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arXiv:2309.00674

Integrating dynamical mean-field theory and diagrammatic Monte Carlo

Johan Carlström

Published Sep 1, 2023• 0 citations
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Abstract

Dynamical mean-field theory (DMFT) is one of the most widely used theoretical methods for electronic structure calculations, providing self-consistent solutions even in low-temperature regimes, which are exact in the limit of infinite dimension. The principal limitation of this method is that it neglects spatial fluctuations, which become important in finite dimensions. Diagrammatic Monte Carlo (diagMC), by contrast, provides results that are asymptotically exact for a convergent or resummable series, but are typically limited to high temperature as they depend on the analytic structure of the expansion. In this work, we present a framework for integrating these two methods so that the diagrammatic expansion is conducted around the DMFT solution. This results in a series expansion conducted only in terms that explicitly depend on nonlocal correlations, and which is asymptotically exact.

DOI
10.48550/arxiv.2309.00674

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Dynamical mean-field theory (DMFT) is one of the most widely used theoretical methods for electronic structure calculations, providing self-consistent solutions even in low-temperature regimes, which are exact in the limit of infinite dimension. The principal limitation of this method is that it neglects spatial fluctuations, which become important in finite dimensions. Diagrammatic Monte Carlo (diagMC), by contrast, provides results that are asymptotically exact for a convergent or resummable series, but are typically limited to high temperature as they depend on the analytic structure of the expansion. In this work, we present a framework for integrating these two methods so that the diagrammatic expansion is conducted around the DMFT solution. This results in a series expansion conducted only in terms that explicitly depend on nonlocal correlations, and which is asymptotically exact.
The dual fermion approach provides a formally exact prescription for calculating properties of a correlated electron system in terms of a diagrammatic expansion around dynamical mean-field theory (DMFT). Most practical implementations, however, neglect higher-order interaction vertices beyond two-particle scattering in the dual effective action and further truncate the diagrammatic expansion in the two-particle scattering vertex to a leading-order or ladder-type approximation. In this work, we compute the dual fermion expansion for the two-dimensional Hubbard model including all diagram topologies with two-particle interactions to high orders by means of a stochastic diagrammatic Monte Carlo algorithm. We benchmark the obtained self-energy against numerically exact diagrammatic determinant Monte Carlo simulations to systematically assess convergence of the dual fermion series and the validity of these approximations. We observe that, from high temperatures down to the vicinity of the DMFT N\'eel transition, the dual fermion series converges very quickly to the exact solution in the whole range of Hubbard interactions considered ($4\ensuremath{\le}U/t\ensuremath{\le}12$), implying that contributions from higher-order vertices are small. As the temperature is lowered further, we observe slower series convergence, convergence to incorrect solutions, and ultimately divergence. This happens in a regime where magnetic correlations become significant. We find, however, that the self-consistent particle-hole ladder approximation yields reasonable and often even highly accurate results in this regime.
The bold diagrammatic Monte Carlo (BDMC) method performs an unbiased sampling of Feynman's diagrammatic series using skeleton diagrams. For lattice models the efficiency of BDMC can be dramatically improved by incorporating dynamical mean-field theory (DMFT) solutions into renormalized propagators. From the DMFT perspective, combining it with BDMC leads to an unbiased method with well-defined accuracy. We illustrate the power of this approach by computing the single-particle propagator (and thus the density of states) in the non-perturbative regime of the Anderson localization problem, where a gain of the order of 104{10}^{4}104 is achieved with respect to conventional BDMC in terms of convergence to the exact answer.
We show how to use diagrammatic techniques to compute the weak-coupling perturbation series of the self-consistent solution to a dynamical mean field theory (DMFT) problem. This approach constitutes an alternative to using diagrammatic techniques directly as an impurity solver. It allows one to bypass the need of multiple perturbative series resummations within the DMFT self-consistency loop. It can be applied at or out of equilibrium, with any diagrammatic formalism, such as real times, imaginary times, or Matsubara frequencies formalisms. As a proof of principle, we illustrate our method with the half-filled Hubbard model on the Bethe lattice in the DMFT approximation, using quantum quasi Monte Carlo (QQMC) to obtain the impurity perturbation series on the real time axis.
We develop a variational perturbation expansion around dynamical mean-field theory (DMFT) that systematically incorporates nonlocal correlations beyond the local correlations treated by DMFT. We apply this approach to investigate how the DMFT critical temperature is suppressed from its mean-field value and how the critical behavior near the finite-temperature phase transition evolves from the mean-field to the Heisenberg universality class. By identifying the symmetry breaking of paramagnetic diagrammatic expansions as a signature of the Néel transition, we accurately predict the Néel temperature of the three-dimensional cubic Hubbard model across all interaction strengths with low computational cost. Introducing a variational order parameter, our method can be applied to both paramagnetic and long-range ordered states, such as antiferromagnetic order. We compute magnetization and antiferromagnetic susceptibility, demonstrating minor corrections to DMFT solutions in the weak-coupling regime while revealing significant modifications to these properties in the intermediate correlation regime. From the analysis of critical exponents, we establish the emergence of Heisenberg critical behavior beyond the mean-field nature of DMFT.
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