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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 Ground state

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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 Ground state Finite temperature Hirsch-Fye. • Special topics (Kondo / Metal-Insulator transition) and outlooks. 21.10.2002 Universität-Stuttgart. MPI-Stuttgart.

  2. Hubbard 6X6 b / t Hubbard. Ground state. Ground state method:CPU V3b Finite temperature. Finite temperature: CPU V3b

  3. The choice of the trial wave function for the Projector method.

  4. Electronic system: X-Y plane. Scaling: Magnetic fields and size effects.

  5. T T Thermodynamic quantities. L = 4,6,...16 L = 4,6,...16 Cv/T T T L = 4,6,...16 L = 4,6,. ..16 L=16: More than an order of magnitude gain in temperature before results get dominated by size effects. FFA PRB 02

  6. I. Basic formalism for the case of the Hubbard model. Magnetic field in z-direction: Trotter. Ground state: Finite temperature:

  7. Hubbard. Breaks SU(2) spin symmetry. Symmetry is restored after summation over HS. Fields. Complex but conservs SU(2) spin symmetry. with: The choice of Hubbard Stratonovich transformation. (Decouples many body propagator into sum of single particle propagator interacting with extermal field.) Generic.

  8. Properties of Slater Determinants. (1) Propagation of a Slater determinant with single body operator remains a Slater determinant. (2) Overlap: (3) Trace over the Fock space:

  9. Ground state. Trial wave function is slater determinant: P is N x Np matrix. Finite temperature.

  10. Observables ground state. For a given HS configuration Wick‘s theorem holds. Thus is suffices to compute Green functions. Observables finite temperature.

  11. Wick´s Theorm

  12. Upgrading, single spin flip. so that This form holds for both the finite temperature and ground state algorithm! Thus the Green function is the central quantity. It allows calculation of observables and determines the Monte Carlo dynamics. Same is valid in the finite temperature approach. If the spin flip is accepted, we will have to upgrade the equal time Green function Upgrading of the Green function is based on the Sherman Morrison formula. Outer product.

  13. II. Comments. (A) Numerical stabilzation T=0. Gram Schmidt. Similarly: • . Green functions remains invariant. Since the algorithm depends only on the equal time Green function everything remains invariant!

  14. The Gram Schmidt orthogonalization.

  15. Numerical stabilization finite T. You cannot throw away scales. Use: Diagonal elements are equal time Green functions. As we will see later, the off-diagonal elements correspond to time displaced Green functions. To calculate the B matrices without mixing scales use.

  16. The inversion.

  17. Measuring time displaced Green functions. (a) Finite temperature. Note:

  18. Thus we have: But we have already calculated the time displace Green functions. See Eq 81.

  19. Time displaced Green functions for ground state (projector) algorithm.

  20. Consider the free electron case: so that

  21. L=8, t-U-W model: W/t =0.35, U/t=2, <n>=1, T=0 (C) Imaginary time displaced correlation functions. • . Gaps. Dynamics (MaxEnt). SU(2) invariant code. SU(2) non-invariant code. SU(2) non-invariant code. Note: Same CPU time for both simulations. tt

  22. (B) Sign problem. <sign> 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 <n> b) Attractive Hubbard, U<0. is real for all band fillings (no magntic field.) General: Models with attractive interactions which couples independently to an internal symmetry with an even number of states leads to no sign problem. a) Repulsive Hubbard. Away from half-filling. Half-filling. Particle-hole symmetry: is real even in the presence of a magnetic field. U/t = 4, 6 X 6

  23. TK/t 0.21 T cI J/t = 1.2 J/t = 1.6 J/t = 2.0 TK/t 0.06 T/TK TK/t 0.12 Dynamical f-spin structure factor J/t=2 T <TK T >TK Impurity models such as Anderson or Kondo model (Hirsch-Fye). No charge fluctuations on f-sites. Numerical (Hirsch-Fye impurity algorithm) CPU:V0(Nimpb)3 is the only low energy scale

  24. The Hirsch Fye Impurity Algorithm.

  25. Single impurity (Hirsch-Fye algorithm) TK/t 0.21 T cI J/t = 1.2 J/t = 1.6 J/t = 2.0 TK/t 0.06 T/TK TK/t 0.12 (IV) Related algorithm: Hirsch Fye Impurity Algorithm. Finite temperature: • We only need Green functions on f-sites • Upgrading (equal time). • Observables. G(s): f-Green function for HS s. (L x L matrix) With in say up spin sector. • Start with D(s) = 1. G(s) is the f-Green function of H0. Exact solution in thermodynamic limit. • From G(s) compute G(s´) at the expense of LxL matrix inversion • CPU time ~ L3 (i.e. b 3).

  26. Recall: 4x4 Hubbard: U/t = 8, <n>=0.625 Second order Trotter. Energy/t First order Trotter. Dt Exact: -17.510t, Extrapolated : -17.520(2)t (III) Approximate strategies to circumvent sign problem Assume that we know and for s´ then we can omit all paths evolving from this point since: But: We do not know ! Approximate it to impose constraint and method becomes approximative. CPQMC (Zhang, Gubernatis). Approximate by a single Slater determinant to impose constraint.

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