Fully algebraic and self-consistent effective dynamics in a static quantum embedding

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 two-dimensional 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.

Figure 1

Schematic structure of the noninteracting part of the cluster Hamiltonian (${\mathcal{H}}_{\mathrm{cl}}^{\left(0\right)}$) and bath orbital couplings in EwDMET after systematic orthogonalization. Bath orbitals are denoted by circles, with their fillings representative of the electron number when the mean-field solution of ${\mathcal{H}}_w$ is projected into the cluster space. Each circle represents another set of bath orbitals, whose maximum dimension is the number of orbitals in the fragment ($n_f$). Lines denote terms coupling the orbitals in the cluster Hamiltonian of Eq. (9). Truncating the moment expansion naturally truncates the chain of bath orbitals at a given $n_{\mathrm{mom}}$.

Figure 2

Schematic of the various Hamiltonians defined in EwDMET, for a two-site fragment (red rectangle) embedded in a one-dimensional chain of sites. Sites are denoted by circles, auxiliary states by stars, and the bath space by a diamond (where its structure and coupling to the fragment space is pictorially described in more detail in Fig. 1). The Hamiltonians shown schematically represent the ‘lattice,’ ‘Weiss,’ ‘cluster,’ and ‘auxiliary’ Hamiltonians from top to bottom. All Hamiltonians are noninteracting, apart from ${\mathcal{H}}_{\text{cl}}$, which features additional interactions over the fragment. At each iteration, $h_{\mathrm{aux}}$ is defined to have the same spectral moments as ${\mathcal{H}}_{\text{cl}}$.

Figure 3

Energy density (E/t), Doublon density (D), and quasiparticle weight (Z) for the Bethe Hubbard lattice, as the interaction strength is increased. Single fragment site EwDMET results are presented with $n_{\mathrm{mom}}=1,3,5$, and 7, and compared to ED-DMFT results with seven bath orbitals.

Figure 4

Density of states of the Bethe Hubbard model for different values of the interaction strength $U/t$, as $n_{\mathrm{mom}}$ is varied in EwDMET. Also included are comparison results from ED-DMFT with seven numerically fitted bath orbitals, which compare very favorably with the EwDMET results at $n_{\mathrm{mom}}=7$. All spectra include an artificial broadening of $0.05t$.

Figure 5

The energies and couplings to the fragment space resulting from the optimization of the auxiliary states for different orders of self-consistency in the spectral moments, as the interaction strength $U/t$ is varied for the Bethe Hubbard model. Only auxiliary states with a coupling magnitude larger than ${10}^{-4}$ is shown. Results for $n_{\mathrm{mom}}=7$ show a larger number of optimized auxiliary states, and is omitted for clarity. Each auxiliary state corresponds to a single pole in the effective self-energy, with Mott transitions characterized by an auxiliary state energy tending to zero, while the increase in coupling to these states results in an opening of the Mott gap. Auxiliary states come in particle-hole symmetric pairs (denoted by dashed and solid lines of the same color), with equal magnitude couplings for both states in the pair. Coupling and energies of the same auxiliary states are denoted by the same color.

Figure 6

The imaginary part of the effective self-energy on the Matsubara axis represented by the self-consistent auxiliary system for different orders of spectral moments for the Bethe lattice at $U/t=1,3,6,9$.

Figure 7

Comparison of the energy per site and double occupancy for the 1D Hubbard chain with different numbers of fragment sites ($n_f$) and number of self-consistent moments ($n_{\mathrm{mom}}$). Also included is a comparison to DMRG values.

Figure 8

Density of states of the 1D Hubbard chain for a selection of interaction strengths ($U/t$), fragment sizes ($n_f$), and moment expansions ($n_{\mathrm{mom}}$). An artificial broadening of $0.05t$ is included for presentation purposes. Also shown as red bars are the Bethe ansatz results for the exact charge gap in the thermodynamic limit, showing improving agreement with the EwDMET results for $n_f=4$.

Figure 9

The energy per site for the 2D Hubbard model on a $48\times 24$ site lattice, via EwDMET and DMFT with a $2\times 2$ fragment plaquette. Results are distinguished by the phase of the converged solution found (metallic or insulating). EwDMET results are converged for $n_{\mathrm{mom}}=1$, with four bath orbitals. Comparison DMFT results are found via exact diagonalization and a numerical fit of the Matsubara hybridization at each iteration to eight bath orbitals.

Figure 10

Relative errors in the moments of the hybridization comparing the “diagonalize-then-project” ($d\to p$) scheme used in previous publications and the “project-then-diagonalize” ($p\to d$) scheme used in this work. The $d\to p$ approach does not lead to an exact matching of hybridization moments. The $p\to d$ method leads to an exact matching of $n_{\mathrm{mom}}$ moments of the Green's function and $n_{\mathrm{mom}}-2$ moments of the hybridization. Only the even moments are shown, since all odd moments of the hybridization are zero for the half-filled one-dimensional Hubbard-model.