1. F.F. Assaad. MPI-Stuttgart. Universität-Stuttgart. 21.10.2002 Numerical approaches to the correlated electron problem: Quantum Monte Carlo.  The Monte Carlo method. Basic.  Spin Systems. World-lines, loops and stochastic series expansions.  The auxiliary field method I  The auxiliary filed method II  Special topics Magnetic impurities Kondo lattices. Metal-Insulator transition

2. One magnetic impurity.  Cu with Fe as impurity.  Fe: 3d 6 4s 2 Hunds rule: S=2 Inverse susceptibility. Free spin. Screened spin. Temperature. Temperature Resistivity. Resistivity minimum. (Normal: a + bT 2 ) Kondo problem: crosover from free to screened impurity spin. Many body non-perturbative problem.

3. Anderson and Kondo models. Conduction electrons. t : creates e - on extended orbital.   UfUf   -Uf-Uf U f : inhibits charge fluctuations. Spin is still active. Coulomb repulsion on localized orbital. + The Kondo limit. V in perturbation. Magnetic scale: UfUf VV + No charge fluctuations on localized-orbital. V Impurity. : creates e - on localized orbital. +

4. The Kondo problem is a many-body problem Impurity spin. Electrons. p k P´ k Spin-flip scattering of p P´ k Spin of p is conserved The scattering of electron k will depend on how electron p scattered. Thus, the impurity spin is a source of correlations between conduction electrons. + k can spin-flip scatter k cannot spin-flip scatter

5. Ground state at J/t >>1 + T <T K T >T K Dynamical f-spin structure factor Ground state:  Spin singlet  J/t = is relevant fixpoint. Wilson (1975) Numerical (Hirsch-Fye impurity algorithm): T/T K J/t = 1.2 J/t = 1.6 J/t = 2.0 T   T K /t 0.21 T K /t 0.06 T K /t 0.12 is the only low energy scale T>>J: Essentially free impurity spin.

6. Lattices of magnetic impurities. Periodic Anderson model (PAM). Kondo lattice model (KLM). Charge fluctuations on f-sites. Charge fluctuations on f-sites frozen. Conduction orbitals: Impurity orbitals:

7. Simulations of the Kondo lattice. Consider: We can simulate this Hamiltonian for all band fillings. No constraint on Hilbert space. How does H relate to the Kondo lattice? Conservation law: Chose so that Let P 0 be projection on Hilbert space with : Then:

8. Mean-field for Kondo lattice. Order parameters: Decoupling: Two energy scales: J/t = 4, =0.5 CvCv T/t TKTK Below T K r>0. Same as for single impurity. T coh Below T coh Fermi liquid T coh TKTK (S. Burdin, A. Georges, D.R. Grempel et al. PRL 01) Mean field Hamiltonian (paramagnetic) Saddle point. Exact for the SU(N) model at N

9. (Ce x La 1-x ) Pb 3 X=1 X=0.6 Crossover to HF state. Finite Temperature: Coherence. Single impurity like E(k) k Ground state (Mean-field). Luttinger volume: n c +1 ZkZk Fermi liquid with large mass or small coherence temperature. Mean-Field Problems:, magnetism, finite T. J/t=2 J/t = 4  tt Periodic table of elements

10. I. Coherence. (FFA. PRB 02) Note: Conduction band is half-filled and particle hole symmetry is present. Allows sign-free QMC simulations but leads to nesting.  At T=0 magnetic insulator. Strong coupling. Fermi line Brillouin zone. Technical constraint: Conduction band has to be half-filled. Otherwise sign problem. Model. Conduction band:Half-filled.

11. T/t = 1/20 T/t = 1/30 T/t = 1/15 T/t = 1/60 T/t = 1/2 T/t = 1/5 T/t = 1/10 tt T/t Optical conductivity  Resistivity Optical conductivity and resistivity. J/t = 1.6 Single impurity like Coherence.  cm -1  Temperature Resistivity Schlabitz et al. 86. Degiorgi et al. 97 L=8 L=6 (L=8: 320 orbitals.)

12. Resistivity T/t Thermodynamics: J/t = 1.6 L=6 L=8 T/t T S T* Scales as a function of J/t. J/t T/t T* TsTs T min /2

13. T/t Specific heat.  c  s L=6 L=4 T S J/t = 0.8 1/8 1/10 1/15 1/20 1/30 1/50 1/80 T/t Resistivity tt  Scales as a function of J/t. J/t T/t T* TsTs T min /2

14. Comparison T* with T K of single impurity probem. TsTs J/t Depleted Kondo lattice. TKTK T*T* T * T K Crossover to the coherent heavy fermion state is set by the single impurity Kondo temperature. Note: Ce x La 1-x Cu 6 T* ~ 5-12 K for x: 0.73-1. T K ~ 3K (Sumiyawa et al. JPSJ 86) T/t  T coh ?  No magnetic order-disorder transition since strong coupling metallic state is unstable towards magnetic ordering.

15. CePd 2 Si 2 (J.D Mathur et al. Nature 98.) [ See also CeCu 6-x Au x ] RKKY J II Magnetism : Order-disorder transitions RKKY Interaction Kondo Effect. T K ~ e -t/J Energy scale Spin susceptibility of conduction electrons. Competition RKKY / Kondo leads to quantum phase transitions.

