Dynamical structure factor of the 
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 Heisenberg model in one dimension: The variational Monte Carlo approach

Ferrari, Francesco; Parola, Alberto; Sorella, Sandro; Becca, Federico

doi:10.1103/physrevb.97.235103

Cited by 53 publications

(65 citation statements)

References 45 publications

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“…The Lehman representation can be evaluated explicitly with ED since we have a complete and exact representation of the Hamiltonian eigenstates, but this technique is limited to small clusters. To evaluate the Green's function for the cases not amenable to the exact diagonalization, we can use a method similar to the approach used to calculate the spin and charge dynamical structure factors by exhausting an important subspace of the Hilbert space for the excitations [24][25][26][27][28]. In this framework, the time evolutions by the Hamiltonian in the N − 1 particle sector for G h σ ðk; ωÞ [Eq.…”

Section: A Green's Functionmentioning

confidence: 99%

“…The recent formulation of the time-dependent variational Monte Carlo method based on the variational principle opened a way to study the long-time dynamics [21,22], but it has not been extensively applied yet to interacting fermion systems except for in a few examples [23]. Meanwhile, methods of calculating the spin and charge dynamical structure factors utilizing the variational wave functions for ground and excited states have been proposed recently [24][25][26][27][28]. Some attempts have been made to calculate the excitation spectrum on larger clusters of the t-J model [29,30].…”

Section: Introductionmentioning

confidence: 99%

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Single-Particle Spectral Function Formulated and Calculated by Variational Monte Carlo Method with Application to d -Wave Superconducting State

Charlebois

Imada

2020

Phys. Rev. X

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A method to calculate the one-body Green's function for ground states of correlated electron materials is formulated by extending the variational Monte Carlo method. We benchmark against the exact diagonalization (ED) for the one-and two-dimensional Hubbard models of 16-site lattices, which proves high accuracy of the method. The application of the method to a larger-sized Hubbard model on the square lattice correctly reproduces the Mott insulating behavior at half-filling and gap structures of the d-wave superconducting state of the hole-doped Hubbard model in the ground state optimized by enforcing the charge uniformity, evidencing a wide applicability to strongly correlated electron systems. From the obtained d-wave superconducting gap of the charge-uniform state, we find that the gap amplitude at the antinodal point is several times larger than the experimental value when we employ a realistic parameter as a model of the cuprate superconductors. The effective attractive interaction of carriers in the d-wave superconducting state inferred for an optimized state of the Hubbard model is as large as the order of the nearest-neighbor transfer, which is far beyond the former expectation in the cuprates. We discuss the nature of the superconducting state of the Hubbard model in terms of the overestimate of the gap and the attractive interaction in comparison to the cuprates.

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Section: A Green's Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Single-Particle Spectral Function Formulated and Calculated by Variational Monte Carlo Method with Application to d -Wave Superconducting State

Charlebois

Imada

2020

Phys. Rev. X

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“…Although extremely powerful, one of the shortcomings of real space QMC methods is that there is less error cancellation than is typically observed in basis set quantum chemistry. This fact, coupled with the recent success of both AFQMC and FCIQMC (both of which work in a finite basis set), has led to renewed interest in development of new QMC algorithms that work in a finite basis set [20][21][22][23][24][25][26][27] . The present work is an attempt in this direction and presents algorithmic improvements for performing orbital space VMC calculations.To put the orbital space VMC in the broader context of wavefunction methods, it is useful to classify the various wavefunctions used in electronic structure theory into three classes.…”

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confidence: 99%

Improved Speed and Scaling in Orbital Space Variational Monte Carlo

Sabzevari

Sharma

2018

J. Chem. Theory Comput.

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In this work, we introduce three algorithmic improvements to reduce the cost and improve the scaling of orbital space variational Monte Carlo (VMC). First, we show that by appropriately screening the one-and two-electron integrals of the Hamiltonian one can improve the efficiency of the algorithm by several orders of magnitude. This improved efficiency comes with the added benefit that the resulting algorithm scales as the second power of the system size O(N 2 ), down from the fourth power O(N 4 ). Using numerical results, we demonstrate that the practical scaling obtained is in fact O(N 1.5 ) for a chain of Hydrogen atoms, and O(N 1.2 ) for the Hubbard model. Second, we introduce the use of the rejection-free continuous time Monte Carlo (CTMC) to sample the determinants. CTMC is usually prohibitively expensive because of the need to calculate a large number of intermediates. Here, we take advantage of the fact that these intermediates are already calculated during the evaluation of the local energy and consequently, just by storing them one can use the CTCM algorithm with virtually no overhead. Third, we show that by using the adaptive stochastic gradient descent algorithm called AMSGrad one can optimize the wavefunction energies robustly and efficiently. The combination of these three improvements allows us to calculate the ground state energy of a chain of 160 hydrogen atoms using a wavefunction containing ∼ 2 × 10 5 variational parameters with an accuracy of 1 mE h /particle at a cost of just 25 CPU hours, which when split over 2 nodes of 24 processors each amounts to only about half hour of wall time. This low cost coupled with embarrassing parallelizability of the VMC algorithm and great freedom in the forms of usable wavefunctions, represents a highly effective method for calculating the electronic structure of model and ab initio systems.Quantum Monte Carlo (QMC) is one of the most powerful and versatile tools for solving the electronic structure problem and has been successfully used in a wide range of problems 1-5 . QMC can be broadly classified into two categories of algorithms, variational Monte Carlo (VMC) 6,7 and projector Monte Carlo (PMC). In VMC one is interested in minimizing the energy of a suitably chosen wavefunction ansatz. The accuracy of VMC is limited by the functional form and the flexibility of the wavefunction employed. PMC on the other hand is potentially exact and several variants exist, such as diffusion Monte Carlo (DMC) 8-10 , Auxiliary field quantum Monte Carlo (AFQMC) 11,12 , Green's function Monte Carlo (GFMC) 13-15 and full configuration interaction quantum Monte Carlo (FCIQMC) [16][17][18][19] . Although exact in principle, in practice all versions of PMC suffer from the fermionic sign problem when used for electronic problems, except in a few special cases. A commonly used technique for overcoming the fermionic sign problem is to employ the fixed-node or fixed-phase approximation that stabilizes the Monte Carlo simulation by eliminating the sign problem at a cost of introduc...

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“…Later developments evidenced that both classical and quantum ground states of the triangular lattice actually displayed long-range order in the so-called 120 • configuration [6]. Among an intense search for quantum spin liquids in models and materials [7,8], considerable attention has been devoted experimentally [9][10][11][12] and theoretically [13][14][15] to the so-called kagome lattice, which is obtained by removing 1/4 of the spins from a triangular lattice, leading to a lower connectivity of four nearest neighbors instead of six, which enhances fluctuations. It has been argued theoretically that the S = 1/2 kagome Heisenberg antiferromagnet shows a spin-liquid ground state [13,16,17], and it is still debated whether the resulting states should be gapped or not [18][19][20].…”

Section: Introductionmentioning

confidence: 99%