G (α) g g g G group character David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 30 / 87

31 Group characters Exact Diagonalizations Lehmann representation 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 31 / 87

32 Exact Diagonalizations Lehmann representation Taking advantage of cluster symmetries (2) Need new basis states, made of sets of binary states related by the group action: ψ = d α G g χ (α) Then matrix elements take the form ψ 2 H ψ 1 = d α G fermionic phase g g b g b = φg(b) gb g χ (α) h φ g (b) gb 2 H b 1 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 c (B) 2 = c 2 c 3 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 32 / 87

33 Exact Diagonalizations Lehmann representation 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, David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 33 / 87

34 Exact Diagonalizations Lehmann representation Large dimensions : need for parallelization Memory needs exceed single cpu capacity beyond L 14 A half-filled 16-site system has dimension 165,636, GB for a state vector. Need to distribute the problem over many processors The main task is matrix-vector multiplication: a a a a a a a a a a a a a a a a a a a a a a a a David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 34 / 87

35 Outline Cluster Perturbation Theory 3 Cluster Perturbation Theory kinematics periodization spectral function and pseudogap summary David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 35 / 87

36 Cluster Perturbation Theory Clusters and superlattices kinematics e 2 e 1 K k k (π, π) (0, 0) ( π, π) 10-site cluster Reduced Brillouin zone David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 36 / 87

37 CPT Green function Cluster Perturbation Theory kinematics lattice Hamiltonian H hopping matrix cluster Hamiltonian = H + V inter-cluster hopping terms cluster hopping matrix t = t + V inter-cluster hopping matrix Treat V at lowest order in Perturbation theory At this order, the Green function is G 1 (ω) = G 1 (ω) V 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 PITP school 37 / 87

38 Cluster Perturbation Theory Interlude : Fourier transforms kinematics Unitary matrices performing Fourier transforms: U k,r = 1 N e ik r V k r = L N e i k r W K,R = 1 L e ik R complete superlattice cluster Various representations of the annihilation operator c(k) = r U kr c r c K ( k) = r,r V k r W KR c r+r c R ( k) = r V k r c r+r c r,k = R W KR c r+r Caveat: U V W The matrix Λ = U(V W) 1 relates (K, k) to k: c( k + K) = Λ K,K ( k)c K ( k) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 38 / 87

39 Cluster Perturbation Theory CPT Green function (cont.) kinematics More accurate notation: G 1 ( k, ω) = G 1 (ω) V( k). But G 1 = ω t Σ G 1 0 = ω t V, Thus : The lattice self-energy is approximated as the cluster self-energy G 1 ( k, ω) = G 1 0 ( k, ω) Σ(ω), Example : 2-site cluster (1D): ( ) t 0 1 = t 1 0 V( k) = t ( ) 0 e 2i k e 2i k 0 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 39 / 87

40 Periodization Cluster Perturbation Theory periodization CPT breaks translation invariance, which needs to be restored: G cpt (k, ω) = 1 ) G RR ( k, ω). L R,R e ik (R R Periodizing the Green function vs the self-energy (1D case): (π) (π) (π/2) (π/2) (0) ω 3 6 Green function periodization (0) ω 3 6 Self-energy periodization David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 40 / 87

41 Periodization (2) Cluster Perturbation Theory periodization G KK ( k, ω) = W KR W K R G RR ( k, ω) or G WGW Converted to the full wavevector basis (k = K + k) with Λ : G( k + K, k ( ) + K ) = Λ( k)gλ ( k) = 1 L 2 R,R,K 1,K 1 KK e i( k+k K 1) R e i( k+k K 1 ) R G K1K 1 = 1 e i( k+k) R e i( k+k ) R G RR ( k, ω). L R,R Then set K = K. Replace k by k = k + K in G RR ( k, ω), since V( k) is unchanged when k is shifted by a reciprocal superlattice vector. G per. (k, ω) = 1 L RR e ik (R R ) G RR (k, ω) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 41 / 87

42 Cluster Perturbation Theory One-dimensional example periodization Evolution of spectral function with increasing U/t: David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 42 / 87

43 Cluster Perturbation Theory Interlude : The spectral function spectral function and pseudogap Lehmann representation: A(k, ω) = 2 lim ImG(k, ω + iη) η 0 + G αβ (ω) = m Ω c α m m c β Ω ω E m + E 0 + n Ω c β n n c α Ω ω + E n E 0 But : lim Im 1 η 0 + ω + iη = lim Therefore : η 0 + η ω 2 + η 2 = πδ(ω) A(k, ω) = m m c k Ω 2 2πδ(ω E m +E 0 )+ n n c k Ω 2 2πδ(ω +E n E 0 ) prob. that particle has energy E m David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 43 / 87

