Calculation of dynamical properties using DMRG

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

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

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

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

8. Lanczos dynamics The Green’s function: Can be written as follows: 0|A†A|0 GA= b12 z-a0- b22 z-a1- z-a2- ……. where

9. Lanczos dynamics In the Krylov |fn basis H is tridiagonal: (with rescaled coefficients an and bn)

10. Lanczos dynamics: target states Target states (TS): a) some eigenstates |n b) some Lanczos vectors |fn Relative importance of these: where 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)

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

12. Lanczos dynamics: extended operators And extended operators like Sq? 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. q=, N=28 and 40 Lanczos dynamics: examples AFM S=1/2 Heisenberg model lower spinon line Hallberg (95) and Kühner and White (99)

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

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 Spin velocity v: q= E(2/N)-E(0)=v sin(2/N) v=3.870.02 =1.28vsw vsw=2S=3 experimental value in CsVCl3: v=1.26vsw (K.H., X. Wang, P. Horsch, A. Moreo, PRL 76, 4955 (1996)

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

22. Lanczos dynamics: applications

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

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

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

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. Lanczos dynamics: application to DMFT Analogy between DMFT and conventional MFT Ising model Hubbard 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: ()=t2G() 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. 1 = b<n b>n +-() a<n a>n Lanczos dynamics: application to DMFT One way of solving the impurity: 1 G0()= +-t2G()

32. 1 +-() Lanczos dynamics: application to DMFT Self-consistent equations: G0()= We will use DMRG

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 IPT insulating DMRG •MC metallic insulating

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

36. IPT NRG U=2.5 Lanczos dynamics: application to DMFT 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 • an and bn) • Improvement: complete the continued fraction for less • finite system “structure”