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arXiv:2002.06866 · Physical review. B./Physical review. B

Continuous momentum dependence in the dynamical cluster approximation

Urs R. Hähner, Thomas Maier, T. C. Schulthess

Published May 11, 2020• 4 citations
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Abstract

The dynamical cluster approximation (DCA) is a quantum cluster extension to the single-site dynamical mean-field theory that incorporates spatially nonlocal dynamic correlations systematically and nonperturbatively. The ${\mathrm{DCA}}^{+}$ algorithm addresses the cluster shape dependence of the DCA and improves the convergence with cluster size by introducing a lattice self-energy with continuous momentum dependence. However, we show that the ${\mathrm{DCA}}^{+}$ algorithm is plagued by a fundamental problem when its self-consistency equations are formulated using the bare Green's function of the cluster. This problem is most severe in the strongly correlated regime at low doping, where the ${\mathrm{DCA}}^{+}$ self-energy becomes overly metallic and local, and persists to cluster sizes where the standard DCA has long converged. In view of the failure of the ${\mathrm{DCA}}^{+}$ algorithm, we propose to complement DCA simulations with a post-interpolation procedure for single-particle and two-particle correlation functions to preserve continuous momentum dependence and the associated benefits in the DCA. We demonstrate the effectiveness of this practical approach with results for the half-filled and hole-doped two-dimensional Hubbard model.

