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Exponential observables, formulated as $\log \langle e^{\hat{X}}\rangle$where $\hat{X}$ is an extensive quantity, play a critical role in study ofquantum many-body systems, examples of which include the free-energy andentanglement entropy. Given that $e^{X}$ becomes exponentially large (or small)in the thermodynamic limit, accurately computing the expectation value of thisexponential quantity presents a significant challenge. In this Letter, wepropose a comprehensive algorithm for quantifying these observables ininteracting fermion systems, utilizing the determinant quantum Monte Carlo(DQMC) method. We have applied this novel algorithm to the 2D half-filledHubbard model. At the strong coupling limit, our method showcases a significantaccuracy improvement compared to conventional methods that are derived from theinternal energy. We also illustrate that this novel approach delivers highlyefficient and precise measurements of the nth R\'enyi entanglement entropy.Even more noteworthy is that this improvement comes without incurring increasesin computational complexity. This algorithm effectively suppresses exponentialfluctuations and can be easily generalized to other models.

An integral algorithm of exponential observables for interacting fermions in quantum Monte Carlo simulation