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					Methods for electronic structure calculations
    with dynamical mean field theory:
  An overview and recent developments




                   Ryotaro ARITA (RIKEN)
                Thanks to …

 S. Sakai (Dept. Applied Phys. Univ. Tokyo)
        H. Aoki (Dept. Phys. Univ. Tokyo)
       K. Held (Max Planck Inst. Stuttgart)
A. V. Lukoyanov (Ural State Technical Univ.)
V. I. Anisimov (Inst. Metal Phys, Ekaterinburg)
Outline
    Introduction
      LDA+DMFT
      Various solvers for DMFT
           IPT, NCA, ED, NRG, DDMRG, QMC, …
    Conventional QMC (Hirsch-Fye 86)
      Algorithm
      Problems
           numerically expensive for low T:
                              numerical effort ~ 1/T3
           sign problem in multi-orbital systems:
                               difficult to treat spin flip terms
    New QMC algorithms
      Projective QMC for T→0 calculations
                (Feldbacher et al 04, Application: Arita et al 07)
      Application of various perturbation series expansions for Z
                 (Sakai et al 06, Rubtsov et al 05, Werner et al 07)



                                                                       R.Arita
LDA+DMFT                              Dr Aryasetiawan July 25, Prof. Savrasov July 27


     Anisimov et al 97,
Lichtenstein, Katsnelson 98



     DFT/LDA                                                         Model Hamiltonians

      material specific, ab initio                          systematic many-body approach
      fails for strong correlations                         input parameters unknown




                           Computational scheme for correlated
                                   electron materials
                                                                                        R.Arita
LDA+DMFT
Application to various correlated materials
                              (reviews) Held et al 03, Kotliar et al 06, etc

   Transition metal oxides                    Organic compounds
     LaTiO3                                     BEDT-TTF
     V2O3, VO2                                  TMTSF
     (Sr,Ca)VO3                               Fullerenes
     LiV2O4                                   Nanostructure materials
     (Sr,Ca)2RuO4                               Zeolites
     NaxCoO2                                  f-electron systems
     Cuprates                                   Rare earths: Ce
     Manganites                                 Actinides: Pu
     …                                          …
   Transition metals
     Fe, Ni
   Heussler alloys


                                                                               R.Arita
LDA+DMFT
    Downfolding: LDA → effective low-energy Hamiltonian




    Expand Ψ+ w.r.t. a localized basis Φilm :




    Supplementing LDA with local Coulomb interactions




                                                           R.Arita
LDA+DMFT

 Solve model by DMFT                    Metzner & Vollhardt 89, Georges & Kotliar 92

   Lattice model:
   DOS D ( ) Self Energy lat (k , in )
   Hlat  t  ci†, c j ,  U  ni ni     ni  ni 
              i, j              i               i



   Effective impurity model:
   Hybridization F   Self Energy  imp (in )
   S    d d 'c ( ) F (   ')c ( ')   d ( (n  n )  Unn )
                                        †

          



   Self-consistency:

    F  imp (in )
                                  D( )                                       F
    Glatt (in )   d                                       1
                                                    F 1  Glatt  imp
                         in      imp (in )


                                                                                        R.Arita
Solvers for the DMFT impurity model




          Iterated perturbation theory
              Perturbation expansion in U
          Non-crossing approximation
              Perturbation expansion in V
          Exact diagonalization for small number of host sites
              Max # of orbitals <2
          Numerical renormalization group
           (logarithmic discretization of host spectrum)
               Max # of orbitals <2
          Dynamical density matrix renormalization group
          Quantum Monte Carlo
          …

                                                                  R.Arita
Auxiliary-field QMC
 Suzuki-Trotter decomposition
                                            L
       Z  Tr e            H
                                 =Tr  e  H 0 e  Hint                      (  1/ T  L )
                                            l 1

 Hubbard-Stratonovich transformation for Hint
            U [ n n  1 ( n  n )]                        s ( n n )
       e               2             
                                             1 e
                                              2                                  (cosh( )  exp[U/2])
                                                    s 1

 Many-particle system
       =                (free one-particle system + auxiliary field)


                                                           

