w2dynamics : operation and applications

Transcription

1 w2dynamics : operation and applications Giorgio Sangiovanni ERC Kick-off Meeting, Hackers Nico Parragh (Uni Wü) Markus Wallerberger (TU) Patrik Gunacker (TU) Andreas Hausoel (Uni Wü)

2 A solver for multi-orbital problems For LDA+DMFT we need to solve an Anderson model containing one correlated impurity site, with typically several d-orbitals a =1,...,N orb H loc = Un a, n a, a + U n a,σ n b, σ +(U J)n a,σ n b,σ a>b,σ a=b J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) spin-flip pair-hopping

3 Hybridization-expansion the best continuous-time quantum Monte Carlo scheme to solve impurity models with complicated interactions (as e.g. the Kanamori Hamiltonian) is the hybridization-expansion ( CT-HYB ) terms leading to a bad sign problem in BSS / Hirsch-Fye -QMC H loc = Un a, n a, a + U n a,σ n b, σ +(U J)n a,σ n b,σ a>b,σ a =1,...,N orb a=b J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) spin-flip pair-hopping

4 References hybridization expansion P. Werner, et al., Phys. Rev. Lett. 97, (2006) P. Werner and A. J. Millis, Phys. Rev. B 74, (2006) K. Haule, Phys. Rev B 75, (2007) E. Gull, et al., Rev. Mod. Phys. 83, 349 (2011) N. Parragh, et al., Phys. Rev. B 86, (2012)! hybridization expansion - Krylov A. M. Laeuchli and P. Werner, Phys. Rev. B 76, (2007) T. J. Park and J. C. Light, J. Chem. Phys. 85, 5870 (1986) Legendre representation L. Boehnke, et al., Phys. Rev. B 84, (2011)

5 The impurity-model (AIM) to be solved FROM SOMEWHERE F (τ) or, equivalently F (iω n ) In Exact Diagonalization : F (iω n )= l V 2 l iω n ε l G 0 CT-HYB calculation F (τ) H loc

6 Hybridization expansion Hamiltonian of the full Anderson model H = H loc + H bath + H hyb H hyb = l V α l c αa l + Vl α a l c α interaction picture perturbation expansion in the hybridization β β Z = 1 dτ 1 k! k 0 0 Tr c e βh loc Tc αk (τ k )c α (τk) c α1 (τ 1 )c k α1(τ1) Tr a e βh bath Ta l k (τ k )a l (τ k k) a l 1 (τ 1 )a l (τ 1 1) dτk α 1 α k α 1 α l k 1 l k l 1 l k V α 1 l 1 V α 1 l 1 V α k l k V α k lk brute force analytically : F(τ)

7 Sampling the local trace Tr c e βh loc Tc αk (τ k )c α (τk) c α1 (τ 1 )c k α1(τ1) after warmup H loc = Un a, n a, a + U n a,σ n b, σ +(U J)n a,σ n b,σ a>b,σ a=b J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) insertion One flavor of creation/ annihilation operators removal generalization to multiple flavors

8 Advantages of the Krylov scheme 7 orbitals with Kanamori interaction doable with Krylov! N. Parragh, et al., PRB 86, (2012) other CT-HYB scheme (so-called matrix-matrix ): H loc cheap exact time evolution (basis is the eigenbasis of ) matrix representation of annihilation/creation operators

9 Krylov implementation β... 0 bottleneck: computation of the local trace : Tr c e βh loc c αk (τ k )c α (τk) c α1 (τ 1 )c k α1(τ1) our basis: occupation numbers sparse matrix in our basis eff. matrix-vector product eigenvectors of H loc example for 1 orbital { 0,,, } e τ 1 H loc c α 1 e τ 1 H loc φ 0 with Lanczos e τh loc v p 1 k=0 ( τ) k H k k! loc v how many outer states are needed for the computation of the trace? how to keep the matrix-size as small as possible?

