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.  Quantum Spin Systems. World-lines, loops and stochastic series expansions.  The auxiliary field method I  The auxiliary filed method II Ground state, finite temperature and Hirsch-Fye.  Special topics (Kondo / Metal-Insulator transition) and outlooks.

2. J J J JJ J J I The World line approach with Loop updates. Two site problem.States. Quantum Numbers. Triplet Singlet Hirsch et. al 81.

3. World Lines Reduce problem to a set of two site problems H 1 and H 2 are sums of commuting two site terms. Easy to solve. Trotter. Error of order Energy/J 4X4 Heisenberg.  is constant.

4. Imaginary time. 123456 1 |n 1  |n 2  |n 3  |n 4  |n 5  |n 6  |n 1  Graphical representation., Bipartite lattices: Canonical trans. and renders. Weight for MC sampling Real space. Weights.

5. Local updates.  Choose a shaded square.  Propose a move.  Accept / reject with prob:  Canonical. (total z-component of spin is conserved.)  Potentially long autocorrelation times.  No winding. One will never reach this configuration:

6. Equivalence to 6-Vertex Model = Identification. Vertex World-Lines  # incoming arrows = # outgoing arrows. (div = 0)  Vertex model lives on a 45 degrees rotated lattice.  Gives an intuitive uderstanding of loop updates. (Barma and Shastry 78.)

7. Loop Updates. Evertz. et al. (93)

8. Loop Updates (more formal). How do we build the loop? World-lines  World-lines + Graphs. S S S GGG GGG G G G

9. Requirements: (Sum runs over all possible G‘s given S) S´ follows from S by flipping arrows according to the rules of graph G. [W(1,2) = W(2,2)] From (3): Flipping probability:, From (1) and (2): Thus: Detailed balance in the space of spins is satisfied.

10. Example: Heisenberg model. Equations are satisfied just by considering graphs G 1 and G 2 From (2): From (3): So that:. L(1) L(2) L(3) L(4) Flip L(1)

11. Loops and magnetic fields. 1 2 -2 3 -3

12. Stochastic series expansion with operator loop updates (A. Sandvick 97). State of the art algorithm for spin systems. No systematic error. No critical slowing down when a magnetic field is introduced. b: is the sum over al the bonds. ( There are M=dN bonds on a d-dimensional hyper-cubic lattice)

13. Rewriting the partition function. Choose (dynamically during the simulation) a maximal value of n: L. Using: All in all,

14. Graphical representation. A configuration X is fully described by the state and the index list Example of a configuration for a L=7 and n = 4 configuration on a 6 site ring. 0 1 2 3 4 5 Bonds 1 2 3 4 5 6 7 L

15. Evaluation of the matrix element: S=0 S=1S=-1 S=2 S=-2 S=3 S=-3 (b,i)=(0,0) W(0) = 1 (b,i)=(b,1) (b,i)=(b,2) Chose C such that W(s)>0 for all S

16. Diagonal moves. (Local updates). n Let M = # bonds. Then. P P Acceptance.

17. Operator loop updates. n remains constant.

18. Building the operator loop. Vertices.Possible graphs. S=1 S=-1 S=2 S=-2 S=3 S=-3 G=1 G=2 G=3 G=1 G=4 G=3 G=1 G=4 G=2

19. 1 2 -2 3 -3 1 2 3 1 4 3 1 4 2 Same requirements as for loop algorithm to satisfy detailed balance.. with From (3): So that (2) reads: a)For finite values of h: so that the magnetic field is included in the construction of the loop via the bounce moves. b) For the Heisenberg model,

20. Magnetic order disorder transitions in planar quantum antiferromagnets. ( M. Troyer, M. Imada and K.Ueda (1997)) Applications. Loop algorithm allows calculation of critical exponents. Sizes up to 10 6 spins. This is possible since the CPU time scales as  V. Result. Same exponents as the O(3) 3D classical sigma model. Berry phase does not alter universality class. J c /J 1 Dimer singlet. J c /J 1 << 1 Plaquette singlett. 1/5 depleted Heiseberg model. Spin gap. Long range order.

21. Sign problem – a simple example. Consider: But: Z´ is the partition funtion of H with fermions replaced by hard core bosons. Thus: World line configuration: has negative weight. For practical purposes we will need: Note:  Had we formulated everything in Fourier space........  Hamiltonian H with t 1 <0 and t 2 <0 and hard core bosons yields a sign problem. This corresponds essentially to a frustrated spin chain. Thus the sign problem is not limited to fermionic systems.

22. Single hole dynamics in non-frustrated quantum spin backgrounds. (M. Brunner, FFA, and A. Muramatsu 2001). Result. Z >0 for the 2D t-J model. Lattice sizes up to 24x24. tt J/t Circles: k =(  /2,  /2) Stars: k =( ,  )