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arXiv:1112.3227 · Physical Review B

Cluster-size dependence in cellular dynamical mean-field theory

志朗 酒井, Giorgio Sangiovanni, Marcello Civelli, Yukitoshi Motome, Karsten Held, Masatoshi Imada

Published Jan 4, 2012• 69 citations
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Abstract

We examine the cluster-size dependence of the cellular dynamical mean-field theory (CDMFT) applied to the two-dimensional Hubbard model. Employing the continuous-time quantum Monte Carlo method as the solver for the effective cluster model, we obtain CDMFT solutions for 4-, 8-, 12-, and 16-site clusters at a low temperature. Comparing various periodization schemes, which are used to construct the infinite-lattice quantities from the cluster results, we find that the cumulant periodization yields the fastest convergence for the hole-doped Mott insulator where the most severe size dependence is expected. We also find that the convergence is much faster around $(0,0)$ and $(\frac{\ensuremath{\pi}}{2},\frac{\ensuremath{\pi}}{2})$ than around $(\ensuremath{\pi},0)$ and $(\ensuremath{\pi},\ensuremath{\pi})$. The cumulant-periodized self-energy seems to be close to its thermodynamic limit already for a 16-site cluster in the range of parameters studied. The 4-site results remarkably agree well with the 16-site results, indicating that the previous studies based on the 4-site cluster capture the essence of the physics of doped Mott insulators.