10 How many outer states? Tr c e βh loc Tc αk (τ k )c α (τk) c α1 (τ 1 )c k α1(τ1) = = φ 0 e βe(0) loc... φ0 + e β(e(1) loc E(0) loc ) φ 1 e βe(0) loc... φ β(e (1) at low-t: loc E(0) loc ) 1 we fix the outer state(s) because of c/c + we can still visit an excited state for a short time inside the trace φ i e βh loc... e τ 1 H loc c α 1 i=0 e τ 1 H loc φ i φ 0 φ 0 at low-t the trace is long easy to accommodate enough visits very controllable and convenient truncation parameter of Krylov CT-HYB at very low T we can restrict ourselves to the lowest multiplet

11 Tanabe-Sugano diagram for d 5 Kanamori full Coulomb high-spin e g 10Dq t 2g low-spin low-spin e g U=5.14eV J=0.71eV t 2g 10Dq low-spin F 0 =4eV F 2 =8.615eV F 4 =5.3846eV high-spin low-spin 10Dq (cubic crystal field) degeneracies high-spin low-spin 10Dq (cubic crystal field)

12 Exploiting conserved quantities good quantum numbers of H loc maximum block-diagonalization eigenvectors of H loc example for 1 orbital { 0,,, } e τ 1 H loc c α 1 e τ 1 H loc φ 0 with Lanczos e τh loc v p 1 k=0 ( τ) k H k k! loc v

13 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b N tot (Sz) tot S 2 tot 1 +½ ½ P. Werner and A. J. Millis, PRB 74, (2006)

14 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b N tot (Sz) tot S 2 tot 1 -½ ½

15 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b N tot (Sz) tot S 2 tot 2 1

16 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b N tot (Sz) tot S 2 tot

17 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b N tot (Sz) tot S 2 tot 2 0 1

18 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b N tot (Sz) tot S 2 tot 2-1 1

19 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b N tot (Sz) tot S 2 tot 2 0 1

20 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b diagonal in the occupation-number basis off-diagonal the pattern of singly-occupied orbitals is conserved! PS -number N. Parragh, et al., PRB 86, (2012)

21 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b N tot (Sz) tot 1 +½

22 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b N tot (Sz) tot PS 1 +½ 1 PS = N orb 1 a=0 2 a (n a, n a, ) 2

23 Example for two orbitals eigenbasis of the Kanamori Hamiltonian H loc = a Un a, n a, + a>b,σ U n a,σ n b, σ +(U J)n a,σ n b,σ J(c a, c b, c b, c a, + c b, c b, c a, c a, + h.c.) a=b N tot (Sz) tot PS 1 +½ 2 PS = N orb 1 a=0 2 a (n a, n a, ) 2

24 Advantages of the PS -number N. Parragh, et al., PRB 86, (2012) PS = N orb 1 a=0 2 a (n a, n a, ) 2

25 Where do the operators go? We sample partition-function configurations 1 k! β β dτk 0 α 1 α k α 1 α l k 1 l k l 1 l k Z = dτ 1 k 0 Tr c e βh loc Tc αk (τ k )c α (τk) c α1 (τ 1 )c k α1(τ1) Tr a e βh bath Ta l k (τ k )a l (τ k k) a l 1 (τ 1 )a l (τ 1 1) V α 1 l 1 V α 1 l 1 V α k l k V α k lk p(k): distribution of the c operators U β Anderson model (βt=100) P. Werner, et al., PRL 97, (2006) Anderson-Holstein model (βt=400, ω 0 =0.2t) E. Gull, et al., RMP 83, 349 (2006)

26 Where do the operators go? We sample partition-function configurations 1 k! β β dτk 0 α 1 α k α 1 α l k 1 l k l 1 l k Z = dτ 1 k 0 Tr c e βh loc Tc αk (τ k )c α (τk) c α1 (τ 1 )c k α1(τ1) Tr a e βh bath Ta l k (τ k )a l (τ k k) a l 1 (τ 1 )a l (τ 1 1) V α 1 l 1 V α 1 l 1 V α k l k V α k lk p(k): distribution of the c operators but we have encountered situations with more complicated p(k): DOUBLE-PEAKED HISTOGRAM???