DOI
10.1103/physrevb.101.195114

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The dynamical cluster approximation (DCA) is a quantum cluster extension to the single-site dynamical mean-field theory that incorporates spatially nonlocal dynamic correlations systematically and nonperturbatively. The DCA+{\mathrm{DCA}}^{+}DCA+ algorithm addresses the cluster shape dependence of the DCA and improves the convergence with cluster size by introducing a lattice self-energy with continuous momentum dependence. However, we show that the DCA+{\mathrm{DCA}}^{+}DCA+ algorithm is plagued by a fundamental problem when its self-consistency equations are formulated using the bare Green's function of the cluster. This problem is most severe in the strongly correlated regime at low doping, where the DCA+{\mathrm{DCA}}^{+}DCA+ self-energy becomes overly metallic and local, and persists to cluster sizes where the standard DCA has long converged. In view of the failure of the DCA+{\mathrm{DCA}}^{+}DCA+ algorithm, we propose to complement DCA simulations with a post-interpolation procedure for single-particle and two-particle correlation functions to preserve continuous momentum dependence and the associated benefits in the DCA. We demonstrate the effectiveness of this practical approach with results for the half-filled and hole-doped two-dimensional Hubbard model.
The dynamical cluster approximation (DCA) is a systematic extension beyond the single-site approximation in dynamical mean field theory, to include spatially nonlocal correlations in quantum many-body simulations of strongly correlated systems. We extend the DCA with a continuous lattice self-energy in order to achieve better convergence with cluster size. This method, which we call DCA+{}^{+}+, cures the cluster-shape dependence problems of the DCA, without suffering from causality violations of previous attempts to interpolate the cluster self-energy. A practical approach based on standard inference techniques is given to deduce the continuous lattice self-energy from an interpolated cluster self-energy. We study the pseudogap region of a hole-doped two-dimensional Hubbard model and find that, in the DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+algorithm, the self-energy and pseudogap temperature T∗{T}^{*}T∗ converge monotonously with cluster size. Introduction of a continuous lattice self-energy eliminates artificial long-range correlations and thus significantly reduces the sign problem of the quantum Monte Carlo cluster solver in the DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+algorithm compared to the normal DCA. Simulations with much larger cluster sizes thus become feasible, which, along with the improved convergence in cluster size, raises hope that precise extrapolations to the exact infinite cluster size limit can be reached for other physical quantities as well.
The DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+algorithm was recently introduced by Stear, Maier, and Schulthess [Phys. Rev. B 88, 115101 (2013)] to extend the dynamic cluster approximation (DCA) with a continuous lattice self-energy in order to achieve better convergence with cluster size. Here we extend the DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+algorithm to the calculation of two-particle correlation functions by introducing irreducible vertex functions with continuous momentum dependence consistent with the DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+self-energy. This enables a significantly more controlled and reliable study of phase transitions than with the DCA. We test the new method by calculating the superconducting transition temperature Tc{T}_{c}Tc​ in the attractive Hubbard model and show that it reproduces previous determinantal quantum Monte Carlo results. We then calculate Tc{T}_{c}Tc​ in the doped repulsive Hubbard model, for which previous DCA calculations could only access the weak-coupling (U=4tU=4tU=4t) regime for large clusters. We show that the new algorithm provides access to much larger clusters and delivers asymptotically converged results for Tc{T}_{c}Tc​ for both the weak (U=4tU=4tU=4t) and intermediate (U=7tU=7tU=7t) coupling regimes, and thereby enables the accurate determination of the exact infinite cluster size result.
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 dynamical cluster approximation (DCA) and its DCA+{\mathrm{DCA}}^{+}DCA+ extension use coarse-graining of the momentum space to reduce the complexity of quantum many-body problems, thereby mapping the bulk lattice to a cluster embedded in a dynamical mean-field host. Here, we introduce a new form of an interlaced coarse-graining and compare it with the traditional coarse-graining. While it gives a more localized self-energy for a given cluster size, we show that it leads to more controlled results with weaker cluster shape and smoother cluster size dependence, which converge to the results obtained from the standard coarse-graining with increasing cluster size. Most importantly, the new coarse-graining reduces the severity of the fermionic sign problem of the underlying quantum Monte Carlo cluster solver and thus allows for calculations on larger clusters. This enables the treatment of correlations longer ranged than those accessible with the standard coarse-graining and thus can allow for the evaluation of the exact infinite cluster size result via finite size scaling. As a demonstration, we study the hole-doped two-dimensional Hubbard model and show that the interlaced coarse-graining in combination with the extended DCA+{\mathrm{DCA}}^{+}DCA+ algorithm permits the determination of the superconducting Tc{T}_{c}Tc​ on cluster sizes for which the results can be fit with a Kosterlitz-Thouless scaling law.
This thesis is dedicated to a theoretical study of the 1-band Hubbard model in the strong coupling limit. The investigation is based on the Dynamical Cluster Approximation (DCA) which systematically restores non-local corrections to the Dynamical Mean Field approximation (DMFA). The DCA is formulated in momentum space and is characterised by a patching of the Brillouin zone where momentum conservation is only recovered between two patches. The approximation works well if k-space correlation functions show a weak momentum dependence. In order to study the temperature and doping dependence of the spin- and charge excitation spectra, we explicitly extend the Dynamical Cluster Approximation to two-particle response functions. The full irreducible two-particle vertex with three momenta and frequencies is approximated by an effective vertex dependent on the momentum and frequency of the spin and/or charge excitations. The effective vertex is calculated by using the Quantum Monte Carlo method on the finite cluster whereas the analytical continuation of dynamical quantities is performed by a stochastic version of the maximum entropy method. A comparison with high temperature auxiliary field quantum Monte Carlo data serves as a benchmark for our approach to two-particle correlation functions. Our method can reproduce basic characteristics of the spin- and charge excitation spectrum. Near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin and single-particle excitations: a collective spin mode emerges at optimal doping and sufficiently low temperatures in the spin response spectrum and exhibits the energy scale of the magnetic exchange interaction J. Simultaneously, the low energy single-particle excitations are characterised by a coherent quasiparticle with bandwidth J. The origin of the quasiparticle can be quite well understood in a picture of a more or less antiferromagnetic ordered background in which holes are dressed by spin-excitations to allow for a coherent motion. By increasing doping, all features which are linked to the spin-polaron vanish in the single-particle as well as two-particle spin response spectrum. In the second part of the thesis an analysis of superconductivity in the Hubbard model is presented. The superconducting instability is implemented within the Dynamical Cluster Approximation by essentially allowing U(1) symmetry breaking baths in the QMC calculations for the cluster. The superconducting transition temperature T_c is derived from the d-wave order parameter which is directly estimated on the Monte Carlo cluster. The critical temperature T_c is in astonishing agreement with the temperature scale estimated by the divergence of the pair-field susceptibility in the paramagnetic phase. A detailed study of the pseudo and superconducting gap is continued by the investigation of the local and angle-resolved spectral function.
Recent years have seen the development of two types of nonlocal extensions to the single-site dynamical mean field theory. On one hand, cluster approximations, such as the dynamical cluster approximation, recover short-range momentum-dependent correlations nonperturbatively. On the other hand, diagrammatic extensions, such as the dual-fermion theory, recover long-ranged corrections perturbatively. The correct treatment of both strong short-ranged and weak long-ranged correlations within the same framework is therefore expected to lead to a quick convergence of results, and offers the potential of obtaining smooth self-energies in nonperturbative regimes of phase space. In this paper, we present an exact cluster dual-fermion method based on an expansion around the dynamical cluster approximation. Unlike previous formulations, our method does not employ a coarse-graining approximation to the interaction, which we show to be the leading source of error at high temperature, and converges to the exact result independently of the size of the underlying cluster. We illustrate the power of the method with results for the second-order cluster dual-fermion approximation to the single-particle self-energies and double occupancies.