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arXiv:cond-mat/0211356

Diagrammatic Quantum Monte Carlo Algorithm in Momentum Representation: Hess-Fairbank Effect and Mesoscopics in 1D BEC with Attractive Interaction

Petr F. Kartsev

Published Nov 18, 2002• 0 citations
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Abstract

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).

DOI
10.48550/arxiv.cond-mat/0211356

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•Jan 1, 2005•0 citations
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).
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 → ∞.
We present the results of an exact numeric simulation of N one-dimensional bosons with attractive \ensuremath{\delta}-functional interaction in a rotating ring. We prove that even at intermediate values of N, the system can be described by conventional methods of weakly interacting gas, the dimensionless parameter of weak interaction being just 1/N.1/N.1/N. When the strength of interaction is less than a certain threshold value, the dependence of angular momentum on the rotation frequency features plateaus characteristic of the irrotational fluid.
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 develop a diagrammatic Monte Carlo method for the real-time dynamics of dissipative quantum impurity models. These are small open quantum systems with interaction and local Markovian dissipation, coupled to a large quantum bath. Our algorithm samples the hybridization expansion formulated on a single real-time contour, rather than on the double Keldysh one, as it naturally arises in the thermofield/vectorized representation of the Lindblad dynamics. We show that local Markovian dissipation generally helps the convergence of the diagrammatic Monte Carlo sampling by reducing the sign problem, thus allowing to reach longer timescales as compared to the conventional unitary case. We apply our method to an Anderson impurity model in the presence of local dephasing and discuss its effect on the charge and spin dynamics of the impurity.
Numerical simulations of the full quantum properties of interacting many-body systems by means of field-theoretic Monte Carlo techniques are often limited due to a sign problem. Here we simulate properties of a dilute two-dimensional Bose gas in the vicinity of the Berezinskii-Kosterlitz-Thouless (BKT) transition by means of the complex Langevin (CL) algorithm, thereby extending our previous CL study of the three-dimensional Bose gas to the lower-dimensional case. The purpose of the paper is twofold. On the one hand, it adds to benchmarking of the CL method and thus contributes to further exploring the range of applicability of the method. With the respective results, the universality of the equation of state is recovered, as well as the long-wave-length power-law dependence of the single-particle momentum spectrum below the BKT transition. Analysis of the rotational part of the current density corroborates vortex unbinding in crossing the transition. Beyond these measures of consistency we compute quantum corrections to the critical density and chemical potential in the weakly coupled regime. Our results show a shift of these quantities to lower values as compared to those obtained from classical field theory. It points in the opposite direction as compared to the shift of the critical density found by means of the path-integral Monte Carlo method at larger values of the coupling. Our simulations widen the perspective for precision comparisons with experiment.
We formulate a quantum Monte Carlo (QMC) method for calculating the ground state of many-boson systems. The method is based on a field-theoretical approach, and is closely related to existing fermion auxiliary-field QMC methods which are applied in several fields of physics. The ground-state projection is implemented as a branching random walk in the space of permanents consisting of identical single-particle orbitals. Any single-particle basis can be used, and the method is in principle exact. We apply this method to an atomic Bose gas, where the atoms interact via an attractive or repulsive contact two-body potential parametrized by the s-wave scattering length. We choose as the single-particle basis a real-space grid. We compare with exact results in small systems, and arbitrarily-sized systems of untrapped bosons with attractive interactions in one dimension, where analytical solutions exist. Our method provides a way to systematically improve upon the mean-field Gross-Pitaevskii (GP) method while using the same framework, capturing interaction and correlation effects with a stochastic, coherent ensemble of non-interacting solutions. to study the role of many-body correlations in the ground state, we examine the properties of the gas, such as the energetics, condensate fraction, and the density and momentum distributions as a function of the number of particles and the scattering length, both in the homogenous and trapped gases. Results are presented for systems with up to 1000 bosons. Comparing our results to the mean-field GP results, we find significant departure from mean field at large positive scattering lengths. The many-body correlations tend to increase the kinetic energy and reduce the interaction energy compared to GP. In the trapped gases, this results in a qualitatively different behavior as a function of the scattering length. Possible experimental observation is discussed.