Diagrammatic Monte Carlo for correlated fermions

Kozik, Evgeny; Houcke, Kris Van; Gull, Emanuel; Pollet, Lode; Prokof’ev, Nikolay; Svistunov, Boris; Troyer, Matthias

doi:10.1209/0295-5075/90/10004

Cited by 147 publications

(143 citation statements)

References 29 publications

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“…The DiagMC technique [4,39,40] evaluates a weakcoupling expansion for the self-energy (k,iω n ) up to relatively high orders by means of stochastic sampling. In contrast to the approximate schemes discussed above, all diagram topologies are included.…”

Section: Diagrammatic Monte Carlomentioning

confidence: 99%

“…By varying N * and monitoring the convergence of the selfenergy, the accessible parameter regime can be determined and the errors can be controlled. We use an expansion in terms of bare propagators which is typically found to converge towards the correct solution in the weak-coupling regime U 4t, wherever numerically exact benchmarks are available [32,40].…”

Section: Diagrammatic Monte Carlomentioning

confidence: 99%

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On the dangers of partial diagrammatic summations: Benchmarks for the two-dimensional Hubbard model in the weak-coupling regime

Gukelberger

Werner

2015

Phys. Rev. B

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We study the two-dimensional Hubbard model in the weak-coupling regime and compare the self-energy obtained from various approximate diagrammatic schemes to the result of diagrammatic Monte Carlo simulations, which sum up all weak-coupling diagrams up to a given order. While dynamical mean-field theory provides a good approximation for the local part of the self-energy, including its frequency dependence, the partial summation of bubble and/or ladder diagrams typically yields worse results than second-order perturbation theory. Even widely used self-consistent schemes such as GW or the fluctuation-exchange approximation (FLEX) are found to be unreliable. Combining the dynamical mean-field self-energy with the nonlocal component of GW in GW + DMFT yields improved results for the local self-energy and nonlocal self-energies of the correct order of magnitude, but here, too, a more reliable scheme is obtained by restricting the nonlocal contribution to the second-order diagram. FLEX + DMFT is found to give accurate results in the low-density regime, but even worse results than FLEX near half-filling.

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Section: Diagrammatic Monte Carlomentioning

confidence: 99%

Section: Diagrammatic Monte Carlomentioning

confidence: 99%

On the dangers of partial diagrammatic summations: Benchmarks for the two-dimensional Hubbard model in the weak-coupling regime

Gukelberger

Werner

2015

Phys. Rev. B

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“…Major recent achievements of DiagMC are the study of the normal phase of the unitary fermi gas [13], the determination of the ground state phase diagram of the Hubbard model up to filling factor 0.7 and interactions U/t ≤ 4 [14], and the settlement of the Bose-metal issue [15]. Given the power and the versatility of the technique, systematic diagrammatic extensions of Dynamical Mean Field Theory are now being studied [16,17].In DiagMC [10,11,18,19] one expresses the perturbative expansion in terms of connected Feynman diagrams, which can be written directly in the TL. One then performs a random walk in the space of topologies and of integration variables of Feynman diagrams.…”

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confidence: 99%

Determinant Diagrammatic Monte Carlo Algorithm in the Thermodynamic Limit

Rossi¹

2017

Phys. Rev. Lett.

118

161

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We present a simple trick that allows to consider the sum of all connected Feynman diagrams at fixed position of interaction vertices for general fermionic models, such that the thermodynamic limit can be taken analytically. With our approach one can achieve superior performance compared to conventional Diagrammatic Monte Carlo, while rendering the algorithmic part dramatically simpler. By considering the sum of all connected diagrams at once, we allow for massive cancellations between different diagrams, greatly reducing the sign problem. In the end, the computational effort increase only exponentially with the order of the expansion, which should be contrasted with the factorial growth of the standard diagrammatic technique. We illustrate the efficiency of the technique for the two-dimensional Fermi-Hubbard model. Finding an efficient method to solve the quantum many-body problem is of fundamental physical importance. A promising strategy to gain insight is represented by quantum simulators. For example, experimentalists working with cold atoms in optical lattices are able to realise one of the most prominent models of stronglycorrelated electrons, the Hubbard model. A study of its equation of state at low-temperature has been reported in [1]. Moreover, short-range antiferromagnetic correlations have been observed [2][3][4][5][6], and, very recently, even a long-range antiferromagnetic state was realized [7].From the theoretical side, making predictions for strongly-correlated fermionic systems is a very challenging problem. Quantum Monte Carlo methods are affected by the infamous sign problem when dealing with fermionic models, which in general precludes reaching low-temperature and large system sizes (see [8,9] and references therein). The simulation time to obtain a given precision scales exponentially with the system size and the inverse of the temperature. In the strongly correlated regime, it is then very difficult to extrapolate to the Thermodynamic Limit (TL). However, the fermionic sign gives an unexpected advantage when considering fermionic perturbation theory. For bosonic theories, typically, the perturbative expansion has a zero radius of convergence. This is due to the factorial number of Feynman diagrams all contributing with the same sign. For fermions, strong cancellations between different diagrams at fixed order lead in general to a finite convergence radius on a lattice at finite temperature. This happens because for sufficiently small interactions of whatever sign (or phase) the system is stable on a lattice at non-zero temperature. This was seen numerically [10,11], and even proved mathematically for the Hubbard model at high enough temperature [12]. Perturbation theory is therefore a powerful tool to study fermionic theories. By * riccardo.rossi@ens.fr analytic continuation, any point of the phase diagram which is in the same phase as the perturbative starting point can be reached in principle, provided that one has a method to compute the diagrammatic series to high order. At the moment...

