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F.F. Assaad.

Numerical approaches to the correlated electron problem: Quantum Monte Carlo. F.F. Assaad. 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

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F.F. Assaad.

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  1. Numerical approaches to the correlated electron problem: Quantum Monte Carlo. F.F. Assaad. • 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 21.10.2002 Universität-Stuttgart. MPI-Stuttgart.

  2. Resistivity. Inverse susceptibility. Temperature. Temperature Resistivity minimum. (Normal: a + bT 2) Free spin. Screened spin. • One magnetic impurity. • Cu with Fe as impurity. • Fe: 3d 64s 2 Hunds rule: S=2 Kondo problem: crosover from free to screened impurity spin. Many body non-perturbative problem.

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

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

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

  6. 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: Let P0 be projection on Hilbert space with : Then: so that Chose

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

  8. X=0.6 Finite Temperature: Zk (CexLa 1-x ) Pb3 Crossover to HF state. Luttinger volume: nc+1 J/t = 4 J/t=2 s(w) w/t Coherence. Fermi liquid with large mass or small coherence temperature. Single impurity like X=1 Mean-Field Problems: , magnetism, finite T. Ground state (Mean-field). E(k) k Periodic table of elements

  9. Model. Strong coupling. Conduction band:Half-filled. Brillouin zone. Fermi line • 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. Note: (FFA. PRB 02) I. Coherence. Technical constraint: Conduction band has to be half-filled. Otherwise sign problem.

  10. T/t Schlabitz et al. 86. Resistivity T/t = 1/20 T/t = 1/30 T/t = 1/15 Temperature T/t = 1/60 Degiorgi et al. 97 T/t = 1/2 T/t = 1/5 T/t = 1/10 s(w) w/t 50 100 150 200 cm -1 Optical conductivity and resistivity. J/t = 1.6 Single impurity like Coherence. Resistivity L=8 L=6 Optical conductivity s(w) (L=8: 320 orbitals.)

  11. T* T S Resistivity T/t T S T* Scales as a function of J/t. Tmin/2 T* T/t Ts J/t Thermodynamics: J/t = 1.6 T/t L=6 L=8 T/t

  12. T S 1/8 1/10 1/15 1/20 T/t 1/30 c s 1/50 1/80 L=6 c L=4 c Scales as a function of J/t. Tmin/2 T* T/t Specific heat. Ts T/t J/t J/t = 0.8 Resistivity s(w) T/t w/t

  13. Depleted Kondo lattice. Crossover to the coherent heavy fermion state is set by the single impurity Kondo temperature. Note: CexLa1-xCu6 T* ~ 5-12 K for x: 0.73-1. TK ~ 3K (Sumiyawa et al. JPSJ 86) TK T*TK T/t T* Ts J/t Comparison T* with TK of single impurity probem. • Tcoh ? • No magnetic order-disorder transition since strong coupling metallic state is unstable towards magnetic ordering.

  14. CePd2Si2 (J.D Mathur et al. Nature 98.) Energy scale Spin susceptibility of conduction electrons. RKKY J [ See also CeCu6-xAux] II Magnetism : Order-disorder transitions RKKY Interaction Kondo Effect. TK ~ e-t/J Energy scale Competition RKKY / Kondo leads to quantum phase transitions.

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

  16. Fit: Spin waves. Fit: Perturbation in t/J. (S. Capponi, FFA PRB 01) Spin Dynamics:S(q,w ) 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).

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

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

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

  20. L = 4 L = 6 L = 10 A(k,w), T/t=1/12 A(k,w), T=0 (p,p) (p,p) T/t TS ~ J2 (0,0) (0,0) (0,p) (0,p) (p,p) (p,p) w/t w/t Origin of quasiparticle gap at weak couplings. J/t = 0.8 Quasiparticle gap of order J is of magnetic origin at J < Jc ~ 1.5 t

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

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

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

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

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

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

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

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

  29. d=1/5,30X8,30x12 d=1/14,30X8 N(w) N(w) (w -m)/t (w -m)/t d=1/6, m = -U/2: L=6 L-1 L-1 N(w) 2 w -m U 0 Single particle: N=4, T=0, U/t=3. 2D. d=0,30X8 N(w) (w -m)/t t/U=0 d=0, m = 0: L=6 L L N(w) w -m -U/2 0 U/2

  30. Spin, S(q), charge, N(q), Structure Factors. S(q) N(q) N=4, T=0, U/t =3, 30X8 N=4, T=0, U/t =3, 30X8 d= 0 d= 1/14 d= 1/7 d= 1/14 d= 1/7 d= 1/5 d= 1/4 d= 1/5 d= 1/4 (0,p) (p,0) (0,0) (p,p) (p,p/2) (p/2,p) Spin. Charge. (p -ex,p) (2ex,0) ex =p d (0,0) (p,p) Real space (caricature). One dimensional d=0 N=4, T=0, U/t =3, 60X1, d=1/5 p - p d = 2kf S(q) p Phase-shift in Spin Structure. p p/2 d=1/5 0 N(q) 2 p d = 4kf Disctance between walls: 1/d p p/2 0

  31. Ly >4: Particle-hole continuum. Transport Optical conductivity: 30 X 8, d=0.2 Ohne Vertex Mit Vertex N(q,w): Dynamical charge Structure factor. w/t Spin-and charge-Dynamics at d=0.2 (T=0,N=4,U/t=3) 60X1 30X4 30X8 30X12 First Spin-excitation at q=(qx,p) First charge-excitation at q=(qx,0) qx qx p - p d 2 p d

  32. Two dimensions Ly=10, Lx=30, d = 0.2 S(q) N(q) qx qx qy Spin. qy (p,p) Charge. (p -ex,p) (0,2ey) (p,p -ey) (2ex,0) (0,0) Two-dimensional metallic with no quasiparticles. Elementary excitations: spin and charge collective modes.

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

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