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.
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.
Establishing the underlying links between the diverse landscape of theoretical frameworks for simulating strongly correlated matter is crucial for advancing our understanding of these systems. In this work, we focus on the ghost Gutzwiller approximation (gGA), an extension of the Gutzwiller approximation (GA) based on the variational principle. We derive a framework called ``ghost density matrix embedding theory'' (gDMET) from quantum embedding (QE) principles similar to those in density matrix embedding theory (DMET), which reproduces the gGA equations for multiorbital Hubbard models with a simpler implementation. This derivation highlights the crucial role of the ghost degrees of freedom, not only as an extension to the GA, but also as the key element in establishing a consistent conceptual connection between DMET and the gGA. This connection further elucidates how gGA overcomes the systematic accuracy limitations of standard GA and achieves results comparable to dynamical mean field theory. Furthermore, it offers an alternative interpretation of the gGA equations, fostering new ideas and generalizations.
We introduce density matrix embedding theory (DMET), a quantum embedding theory for computing frequency-independent quantities, such as ground-state properties, of infinite systems. Like dynamical mean-field theory, DMET maps the bulk interacting system to a simpler impurity model and is exact in the noninteracting and atomic limits. Unlike dynamical mean-field theory, DMET is formulated in terms of the frequency-independent local density matrix, rather than the local Green's function. In addition, it features a finite, algebraically constructible bath of only one bath site per impurity site, with no bath discretization error. Frequency independence and the minimal bath make DMET a computationally simple and efficient method. We test the theory in the one-dimensional and two-dimensional Hubbard models at and away from half filling, and we find that compared to benchmark data, total energies, correlation functions, and metal-insulator transitions are well reproduced, at a tiny computational cost.
This thesis develops a theoretical framework for quantum embedding based on compact sum-over-poles (SOP) representations of Green's functions, with a particular focus on dynamical mean-field theory (DMFT) formulated close to the real-frequency axis. The work starts from the algorithmic inversion method (AIM-SOP) and generalizes it to many-body lattice systems, proving that any interacting fragment embedded in an arbitrary bath can be represented by an auxiliary Hamiltonian with a finite set of non-interacting bath sites. The author embeds this formalism into a DMFT self-consistency loop formulated on complex contours close to the real axis, complemented by a SOP-based fitting strategy and exact-diagonalization (ED) impurity solvers, constructing a DMFT scheme that does not rely on approximate analytic continuation techniques. In particular, the framework enables us to work on compact (SOP) representations of Green's functions and self-energies alone, and leads to a frequency-independent formulation of DMFT, grounded in effective free energy (Baym-Kadanoff and Klein) functionals. Applications to a 1000-site one-dimensional Hubbard ring demonstrate that compact SOP forms reproduce state-of-the-art ED-DMFT results on the Matsubara frequencies and retain key spectral features as the contour approaches the real axis, while also revealing convergence trends and multiple-solution behavior near challenging correlation regimes. Beyond its numerical validation, this approach paves the way for novel variational implementations of DMFT and realistic applications to materials, harnessing its promising ability to resolve fine spectral features.
Density matrix embedding theory (DMET) [Phys. Rev. Lett. 109, 186404 (2012)], introduced an approach to quantum cluster embedding methods whereby the mapping of strongly correlated bulk problems to an impurity with finite set of bath states was rigorously formulated to exactly reproduce the entanglement of the ground state. The formalism provided similar physics to dynamical mean-field theory at a tiny fraction of the cost but was inherently limited by the construction of a bath designed to reproduce ground-state, static properties. Here, we generalize the concept of quantum embedding to dynamic properties and demonstrate accurate bulk spectral functions at similarly small computational cost. The proposed spectral DMET utilizes the Schmidt decomposition of a response vector, mapping the bulk dynamic correlation functions to that of a quantum impurity cluster coupled to a set of frequency-dependent bath states. The resultant spectral functions are obtained on the real-frequency axis, without bath discretization error, and allows for the construction of arbitrary dynamic correlation functions. We demonstrate the method on the one- (1D) and two-dimensional (2D) Hubbard model, where we obtain zero temperature and thermodynamic limit spectral functions, and show the trivial extension to two-particle Green's functions. This advance therefore extends the scope and applicability of DMET in condensed-matter problems as a computationally tractable route to correlated spectral functions of extended systems and provides a competitive alternative to dynamical mean-field theory for dynamic quantities.
In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations-a high level calculation on the active region of interest, and a low level calculation on its environment-formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: density functional embedding, Green's function embedding, and density matrix embedding, respectively, use the single-particle density, single-particle Green's function, and single-particle density matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used, functional equations for density functional embedding, diagrammatics for Green's function embedding, and entanglement arguments for density matrix embedding, make the three formulations appear vastly different. In this Account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theoretical perspective, we also give a brief overview of recent applications and possible future developments. The first section starts with density functional embedding, where we introduce the key embedding potential via the Euler equation. We then discuss recent work concerning the treatment of the nonadditive kinetic potential, before describing mean-field density functional embedding and wave function in density functional embedding. We finish the section with extensions to time-dependence and excited states. The second section is devoted to Green's function embedding. Here, we use the Dyson equation to obtain equations that parallel as closely as possible the density functional embedding equations, with the hybridization playing the role of the embedding potential. Embedding a high-level self-energy within a low-level self-energy is treated analogously to wave function in density functional embedding. The numerical computation of the high-level self-energy allows us to briefly introduce the bath representation in the quantum impurity problem. We then consider translationally invariant systems to bring in the important dynamical mean-field theory. Recent developments to incorporate screening and long-range interactions are discussed. The third section concerns density matrix embedding. Here, we first highlight some mathematical complications associated with a simple Euler equation derivation, arising from the open nature of fragments. This motivates the density matrix embedding theory, where we use the Schmidt decomposition to represent the entanglement through bath orbitals. The resulting impurity plus bath formulation resembles that of dynamical mean-field theory. We discuss the numerical self-consistency associated with using a high-level correlated wave function with a mean-field low-level treatment, and connect the resulting numerical inversion to that used in density functional embedding. We finish with perspectives on the future of all three methods.
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.