By P. de Forcrand, M. Troyer
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Extra info for Computational Quantum Physics
The Metropolis algorithm. 23) This represents the desired stationary distribution of the Markov chain corresponding to our Monte Carlo process. g. xk = 0 ∀k), perform many sweeps, where one sweep consists of updating once each xk0 , keeping all xk , k = k0 fixed. 24) Therefore, a Metropolis update of xk0 is implemented as: Step 1. g. uniform in [−∆, +∆]). 2: Comparison between Diffusion Monte Carlo (left) and Path integral Monte Carlo (right): many paths or many time-steps. Step 2. 24). 24) cancels in the ratio.
2. The size of the Hilbert space of an N-site system [4N for a Hubbard model , 3N for a t-J model and (2S + 1)N for a spin-S model] can be reduced by making use of symmetries. Translational symmetries can be employed by using Bloch waves with fixed momentum as basis states. Conservation of 51 particle number and spin allows to restrict a calculation to subspaces of fixed particle number and magnetization. ) + V i=1 ni ni+1 . 5) i=1 The first step is to construct a basis set. We describe a basis state using “multi-bit coding”.
Let us check the effect of a good trial wave-function on the walkers’ weight, by choosing for ψT the groundstate ψ0 (assuming ψ0 is known). 22) is equal to (−V + E0 ). The new potential is simply E0 , which is a constant independent of x. In Step 2, the weights of all walkers are multiplied by the same factor. As a result, in Step 3, no walker is either cloned or killed. 2 Fermionic systems A major application of Diffusion Monte Carlo is to compute the energy of electrons in a molecule or a crystal, given some positions for the atomic nuclei.