World-Line Monte Carlo

The general idea of the world-line Monte Carlo algorithm is similar to that of the quantum-to-classical mapping discussed in the last section. The Hamiltonian is split into two or more terms H = ft such that the matrix 

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N. Kawashima and K. Harada (2004) Recent developments of world-line Monte Carlo methods. J. Phys. Soc. Jpn. 73, p. 1379... 

Inserting complete sets of basis states between the different Hi factors then leads to a similar representation of the partition function and a similar world-line picture as in the world-line Monte Carlo method. Because there is no Trotter decomposition involved, the method is free of time discretization errors from the outset. Early applications of the SSE method employed local updates, but more efficient cluster-type updates have been developed more recently to overcome the critical slowing down. They include the operator-loop update and the previously mentioned directed-loop algorithm. ... 

Figure 12 World-line Monte Carlo results for the square lattice Heisenberg antiferro-magnet Structure factor S and the long-distance limit C of the correlation function as functions of the linear system size L. The intercept on the vertical axis can be used to find the staggered magnetization. (Taken with permission from Ref. 104.)...

To provide quantitative predictions about how to detect the superfluid-insulator transition in these experiments, Kashurnikov, Prokofev and Svistunov performed quantum Monte Carlo simulations of the singleparticle density matrix p,y = (I I /). They used the Bose-Hubbard model with harmonic confining potential and carried out world-line Monte Carlo simulations with the continuous-time Worm algorithm. The diagonal elements of the density matrix provide the real-space particle density, and... 

Quantum Monte Carlo World Line Algorithms.613... 

Equation (47) is now just a classical sum of integrals and can be evaluated by Monte Carlo sampling in the intitial states i), the order of the perturbation n and the times Ti i = 1,..., n). This is best done by considering a graphical world line representation of the partition function (47) shown in Fig. 9. The zero-th order terms in the sum exp(—/ 77o) ) are given by straight world... 

To perform a quantum Monte Carlo simulation on the world line representation, update moves that are ergodic and fulfill detailed balance are required. The simplest types of moves are again local updates. Since magnetization conservation prohibits the breaking of world lines, the local updates need to move world lines instead of just changing local states as in a classical model. 

In this section we will present generalizations of extended ensemble simulations to world line quantum Monte Carlo simulations, in particular ... 

In the second part we have given a short introduction and review of modern world line quantum Monte Carlo algorithms for quantum systems, which are free of any time discretization errors. We have highlighted the fact that... 

We will use our first example, the spin- quantum Heisenberg magnet, to further illustrate the world-line quantum Monte Carlo method. The model we present is the quantum XXZ model ... 

Figure 11 World-line configuration for the XXZ Hamiltonian of [38]. The world lines (thick lines) connect space-time points where the component of the spin points up. They can be either straight or cross the shaded squares, which show where the imaginary time evolution operators and act. The dotted line shows the configuration change after a local Monte Carlo update.

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