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

Ground state method:CPU V 3  t Hubbard 6X6 Finite temperature: CPU V 3  Ground state. Finite temperature. Hubbard.

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

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

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

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

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

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

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

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

Wick´s Theorm

Upgrading, single spin flip. so that 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. This form holds for both the finite temperature and ground state algorithm! 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.

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

The Gram Schmidt orthogonalization.

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

The inversion.

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

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

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

Consider the free electron case: so that

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

(B) Sign problem U/t = 4, 6 X 6 a) Repulsive Hubbard. Particle-hole symmetry: is real even in the presence of a magnetic field. 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. Away from half-filling. Half-filling.

Impurity models such as Anderson or Kondo model (Hirsch-Fye). No charge fluctuations on f-sites. T <T K T >T K Dynamical f-spin structure factor J/t=2 Numerical (Hirsch-Fye impurity algorithm) CPU: V 0 (N imp  3 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

The Hirsch Fye Impurity Algorithm.

Finite temperature: We only need Green functions on f-sites (1)Upgrading (equal time). (2)Observables. G(s): f-Green function for HS s. (L x L matrix)  Start with D(s) = 1. G(s) is the f-Green function of H 0. Exact solution in thermodynamic limit.  From G(s) compute G(s´) at the expense of LxL matrix inversion  CPU time ~ L 3 (i.e.  3 ). With in say up spin sector. T/T K J/t = 1.2 J/t = 1.6 J/t = 2.0 T   Single impurity (Hirsch-Fye algorithm) T K /t 0.21 T K /t 0.06 T K /t 0.12 (IV) Related algorithm: Hirsch Fye Impurity Algorithm.

(III) Approximate strategies to circumvent sign problem Recall: 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. 4x4 Hubbard: U/t = 8, =0.625  Energy/t First order Trotter. Second order Trotter. Exact: t, Extrapolated : (2)t