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arXiv:1303.6813 · Physical Review B

Semiclassical approximation solved by Monte Carlo integration as an efficient impurity solver for dynamical mean field theory and its cluster extensions

Hunpyo Lee, Yu‐Zhong Zhang, Hoonkyung Lee, Yongkyung Kwon, Harald O. Jeschke, Roser Valentí

Published Oct 18, 2013• 5 citations
arXivHTMLPDF

Abstract

We propose that a combination of the semiclassical approximation with Monte Carlo simulations can be an efficient and reliable impurity solver for dynamical mean field theory equations and their cluster extensions with large cluster sizes. In order to show the reliability of the method, we consider two test cases: (i) the single-band Hubbard model within the dynamical cluster approximation with four- and eight-site clusters and (ii) the anisotropic two-orbital Hubbard model with orbitals of different bandwidth within the single-site dynamical mean field theory. We compare the critical interaction ${U}_{c}/t$ with those obtained from solving the dynamical mean field equations with the continuous-time and Hirsch-Fye quantum Monte Carlo approaches. In both test cases we observe reasonable values of the metal-insulator critical interaction strength ${U}_{c}/t$ and the nature of Mott physics in the self-energy behavior. While some details of the spectral functions cannot be captured by the semiclassical approximation due to the freezing of dynamical fluctuations, the main features are reproduced by the approach.

DOI
10.1103/physrevb.88.165126

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We propose that a combination of the semiclassical approximation with Monte Carlo simulations can be an efficient and reliable impurity solver for dynamical mean field theory equations and their cluster extensions with large cluster sizes. In order to show the reliability of the method, we consider two test cases: (i) the single-band Hubbard model within the dynamical cluster approximation with four- and eight-site clusters and (ii) the anisotropic two-orbital Hubbard model with orbitals of different bandwidth within the single-site dynamical mean field theory. We compare the critical interaction Uc/t{U}_{c}/tUc​/t with those obtained from solving the dynamical mean field equations with the continuous-time and Hirsch-Fye quantum Monte Carlo approaches. In both test cases we observe reasonable values of the metal-insulator critical interaction strength Uc/t{U}_{c}/tUc​/t and the nature of Mott physics in the self-energy behavior. While some details of the spectral functions cannot be captured by the semiclassical approximation due to the freezing of dynamical fluctuations, the main features are reproduced by the approach.
An investigation is presented of the utility of semiclassical approximations for solving the quantum-impurity problems arising in the dynamical-mean-field approach to correlated-electron models. The method is based on performing an exact numerical integral over the zero-Matsubara-frequency component of the spin part of a continuous Hubbard-Stratonovich field, along with a spin-field-dependent steepest descent treatment of the charge part. We test this method by applying it to one- or two-site approximations to the single-band Hubbard model with different band structures, and comparing the results to quantum Monte Carlo and simplified exact diagonalization calculations. The resulting electron self-energies, densities of states, and magnetic transition temperatures show reasonable agreement with the quantum Monte Carlo simulation over wide parameter ranges, suggesting that the semiclassical method is useful for obtaining a reasonable picture of the physics in situations where other techniques are too expensive.
We develop a projective quantum Monte Carlo algorithm of the Hirsch-Fye type for obtaining ground state properties of the Anderson impurity model. This method is employed to solve the self-consistency equations of dynamical mean field theory. It is shown that the approach converges rapidly to the ground state so that reliable zero-temperature results are obtained. As a first application, we study the Mott-Hubbard metal-insulator transition of the frustrated one-band Hubbard model, reconfirming the numerical renormalization group results.
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Accurate and reliable algorithms to solve complex impurity problems are instrumental to a routine use of quantum embedding methods for material discovery. In this context, we employ an efficient selected configuration-interaction impurity solver to investigate the role of bath discretization---specifically, bath size and parametrization---in Hamiltonian-based cluster dynamical mean-field theory (CDMFT) for the one- and two-orbital Hubbard models. We consider two- and four-site clusters for the single-orbital model and a two-site cluster for the two-orbital model. Our results demonstrate that, for small bath sizes, the choice of parametrization can significantly influence the solution, highlighting the importance of systematic convergence checks. Comparing different bath parametrizations not only reveals the robustness of a given solution but can also provide insights into the nature of different solutions and potential instabilities of the paramagnetic state. We present an extensive analysis of the zero-temperature Mott transition of the paramagnetic half-filled single-band Hubbard model, benchmarking our findings against previous literature. We find that, for the single-band model the dependence on parametrization is weak for the largest bath sizes accessible with ASCI, while a tendency towards a nematic solution can be seen when the bath size is small. Building on this, we extend our study to the multiband regime, where we present a systematic analysis at zero temperature for two orbitals and a two-site cluster. This setup allows us to assess the effect of nearest-neighbor dynamical correlations on the multi-orbital Mott transition. In this case, some quantitative dependence on the parametrization is retained for the two-orbital model, for instance in the value of the critical interaction for a Mott transition.
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