F.F. Assaad. MPI-Stuttgart. Universität-Stuttgart 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.

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.

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.

Imaginary time |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.

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:

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.)

Loop Updates. Evertz. et al. (93)

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

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.

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)

Loops and magnetic fields

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)

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

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 Bonds L

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

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

Operator loop updates. n remains constant.

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

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,

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.

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.

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 =( ,  )