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

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Daniel Garcia Marcelo Rozenberg

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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

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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)

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

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

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

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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 - …….

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

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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

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

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

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

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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

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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)

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

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

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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

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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

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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

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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

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Lanczos dynamics: applications

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

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

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

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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:

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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)

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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

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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

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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

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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( )

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

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

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

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

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

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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”

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