44 Cluster Perturbation Theory spectral function and pseudogap h-doped cuprates : Pseudogap from CPT Sénéchal and Tremblay., Phys. Rev. Lett. 92, (2004) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 44 / 87

45 Cluster Perturbation Theory spectral function and pseudogap e-doped cuprates : Pseudogap from CPT Sénéchal and Tremblay., Phys. Rev. Lett. 92, (2004) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 45 / 87

46 Cluster Perturbation Theory spectral function and pseudogap Model for the cuprates: Fermi surface maps U= 2, n = 5/6 (π,π) (0,0) U= 8, n = 5/6 0% U= 4, n = 7/6 90% U= 8, n = 7/6 0% 90% Se ne chal and Tremblay., Phys. Rev. Lett. 92, (2004) David Se ne chal (Sherbrooke) Quantum Cluster Methods PITP school 46 / 87

47 Cluster Perturbation Theory summary CPT : features Exact at U = 0 Exact at t ij = 0 Exact short-range correlations Allows all values of the wavevector But : No long-range order Controlled by the size of the cluster David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 47 / 87

48 Outline The self-energy functional approach 4 The self-energy functional approach The variational principle Type III approximation: The Reference system Calculating the Potthoff functional David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 48 / 87

49 The self-energy functional approach 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 (π, π) e iq 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 49 / 87

50 The self-energy functional approach The Luttinger-Ward functional The variational principle Luttinger-Ward (or Baym-Kadanoff) functional: Φ[G] = Relation with self-energy: Legendre transform: δf [Σ] δσ M. Potthoff, Eur. Phys. J. B 32, 429 (2003) δφ[g] δg = Σ functional trace F [Σ] = Φ[G] Tr(ΣG) = δφ[g] δg[σ] δg δσ ΣδG[Σ] δσ G = G David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 50 / 87

51 The self-energy functional approach The variational principle The variational principle Free energy functional: Ω t [Σ] = F [Σ] Tr ln( G 1 0t + Σ) The functional is stationary at the physical self-energy (Euler eq.): δω t [Σ] δσ = G + (G 1 0t Σ) 1 = 0 At the physical self-energy, Ω t [Σ] is the thermodynamic 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 51 / 87

52 The self-energy functional approach Type III approximation: The Reference system 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 ) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 52 / 87

53 The self-energy functional approach The Potthoff functional Type III approximation: The Reference system Making the trace explicit, one finds Ω t [Σ] = Ω T ω = Ω T ω k k [ ] tr ln 1 V( k)g ( k, ω) [ ] ln det 1 V( k)g ( k, ω) 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 = 0. David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 53 / 87

54 The self-energy functional approach Calculating the Potthoff functional Calculating the functional I : exact form It can be shown that Tr ln( G) = T m poles of G ln(1 + e βω m) + T m ln(1 + e βζ m) zeros of G Use the Lehmann representation of the GF: G αβ(ω) = r M. Potthoff, Eur. Phys. J. B, 36:335 (2003) Lehmann representation Q αr Q βr G (ω) = Q 1 ω ω r ω Λ Q diagonal(ω r) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 54 / 87

55 The self-energy functional approach Calculating the Potthoff functional 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 [ 1 = Q ω L( k) Q Q 1 ω Λ Q ] 1 V( k) 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 + L N ω r <0 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) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 55 / 87

56 The self-energy functional approach Calculating the Potthoff functional Calculating the functional II : numerical integral Except for very small clusters (L 4), it is much faster to perform a numerical integration over frequencies: Z dω L X Ω(x) = Ω0 (x) ln det(1 V (k )G0 (iω)) L(µ µ0 ) π N 0 k 1 The integral must be done using an adaptive method that refines a mesh where necessary. For instance: The CUBA library k y /(2π) Grid of 17,095 points used in an adaptive integration over wavevectors k x /(2π) David Se ne chal (Sherbrooke) Quantum Cluster Methods PITP school 56 / 87

57 Outline The Variational Cluster Approximation 5 The Variational Cluster Approximation Néel Antiferromagnetism Superconductivity in VCA Thermodynamic consistency Optimization Examples Summary David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 57 / 87

