The ALF (Algorithms for Lattice Fermions) project release 1.0.  Documentation for the auxiliary field quantum Monte Carlo code

Bercx, Martin; Goth, Florian; Hofmann, Johannes S.; Assaad, Fakher F.

doi:10.21468/scipostphys.3.2.013

Cited by 84 publications

(86 citation statements)

References 89 publications

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“…[51]. For the implementation we have used the ALF library [52]. To carry out the analytic continuation, we have used the stochastic MaxEnt implementation [53,54] of the ALF library [52].…”

Section: Lattice Quantum Monte Carlo (Qmc)mentioning

confidence: 99%

Hubbard model on the honeycomb lattice: From static and dynamical mean-field theories to lattice quantum Monte Carlo simulations

et al. 2020

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We study the one-band Hubbard model on the honeycomb lattice using a combination of quantum Monte Carlo (QMC) simulations and static as well as dynamical mean-field theory (DMFT). This model is known to show a quantum phase transition between a Dirac semi-metal and the antiferromagnetic insulator. The aim of this article is to provide a detailed comparison between these approaches by computing static properties, notably ground-state energy, single-particle gap, double occupancy, and staggered magnetization, as well as dynamical quantities such as the single-particle spectral function. At the static mean-field level local moments cannot be generated without breaking the SU(2) spin symmetry. The DMFT approximation accounts for temporal fluctuations and captures the local moment formation in the paramagnetic phase. As a consequence, the DMFT approximation is found to be very accurate in the Dirac semi-metallic phase where local moment formation is present and the spin correlation length small. However, in the vicinity of the fermion quantum critical point the spin correlation length diverges and the spontaneous SU(2) symmetry breaking leads to low-lying Goldstone modes in the magnetically ordered phase. The impact of these spin fluctuations on the single particle spectral function -waterfall features and narrow spin-polaron bands -is only visible in the lattice QMC approach. arXiv:1908.04307v1 [cond-mat.str-el]

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“…[51]. For the implementation we have used the ALF library [52]. To carry out the analytic continuation, we have used the stochastic MaxEnt implementation [53,54] of the ALF library [52].…”

Section: Lattice Quantum Monte Carlo (Qmc)mentioning

confidence: 99%

Hubbard model on the honeycomb lattice: From static and dynamical mean-field theories to lattice quantum Monte Carlo simulations

et al. 2020

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“…In order to make a reasonable comparison, we developed a discrete time version of our method. Fakher Assaad was willing to share with us the results for the t-V using the Algorithms for Lattice Fermions (ALF) project [11], which employs the finite temperature ), and the auxiliary field algorithm (ALF-AUX). The solid lines are τ 1 " 7.57467ˆ10´8L 7 for the fermion bag algorithm and τ 2 " 3.93116ˆ10´7L 7 for the auxiliary field algorithm.…”

Section: Comparison With Auxiliary Field Methodsmentioning

confidence: 99%

“…While this re-expression is a simple way to always allow for Monte Carlo calculations, it leads to a method that requires computations that scale exponentially with the system size. To see why, recall that a partition function, Z, is related to a system's free energy density, f , by 11) where N is the number of spatial sites in the system [67]. Thus we can also expect…”

Section: The Sign Problemmentioning

confidence: 99%

Fermion Bag Approach to Lattice Hamiltonian Field Theories

Huffman

2018

EPJ Web Conf.

