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Journal of Experimental and Theoretical Physics

A new quantum Monte Carlo algorithm in the momentum representation: The sign problem and the Hess-Fairbank effect

Petr F. Kartsev

Published Oct 1, 2003• 1 citation
Publisher

Abstract

An exact numerical algorithm based on the diagrammatic quantum Monte Carlo method in the momentum representation is proposed; in many cases, this algorithm is free of the sign problem and extends the class of models that can be analyzed by cluster methods. The weakening of the sign problem is demonstrated via the determination of the ground state of electrons on a chain in the Hubbard model. The algorithm is applied to the investigation of the behavior of a one-dimensional boson system with attraction in a rotating ring in the region of the Hess-Fairbank effect predicted by Ueda and Leggett. The existence of this effect for a comparatively small number of particles N ∼ 10 is confirmed. An analytic boundary of this effect is determined in the limit as N → ∞.

DOI
10.1134/1.1625074

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An exact numerical algorithm based on the diagrammatic quantum Monte Carlo method in the momentum representation is proposed; in many cases, this algorithm is free of the sign problem and extends the class of models that can be analyzed by cluster methods. The weakening of the sign problem is demonstrated via the determination of the ground state of electrons on a chain in the Hubbard model. The algorithm is applied to the investigation of the behavior of a one-dimensional boson system with attraction in a rotating ring in the region of the Hess-Fairbank effect predicted by Ueda and Leggett. The existence of this effect for a comparatively small number of particles N ∼ 10 is confirmed. An analytic boundary of this effect is determined in the limit as N → ∞.
A novel algorithm of Diagrammatic Quantum Monte Carlo in momentum representation is reported in details. New models can be studied with this algorithm. For Bose systems with attractive interaction, the algorithm is free of the well-known minus-sign problem, while in other models it is weaker than in real-space methods. Using this algorithm, we present the results of an exact numeric simulation of N one-dimensional bosons with attractive delta-functional interaction in a rotating ring. We prove that in the large-N limit the system can be described by conventional methods of weakly interacting gas, the dimensionless parameter of weak interaction being just 1/N. When the strength of interaction is less then a certain threshold value, the dependence of angular momentum on the rotation frequency features plateaus characteristic of the irrotational fluid (the Hess-Fairbank effect).
The quantum Monte Carlo method on asymptotic Lefschetz thimbles is a numerical algorithm devised specifically for alleviation of the sign problem appearing in the simulations of quantum many-body systems. In this method, the sign problem is alleviated by shifting the integration domain for the auxiliary fields, appearing, for example, in the conventional determinant quantum Monte Carlo method, from real space to an appropriate manifold in complex space. Here, we extend this method to quantum spin models with generic two-spin interactions, by using the Hubbard-Stratonovich transformation to decouple the exchange interactions and the Popov-Fedotov transformation to map the quantum spins to complex fermions. As a demonstration, we apply the method to the Kitaev model in a magnetic field whose ground state is predicted to deliver a topological quantum spin liquid with non-Abelian anyonic excitations. To illustrate how the sign problem is alleviated in this method, we visualize the asymptotic Lefschetz thimbles in complex space, together with the saddle points and the zeros of the fermion determinant. We benchmark our method in the low-temperature region in a magnetic field and show that the sign of the action is recovered considerably and unbiased numerical results are obtained with sufficient precision.
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.
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 introduce the adiabatic quantum Monte Carlo (AQMC) method, where we gradually crank up the interaction strength, as an amelioration of the sign problem. It is motivated by the adiabatic theorem and will approach the true ground state if the evolution time is long enough. We demonstrate that the AQMC algorithm enhances the average sign exponentially such that low enough temperatures can be accessed and ground-state properties probed. It is a controlled approximation that satisfies the variational theorem and provides an upper bound for the ground-state energy. We first benchmark the AQMC algorithm vis-à-vis the undoped Hubbard model on the square lattice which is known to be sign-problem-free within the conventional quantum Monte Carlo formalism. Next, we test the AQMC algorithm against the density-matrix-renormalization-group approach for the doped four-leg ladder Hubbard model and demonstrate its remarkable accuracy. As a nontrivial example, we apply our method to the Hubbard model at p=1/8 doping for a 16×8 system and discuss its ground-state properties. We finally utilize our method and demonstrate the emergence of U(1)_{2}∼SU(2)_{1} topological order in a strongly correlated Chern insulator.