16. Half-filled Kondo lattice. One conduction electron per impurity spin. (FFA PRL. 99) Model Strong coupling limit. J/t >> 1 Spin Singlet 1) Spin gap Energy J ss 2) Quasiparticle gap. Energy 3J/4  qp QMC, T=0, L m > 0, Q=( ,  ): long range antiferromagnetic order. 3) Magnetism. ( m f ) 2 = 1D (Tsunetsugu et. al. RMP 97)

17. Spin Dynamics: S(q, ) Fit: Perturbation in t/J. Fit: Spin waves. Excitations of disordered phase condense to form the order of the ordered state. Bond mean-field of Kondo necklace (G.M. Zhang et. al. PRB 00). (S. Capponi, FFA PRB 01)

18. Single particle spectral function. A(k, Fit: Strong coupling. Weak coupling ? (S. Capponi, FFA PRB 01)

19. JcJc TKTK msms f-Spins are frozen. a) E(k) Magnetic BZ. JcJc TKTK msms b) Partial Kondo screening, remnant magnetic moment orders. (M. Feldbacher, C. Jureka, F.F.A., W. Brenig PRB submitted.)

20. J < 0 No Kondo effect. Single particle spectral function. A(k, Fit: Strong coupling.  In ordered phase impurity spins are partially screened. Remnant moment orders. Mean-field interpretation: Coexistence of Kondo screening and magnetism. (Zhang and Yu PRB 00) (S. Capponi, FFA PRB 01)

21.      /t k,      /t k,/t= t L = 4 L = 6 L = 10 T S ~ J 2 Origin of quasiparticle gap at weak couplings. J/t = 0.8 Quasiparticle gap of order J is of magnetic origin at J < J c ~ 1.5 t

22. Conclusions.  QMC algorithm for Kondo lattices.  Restriction. Particle-hole symmetric conduction bands.  Depleted lattices.  T * T K  Half-filled Kondo lattice in 2D.  Pairing. No.

23. II.Doped Mott insulators. MPI-Stuttgart. Universität-Stuttgart.

24. Metal Half filling: Insulator. Scale U Charge is localized. Internal degree of freedom (spin) is still active. t Hubbard Model. U U Strong coupling U/t >>1 (Half filling) t t Magnetic scale: Heisenberg Model.

25. The Mott Insulator. Half filling (2D,T=0) Charge. Quasiparticle gap > 0 F.F. Assaad M. Imada JPSJ 95. Spin. Long range magnetic order. Goldstone mode: Spin-waves.

26.  Mott Insulator U/W Bandwidth W.  -(BEDT-TTF) 2 CU[N(CN) 2 ]C (2D) V 2 O 3 (3D) F.F. Assaad, M. Imada und D.J. Scalapino Phys. Rev. Lett. 77, 4592, (1996) The Metal-Insulator Transition. Metal Doping Cuprates. (2D) [ (La Nd) 2-x Sr x Cu O 4 ] Superconductivity-Stripes. Titanates (3D) (La x Sr 1-x Ti O 3 ) F.F. Assaad und M. Imada Phys. Rev. Lett. 76, 3176, (1996). Phys. Rev. Lett. 74, 3868, (1995).

27. How can we avoid the sign problem? N = 4 n. No sign problem irrespective of lattice topology and doping. N=2: H N=2 = Hubbard N = Mean-field N > 2 Symmetry: SU(N/2) SU(N/2) Orbital Picture. Elementary Cell NZ. 2 4 6 8 0 2 3 1 N N/2-1

28. N = : SDW Mean field. so that with Langevin: More Formal.

29. 1/N E(N)/E(N=2) D()(N)/D()(N=2) T=0, 4 X 4, U/t = 4, 2 Löcher. Lanczos. Mean-field. F.F. Assaad et al. PRL submitted. Test. Note: =1, U/t >> 1

30. Single particle: N=4, T=0, U/t=3. 2D. ()/t N() =0, 30X8 t/U=0 N() =0,  = 0: U/2 -U/2 L=6 L L  0 N() =1/6,  = -U/2: U L=6 L-1 2  0 ()/t N() =1/14, 30X8 =1/5, 30X8, 30x12

31. Spin, S(q), charge, N(q), Structure Factors. Real space (caricature). Disctance between walls: 1/   =1/5  =0  Phase-shift in Spin Structure. One dimensional    /2   S(q) N(q) N=4, T=0, U/t =3, 60X1, =1/5  4k f  2k f N=4, T=0, U/t =3, 30X8 (/2,) (,)(,) (,/2) S(q) = 0 = 1/14 = 1/7 = 1/5 = 1/4 (,)(,) () (,)(,) N=4, T=0, U/t =3, 30X8 N(q) = 1/14= 1/7 = 1/5 = 1/4 (  ) ( x,) Spin. (0,0) Charge. (2 x,0)  x = 

32. Spin-and charge-Dynamics at  =0.2 (T=0,N=4,U/t=3) qxqx qxqx   60X1 30X4 30X8 30X12 First charge-excitation at q=(q x,0) First Spin-excitation at q=(q x,  ) Optical conductivity: 30 X 8,  =0.2 Ohne Vertex Mit Vertex  /t N(q,  ): Dynamical charge Structure factor. Transport L y >4: Particle-hole continuum.

33. S(q) N(q) (  ) ( x,) ( y ) (0,0) (2 x,0) (0,2 y ) Charge. Spin. Two dimensions L y =10, L x =30,  = 0.2 Two-dimensional metallic with no quasiparticles. Elementary excitations: spin and charge collective modes. qyqy qyqy qxqx qxqx

34. Interpretation of collective modes. 1) Analogy to 1D ? 2) Goldstone Modes. a) SU(2) SU(2) Symmetry is not broken.  Energy is invariant under Translation:   b) Phasons.