DOI
10.1103/physrevb.85.035102

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We examine the cluster-size dependence of the cellular dynamical mean-field theory (CDMFT) applied to the two-dimensional Hubbard model. Employing the continuous-time quantum Monte Carlo method as the solver for the effective cluster model, we obtain CDMFT solutions for 4-, 8-, 12-, and 16-site clusters at a low temperature. Comparing various periodization schemes, which are used to construct the infinite-lattice quantities from the cluster results, we find that the cumulant periodization yields the fastest convergence for the hole-doped Mott insulator where the most severe size dependence is expected. We also find that the convergence is much faster around (0,0)(0,0)(0,0) and (\ensuremathπ2,\ensuremathπ2)(\frac{\ensuremath{\pi}}{2},\frac{\ensuremath{\pi}}{2})(2\ensuremathπ​,2\ensuremathπ​) than around (\ensuremathπ,0)(\ensuremath{\pi},0)(\ensuremathπ,0) and (\ensuremathπ,\ensuremathπ)(\ensuremath{\pi},\ensuremath{\pi})(\ensuremathπ,\ensuremathπ). The cumulant-periodized self-energy seems to be close to its thermodynamic limit already for a 16-site cluster in the range of parameters studied. The 4-site results remarkably agree well with the 16-site results, indicating that the previous studies based on the 4-site cluster capture the essence of the physics of doped Mott insulators.
Even after decades of intense research, the single band Hubbard model representing the fundamental model for interacting quantum systems and electronic correlations remains an unsolved cornerstone paradigm in theoretical solid state physics. Within the non-perturbative dynamical mean-field theory (DMFT), the lattice problem is mapped to a self-consistent auxiliary quantum impurity model leading to a local approximation of the self-energy. Nonlocal correlations can be included by the real-space cluster extension of the DMFT (CDMFT). Enlarging the unit cell makes the calculation of a numerically exact solution in CDMFT very challenging. We here use a quantum Monte-Carlo approach in the imaginary-time space in order to solve the self-consistency equations for different cluster sizes. In order to restore the broken translational symmetry re-periodization schemes for the Green function, the self-energy, or its cumulants have been introduced. However, these suffer from ambiguity and may even lead to convergence problems when attempted inside the self-consistency loop. The comparison to numerically exact diagrammatic quantum Monte Carlo calculations shows that introducing a so-called centerfocused extrapolation (CFE) to approximate the lattice self-energy yields very accurate results. Moreover, the CFE converges faster with the cluster size than previous periodization schemes. We here perform a detailed CDMFT analysis for the single-band Hubbard model on the 2D square lattice, reaching real-space cluster sizes of up to 9x9 sites. In addition to spectral properties, we also compute two-particle correlation functions which are not accessible in DMFT. Using benchmarks against diagrammatic Monte Carlo at high temperature, we show that the cluster spin susceptibility can be extrapolated to the exact result at large cluster size. In particular, the exponential decay of spin-spin correlations is very well captured by CDMFT calculations, even when correlations extend beyond the size of the cluster. We further present results at lower temperature T and larger UUU than the range currently accessible with diagrammatic Monte Carlo methods techniques. The CDMFT+CFE represents therefore a powerful computational tool to access the physics of non-local correlations beyond dynamical mean-field theory.
We revisit the cellular dynamical mean-field theory (CDMFT) for the single-band Hubbard model on the square lattice at half filling, reaching real-space cluster sizes of up to 9 × 9 sites.Using benchmarks against direct lattice diagrammatic Monte Carlo at high temperature, we show that the self-energy obtained from a cluster center-focused extrapolation converges faster with the cluster size than the periodization schemes previously introduced in the literature.The same benchmark also shows that the cluster spin susceptibility can be extrapolated to the exact result at large cluster size, even though its spatial extension is larger than the cluster size.
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Two widely-used non-local extensions of dynamical mean-field theory (DMFT), namely cellular DMFT (CDMFT) and the dynamical cluster approximation (DCA), both yield self-energies marred by having some unphysical properties: CDMFT yields real-space self-energies that are not translationally invariant, and DCA yields momentum-space self-energies with discontinuities in their momentum dependence. It is often desirable to remove these flaws by post-processing cluster DMFT results, using strategies called periodization for CDMFT and interpolation for DCA---for brevity, we refer to both cases as \textit{interpolation}. However, traditional interpolation approaches struggle to reliably capture intricate structures such as hole pockets in the hole-doped square-lattice Hubbard model. Further, these approaches interpolate frequency-dependent functions, which may lead to causality violations. Here, we propose \textit{Liouvillian interpolation}, a novel, intuitive, and robust scheme for interpolating cluster DMFT results. Our key idea is to interpolate frequency-\textit{independent} matrix elements of the single-particle irreducible part of the Liouvillian, obtained from a continued-fraction expansion of the cluster DMFT self-energy. We demonstrate that the ingredients of such an expansion possess a more local Fourier expansion than the functions involved in traditional interpolation schemes, and that Liouvillian interpolation conserves causality by construction. We illustrate our method for the one-dimensional Hubbard model using CDMFT, and for the two-dimensional Hubbard model using four-site CDMFT and four-patch DCA. For the latter, L\mathcal{L}L-interpolation can (depending on doping) yield Fermi and Luttinger arcs which together seem to form a closed surface.
The one-dimensional half-filled Hubbard model is considered at zero temperature within the cellular dynamical mean-field theory (CDMFT). By the computation of the spectral gap and the energy density with various cluster and bath sizes we examine the accuracy of the CDMFT in a systematic way, which proves the accurate description of the one-dimensional systems by the CDMFT with small clusters. We also calculate the spectral weights in a full range of the momentum for various interaction strengths. The results do not only account for the spin-charge separation, but they also reproduce all the features of the Bethe ansatz dispersions, implying that the CDMFT provides an excellent description of the spectral properties of low-dimensional interacting systems.
The one-dimensional Hubbard model is investigated by means of two different cluster schemes suited to introduce short-range spatial correlations beyond the single-site dynamical mean-field theory, namely, the cellular dynamical mean-field theory, which does not impose the lattice symmetries, and its periodized version in which translational symmetry is recovered. It is shown that both cluster schemes are able to describe with extreme accuracy the evolution of the density as a function of the chemical potential from the Mott insulator to the metallic state. Using exact diagonalization to solve the cluster-impurity model, we discuss the role of the truncation of the Hilbert space of the bath, and propose an algorithm that gives higher weights to the low-frequency hybridization matrix elements and improves the speed of the convergence of the algorithm.