Fully Algebraic and Self-consistent Effective Dynamics in a Static Quantum Embedding

Abstract

Quantum embedding approaches involve the self-consistent optimization of a local fragment of a strongly correlated system, entangled with the wider environment. The 'energy-weighted' density matrix embedding theory (EwDMET) was established recently as a way to systematically control the resolution of the fragment-environment coupling, and allow for true quantum fluctuations over this boundary to be self-consistently optimized within a fully static framework. In this work, we reformulate the algorithm to ensure that EwDMET can be considered equivalent to an optimal and rigorous truncation of the self-consistent dynamics of dynamical mean-field theory (DMFT). A practical limitation of these quantum embedding approaches is often a numerical fitting of a self-consistent object defining the quantum effects. However, we show here that in this formulation, all numerical fitting steps can be entirely circumvented, via an effective Dyson equation in the space of truncated dynamics. This provides a robust and analytic self-consistency for the method, and an ability to systematically and rigorously converge to DMFT from a static, wave function perspective. We demonstrate that this improved approach can solve the correlated dynamics and phase transitions of the Bethe lattice Hubbard model in infinite dimensions, as well as one-and twodimensional Hubbard models where we clearly show the benefits of this rapidly convergent basis for correlation-driven fluctuations. This systematically truncated description of the effective dynamics of the problem also allows access to quantities such as Fermi liquid parameters and renormalized dynamics, and demonstrates a numerically efficient, systematic convergence to the zero-temperature dynamical mean-field theory limit.

High harmonic generation in two-dimensional Mott insulators

“…These fictitious degrees of freedom hybridize with the physical lattice in such a way as to mimic the band splitting caused by the correlations of the interacting lattice. The parameter V (U ) is optimized for each interaction strength by matching the first spectral moment of the particle and hole density of states of the ground state, given by Ψ|ĉ ( †) ( Ĥ − E)ĉ ( †) |Ψ , between the noninteracting model and the correlated system as described by variational Monte Carlo (VMC) [54][55][56]. After tracing out the fictitious degrees of freedom, this can introduce a second band and open an effective Mott gap in the bandstructure of the non-interacting model, ∆(U ), with the fictitious space representing a local self-energy, comprising a single pole with spectral weight |V (U )| 2 at the Fermi level.…”

“…The mapping between interacting (at a given U/t 0 ) and non-interacting systems is achieved by choosing V (U ) to match the mean of the particle and hole distributions of the density of states. This is first calculated for the correlated system using VMC, by writing it as an expectation of the ground state wave function [54][55][56]…”

Scalable semidefinite programming approach to variational embedding for quantum many-body problems

“…Local problems are then stitched together via a reduced set of global quantities or a lessaccurate 'low-level' method that operates on the global scale, and the local and global perspectives are constrained to be compatible via some self-consistency condition. Such approaches include dynamical mean-field theory (DMFT) [12,18] and density matrix embedding theory (DMET) [16,17], as well as variants such as the energy-weighted DMET (EwDMET) [10,11] which in a certain sense interpolates between DMFT and DMET [28].…”