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“…In experiments, a Mott insulating phase has been realized 8,9 , and the detection of a Néel state seems feasible in the not too distant future. Progress in simulation methods 10,11 13 .Even more intriguing physics than in the Hubbard model is expected in shallow optical lattices, where-in analogy to materials-a rich variety of phases is expected as the bandgap closes and multi-band and orbital effects become important. Although shallow lattices are easy to achieve experimentally by reducing the laser amplitude, they pose a challenge to theoretical treatments.…”

mentioning

confidence: 99%

“…In experiments, a Mott insulating phase has been realized 8,9 , and the detection of a Néel state seems feasible in the not too distant future. Progress in simulation methods 10,11 13 .…”

mentioning

confidence: 99%

Density functional theory for atomic Fermi gases

et al. 2012

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The interplay between interaction and inhomogeneity for electrons in solids generates many interesting phenomena, including insulating and metallic behaviour, magnetism, superconductivity, quantum criticality and more exotic phases 1 . Many of the same phenomena appear in ultracold fermionic atoms in optical lattices 2 , which provide clean, controlled and tunable 'quantum simulators' to explore the intriguing physics of fermionic systems. Although density functional theory 3-5 (DFT) is widely used to calculate material properties 6 , it has not yet been applied to cold atomic gases in optical lattices. Here we present a new density functional for short-range interactions (as opposed to Coulomb interactions of electrons), which renders DFT suitable for atomic Fermi gases. This grants us access to an extensive toolset, previously developed for materials simulations, to calculate the static and dynamic properties of atomic Fermi gases in optical lattices and external potentials. Ultracold atom quantum simulators can in turn be used to explore limitations of DFT functionals, and to further improve hybrid functionals, thus forming a bridge between materials simulations and atomic physics.Ultracold atomic gases have several advantages over materials for systematically exploring fermionic quantum systems. In materials, strong interactions go hand in hand with increasing localization of electrons as one moves from weakly correlated materials with s and p electrons to transition metal compounds with d electrons, and lanthanides and actinides with even more localized f orbitals. In contrast, all relevant parameters can be almost arbitrarily changed for fermionic gases in optical lattices: increasing the intensity of the optical lattice tunes continuously from shallow lattices with extended wave functions to deep lattices with almost localized Wannier functions, and the band structure can be further modified by changing the laser wave form. The strength of the inter-atomic interactions can independently be varied by tuning an external magnetic field across Feshbach resonances 7 . This facilitates a controlled realization of intriguing quantum phenomena in fermionic systems, and the systematic exploration of interaction and localization effects. The simplicity of atomic gases, the absence of core electrons and the ability to realize strongly interacting systems already in the lowest band makes optical lattice systems also much easier to treat from a theoretical point of view, avoiding the need for expensive all-electron calculations including the core electrons or the use of less reliable pseudopotentials incorporating their effects.So far, the experimental and theoretical focus has been on deep optical lattices, which are described well by single-band Hubbard models and where exciting progress has been achieved. In experiments, a Mott insulating phase has been realized 8,9 , and the detection of a Néel state seems feasible in the not too distant future. Progress in simulation methods 10,11 13 .Even more intriguing physics...

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Diagrammatic Monte Carlo for correlated fermions

Cited by 147 publications

References 29 publications

On the dangers of partial diagrammatic summations: Benchmarks for the two-dimensional Hubbard model in the weak-coupling regime

On the dangers of partial diagrammatic summations: Benchmarks for the two-dimensional Hubbard model in the weak-coupling regime

Determinant Diagrammatic Monte Carlo Algorithm in the Thermodynamic Limit

Density functional theory for atomic Fermi gases

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