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arXiv:1601.03838 · Physical review. B./Physical review. B

Interlaced coarse-graining for the dynamic cluster approximation

Peter Staar, Mi Jiang, Urs R. Hähner, T. C. Schulthess, Thomas Maier

Published Apr 27, 2016• 12 citations
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Abstract

The dynamical cluster approximation (DCA) and its ${\mathrm{DCA}}^{+}$ 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 ${\mathrm{DCA}}^{+}$ algorithm permits the determination of the superconducting ${T}_{c}$ on cluster sizes for which the results can be fit with a Kosterlitz-Thouless scaling law.

DOI
10.1103/physrevb.93.165144

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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.
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 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 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.
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.
We derive a cluster mean-field theory for an Ising Hamiltonian using a cluster-lattice Fourier transform with a cluster of size Nc{N}_{\mathrm{c}}Nc​ and a coarse-grained (CG) lattice into cells of size Ncell{N}_{\mathrm{cell}}Ncell​. We explore forms with ${N}_{\mathrm{cell}}\ensuremath{\geqslant}{N}_{\mathrm{c}}$, including a non-CG (NCG) version with ${N}_{\mathrm{cell}}\ensuremath{\rightarrow}\ensuremath{\infty}$. For Nc=Ncell{N}_{\text{c}}={N}_{\mathrm{cell}}Nc​=Ncell​, the set of static, self-consistent equations relating cluster and CG lattice correlations is analogous to that in dynamical cluster approximation and cellular dynamical mean-field theory used in correlated electron physics. A variational Nc{N}_{\mathrm{c}}Nc​-site cluster grand potential based on Nc=Ncell{N}_{\mathrm{c}}={N}_{\mathrm{cell}}Nc​=Ncell​ CG lattice maintains thermodynamic consistency and improves predictions, recovering Monte Carlo and series expansion results upon finite-size scaling; notably, the Nc=1{N}_{\mathrm{c}}=1Nc​=1 CG results already predict well the first- and second-order phase boundary topology and transition temperatures for frustrated lattices. The NCG version is significantly faster computationally than the CG case and more accurate at fixed Nc{N}_{\mathrm{c}}Nc​ for ferromagnetism, which is potentially useful for cluster expansion and quantum cluster applications.
We have designed a multiscale approach for strongly correlated systems by combining the dynamical cluster approximation (DCA) and the recently introduced dual fermion formalism. This approach employs an exact mapping from a real lattice to a DCA cluster of linear size Lc{L}_{c}Lc​ embedded in a dual fermion lattice. Short-length-scale physics is addressed by the DCA cluster calculation, while longer-length-scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as O(1/Lc)\mathcal{O}(1/{L}_{c})O(1/Lc​), so perturbation theory on the dual lattice converges very quickly, e.g., the dual Fermion self-energy calculated with simple second-order perturbation theory is of order O(1/Lc3)\mathcal{O}(1/{L}_{c}^{3})O(1/Lc3​) with third-order and three-body corrections down by an additional factor of O(1/Lc)\mathcal{O}(1/{L}_{c})O(1/Lc​).