MONTE CARLO DIAGONALIZATION OF VERY LARGE MATRICES: APPLICATION TO FERMION SYSTEMS

1992 ◽  
Vol 03 (01) ◽  
pp. 97-104 ◽  
Author(s):  
H. DE RAEDT ◽  
W. VON DER LINDEN
Keyword(s):  
Monte Carlo ◽  
Quantum Monte Carlo ◽  
General Feature ◽  
Degrees Of Freedom ◽  
Fundamental Problem ◽  
Simulation Methods ◽  
Fermion Systems ◽  
Monte Carlo Algorithms ◽  
Minus Sign ◽  
Sign Problem

All known Quantum-Monte-Carlo algorithms for fermions suffer from the so-called “minus-sign-problem” which is detrimental to the application of these simulation methods to fermion systems at very low temperatures and/or of very many lattice sites. We identify the origin of this fundamental problem, demonstrate that it is a very general feature, not necessarily related to the presence of fermionic degrees of freedom. We describe an novel algorithm which does not suffer from the minus-sign problem. Illustrative results for the two-dimensional Hubbard model are presented

scholarly journals Comparison of Calculations for the Hubbard Model Obtained with Quantum-Monte-Carlo, Exact, and Stochastic Diagonalization

1997 ◽  
Vol 08 (02) ◽  
pp. 397-415 ◽  
Author(s):  
Thomas Husslein ◽  
Werner Fettes ◽  
Ingo Morgenstern
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Hubbard Model ◽  
Ground State ◽  
Quantum Monte Carlo ◽  
Minus Sign ◽  
Sign Problem ◽  
Quantum Monte Carlo Method ◽  
Convergence Problems ◽  
Superconducting Correlations

In this paper we compare numerical results for the ground state of the Hubbard model obtained by Quantum-Monte-Carlo simulations with results from exact and stochastic diagonalizations. We find good agreement for the ground state energy and superconducting correlations for both, the repulsive and attractive Hubbard model. Special emphasis lies on the superconducting correlations in the repulsive Hubbard model, where the small magnitude of the values obtained by Monte-Carlo simulations gives rise to the question, whether these results might be caused by fluctuations or systematic errors of the method. Although we notice that the Quantum-Monte-Carlo method has convergence problems for large interactions, coinciding with a minus sign problem, we confirm the results of the diagonalization techniques for small and moderate interaction strengths. Additionally we investigate the numerical stability and the convergence of the Quantum-Monte-Carlo method in the attractive case, to study the influence of the minus sign problem on convergence. Also here in the absence of a minus sign problem we encounter convergence problems for strong interactions.

scholarly journals Sign-Problem-Free Fermionic Quantum Monte Carlo: Developments and Applications

2019 ◽  
Vol 10 (1) ◽  
pp. 337-356 ◽  
Author(s):  
Zi-Xiang Li ◽  
Hong Yao
Keyword(s):  
Monte Carlo ◽  
Quantum Monte Carlo ◽  
Hot Spot ◽  
High Temperature Superconductors ◽  
System Size ◽  
Many Body ◽  
Broken Symmetries ◽  
Sign Problem ◽  
Universal Quantum ◽  
Many Body Systems

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.

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