Excitation gap from optimized correlation functions in quantum Monte Carlo simulations

We give a prescription for finding optimized correlation functions for the extraction of the gap to the first excited state within quantum Monte Carlo simulations. We demonstrate that optimized correlation functions provide a more accurate reading of the gap when compared to other “nonoptimized” correlation functions and are generally characterized by considerably larger signal-to-noise ratios. We also analyze the cost of the procedure and show that it is not computationally demanding. We illustrate the effectiveness of the proposed procedure by analyzing several exemplary many-body systems of interacting spin-$1/2$ particles.

Figure 1

A log-linear plot of a time-dependent correlation function for an $N=64$ spins system of one instance of the 1-in-3SAT problem with $\beta =1024$ (further details of the problem can be found in Sec. 4). While at early times there are contributions from several energy levels, there is a region where the correlation function may be fit with a simple straight line (on a log-linear scale) from which the energy gap which is the negative of the slope could be obtained (giving here $\Delta E_1=0.0364$). At still later times where the correlation function is significantly attenuated, the statistical errors become huge.

Figure 2

Optimized (red triangles), partially optimized (green squares) and nonoptimized (i.e., a randomly generated operator, blue circles) coefficient values of the operator $\widehat{O}$ whose different-time correlations are used to extract the gap. The values shown are normalized according to $\left|\right|\widehat{O}\left|\right|=1$ or, equivalently, $\langle \mathit{\alpha }\left|\mathit{\eta }\right|\mathit{\alpha }\rangle =1$. The horizontal line indicates the indices of the basic operators as they are defined in the text.

Figure 3

Optimized (red triangles), partially optimized (green squares), and nonoptimized (blue circles) correlation functions as functions of imaginary time on a log-linear scale for an instance of a 1-in-3SAT problem with $N=16$ interacting spin-$1/2$ particles. As the figure indicates, the optimization process substantially reduces the contributions from higher-order coefficients leaving even in relatively short times only the slowest-decaying exponent. The resulting optimal correlation function was easily fit with a linear function at the appropriate region, producing an accurate prediction of the excitation gap (the various linear fits are the solid lines). The partially optimized correlation function is also a vast improvement over the nonoptimized correlation function, which, in turn, yields results of a much poorer quality. For the latter correlation function, there seems to be no region where a linear behavior is obvious, as higher-order contributions are evident throughout the (also very noisy) examined region.

Figure 4

Optimized (red triangles), partially optimized (green squares), and nonoptimized (blue circles) correlation functions as functions of imaginary time on a log-linear scale for an instance of the 1-in-3SAT problem with $N=48$ interacting spin-1/2 particles. The figure shows that the optimal correlation function is easily fit with a linear function at the appropriate region, producing an accurate prediction of the excitation gap (the various linear fits are the solid lines). The partially optimized correlation function is also a vast improvement over the nonoptimized correlation function, which, in turn, yields results of a much poorer quality.

Figure 5

Same as Fig. 4 but for a system with $N=64$ spins. Here, too, the optimal correlation function (red triangles) is much less noisy than the other partially optimized (green squares) and nonoptimized (blue circles) correlation functions, eventually leading to a more accurate prediction of the system gap. Of the two latter correlation functions, the nonoptimized one is particularly noisy in this example and yields an extremely poor estimation of the gap.