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arXiv:1712.10304 · Physical review. B./Physical review. B

Determinant Monte Carlo algorithms for dynamical quantities in fermionic systems

Alice Moutenet, Wéi Wú, Michel Ferrero

Published Feb 12, 2018• 49 citations
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Abstract

We introduce and compare three different Monte Carlo determinantal algorithms that allow one to compute dynamical quantities, such as the self-energy, of fermionic systems in their thermodynamic limit. We show that the most efficient approach expresses the sum of a factorial number of one-particle-irreducible diagrams as a recursive sum of determinants with exponential complexity. By comparing results for the two-dimensional Hubbard model with those obtained from state-of-the-art diagrammatic Monte Carlo, we show that we can reach higher perturbation orders and greater accuracy for the same computational effort.

DOI
10.1103/physrevb.97.085117

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We introduce and compare three different Monte Carlo determinantal algorithms that allow one to compute dynamical quantities, such as the self-energy, of fermionic systems in their thermodynamic limit. We show that the most efficient approach expresses the sum of a factorial number of one-particle-irreducible diagrams as a recursive sum of determinants with exponential complexity. By comparing results for the two-dimensional Hubbard model with those obtained from state-of-the-art diagrammatic Monte Carlo, we show that we can reach higher perturbation orders and greater accuracy for the same computational effort.
We introduce a Monte Carlo determinantal algorithm that allows to efficiently sample dynamical quantities, such as the self-energy, of fermionic quantum systems in their thermodynamic limit. The method expresses the sum of a factorial number of one-particle-irreducible diagrams as a recursive sum of determinants with exponential complexity. By comparing results of this algorithm for the lattice Hubbard model with those obtained from diagrammatic Monte Carlo and the recently introduced connected determinant approach, we show that we can reach higher perturbation orders and greater accuracy for the same computational effort.
We present a simple trick that allows us 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 algorithm, 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 increases 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.
We introduce a numerically exact method for efficient summation of irreducible Feynman diagrams for the fermionic self-energy in the thermodynamic limit. The technique, based on the Diagrammatic Determinant Monte Carlo, allows us to reach unprecedentedly high (∼10\sim 10∼10) orders of the weak-coupling expansion for the self-energy of the two-dimensional Hubbard model. Access to high orders reveals a non-trivial analytic structure of the self-energy and enables its controlled reconstruction with arbitrary momentum resolution in the non-perturbative regime of essentially strong correlations, which has recently been reached with ultracold atoms in optical lattices.
We develop a numerically exact method for the summation of irreducible Feynman diagrams for fermionic self-energy in the thermodynamic limit. The technique, based on the diagrammatic determinant Monte Carlo and its recent extension to connected diagrams, allows us to reach high $(\ensuremath{\sim}10)$ orders of the weak-coupling expansion for the self-energy of the two-dimensional Hubbard model. Access to high orders reveals a nontrivial analytic structure of the self-energy and enables its controlled reconstruction with arbitrary momentum resolution in the nonperturbative regime of essentially strong correlations, which has recently been reached with ultracold atoms in optical lattices.
Exponential observables, formulated as $ln\ensuremath{\langle}{e}^{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}}\ensuremath{\rangle}$ where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}$ is an extensive quantity, play a critical role in the study of quantum many-body systems, examples of which include the free energy and entanglement entropy. Given that eX{e}^{X}eX becomes exponentially large (or small) in the thermodynamic limit, the accurate computation of the expectation value of this exponential quantity presents a significant challenge. In this paper, we propose a comprehensive algorithm to quantify these observables in interacting fermion systems, utilizing the determinant quantum Monte Carlo method. We have applied this algorithm to the two-dimensional square-lattice half-filled Hubbard model and $\ensuremath{\pi}$-flux t-V model. In the Hubbard model case at the strong-coupling limit, our method showcases a significant accuracy improvement on free energy compared to conventional methods that are derived from the internal energy, and in the t-V model, we indicate that the free energy offers a precise determination of the second-order phase transition. We also illustrate that this approach delivers highly efficient and precise measurements of the nthn\mathrm{th}nth R\'enyi entanglement entropy. Even more noteworthy is that this improvement comes without incurring increases in computational complexity. This algorithm effectively suppresses exponential fluctuations and can be easily generalized to other models.
This thesis develops diagrammatic methods for thermodynamic and subsystem quantities of interacting lattice fermions, providing reference calculations for quantum simulators and many-body approximations. I formulate an inverse-flavour expansion about a self-consistent Hartree saddle, enumerate non-Gaussian diagrams, and use a discrete Lehmann representation to integrate selected internal vertices analytically. Density results cover 31 state points of the six-flavour Hubbard model on a 70×7070\times7070×70 lattice at T/t=0.15T/t=0.15T/t=0.15. I generalize combinatorial summation to flavour-dependent propagators and expansion shifts. CoS++ combines partial contractions to obtain connected sums for interaction subsets. A determinant-polynomial sieve evaluates the same sums through local coefficient filters; at ninth order it is 24--25 times faster than the specified CoS baselines in three-flavour dense-random-matrix benchmarks. Thermodynamic applications include a 2048-site honeycomb lattice and a three-flavour square lattice with 1024 sites. For the second Renyi entropy, I derive a fermionic replica path integral and a stable evaluation of its two-time Green functions. The connected-determinant calculation reaches tenth order on a 3×33\times33×3 cluster and ninth order on larger lattices. On 50×5050\times5050×50 sites at (U/t,μ/t,βt)=(3,1,8)(U/t,μ/t,βt)=(3,1,8)(U/t,μ/t,βt)=(3,1,8), the ninth-order continuation at shift α=1.3α=1.3α=1.3 gives S2/Ns=0.06695S_2/N_s=0.06695S2​/Ns​=0.06695 with propagated Monte Carlo uncertainty 3.0×10−53.0\times10^{-5}3.0×10−5. Comparing two expansion shifts at their respective retained orders gives an endpoint difference of 2.6×10−42.6\times10^{-4}2.6×10−4.