A Hybrid Method for Global Updates in Monte Carlo Study

Tomo Munehisa, Yasuko Munehisa
1995 Progress of theoretical physics  
We propose a new algorithm which works effectively in global updates in Monte Carlo study. We apply it to the quantum spin chain with next-nearest-neighbor interactions. We observe that Monte Carlo results are in excellent agreement with numerically exact ones obtained by the transfer matrix method. The Monte Carlo method has been an indispensable tool for theoretical study in physics. An essential point of the method is to generate configurations according to their Boltzmann weights. For this
more » ... urpose one usually starts with an initial configuration and obtains a new configuration sequentially by some rule that guarantees the ergodicity and satisfies the condition of the detailed balance. Widely used rules are the heat bath method and the Metropolis algorithm. The heat bath method is useful for systems which have a small number of candidates for updating because one has to calculate weights for all candidates to determine which one should make the new configuration. The Metropolis algorithm is suitable for complicated systems since in this algorithm it is enough to know the weight just for one candidate to decide whether one should accept or reject that candidate for updating. The problem in the latter is that the efficiency of updating would be bad if the selection of the candidate is not appropriate. In the previous papers 0 • 2 > for quantum Monte Carlo study using the Suzuki-Trotter formula we proposed re-structuring method to improve the negative sign problem and applied it to one dimensional spin-1/2 system with next-nearest-neighbor interactions. The Hamiltonian of this system is where N is the number of sites on the chain and <1N+i= <1;. It turned out that this method is very useful for this model. We found however that it was difficult to make Monte Carlo simulations with large values of the Trotter number for technical reasons on the so-called global updates, which are necessary in the quantum Monte Carlo method in addition to the usual local updates. In the global updates one needs to change spin states on all sites along some lines connecting these sites, which usually depend on the size of the system, while in the local updates limited part of the system irrelevant to whole size of the system should be updated. In the conventional apporach to the model, where the state on each site is represented spin up or spin down, conservation property of the quantum number ]z(z-component of the total spin !) is helpful to carry out global updates for large sizes of the system. In the restructuring approach, on the contrary, we employ eigenstates of Ou-10u, that is L, EB;,
doi:10.1143/ptp.93.251 fatcat:lamieissrzebrn5bgd6pbpd6te