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arXiv:2303.13403 · Physical review. B./Physical review. B

Boosting quantum Monte Carlo and alleviating sign problem by Gutzwiller projection

Wei-Xuan Chang, Zixiang Li

Published Aug 27, 2024• 6 citations
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Abstract

We present a scheme for projective quantum Monte Carlo (QMC) simulation that combines unbiased zero-temperature (projective) determinant QMC with variational Monte Carlo based on the Gutzwiller projected wave function. This approach is dubbed as Gutzwiller-projection QMC. Our numerical results demonstrate that the use of the Gutzwiller projected trial wave function significantly accelerates the convergence of computational results, thereby greatly reducing computational time in the simulation. Moreover, we provide an illustrative example showing that the sign problem is substantially mitigated in the Gutzwiller-projection QMC. We believe that the Gutzwiller-projection QMC opens up another pathway for enhancing efficiency and alleviating the sign problem in QMC simulations of interacting fermionic systems.

DOI
10.1103/physrevb.110.085152

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We present a scheme for projective quantum Monte Carlo (QMC) simulation that combines unbiased zero-temperature (projective) determinant QMC with variational Monte Carlo based on the Gutzwiller projected wave function. This approach is dubbed as Gutzwiller-projection QMC. Our numerical results demonstrate that the use of the Gutzwiller projected trial wave function significantly accelerates the convergence of computational results, thereby greatly reducing computational time in the simulation. Moreover, we provide an illustrative example showing that the sign problem is substantially mitigated in the Gutzwiller-projection QMC. We believe that the Gutzwiller-projection QMC opens up another pathway for enhancing efficiency and alleviating the sign problem in QMC simulations of interacting fermionic systems.
Sign problem in quantum Monte Carlo (QMC) simulation appears to be an extremely hard yet interesting problem. In this article, we present a pedagogical overview on the origin of the sign problem in various quantum Monte Carlo simulation techniques, ranging from the world-line and stochastic series expansion Monte Carlo for boson and spin systems to the determinant and momentum-space quantum Monte Carlo for interacting fermions. We point out the basis dependency of the sign problem and summarize the progresses to cure, ease and even make use of the sign problem over the years, such as symmetry analysis of the underlying Hamiltonian, basis optimization in writting down the partition functions and many others. Moreover, we state that although traditional lore saying that in case of sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space, there are recent breakthroughs showing this is not always the case and based on the properties of the partition function for finite size systems, one could distinguish when the average sign has the usual exponential scaling and when it is bestowed with an algebraic scaling at the low temperature limit. Fermionic QMC simulations with such algebraic sign problems have been successfully carried out for extended Hubbard-type and quantum Moiré lattice models.
We present a unified theory of the variational Monte Carlo (VMC) and determinant quantum Monte Carlo (DQMC) methods using a novel density matrix formulation of VMC. We introduce an efficient algorithm for VMC to compute correlation functions and expectation values based on the auxiliary field Hirsch-Hubbard-Stratonovic transformation. We show that this new approach to VMC converges significantly faster than its traditional implementations. Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginary time steps τ∼O(1)τ\sim O(1)τ∼O(1) and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.
The projective quantum Monte Carlo (PQMC) algorithms are among the most powerful computational techniques to simulate the ground-state properties of quantum many-body systems. However, they are efficient only if a sufficiently accurate trial wave function is used to guide the simulation. In the standard approach, this guiding wave function is obtained in a separate simulation that performs a variational minimization. Here we show how to perform PQMC simulations guided by an adaptive wave function based on a restricted Boltzmann machine. This adaptive wave function is optimized along the PQMC simulation via unsupervised machine learning, avoiding the need of a separate variational optimization. As a byproduct, this technique provides an accurate ansatz for the ground-state wave function, which is obtained by minimizing the Kullback-Leibler divergence with respect to the PQMC samples, rather than by minimizing the energy expectation value as in standard variational optimizations. The high accuracy of this self-learning PQMC technique is demonstrated for a paradigmatic sign-problem-free model, namely, the ferromagnetic quantum Ising chain, showing very precise agreement with the predictions of the Jordan-Wigner theory and of loop quantum Monte Carlo simulations performed in the low-temperature limit.
We explore different ways of incorporating accurate trial wave functions into free projection auxiliary field quantum Monte Carlo (fp-AFQMC). Trial states employed include coupled cluster singles and doubles, multi-Slater, and symmetry projected mean-field wave functions. We adapt a recently proposed fast multi-Slater local energy evaluation algorithm for fp-AFQMC, making the use of long expansions from selected configuration interaction methods feasible. We demonstrate how these trial wave functions serve to mitigate the sign problem and accelerate convergence in quantum chemical problems, allowing the application of fp-AFQMC to systems of substantial sizes. Our calculations on the widely studied Cu2O22+\text{Cu}_2\text{O}_2^{2+}Cu2​O22+​ model system show that many previously reported isomerization energies differ substantially from the near-exact fp-AFQMC value.
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.
We propose a projection approach to perform quantum Monte Carlo (QMC) simulation on the infinite-UUU Hubbard model at some integer fillings where either it is sign problem free or surprisingly has an algebraic sign structure---a power law dependence of average sign on system size. We demonstrate our scheme on the infinite-USU(2N)U\phantom{\rule{4pt}{0ex}}SU(2N)USU(2N) fermionic Hubbard model on both a square and honeycomb lattice at half filling, where it is sign problem free, and suggest possible correlated ground states. The method can be generalized to study certain extended Hubbard models applying to cluster Mott insulators or two-dimensional Moir\'e systems; among one of them at certain non-half-integer filling, the sign has an algebraic behavior such that it can be numerically solved within a polynomial time. Further, our projection scheme can also be generalized to implement the Gutzwiller projection to spin basis such that SU(2N)SU(2N)SU(2N) quantum spin models and Kondo lattice models may be studied in the framework of fermionic QMC simulations.