              
                                                                                   L
     Z                  Z s1s2 ...sL              Z s s ...s 
                                                     1 2    L
                                                                     1
                                                                    2L    Tr [exp( H            
                                                                                                     0
                                                                                                         ) exp( sl n )]
            s1s2 ...sL                                                           l 1

                                 Z s1s2 ...sL
      A           
                  s1s2 ... sL           Z
                                                    A s1s2 ...sL               Monte Carlo sampling
                                                                                                                             R.Arita
QMC for the Anderson impurity model                    ( Hirsch-Fye 86 )




                  Integrate out the conduction bands



   Calculate                                          0<1,2<1/T, =L


      G0(1,2)      G{s}(1,2), w{s}         G{s}(1,2), w{s}      …

                                         Updating: numerical effort ~L2




                                                                             R.Arita
Problems & Recent developments

   Numerically expensive for low T: numerical effort ~ 1/T3
       Projective QMC (Feldbacher et al 04): A new route to T→0


   Sign problem in multi-orbital systems: difficult to treat spin flip terms


                                                 Zs can be negative:
                                                 Norm can be small
                                     norm        → <A>=0/0


       Application of various perturbation series expansions (Rombouts et al, 99):
                           less severe sign problem
         Combination with HF algorithm (Sakai et al, 06)
         Continuous time QMC
             weak coupling expansion (Rubtsov et al, 05)
             hybridization expansion (Werner et al, 06)
                                                                                R.Arita
Projective QMC and its application to
          DMFT calculation
Projective QMC            Feldbacher et al, PRL 93 136405(2004)


 Conventional QMC                Projective QMC




 • Thermal fluctuations
 • effort: ~1/T3




                                       Interaction
                                       Ising fields       no interaction
                                                                           
                                   0                                
                                                      →∞             →∞



                                                                               R.Arita
Projective QMC
                            Tre   H 0 e  H / 2Oe  H / 2           O  c( 1 )c † ( 2 )  e1H ce  (1  2 ) H c †e  2 H
        O            lim
            T 0               Tre   H 0 e  H

                                                       Tre   H 0 e  H / 21H ce  (1  2 ) H c †e  2 H  H / 2
                                G ( 1 , 2 )   lim
                                                                         Tre   H 0 e  H

 -/2                     /2                                   /2+




                                                                                  Interaction U only in red part

                                                                                  for sufficiently large P:
                                                                                  Accurate information on
                                                                                  G for light red part




                                                                                                                                       R.Arita
   Application of PQMC to DMFT (1)

        DMFT self-consistent loop                           Maximum Entropy Method
                                                                      1
                  PQMC
                                                            G( ) 
                                                                          A( ) exp( )d
                                                                          1    A( )
                                                                            in  
G0 ( 1 , 2 )                     G ( 1 , 2 ) (T=0)      G (in )                 d
 (T=0)                   (i )  G01 (i )  G 1 (i )


                               D( )
      G (i )   d 
                        in    (in )  
           G01 (i )  G 1 (i )  (i )



    Problem: How to obtain (i)?                                                P
         G()→FT→G(i)? No
   only G(),<P obtained by PQMC                          Calculate G only for <P
                                                            Large : Extrapolation by
                                                            Maximum Entropy Method


                                                                                                R.Arita
Application of PQMC to DMFT (2)


                Single band Hubbard model




                       M           I


              HF-QMC

                           =16             =40




                 insulating         metallic



                                                   R.Arita
Application of PQMC to DMFT (2)


                 Single band Hubbard model




                        M                I


               PQMC
                            =16                =40




                                                           Application to
                  Metallic solution obtained for =16       LDA+DMFT
             (same numerical effort as HF-QMC with =16)      at T→0

                                                                      R.Arita
Application of PQMC to LDA+DMFT
              for LiV2O4




            RA-Held-Lukoyanov-Anisimov
               PRL 98 166402 (2007)
LiV2O4: 3d heavy Fermion system




                                    Crossover at T*~20K
     FL(T2 law)
                                    ・ resistivity:  =0+AT2 with an enhanced A

