Unifying Variational and Dynamical Quantum Embedding: From Ghost Gutzwiller Approximation to Dynamical Mean-Field Theory

Abstract

Dynamical and variational frameworks have long been viewed as distinct paradigms. In particular, in quantum embedding (QE) frameworks, dynamical mean-field theory (DMFT) captures nonperturbative dynamical correlations through a frequency-dependent self-energy, while the Gutzwiller approximation (GA) is formulated in terms of a variationally optimized ground-state wavefunction. Here we bridge these perspectives, proving that the ghost-Gutzwiller approximation (ghost-GA), which also admits a density-matrix-matching QE formulation known as ghost density matrix embedding theory (ghost-DMET), becomes strictly equivalent to DMFT in the limit of infinitely many auxiliary bath modes. This formal unification has immediate consequences. In particular, it yields a rigorous finite-temperature extension of ghost-GA and shows that the physical Green's function can be determined from static expectation values of the embedding Hamiltonians, providing a route to computational studies of competing phases in strongly correlated matter with DMFT-level accuracy, while bypassing the need to calculate dynamical spectra with conventional impurity solvers. More broadly, it shows that the variational ghost-GA, the density-matrix-matching ghost-DMET formulation, and the dynamical DMFT description are not separate constructions, but complementary formulations of the same QE structure, thereby providing a concrete formal basis for future controlled extensions beyond DMFT.

Figures (8)

• Figure 1: Schematic representation of the multi-orbital Hubbard lattice Hamiltonian.
• Figure 2: Schematic representation of the correlated embedding Hamiltonian $\hat{H}^{i}_{\mathrm{emb}}[\mathcal{D}_i,\Lambda_i^c]$ [Eq. \ref{['eq:Hemb']}]. The physical local degrees of freedom of fragment $i$ (governed by $\hat{H}^{i}_{\mathrm{loc}}$) are hybridized with $B\nu_i$ auxiliary bath modes $\{b_{ia}\}$ through the matrix $\mathcal{D}_i$, while the quadratic bath term is parametrized by $\Lambda_i^c$.
• Figure 3: Schematic representation of the ghost-GA excitation construction. A quasiparticle excitation $\xi_n^\dagger|\Psi_0\rangle$ of the auxiliary reference state $|\Psi_0\rangle$ in the ghost Hilbert space $\tilde{H}_{\text{ghost}}$ is mapped into the physical Hilbert space $\tilde{H}$ by the same variationally-optimized local map $\hat{\mathcal{P}}_G$ that defines the ground-state $\vert\Psi_G\rangle$, yielding the corresponding physical excited state $\vert\Psi^{(+)}_{G,n}\rangle$.
• Figure 4: Schematic representation of the auxiliary quadratic embedding Hamiltonian $\hat{H}^{i}_{0,\mathrm{emb}}[\mathcal{D}_i,\Lambda_i^c;\mathcal{R}_i,\Lambda_i]$ for a fixed fragment $i$. The impurity modes $\{\tilde{c}_{i\alpha}\}_{\alpha=1,\dots,\nu_i}$ are embedded within the local $f$ sector and hybridize with the bath modes $\{b_{ia}\}_{a=1,\dots,B\nu_i}$ through $\mathcal{D}_i$. The bath one-body term is controlled by $\Lambda_i^c$ [Eq. \ref{['eq:setup_Delta_fit']}], while the local quadratic term $\Lambda_i$ acts in the full local $f$ space, mixing $\tilde{c}_{i\alpha}$ with the remaining local degrees of freedom (complement within the $f$ sector).
• Figure 5: Schematic representation of the quasiparticle Hamiltonian $\hat{H}_{\mathrm{qp}}[\mathcal{R},\Lambda]$. Inter-fragment hopping is restricted to the $\{\tilde{c}_{i\alpha}\}$ subspace and is governed by the physical hopping matrices $t_{ij}$, whereas the local quadratic term $\Lambda_i$ acts on the full local $f$ sector and therefore couples $\tilde{c}_{i\alpha}$ to the remaining local degrees of freedom within $\{f_{ia}\}$.