SciPost Phys. Core 8, 036 (2025) ·
published 8 April 2025
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Simulating long-range interacting systems is a challenging task due to its computational complexity that the computational effort for each local update is of order $\mathcal{O}(N)$, where $N$ is the size of the system. In this work, we introduce the clock factorized quantum Monte Carlo method, an efficient technique for simulating long-range interacting quantum systems. The method is developed by generalizing the clock Monte Carlo method for classical systems [Phys. Rev. E 99 010105 (2019)] to the path-integral representation of long-range interacting quantum systems, with some specific treatments for quantum cases and a few significant technical improvements in general. We first explain how the clock factorized quantum Monte Carlo method is implemented to reduce the computational overhead from $\mathcal{O}(N)$ to $\mathcal{O}(1)$. In particular, the core ingredients, including the concepts of bound probabilities and bound rejection events, the recursive sampling procedure, and the fast algorithms for sampling an extensive set of discrete and small probabilities, are elaborated. Next, we show how the clock factorized quantum Monte Carlo method can be flexibly implemented in various update strategies, like the Metropolis and worm-type algorithms. Finally, we demonstrate the high efficiency of the clock factorized quantum Monte Carlo algorithms using examples of three typical long-range interacting quantum systems, including the transverse field Ising model with long-range $z$-$z$ interaction, the extended Bose-Hubbard model with long-range density-density interactions, and the XXZ Heisenberg model with long-range spin interactions. We expect that the clock factorized quantum Monte Carlo method would find broad applications in statistical and condensed-matter physics.
Yu-Feng Song, Jesper Lykke Jacobsen, Bernard Nienhuis, Andrea Sportiello, Youjin Deng
SciPost Phys. 18, 057 (2025) ·
published 18 February 2025
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Recent work on percolation in $d=2$ [J. Phys. A: Math. Theor. 55, 204002 (2015)] introduced an operator that gives a weight $k^{\ell}$ to configurations with $\ell$ 'nested paths' (NP), i.e.\ disjoint cycles surrounding the origin, if there exists a cluster that percolates to the boundary of a disc of radius $L$, and weight zero otherwise. It was found that $\mathbb{E}(k^{\ell}) \sim L^{-X_{NP}(k)}$, and a formula for $X_{NP}(k)$ was conjectured. Here we derive an exact result for $X_{NP}(k)$, valid for $k ≥ -1$, replacing the previous conjecture. We find that the probability distribution $\mathbb{P}_\ell (L)$ scales as $ L^{-1/4} (\ln L)^\ell [(1/\ell!) \Lambda^\ell]$ when $\ell ≥ 0$ and $L \gg 1$, with $\Lambda = 1/\sqrt{3} \pi$. Extensive simulations for various critical percolation models confirm our theoretical predictions and support the universality of the NP observables.
Kris Van Houcke, Evgeny Kozik, Riccardo Rossi, Youjin Deng, Félix Werner
SciPost Phys. 16, 133 (2024) ·
published 27 May 2024
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In the standard framework of self-consistent many-body perturbation theory, the skeleton series for the self-energy is truncated at a finite order N and plugged into the Dyson equation, which is then solved for the propagator $G_N$. We consider two examples of fermionic models, the Hubbard atom at half filling and its zero space-time dimensional simplified version. First, we show that $G_N$ converges when $N\to∞$ to a limit $G_∞\,$, which coincides with the exact physical propagator $G_{exact}$ at small enough coupling, while $G_∞ ≠ G_{exact}$ at strong coupling. This follows from the findings of [Phys. Rev. Lett. 114, 156402 (2015)] and an additional subtle mathematical mechanism elucidated here. Second, we demonstrate that it is possible to discriminate between the $G_∞=G_{exact}$ and $G_∞≠G_{exact}$ regimes thanks to a criterion which does not require the knowledge of $G_{exact}$, as proposed in [Phys. Rev. B 93, 161102 (2016)].
SciPost Phys. 16, 121 (2024) ·
published 6 May 2024
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We show by extensive simulations that the whole supercritical phase of the three-dimensional uniform forest model simultaneously exhibits an infinite tree and a rich variety of critical phenomena. Besides typical scalings like algebraically decaying correlation, power-law distribution of cluster sizes, and divergent correlation length, a number of anomalous behaviors emerge. The fractal dimensions for off-giant trees take different values when being measured by linear system size or gyration radius. The giant-tree size displays two-length scaling fluctuations, instead of following the central-limit theorem.