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Fermion Bag Approach for Hamiltonian Lattice Field TheoriesAbstract Understanding the critical behavior near quantum critical points for strongly correlated quantum many-body systems remains intractable for the vast majority of scenarios. Challenges involve determining if a quantum phase transition is first-or second-order, and finding the critical exponents for second-order phase transitions.Learning about where second-order phase transitions occur and determining their critical exponents is particularly interesting, because each new second-order phase transition defines a new quantum field theory.Quantum Monte Carlo (QMC) methods are one class of techniques that, when applicable, offer reliable ways to extract the nonperturbative physics near strongly coupled quantum critical points. However, there are two formidable bottlenecks to the applicability of QMC: (1) the sign problem and (2) algorithmic update inefficiencies. In this thesis, I overcome both these difficulties for a class of problems by extending the fermion bag approach recently developed by Shailesh Chandrasekharan to the Hamiltonian formalism and by demonstrating progress using the example of a specific quantum system known as the t-V model, which exhibits a transition from a semimetal to an insulator phase for a single flavor of four-component Dirac fermions.I adapt the fermion bag approach, which was originally developed in the context of Lagrangian lattice field theories, to be applicable within the Hamiltonian formalism, and demonstrate its success in two ways: first, through solutions to new sign iv problems, and second, through the development of new efficient QMC algorithms. In addressing the first point, I present a solution to the sign problem for the t-V model.While the t-V model is the simplest Gross-Neveu model of the chiral Ising universality class, the specter of the sign problem previously prevented its simulation with QMC for 30 years, and my solution initiated the first QMC studies for this model.The solution is then extended to many other Hamiltonian models within a class that involves fermions interacting with quantum spins. Some of these models contain an interesting quantum phase transition between a massless/semimetal phase to a massive/insulator phase in the so called Gross-Neveu universality class. Thus, the new solutions to the sign problem allow for the use of the QMC method to study these universality classes.The second point is addressed through the construction of a Hamiltonian fermion bag algorithm. The algorithm is then used to compute the critical exponents for the second-order phase transition in the t-V model. By pushing the calculations to significantly larger lattice sizes than previous recent computations (64 2 sites versus 24 2 sites), I am able to compute the critical exponents more reliably here compared to earlier work. I show that the inclusion of these larger lattices causes a significant shift in the values of the critical exponents that was not evident for the smaller lattices. This shift puts the critic...

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“…We find that N τ = 256 is large enough to probe both the low-temperature regime as well as the continuum limit in Euclidean time simultaneously. We further note that the state-of-the-art QMC algorithm for condensed matter systems, BSS-QMC, taken from the ALF package [41], experiences exponential decay of the average sign at these parameters, even in the optimal regime where the discrete auxiliary field is coupled to spin. It is thus apparent that the sign problem is already strong in this regime.…”

Section: Hmc With Gradient Flowmentioning

confidence: 96%

Lefschetz thimbles decomposition for the Hubbard model on the hexagonal lattice

2020

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In this article we study the properties of the Lefschetz thimbles decomposition for the Hubbard model on the hexagonal lattice both at zero chemical potential and away from half-filling. We start from the detection of saddle points, for real and complex-valued fields. A Schur complement solver and the exact computation of the derivatives of the fermion determinant are employed. These technical improvements allows us to look at lattice volumes as large as 12 × 12 with Nτ = 256 steps in Euclidean time, in order to capture the properties of the thimbles decomposition as the thermodynamic, low temperature and continuum limits are approached. Different versions of the Hubbard-Stratonovich (HS) decomposition were studied and we show that the complexity of the thimbles decomposition is very dependent on its specific form. In particular, we demonstrate the existence of an optimal regime, with a reduced number of thimbles becoming important in the overall sum. In order to check these findings, we have performed quantum Monte Carlo (QMC) simulations using the holomorphic gradient flow to deform the integration contour into the complex plane. Several benchmark calculations were made on small volumes (Ns = 8 sites in space), albeit still at low temperatures and with the chemical potential tuned to the van Hove singularity, thus entering into a regime where standard QMC techniques exhibit exponential decay of the average sign. The results are compared versus exact diagonalization (ED), and we demonstrate the importance of choosing an optimal form for the HS transformation to avoid issues associated with ergodicity. We compare the residual sign problem with the state-of-the-art BSS(Blankenbecler, Scalapino and Sugar)-QMC and show that the average sign can be kept substantially higher using the Lefschetz thimbles approach.

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The ALF (Algorithms for Lattice Fermions) project release 1.0. Documentation for the auxiliary field quantum Monte Carlo code

Cited by 84 publications

References 89 publications

Hubbard model on the honeycomb lattice: From static and dynamical mean-field theories to lattice quantum Monte Carlo simulations

Hubbard model on the honeycomb lattice: From static and dynamical mean-field theories to lattice quantum Monte Carlo simulations

Fermion Bag Approach to Lattice Hamiltonian Field Theories

Lefschetz thimbles decomposition for the Hubbard model on the hexagonal lattice

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