A new variational theory is proposed, based on the sequential product density matrix (SPD) ansatz, which is characterized by an integer and monotonically approaches the exact solution. The authors introduce the discrete action theory to evaluate the SPD, yielding the variational discrete action theory (VDAT). VDAT can be exactly evaluated for the multiband Anderson impurity model and the multiband Hubbard model in infinite dimensions. This provides an unprecedented combination of efficiency and precision within a single framework.
A new variational theory is proposed, based on the sequential product density matrix (SPD) ansatz, which is characterized by an integer and monotonically approaches the exact solution. The authors introduce the discrete action theory to evaluate the SPD, yielding the variational discrete action theory (VDAT). VDAT can be exactly evaluated for the multiband Anderson impurity model and the multiband Hubbard model in infinite dimensions. This provides an unprecedented combination of efficiency and precision within a single framework.
Here we propose the variational discrete action theory (VDAT) to study the ground state properties of quantum many-body Hamiltonians. VDAT is a variational theory based on the sequential product density matrix (SPD) ansatz, characterized by an integer N, which monotonically approaches the exact solution with increasing N. To evaluate the SPD, we introduce a discrete action and a corresponding integer time Green's function. We use VDAT to exactly evaluate the SPD in two canonical models of interacting electrons: the Anderson impurity model and the d=∞ Hubbard model. For the latter, we evaluate N=2-4, where N=2 recovers the Gutzwiller approximation (GA), and we show that N=3, which exactly evaluates the Gutzwiller-Baeriswyl wave function, provides a truly minimal yet precise description of Mott physics with a cost similar to that of the GA. VDAT is a flexible theory for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single framework.
Determining the ground state of multiorbital Hubbard models is critical for understanding strongly correlated electron materials, yet, existing methods struggle to simultaneously reach zero temperature and infinite system size. The de facto standard is to approximate a finite dimension multiorbital Hubbard model with a $d=\ensuremath{\infty}$ version, which can then be formally solved via the dynamical mean-field theory (DMFT), although the DMFT solution is limited by the state of unbiased impurity solvers for zero temperature and multiple orbitals. The recently developed variational discrete action theory (VDAT) offers a new approach to solve the $d=\ensuremath{\infty}$ Hubbard model, with a variational ansatz that is controlled by an integer N, and monotonically approaches the exact solution at an increasing computational cost. Here we propose a decoupled minimization algorithm to implement VDAT for the multiorbital Hubbard model in $d=\ensuremath{\infty}$ and study $\mathcal{N}=2\ensuremath{-}4$. At N=2, VDAT rigorously recovers the multiorbital Gutzwiller approximation, reproducing known results. At N=3, VDAT precisely captures the competition between the Hubbard U, Hund J, and crystal field $\mathrm{\ensuremath{\Delta}}$ in the two-orbital Hubbard model over all parameter space, with a negligible computational cost. For sufficiently large U/t and J/U, we show that $\mathrm{\ensuremath{\Delta}}$ drives a first-order transition within the Mott insulating regime. In the large orbital polarization limit with finite J/U, we find that interactions have a nontrivial effect even for small U/t. VDAT will have far ranging implications for understanding multiorbital model Hamiltonians and strongly correlated electron materials.
The recently developed variational discrete action theory (VDAT) provides a systematic variational approach to the ground state of the quantum many-body problem, where the quality of the solution is controlled by an integer N, and increasing N monotonically approaches the exact solution. VDAT can be exactly evaluated in the $d=\ensuremath{\infty}$ multiorbital Hubbard model using the self-consistent canonical discrete action theory (SCDA), which requires a self-consistency condition for the integer time Green's functions. Previous work demonstrates that N=3 accurately captures multiorbital Mott/Hund physics at a cost similar to the Gutzwiller approximation. Here we employ a gauge constraint to automatically satisfy the self-consistency condition of the SCDA at N=3, yielding an even more efficient algorithm with enhanced numerical stability. We derive closed form expressions of the gauge constrained algorithm for the multiorbital Hubbard model with general density-density interactions, allowing VDAT at N=3 to be straightforwardly applied to the seven-orbital Hubbard model. We present results and a performance analysis using N=2 and N=3 for the SU(2Norb) Hubbard model in $d=\ensuremath{\infty}$ with Norb=2−−8, and compare to numerically exact dynamical mean-field theory solutions where available. The developments in this work will greatly facilitate the application of VDAT at N=3 to strongly correlated electron materials.