27 Where does the AIM come from? FROM SOMEWHERE F (τ) or, equivalently F (iω n ) plain vanilla DMFT LDA+DMFT H(k) in wannier90-format long-range ordered magnetic solutions enlarged basis ( PAM /dp-models) and DMFT+Hartree flexible and user-defined Python classes in w2dynamics various flavors of non-equivalent atoms/layers/ real-space schemes ( nanodmft and nanodγa )

28 plain vanilla DMFT LDA+DMFT H(k) in wannier90-format long-range ordered magnetic solutions enlarged basis ( PAM /dp-models) and DMFT+Hartree various flavors of non-equivalent atoms/layers/ real-space schemes ( nanodmft and nanodγa )

29 Periodic Anderson Model ( t dd -PAM )

30 Hole-doped Mott insulator 10% doping with t pd =0 n d =0.9 10% doping with t pd 0 n tot =n d +n p =2.9 all energies in units of t pp t dd =1/4

31 Histogram in the t dd -PAM spectral representation of the hyb. function CT-HYB histogram t pd =0.00 t pd =0.25 t pd =0.50 t pd =0.75 t pd =1.00 expansion order

32 How can we label the histogram peaks? Hubbard model (t pd =0) below half-filling σ = one possible way: order-resolved density-matrix (in Fock space) k σ m ρ α, α = αα α = 0,,, ρ 0, 0 ρ, ρ, ρ, φ i e βh loc......c α1(τ1) φ i i=0 a bit more of hybridization events with many spin up in order to empty the impurity

33 How can we label the histogram peaks? t dd -PAM (t pd =0.75) 10% hole doping ρ 0, 0 σ = ρ, order-resolved density-matrix: k σ m ρ α, α = αα now diagrams making the impurity doubly occupied belong almost fully to the second peak in our case the p-band is almost full, it can only give electrons second peak has p character ρ, ρ, hybridization events filling the impurity φ i e βh loc......c α1(τ1) φ i i=0

34 Simple way of getting two peaks? coherent state of the harmonic oscillator a α = α α α =e α2 /2 n=0 α n n! n which n contribute? P (n) = n α 2 = e α2 n! α 2n two components with different strength of the hybridization

35 Kinetic energy argument

36 Kinetic energy argument In CT-HYB the kinetic energy can be calculated from the average exp. order Z = D[ψ ψ]e S ( 1)k 0 ( S) k S = hybridization part of the action k! k k = 1 D[ψ ψ]e S 0 k ( 1)k ( S) k = 1 D[ψ ψ]e S ( 1) k 1 0 Z k! Z (k 1)! ( S)k 1 S = S S = β 0 dτ l,σ a lσ k τ µ + ε l integrating out the bath fermions a lσ + V l c σa lσ + V l a lσ c σ S = β 0 β 0 k dτdτ σ k = 1 T E kin S =Tr[F G] c σ(τ)f (τ τ )c σ (τ )

37 Kinetic energy argument Two components with different mobility? k = 1 T E kin two different screening channels? ongoing study of the spin susceptibility

38 Second moment What does the second moment of the distribution correspond to? Z = D[ψ ψ]e S ( 1)k 0 ( S) k S = hybridization part of the action k! k k 2 = 1 Z D[ψ ψ]e S 0 k k 2 ( 1)k k! ( S) k = 1 Z D[ψ ψ]e S 0 k k ( 1) k 2 k 1 (k 2)! ( S)k 2 ( S) 2 ( S) 2 connected to a four-point T-product does a susceptibility get large when the distribution becomes bi-modal? see also yesterday s talk by Sergio Ciuchi ongoing project with AG Held supervised by M. Wallerberger and A. Toschi

39 Conclusions plain vanilla DMFT LDA+DMFT H(k) in wannier90-format long-range ordered magnetic solutions enlarged basis ( PAM /dp-models) and DMFT+Hartree w2dynamics various flavors of non-equivalent atoms/layers/ real-space schemes ( nanodmft and nanodγa ) talk by Nico about future developments and talk by Markus about two-particle results

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