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 $\mathcal{N}$ , 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 $\mathcal{N}=2$ , VDAT rigorously recovers the multiorbital Gutzwiller approximation, reproducing known results. At $\mathcal{N}=3$ , 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.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 $\mathcal{N}$, 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 $\mathcal{N}=2$, VDAT rigorously recovers the multiorbital Gutzwiller approximation, reproducing known results. At $\mathcal{N}=3$, 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.

arXiv:2206.08433Physical review. B./Physical review. BNov 18, 20227 citations

Ground States of the Hubbard Model

Assa Auerbach

Graduate texts in contemporary physicsJan 1, 19940 citations

The ground state of the two-dimensional Hubbard model

Yoshihiro Asai

arXiv:cond-mat/9909074Physica B Condensed MatterJun 1, 20002 citations

Pair theory of the Hubbard Hamiltonian: variational approach

H. Barentzen, Christo I. Ivanov, Marvin D. Girardeau

A novel variational approach is outlined for treating electron correlation in many-particle systems and is applied to the one-band Hubbard model. The authors' treatment is based on the concept of fermion pairs and their representation in terms of ideal-boson operators. In this representation approximate eigen-states of the Hamiltonian are obtained in the form of an antisymmetrised product of pair orbitals. For the ground state of the Hubbard model these orbitals are then determined by the use of the variational principle. Because of the complexity of the resulting equation a simplified ansatz is used for the orbitals, which leads to the same energy expression as is obtained by the traditional Bogolubov method. The trial states used in both approaches are, however, quite different. They find reasonable qualitative agreement with well established results for the Hubbard model in one dimension even in those cases where the Bogolubov or related approaches fail. They are also able to show that in the atomic limit the Bogolubov energy bound is the best upper bound that can be obtained by using a single determinant of pair orbitals. Therefore, the only way to improve the ground-state energy in the atomic limit is by means of configuration interaction.A novel variational approach is outlined for treating electron correlation in many-particle systems and is applied to the one-band Hubbard model. The authors' treatment is based on the concept of fermion pairs and their representation in terms of ideal-boson operators. In this representation approximate eigen-states of the Hamiltonian are obtained in the form of an antisymmetrised product of pair orbitals. For the ground state of the Hubbard model these orbitals are then determined by the use of the variational principle. Because of the complexity of the resulting equation a simplified ansatz is used for the orbitals, which leads to the same energy expression as is obtained by the traditional Bogolubov method. The trial states used in both approaches are, however, quite different. They find reasonable qualitative agreement with well established results for the Hubbard model in one dimension even in those cases where the Bogolubov or related approaches fail. They are also able to show that in the atomic limit the Bogolubov energy bound is the best upper bound that can be obtained by using a single determinant of pair orbitals. Therefore, the only way to improve the ground-state energy in the atomic limit is by means of configuration interaction.

Journal of Physics C Solid State PhysicsOct 20, 19873 citations

Correlated ground state of the Hubbard model

MANFRED L. RISTIG

The European Physical Journal BOct 1, 199012 citations

\mathcal{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

\mathcal{N}=2

, VDAT rigorously recovers the multiorbital Gutzwiller approximation, reproducing known results. At

\mathcal{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.