A gauge constrained algorithm of VDAT at N=3 for the multi-orbital Hubbard model
The recently developed variational discrete action theory (VDAT) provides a systematic variational approach to the ground state of the quantum many-body problem, where the quality of the solution is controlled by an integer N, and increasing N monotonically approaches the exact solution. VDAT can be exactly evaluated in the d=∞ multi-orbital Hubbard model using the self-consistent canonical discrete action theory (SCDA), which requires a self-consistency condition for the integer time Green's functions. Previous work demonstrates that N=3 accurately captures multi-orbital Mott/Hund physics at a cost similar to the Gutzwiller approximation. Here we employ a gauge constraint to automatically satisfy the self-consistency condition of the SCDA at N=3, yielding an even more efficient algorithm with enhanced numerical stability. We derive closed form expressions of the gauge constrained algorithm for the multi-orbital Hubbard model with general density-density interactions, allowing VDAT at N=3 to be straightforwardly applied to the seven orbital Hubbard model. We present results and a performance analysis using N=2 and N=3 for the SU(2Norb) Hubbard model in d=∞ with Norb=2−8, and compare to numerically exact dynamical mean-field theory solutions where available. The developments in this work will greatly facilitate the application of VDAT at N=3 to strongly correlated electron materials.
The variational discrete action theory (VDAT) at \mathcal{N}=3 is a potent tool for accurately capturing Mott and Hund physics at zero temperature in d=\infty at a cost comparable to the Gutzwiller approximation, which is recovered by VDAT at \mathcal{N}=2. Here we develop a qubit parametrization of the gauge constrained algorithm of VDAT at \mathcal{N}=3 for the multiorbital Hubbard model with general density-density interactions. The qubit parametrization yields an explicit variational trial energy, and the variational parameters consist of the momentum density distribution, the shape of a reference fermi surface, and the pure state of a qubit system with dimension of the local Hilbert space. To illustrate the power of the qubit parametrization, we solve for the ground state properties of the multiorbital Hubbard model with Hund coupling for local orbital number N_{orb}=2-7. A Taylor series expansion of the partially optimized trial energy is used to explain how the Hund's coupling changes the order of the Mott transition. For the case of the SU(2N_{orb}) Hubbard model, an explicit approach for computing the critical U_{c} for the Mott transition is provided, yielding an analytical expression for U_{c} in the large N_{orb} limit. Additionally, we provide an analytical solution for the ground state properties of the single band Hubbard model with a special density of states. Finally, we demonstrate that the qubit parametrization can also be applied to \mathcal{N}=2, for both G-type and B-type variants, where the G-type yields an identical expression to the slave spin mean-field theory. The qubit parametrization not only improves the efficiency and transparency of VDAT at \mathcal{N}=3, but also provides the key advances for the construction of a one-body reduced density matrix functional capable of capturing Mott and Hund physics.
We present a self-consistent numerical approach to solve the Gutzwiller variational problem for general multiband models with arbitrary on-site interaction. The proposed method generalizes and improves the procedure derived by Deng et al. [Phys. Rev. B 79, 075114 (2009)], overcoming the restriction to density-density interaction without increasing the complexity of the computational algorithm. Our approach drastically reduces the problem of the high-dimensional Gutzwiller minimization by mapping it to a minimization only in the variational density matrix, in the spirit of the Levy and Lieb formulation of density functional theory (DFT). For fixed density the Gutzwiller renormalization matrix is determined as a fixpoint of a proper functional, whose evaluation requires only ground-state calculations of matrices defined in the Gutzwiller variational space. Furthermore, the proposed method is able to account for the symmetries of the variational function in a controlled way, reducing the number of variational parameters. After a detailed description of the method we present calculations for multiband Hubbard models with full (rotationally invariant) Hund's rule on-site interaction. Our analysis shows that the numerical algorithm is very efficient, stable, and easy to implement. For these reasons this method is particularly suitable for first-principles studies (e.g., in combination with DFT) of many complex real materials, where the full intra-atomic interaction is important to obtain correct results.