Integral algorithm of exponential observables for interacting fermions in quantum Monte Carlo simulations

Abstract

<p>Exponential observables, formulated as ln⁡⟨𝑒ˆ𝑋⟩ where ˆ𝑋 is an extensive quantity, play a critical role in the study of quantum many-body systems, examples of which include the free energy and entanglement entropy. Given that 𝑒𝑋 becomes exponentially large (or small) in the thermodynamic limit, the accurate computation of the expectation value of this exponential quantity presents a significant challenge. In this paper, we propose a comprehensive algorithm to quantify these observables in interacting fermion systems, utilizing the determinant quantum Monte Carlo method. We have applied this algorithm to the two-dimensional square-lattice half-filled Hubbard model and 𝜋-flux t-V model. In the Hubbard model case at the strong-coupling limit, our method showcases a significant accuracy improvement on free energy compared to conventional methods that are derived from the internal energy, and in the t-V model, we indicate that the free energy offers a precise determination of the second-order phase transition. We also illustrate that this approach delivers highly efficient and precise measurements of the 𝑛⁢th Rényi entanglement entropy. Even more noteworthy is that this improvement comes without incurring increases in computational complexity. This algorithm effectively suppresses exponential fluctuations and can be easily generalized to other models.<br></p>