Deconvolving the components of the sign problem

Tarat, Sabyasachi; Xiao, Bo; Mondaini, Rubem; Scalettar, Richard

doi:10.1103/physrevb.105.045107

Cited by 11 publications

(6 citation statements)

References 53 publications

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“…This is similar to the case displayed in Fig. 1, and also explains why the minimum of sign can be a criterion judging the phase transition in Ref [34].…”

supporting

confidence: 85%

“…Or more strictly speaking, those positive examples where the average sign can probe the critical value can be excep- tional cases. On the other hand, in previous studies, the phase transition can be related to the sign value itself or its derivative in different circumstances without an explicit criterion [32][33][34][35][36][37][38]. This non-conformity has also been discussed in the present work.…”

mentioning

confidence: 66%

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How universal is the relation between sign problem and phase transition

Zheng¹,

Sun²,

Pan³

et al. 2023

Preprint

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The mystery of the infamous sign problem in quantum Monte Carlo simulations mightily restricts applications of the method in fermionic and frustrated systems. A recent work [Science 375, 418 (2022)] made a remarkable breakthrough in the sign problem and suggested a positive connection between the sign and phase transition. How general this argument is can be crucial in various fields related to many-body systems, such as condensed matter physics, quantum chemistry, and nuclear physics. Based on universal analyses of typical examples and numerical simulations from different approaches, we discuss when and how studying the sign can provide helpful information on phase transitions in general systems independent of specific models and algorithms. While our results support that the notorious sign offers new angles in exploring quantum many-body problems, we also notice that taking advantage of the sign can even be as challenging as neutralizing the sign problem itself in unknown systems.

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“…This is similar to the case displayed in Fig. 1, and also explains why the minimum of sign can be a criterion judging the phase transition in Ref [34].…”

supporting

confidence: 85%

mentioning

confidence: 66%

How universal is the relation between sign problem and phase transition

Zheng¹,

Sun²,

Pan³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…But Recent developments have changed this viewpoint, there are cases in Hubbard model at certain filling the average sign is not exponential decay [61,62] and we have found that it is possible to prove the average sign of quantum many-body lattice models accquire algebraic sign structure [32,33], if their finite size partition functions satisfy the Sign bound theory. The details on the theory and its application on the systems with extended or long-range interaction such as the extended Hubbard [32,63] and quantum Moiré lattice models [64,65,66,30,67,33,68], are given in Sec.…”

Section: Introductionmentioning

confidence: 98%

“…At the same time, there are also proposals that the negative sign of a configuration is a topological invariant, which is an imaginary time counterpart of the Aharonov-Anandan phase and can be reduced to a Berry phase in the adiabatic limit [12]. And the sign problem has even been linked to quantum phase transitions [13,14]. Moreover, it was shown recently that some interacting models may have intrinsic sign problem, which cannot be cured [15,16,17,18].…”

Section: Introductionmentioning

confidence: 99%

Sign Problem in Quantum Monte Carlo Simulation

Pan¹,

Meng²

2022

Preprint

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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.

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“…Perhaps most notably, DQMC has enabled the study of electron-electron interactions in the repulsive Hubbard model, where Mott insulator physics, magnetic order, unconventional superconductivity, and various additional correlation effects have been observed [2][3][4][5][6][7][8][9][10]. The sign problem, however, has severely limited our ability to simulate systems absent particle-hole or other symmetries, giving rise to an effective computational cost that scales exponentially with system size and inverse temperature [11][12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Fast and scalable quantum Monte Carlo simulations of electron-phonon models

Cohen-Stead,

Bradley,

Miles

et al. 2022

Preprint

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We introduce methodologies for highly scalable quantum Monte Carlo simulations of electronphonon models, and report benchmark results for the Holstein model on the square lattice. The determinant quantum Monte Carlo (DQMC) method is a widely used tool for simulating simple electron-phonon models at finite temperatures, but incurs a computational cost that scales cubically with system size. Alternatively, near-linear scaling with system size can be achieved with the hybrid Monte Carlo (HMC) method and an integral representation of the Fermion determinant. Here, we introduce a collection of methodologies that make such simulations even faster. To combat "stiffness" arising from the bosonic action, we review how Fourier acceleration can be combined with time-step splitting. To overcome phonon sampling barriers associated with strongly-bound bipolaron formation, we design global Monte Carlo updates that approximately respect particle-hole symmetry. To accelerate the iterative linear solver, we introduce a preconditioner that becomes exact in the adiabatic limit of infinite atomic mass. Finally, we demonstrate how stochastic measurements can be accelerated using fast Fourier transforms. These methods are all complementary and, combined, may produce multiple orders of magnitude speedup, depending on model details.

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Deconvolving the components of the sign problem

Cited by 11 publications

References 53 publications

How universal is the relation between sign problem and phase transition

How universal is the relation between sign problem and phase transition

Sign Problem in Quantum Monte Carlo Simulation

Fast and scalable quantum Monte Carlo simulations of electron-phonon models

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