                                    ・ specific heat coefficient: anomalously large
                                         g(T→0)~190mJ/V mol・K2
    g(T→0)~190mJ/Vmol・K2
                                          cf) CeRu2Si2 ~350mJ/Ce mol・K2
                                              UPt3    ~420mJ/U mol・K2
                                         (Kadowaki-Woods relation satisfied)
          CW law at HT
          S=1/2 per V ion           ・ c: broad maximum (Wilson ratio~1.8)
     T*
                                     heavy mass quasiparticles (m*~25mLDA)
 (Urano et al. PRL85, 1052(2000))

                                                                                  R.Arita
 LiV2O4: 3d heavy Fermion system
     PhotoEmission Spectroscopy                  LDA+DMFT(HF-QMC)
                                           (Nekrasov et al, PRB 67 085111 (2003))
(Shimoyamada et al. PRL 96 026403(2006))

                                             T=750K




         A sharp peak appears
              for T<26K
                                                  LDA+DMFT(PQMC)
          =4meV, ~10meV

                                                                              R.Arita
Results
                U     U’         U’-J   (Hund coupling = Ising)

          U=3.6, U’=2.4, J=0.6


          a1g        PQMC
                     T=300K
                      T=1200K
                       T=300K




          eg




                                                                  R.Arita
 FAQ
           Why can we discuss A(→0) without calculating G(→∞) explicitly?

            Large T                                         T→0




                                                 A()
 A()




        0                  0                           0                0
G()




             ~exp(-00)                        G()
                                                              Slow-decay component



       0                                               0
                                                                                   R.Arita
Results: G() & A()




                       R.Arita
            Application of
perturbation series expansions to QMC

・Combination with Hirsch-Fye’s algorithm
                (Sakai, RA, Held, Aoki PRB 74 155102 (2006))


・Continuous time QMC
    weak coupling expansion
    (Rubtsov et al, JETP Lett 80 61 (2004), PRB 72 035122 (2005))
    hybridization expansion
    (Werner et al, PRL 97 076405 (2006), PRB 74 155107 (2006))
QMC for multi-orbital systems




                    -J                                             -J


                                 HJ : usually neglected
                 sign problem
                 difficult to treat for multi-orbital systems
              ˆ      ˆ 23 ˆ 31
              H 12  H J  H J        ˆ          ˆ 23       ˆ 31      ˆ      ˆ 23
          e     J
                                 e   H 12
                                        J
                                             e   HJ
                                                        e   HJ
                                                                    [ H 12 , H J ]  0
                                                                        J


                  ⇒ Non-trivial Suzuki-Trotter decomposition?

                                                                                         R.Arita
Ising-type vs Heisenberg-type interaction

      DMFT study for ferromagnetism in the 2-band Hubbard model

       n=1.25, Bethe-lattice,
     W=4, U=9, U’=5, J=2 (Ising)




                   J
                                               Ising-type couling:
             Held-Vollhardt, 98       Ferromagnetic instability overestimated

                                            Sakai, RA, Held, Aoki 06


                                                                          R.Arita
PSE + Hirsch-Fye QMC              Sakai, RA, Held, Aoki PRB 74 155102 (2006)

 PSE with respect to V (V: interaction term) (Rombouts et al, 99)




                                                           Same Algorithm as
                                                           Hirsch-Fye




 For spin flip & pair hopping term:



 extention to m>2 straightforward:
                                                                               R.Arita
PSE + Hirsch-Fye QMC                                            Sakai, RA, Held, Aoki PRB 74 155102 (2006)


Large U,U’,J                         <k> becomes large                Large L needed
                                                   2
e     H 0  (   V )
                             (  ) k  d k  d 1e1H 0 (1   V )e  ( 2 1 ) H 0  (1   V )e  (   k ) H 0
                                 
                                                                                                   
                                          0          0
                              k 0

It is not a good idea to treat all U,U’,J terms as V

H0+HU+HU‘+HIsing≡ H0+H1 → standard HF                                              e ( H0  H1 )  e H0 e H1
HJ → PSE (<k> is small for HJ)


                      2-band Hubbard model, n=1.9, =8, U=4.4, U‘=4, J=0.2, W=2
                                                  PSE only                                              PSE+HF
                                                                          Nk
                 Nk




