1. Calculation of dynamical properties using DMRG Karen A. Hallberg Centro Atómico Bariloche and Instituto Balseiro, Bariloche, Argentina Leiden, August 2004

2. Daniel Garcia Marcelo Rozenberg

3. INTRODUCTION Importance of the study of dynamical properties e.g. nuclear magnetic resonance (NMR) electron paramagnetic resonance (EPR) neutron scattering optical absorption photoemission and even transport behaviour Dynamics @ T=0  basic properties of quantum systems

4. Basic facts Lanczos method Target states Extended operators Applications Other methods Application to the Dynamical Mean Field Theory Conclusions OUTLINE K. H., PRB 52, 9827 (1995)

5. BASIC FACTS We want to calculate the following dynamical correlation function: Fourier transforming: Subspace A Subspace B

6. BASIC FACTS Defining the Green’s function: The correlation in frequency space reads: Where  defines casuality and a finite broadening of the peaks

7. BASIC FACTS We could diagonalize H and obtain the Green’s function as: And the dynamical correlation function as:

8. Lanczos dynamics The Green’s function: Can be written as follows: GA=GA=  0 |A † A|  0  z-a 0 - b12b12 where z-a 1 - b22b22 z-a 2 - …….

9. Lanczos dynamics In the Krylov |f n  basis H is tridiagonal: (with rescaled coefficients a n and b n )

10. Lanczos dynamics: target states Target states (TS): a) some eigenstates |n  b) some Lanczos vectors |f n  Relative importance of these: e.g. for the S=1 Heisenberg model, where A=S + q at q=  3 TS have 98% weight S=1/2 Heisenberg model, instead, it is only 28% Kühner and White (99) where

11. Lanczos dynamics: target states L=320 m=128 (S=1) m=256 (S=1/2) Weight of the target states at q= 

12. Lanczos dynamics: extended operators And extended operators like S q ? N=28, 44, 60, 72 (pbc)

13. Lanczos dynamics: extended operators Filter for open BC: smooths Fourier transform, for example: Kühner and White (99)

14. Lanczos dynamics: precision Some considerations: Higher precision: Local operators Open boundary conditions Finite-size DMRG Checks: Sum rules for momenta, e.g.:  0 lim  0

15. Lanczos dynamics: applications Some applications: Spin chain structure factors (K.H., 1995; Kühner and White 1999) The spin-boson model (Nishiyama, 1999) Spin-orbital chains (Yu et al, 2000) General spin chain dispersion relations (Okunishi et al, 2001) Dynamics of spin ladders (Nunner et al, 2002) Spectral functions in the U  Hubbard model (Penc et al, 1996) Critical behaviour of spin chains (K.H. et al, 1996) Optical response in 1D Mott insulators (Kancharla et al, 2001) Impurity-solver in the DMFT method (Garcia et al, 2004)

16. Lanczos dynamics: examples AFM S=1/2 Heisenberg model q= , N=28 and 40 Hallberg (95) and Kühner and White (99) lower spinon line

17. Lanczos dynamics: examples AFM S=1 Heisenberg model (single magnon line) Kühner and White (99) Truncation error: 1 st peak’s weight

18. Lanczos dynamics: applications Spin chain dispersion relations (Okunishi et al., 2001) BLBQ S=1 spin chain Heisenberg model:  =0 VBS chain:  =1/3 Relationship between dispersion relation and correlation length for gapped spin chains  6.03 for  =0 and 0.92 for  =1/3

19. Lanczos dynamics: applications The S=1/2 zig-zag ladder:  5.71 for  =0.48 and 4.35 for  =0.6, confirmed with  from static correlation functions

20. Lanczos dynamics: applications Critical behaviour of spin chains, e.g. S=3/2 Heisenberg model (K.H., X. Wang, P. Horsch, A. Moreo, PRL 76, 4955 (1996) Spin velocity v:  q = E(2  /N)-E(0)=v sin(2  /N) v=3.87  0.02 =1.28v sw experimental value in CsVCl 3 : v=1.26v sw v sw =2S=3

21. Lanczos dynamics: applications Cyclic spin exchange in cuprate ladders (Nunner et al, 2002) Lowest excitation behaviour  strong reduction of the dispersion of the S=0 bound-triplet excitation with J cyc. Good agreement with experiments. 1 and 2 triplet excitations

22. Lanczos dynamics: applications

23. Spectral functions for the U  Hubbard model (Penc et al., 1996)

24. Lanczos dynamics: applications where the lower Hubbard band spectra are: charge part spin part

25. Lanczos dynamics: applications k F =  /4 empty band full band Shadow bands

26. Correction vector dynamics Target: a particular energy z=  +i  So that the Green’s function is a product of two vectors: where Use as target states:

27. Correction vector dynamics z=  +i  The corrector vector |x(z)  is complex: Multiplying and dividing by (  +i  -H) we obtain: and (Ramasesha et al., 1989 & succ.; Kühner and White, 1999; Jeckelmann, 2002)

28. Analogy between DMFT and conventional MFT Lanczos dynamics: application to DMFT Hubbard model Ising model D. Garcia, K. H. and M. Rozenberg, cond-mat/0403169

29. Lanczos dynamics: application to DMFT DMFT  mapping of the original Hubbard model onto a SIAM “in a self-consistent bath” Hibridization for the Bethe lattice:  (  )=t 2 G(  ) Where G(  ): impurity Green’s function At the self-consistent point, G(  ) coincides with the local G of the original model

30. Lanczos dynamics: application to DMFT Several numerical impurity-solver methods: Quantum Monte Carlo (needs analytic continuation for real  ) Wilson’s NRG (precise at low  ) Exact Diagonalization for small effective 1D chains

31. Lanczos dynamics: application to DMFT a<na<n a>na>n b<nb<n b>nb>n One way of solving the impurity: G 0 (  )= 1 +-()+-() = 1  +  -t 2 G(  )

32. Lanczos dynamics: application to DMFT Self-consistent equations: We will use DMRG G 0 (  )= 1 +-()+-()

33. Lanczos dynamics: application to DMFT MR, XY Zhang & G Kotliar, PRB ‘94 A Georges & W Krauth ‘93

34. Lanczos dynamics: application to DMFT metallic insulating IPT DMRG metallic insulating MC

35. Lanczos dynamics: application to DMFT Important finite-size behaviour: “Kondo physics” in finite systems

36. Lanczos dynamics: application to DMFT U=2.5 NRG IPT Band substructure:

37. CONCLUSIONS : Several ways of calculating dynamics within DMRG With Lanczos, one can obtain a broad portion of the spectra which is reliable especially for the low-lying states (first excitation peaks) It has been applied to several systems Application to DMFT: it’s one way of solving the “impurity” part which leads to the self-consistent Hamiltonian directly without approximations (except for the truncation at finite a n and b n ) Improvement: complete the continued fraction for less finite system “structure”