58 The Variational Cluster Approximation 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: Ω t [Σ] = Ω T L N ω k [ ] ln det 1 V( k)g ( k, ω) 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 6 Adopt the value of Ω as the best estimate of the grand potential David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 58 / 87

59 The Variational Cluster Approximation Néel Antiferromagnetism Example : Néel Antiferromagnetism Used the Weiss field (π, π) H M = M r e iq 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 59 / 87

60 The Variational Cluster Approximation Néel Antiferromagnetism Example : Néel Antiferromagnetism (2) 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 60 / 87

61 Example clusters The Variational Cluster Approximation Néel Antiferromagnetism T9 B T15 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 61 / 87

62 The Variational Cluster Approximation VCA and Superconductivity Superconductivity in VCA Need to add a pairing field O sc = r,r rr c r c r + H.c extended s-wave s-wave pairing: rr = δ rr d x 2 y 2 pairing: { 1 if r r = ±a Ω d xy rr = 1 if r r = ±b -1.8 d x 2 y 2 d xy pairing: { 1 if r r = ±(a + b) rr = 1 if r r = ±(a b) cluster s-wave U = 8, µ = David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 62 / 87

63 The Variational Cluster Approximation The Nambu representation Superconductivity in VCA 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, i.e., a particle-hole transformation on the spin-down sector : c r = c r and d r = c r Then the Hamiltonian looks like it conserves total particle number N d + N c : H = ( t rr c r c r d r d r ) U n c r n d r r,r (µ U)N c µn d ) ( rr d r c r + H.c r,r r David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 63 / 87

64 The Variational Cluster Approximation Competing SC and AF orders Superconductivity in VCA 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 M. Guillot, MSc thesis, Univ. de Sherbrooke (2007) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 64 / 87

65 The Variational Cluster Approximation Thermodynamic consistency 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 65 / 87

66 The Variational Cluster Approximation Optimization Optimization procedure Need to find the stationary points of Ω(x) with as few evaluations as possible Use 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 The NR method is 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) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 66 / 87

67 The Variational Cluster Approximation Examples Example: Homogeneous coexistence of dsc and AF orders David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 67 / 87

68 The Variational Cluster Approximation VCA vs Mean-Field Theory Summary 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 68 / 87

69 Outline Cluster Dynamical Mean Field Theory 6 Cluster Dynamical Mean Field Theory The hybridization function The DIA The CDMFT self-consistency condition Application to the cuprates The Mott transition The Dynamical Cluster Approximation (DCA) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 69 / 87

70 Dynamical mean field Cluster Dynamical Mean Field Theory The hybridization function Action associated with a correlated lattice model, in imaginary time: S[c, c ] = β Intra- and inter-cluster terms: 0 { } dτ c α(τ) [δ αβ τ + t αβ ] c β (τ) + H 1 (c, c ) α,β S = m Γ S (m) + m,n Γ S (m,n) Effect of inter-cluster terms represented by dynamical mean field G 0 : S eff [c, c ] = β 0 dτdτ α,β interaction part β c α(τ)g 1 0,αβ (τ τ )c β (τ ) + dτ H 1 (c, c ) 0 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 70 / 87

71 Cluster Dynamical Mean Field Theory The hybridization function The hybridization function In the frequency domain: G 1 hybridization function 0 (iω n ) = iω n t Γ(iω n ) where G 0 (iω n ) = Spectral representation of Γ: Γ αβ (iω n ) = N b µ θ αµ θ βµ iω n ε µ Corresponding Hamiltonian: Anderson impurity model H AIM = α,β t αβc αc β + α,µ ( θ αµ c αa µ ) + H.c. β 0 e iωnτ G 0 (τ) N b + ε µ a µa µ µ fictitious bath degree of freedom David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 71 / 87

72 Cluster Dynamical Mean Field Theory The hybridization function (2) The hybridization function Proof: (U = 0) G 1 full (ω) = Need to find G = B 11, where Simple manipulations lead to ( ω t θ ) θ = A ω ε ( ) ( A 11 A 12 B 11 B 12 = A 21 A 22 B 21 B 22 ) 1 ( A11 A 12 A 1 22 A ) 21 B11 = 1 G 1 = ω t θ 1 ω ε θ U 0 : simply add the free energy (by definition) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 72 / 87

