Wave function optimization in the variational Monte Carlo method

An appropriate iterative scheme for the minimization of the energy, based on the variational Monte Carlo (VMC) technique, is introduced and compared with existing stochastic schemes. We test the various methods for the one-dimensional Heisenberg ring and the two-dimensional $t\text{−}J$ model and show that, with the present scheme, very accurate and efficient calculations are possible, even for several variational parameters. Indeed, by using a very efficient statistical evaluation of the first and the second energy derivatives, it is possible to define a very rapidly converging iterative scheme that, within VMC, is much more convenient than the standard Newton method. It is also shown how to optimize simultaneously both the Jastrow and the determinantal part of the wave function.

Figure 1

(Color online) Left: convergence of the SRH technique for different sizes. Right: the same for all the 149 independent VMC parameters for the maximum size studied. For all simulations the bin length [the control parameter $r$ defined in Eq. (11)] $M$ was taken proportional to $L^2$ $\left(L\right)$, starting from $M=500$ $(r=2)$ for $L=30$.

Figure 2

(Color online) Convergence of the VMC nearest-neighbor spin Jastrow for various methods (see text). For the SR technique $\Delta tJ=0.125$. For the NM the first four iterations are with $a_{\mathit{diag}}=0.02$, for the next six $a_{\mathit{diag}}=0.002$, and the rest with $a_{\mathit{diag}}=0$. For $M=2500$ the NM is unstable after 30 iterations due to a big fluctuation of the matrix $G$ (see text) not used in SRH. For the SRH technique $r=5$ for $M=2500$ and $r=10$ for $M=5000$. After the third iteration $\mid \Delta \mathrm{WF}\mid <r$ is always satisfied, even after further 1000 iterations (not shown).

Figure 3

(Color online) Parameters in $\mid D⟩$, while optimizing the Jastrow part simultaneously for the $t\text{−}J$ model at $J∕t=0.4$ with bin $M=5000$. For SRH the control parameter is $r=0.5$ $(r=0.1)$ for $L=242$ $(L=50)$, for SR $J\Delta t=0.125$. The hybrid method is also shown (left) for ${\mu }_0$ $(r=1,J\Delta t=0.3125)$.