                       0                60                120                  0              40              80
                                                                                                                              R.Arita
PSE + Hirsch-Fye QMC               Sakai, RA, Held, Aoki PRB 74 155102 (2006)


   Sign problem: less severe                Conventional HF:
                                                             s 1


    Wide region of norm >0.01
                                             e    H
                                                             e
                                                             s1 sL
                                                                         H 0
                                                                                  Qs1 e H0 Qs2  e H0 QsL

  2-band, n=2, W=2, U=U’+2J, U’=4
                                                                                                            

                                            We have to consider sn=±1 for every n,

                                            PSE+HF:
                                                           0,1
                    (Sakai et al 04)         e    H
                                                          e H0 Qs1 e H0 Qs2  e H0 QsL
                                                          s1 sL


                                                                                                                
                                             For small HJ, small number of n have sn≠0

Lower T, large J can be explored
                                            Expansion with respect to HJ :
                                            ~ <HJ> negative sign problem relaxed


                                                                                                                    R.Arita
Application to LDA+DMFT calculation for Sr2RuO4

   Ising-type Hund, =70                         SU(2) Hund + pair hopping, =40
             U=1.2, U’=0.8, J=0.2 [eV]
                                             1
                                                                       dxy


                                                              dyz/zx




                                             0-3        -2       -1          0       1
                                                              Energy [eV]
[Liebsch-Lichtenstein, PRL 84,1591 (2000)]

                                                   SU(2) symmetric 3-band LDA+DMFT




                                                                                 R.Arita
Continuous time QMC
 Weak coupling expansion:                 Rubtsov et al, JETP Lett 80 61 (2004),
                                          PRB 72 035122 (2005)


                                                                                                               r   , i, s
 S   t c c drdr '   w
          r' † r
          r r'
                                          r '1 r '2 † r1 †
                                          r1r2
                                                                         r2
                                                   c c c c dr1dr '1 dr2 dr '2
                                                    r '1 r '2
                                                                                                                     
                                                                                                             dr   d 
                                       Non-local in time & space                                                     0        i     s
                              
  Z  Tr T exp( S )    dr1  dr '1                          dr  dr '
                                                                    2k               2k   k (r1 , r '1 , , r2 k , r '2 k )
                             k 0

          (1) k r '1 r '2
 k  Z 0       wr1r2        wrr2'k2k1r12rk '2 k Tcr†'1 c r1      cr†'2 k c r2 k
            k!

   Perform a random walk in the space of K={k, (arguments of integrals)}




                  (cf. K={auxiliary spins} for Hirsch-Fye scheme)

                                                                                                                                  R.Arita
Applications

 LDA+DMFT study for V2O3
 (Ising type of Hund coupling)




                                 Poteryaev et al, cond-mat/0701263

 Correlated Adatom Trimer
    on a Metal Surface




                                 Savkin et al, PRL 94 026402 (2005) R.Arita
Continuous time QMC (2)
 Hybridization expansion: Werner et al, PRL 97 076405 (2006),
                                     PRB 74 155107 (2006)
 Z  Tr T exp(S )               S  S0  S1
                                                               
 S0    d    b  U
                    ab   †
                         a
                                  abcd
                                           c d  S1   d d ' a ( ) a (   ') a ( ')
                                          †
                                          a
                                              †
                                              b
                                                                                           †

               0                                                0

                                                                        Impurity-bath hybridization


                     =100
 Matrix size




                                                    Numerical effort decreases
                   (~5U)                           with increasing U
                                                    Allows access to low T, even
                             (~0.5U)               at large U




                             U
                                                                                                R.Arita
Summary


   QMC: A powerful tool for LDA+DMFT, but
     low T not accessible
     sign problem in multi-orbital systems
     …

   Recent developments
     Access to low T, strong coupling, multi-orbital systems
        Projective QMC for T→0 calculations
        Application of various perturbation series expansions for Z

   Future Problems
     Spatial fluctuations (cluster extensions)
     Coupling to bosonic baths
     …


                                                                       R.Arita

				
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