Introduction to quantum Monte Carlo simulations for fermionic systems

Santos, Raimundo R. dos

doi:10.1590/s0103-97332003000100003

Cited by 144 publications

(144 citation statements)

References 55 publications

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“…Previous work, comparing 2D 4 × 4, 6 × 6, and 8 × 8 Hubbard lattices 7,10 and 3D 4 × 4 × 4 and 6 × 6 × 6 lattices 7 was inconclusive. In both the 2D and 3D cases, for example, for some parameter ranges S was actually worst for the smallest systems studied.…”

Section: Discussionmentioning

confidence: 92%

“…Although one can formally use the absolute value as a weight, and include the sign in the measurement, in practice one ends up evaluating the ratio of two numbers, which becomes dominated by statistical error at low temperatures as they both become very small. This situation is known as the "fermion sign problem", 6,7 whose solution is conjectured to be NPhard. 8 At present, therefore, there is no known method of accessing the low temperature properties of Hamiltonians like the Fermi-Hubbard model with QMC.…”

Section: Introductionmentioning

confidence: 99%

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Geometry dependence of the sign problem in quantum Monte Carlo simulations

2015

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The sign problem is the fundamental limitation to quantum Monte Carlo simulations of the statistical mechanics of interacting fermions. Determinant quantum Monte Carlo (DQMC) is one of the leading methods to study lattice models such as the Hubbard Hamiltonian, which describe strongly correlated phenomena including magnetism, metal-insulator transitions, and (possibly) exotic superconductivity. Here, we provide a comprehensive dataset on the geometry dependence of the DQMC sign problem for different densities, interaction strengths, temperatures, and spatial lattice sizes. We supplement these data with several observations concerning general trends in the data, including the dependence on spatial volume and how this can be probed by examining decoupled clusters, the scaling of the sign in the vicinity of a particle-hole symmetric point, and the correlation between the total sign and the signs for the individual spin species.

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Section: Discussionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Geometry dependence of the sign problem in quantum Monte Carlo simulations

2015

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“…[26][27][28] There are many different approximate techniques that can be used, but the intention of this paper is to establish fundamentally exact procedures that avoid the Fermi sign problem. As reported earlier, [29][30][31] the Gaussian method has been successfully applied to the difficult case of the repulsive Hubbard model.…”

Section: Introductionmentioning

confidence: 99%

Gaussian phase-space representations for fermions

2006

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We introduce a positive phase-space representation for fermions, using the most general possible multimode Gaussian operator basis. The representation generalizes previous bosonic quantum phase-space methods to Fermi systems. We derive equivalences between quantum and stochastic moments, as well as operator correspondences that map quantum operator evolution onto stochastic processes in phase space. The representation thus enables first-principles quantum dynamical or equilibrium calculations in many-body Fermi systems. Potential applications are to strongly interacting and correlated Fermi gases, including coherent behavior in open systems and nanostructures described by master equations. Examples of an ideal gas and the Hubbard model are given, as well as a generic open system, in order to illustrate these ideas.

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“…We show that for the Hubbard model the Gaussian representation leads to imaginarytime equations with no negative probabilities or weights. We demonstrate that this removes the well-known Fermi sign problem [1,2,3], by first principles numerical simulation without fixed-node[4] or variational approximations.Phase-space methods[5] provide a way to simulate quantum many-body systems both dynamically and at finite temperature, and have proved useful in bosonic cases. These methods sample the time evolution of a positive distribution on an overcomplete basis set, which is usually the set of coherent states.…”

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confidence: 99%

“…As an application of the method to a dynamical calculation, in particular to a composite Bose/Fermi system, , σ(g (2) ) x10 3 Figure 1: Second-order correlation function g (2) versus inverse temperature τ for t = 0, U = 2 and µ = 1, for which nj = 0.5. The solid curve gives the simulation result, and the dashed and dot-dashed line show the estimated sampling error and deviation from the analytic result, respectively (on a ×1000 scale).…”

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confidence: 99%

Gaussian Quantum Monte Carlo Methods for Fermions and Bosons

2004

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We introduce a new class of quantum Monte Carlo methods, based on a Gaussian quantum operator representation of fermionic states. The methods enable first-principles dynamical or equilibrium calculations in many-body Fermi systems, and, combined with the existing Gaussian representation for bosons, provide a unified method of simulating Bose-Fermi systems. As an application, we calculate finite-temperature properties of the two dimensional Hubbard model.Calculating the quantum many-body physics of interacting Fermi systems is one of the great challenges in modern theoretical physics. These issues appear in physical problems at all energy scales, from ultra-cold atomic physics to high-energy lattice QCD. In even the simplest cases, first-principles calculations are inhibited by the complexity of the fermionic wavefunction, manifest notoriously in the Fermi sign problem. In previous quantum Monte Carlo (QMC) techniques, the sign problem appears as trajectories with negative weights, which contribute to a large sampling error [1]. QMC methods are also complicated by the calculation of large determinants.In this letter, we introduce a new QMC method for simulating many-body fermion systems, based on a Gaussian phase-space representation. As an application to condensed matter and AMO physics, we study the well-known Hubbard model. Although it is the simplest model of interacting fermions on a lattice, it is rich in physics and may even describe high-temperature superconductivity [2]. We show that for the Hubbard model the Gaussian representation leads to imaginarytime equations with no negative probabilities or weights. We demonstrate that this removes the well-known Fermi sign problem [1,2,3], by first principles numerical simulation without fixed-node[4] or variational approximations.Phase-space methods[5] provide a way to simulate quantum many-body systems both dynamically and at finite temperature, and have proved useful in bosonic cases. These methods sample the time evolution of a positive distribution on an overcomplete basis set, which is usually the set of coherent states. However, whereas coherent state representations are well-defined in the bosonic case, the only known coherent state techniques for fermions involve Grassmann algebra [6], which has an enormous computational complexity.Here we introduce a phase-space method that overcomes the problem of Grassmann complexity, using a Gaussian expansion for fermions. The operator basis is constructed from pairs of Fermi operators. Because these pairs obey commutation relations, a natural solution of the Grassmann problem is achieved. Furthermore, the resulting equations obviate the need to evaluate large determinants. The elimination of anti-commutators means that the technique is far more efficient than previous QMC and stochastic fermion methods [7]. We give examples in cases of experimental relevance involving the dynamical problem of Pauli blocking in molecular dissociation, and finite temperature correlations of fermions in an optical lattice, where the ...

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Introduction to quantum Monte Carlo simulations for fermionic systems

Cited by 144 publications

References 55 publications

Geometry dependence of the sign problem in quantum Monte Carlo simulations

Geometry dependence of the sign problem in quantum Monte Carlo simulations

Gaussian phase-space representations for fermions

Gaussian Quantum Monte Carlo Methods for Fermions and Bosons

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