73 Reference system Cluster Dynamical Mean Field Theory The hybridization function Reference Hamiltonian: hybridization matrix H = α,β t αβc αc β + U a n a n a + µ,α θ µα (c αa µ + H.c.) + µ ε µ a µa µ Example bath systems: bath energies 7 8 ε 2 t 2 t 2 ε 2 ε 2 t 2 t 2 ε ε 1 t 1 t 1 ε 1 ε 1 t 1 t 1 ε David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 73 / 87

74 Cluster Dynamical Mean Field Theory The DIA The Dynamical Impurity Approximation (DIA) The bath parameters (ε µ, θ αµ, etc) are variational parameters The bath makes a contribution to the Potthoff functional: Ω bath = ε α<0 On 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 1 (iω n ) Ḡ 1 (iω n ) Σ (iω n ) = 0. θ Potthoff functional ω n ε α defined below David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 74 / 87

75 Cluster Dynamical Mean Field Theory The CDMFT Procedure The CDMFT self-consistency condition Self-consistency is generally used instead of the DIA: 1 Start with a guess value of (θ αµ, ε µ ). 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 G 1 0 (ω) = Ḡ 1 + Σ(ω) 4 Minimize the following distance function: d(θ, ε) = ω n W (iω n ) tr G 1 (iω n ) Ḡ 1 (iω n ) 2 over the set of bath parameters. Thus obtain a new set (θ αµ, ε µ ). 5 Go back to step (2) until convergence. David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 75 / 87

76 Cluster Dynamical Mean Field Theory The CDMFT self-consistency condition The CDMFT Procedure (2) Initial guess for Γ Cluster Solver: Compute G Ḡ = L N k [G 1 0 ( k) Σ(ω)] 1 G 1 0 = Ḡ 1 + Σ update Γ by minimizing d No Γ converged? Yes Exit David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 76 / 87

77 Cluster Dynamical Mean Field Theory The 1D Hubbard model The CDMFT self-consistency condition ε1 & ε2 θ1 & θ2 n 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 77 / 87

78 Cluster Dynamical Mean Field Theory The CDMFT self-consistency condition 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 78 / 87

79 Cluster Dynamical Mean Field Theory Application to the cuprates Example : dsc and AF in the 2D Hubbard model Nine bath parameters Homogeneous coexistence of d x 2 y 2 SC and Néel AF 0.04 ψ ψ/j 0.04 U = 4t U = 8t U = 12t U = 16t U = 24t M, ψ U = 8t t = 0.3t t = 0.2t ψ (10 ) M n n David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 79 / 87

80 Cluster Dynamical Mean Field Theory Application to the cuprates Effect of the distance function The 1/ω weight is better in the underdoped region The sharp cutoff is 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 David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 80 / 87

81 The Mott transition Cluster Dynamical Mean Field Theory 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 U U M. Balzer et al., Europhys. Lett. 85, (2009) 1 site 4 sites U=t U=8t U=12t U=14t U=t U=5t U=7t U=10t 6 /t 6 /t 6 6 Y.Z. Zhang, M. Imada, Phys. Rev. B 76, (2007) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 81 / 87

82 Cluster Dynamical Mean Field Theory The Mott transition First-order character of the Mott transition (DIA) 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) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 82 / 87

83 Cluster Dynamical Mean Field Theory The Mott transition Mott transition (DIA vs CDMFT) The Mott transition may not show up as first-order in CDMFT, even though it is in the DIA The choice of distance function matters D. Sénéchal, Phys. Rev. B 81 (2010), θ ε SFA gradient (A) Uc1 Uc2 (B) (C) 1/ω tr Σ 2 sharp VCA U David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 83 / 87

84 Cluster Dynamical Mean Field Theory The Mott transition 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 B. Kyung, A.M.S. Tremblay, Phys. Rev. Lett. 97, (2006) U/t David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 84 / 87

85 Cluster Dynamical Mean Field Theory The Mott transition Mott transition and superconductivity T The Mott transition occurs also upon doping 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 (µ p,t p ) µ T d c T µ c1 µ c2 MI (U MIT,T MIT ) Tc d U c2 U c1 U Sordi et al., arxiv: David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 85 / 87

86 Cluster Dynamical Mean Field Theory The Dynamical Cluster Approximation (DCA) 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 ε( k + K) N Not derivable from the SFA For large clusters: DCA converges better for k = 0 (average) quantities CDMFT converges better for r = 0 (local) quantities k 1 iω n ε( k + K) Σ K (ω) David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 86 / 87

87 Questions QUESTIONS? David Sénéchal (Sherbrooke) Quantum Cluster Methods PITP school 87 / 87