The ground-state phase diagrams of the three-orbital t2g Hubbard model are studied using a Hartree-Fock approximation. First, a complete set of multipolar order parameters for t2g models defined in terms of the effective total angular momentum jeff are theoretically derived. These order parameters can classify off-diagonal orders between jeff = 1/2 and jeff = 3/2 manifolds. Second, through extensive Hartree-Fock calculations, the ground-state phase diagrams in the space of (1) the onsite Coulomb repulsion U, (2) the spin-orbit coupling (SOC), and (3) the number of electrons are mapped out. A variety of nontrivial quantum phases with jeff-diagonal and jeff-off-diagonal multipole orders are found. Finally, future studies using more numerically expensive methods, such as dynamical mean-field theory are discussed.
This lecture note reviews recently proposed sparse-modeling approaches for efficient ab initio many-body calculations based on the data compression of Greens functions. The sparse-modeling techniques are based on a compact orthogonal basis representation, intermediate representation (IR) basis functions, for imaginary-time and Matsubara Greens functions. A sparse sampling method based on the IR basis enables solving diagrammatic equations efficiently. We describe the basic properties of the IR basis, the sparse sampling method and its applications to ab initio calculations based on the GW approximation and the Migdal-Eliashberg theory. We also describe a numerical library for the IR basis and the sparse sampling method, irbasis, and provide its sample codes. This lecture note follows the Japanese review article [H. Shinaoka et al., Solid State Physics 56(6), 301 (2021)].
The Bethe-Salpeter equation plays a crucial role in understanding the physics of correlated fermions, relating to optical excitations in solids as well as resonances in high-energy physics. Yet, it is notoriously difficult to control numerically, typically requiring an effort that scales polynomially with energy scales and accuracy. This puts many interesting systems out of computational reach. Using the intermediate representation and sparse modelling for two-particle objects on the Matsubara axis, we develop an algorithm that solves the Bethe-Salpeter equation in $O(L^8)$ time with $O(L^4)$ memory, where $L$ grows only logarithmically with inverse temperature, bandwidth, and desired accuracy, This opens the door for computations in hitherto inaccessible regimes. We benchmark the method on the Hubbard atom and on the multi-orbital weak-coupling limit, where we observe the expected exponential convergence to the analytical results. We then showcase the method for a realistic impurity problem.
Quantum embedding theories provide a feasible route for obtaining quantitative descriptions of correlated materials. However, a critical challenge is solving an effective impurity model of correlated orbitals embedded in an electron bath. Many advanced impurity solvers require the approximation of a bath continuum using a finite number of bath levels, producing a highly nonconvex, ill-conditioned inverse problem. To address this drawback, this study proposes an efficient fitting algorithm for matrix-valued hybridization functions based on a data-science approach, sparse modeling, and a compact representation of Matsubara Greens functions. The efficiency of the proposed method is demonstrated by fitting random hybridization functions with large off-diagonal elements as well as those of a 20-orbital impurity model for a high-Tc compound, LaAsFeO, at low temperatures (T). The results set quantitative goals for the future development of impurity solvers toward quantum embedding simulations of complex correlated materials.
We present a new open-source program, DCore, that implements dynamical mean-field theory (DMFT). DCore features a user-friendly interface based on text and HDF5 files. It allows DMFT calculations of tight-binding models to be performed on predefined lattices as well as textit{ab initio} models constructed by external density functional theory codes through the Wannier90 package. Furthermore, DCore provides interfaces to many advanced quantum impurity solvers such as quantum Monte Carlo and exact diagonalization solvers. This paper details the structure and usage of DCore and shows some applications.
This review paper describes the basic concept and technical details of sparse modeling and its applications to quantum many-body problems. Sparse modeling refers to methodologies for finding a small number of relevant parameters that well explain a given dataset. This concept reminds us physics, where the goal is to find a small number of physical laws that are hidden behind complicated phenomena. Sparse modeling extends the target of physics from natural phenomena to data, and may be interpreted as physics for data. The first half of this review introduces sparse modeling for physicists. It is assumed that readers have physics background but no expertise in data science. The second half reviews applications. Matsubara Greens function, which plays a central role in descriptions of correlated systems, has been found to be sparse, meaning that it contains little information. This leads to (i) a new method for solving the ill-conditioned inverse problem for analytical continuation, and (ii) a highly compact representation of Matsubara Greens function, which enables efficient calculations for quantum many-body systems.
Many-body calculations at the two-particle level require a compact representation of two-particle Greens functions. In this paper, we introduce a sparse sampling scheme in the Matsubara frequency domain as well as a tensor network representation for two-particle Greens functions. The sparse sampling is based on the intermediate representation basis and allows an accurate extraction of the generalized susceptibility from a reduced set of Matsubara frequencies. The tensor network representation provides a system independent way to compress the information carried by two-particle Greens functions. We demonstrate efficiency of the present scheme for calculations of static and dynamic susceptibilities in single- and two-band Hubbard models in the framework of dynamical mean-field theory.
Computing momentum-dependent susceptibilities in the dynamical mean-field theory (DMFT) requires solving the Bethe-Salpeter equation, which demands large computational cost. Exploiting the strong-coupling feature of local fluctuations, we derive a simplified formula that can be solved at a considerably lower cost. The validity and the physical meaning of the formula are confirmed by deriving the effective intersite interactions in the strong-coupling limit, such as the kinetic exchange and RKKY interactions. Furthermore, numerical calculations for single-orbital and multiorbital models demonstrate surprisingly wider applicability including weak-coupling region. Based on this formula, we propose three levels of practical approximations that can be chosen depending on complexity of problems. Simpler evaluations of spin and orbital susceptibilities in multiorbital systems thus become possible within DMFT.
The pyrochlore oxides $A_2B_2$O$_7$ exhibit a complex interplay between geometrical frustration, electronic correlations, and spin-orbit coupling, due to the lattice structure and active charge, spin, and orbital degrees of freedom. Understanding the properties of these materials is a theoretical chalenge, because their intricate nature depends on material-specific details and quantum many-body effects. Here we review our recent studies based on first-principles calculations and quantum many-body theories for 4$d$ and 5$d$ pyrochlore oxides with $B$=Mo, Os, and Ir. In these studies, the spin-orbit coupling and local electron correlations are treated within the LDA+$U$ and LDA+dynamical mean-field theory formalisms. We also discuss the technical aspects of these calculations.
We check the accuracy of the constrained random phase approximation (cRPA) downfolding scheme by considering one-dimensional two- and three-orbital Hubbard models with a target band at the Fermi level and one or two screening bands away from the Fermi level. Using numerically exact quantum Monte Carlo simulations of the full and downfolded model we demonstrate that depending on filling the effective interaction in the low-energy theory is either barely screened, or antiscreened, in contrast to the cRPA prediction. This observation is explained by a functional renormalization group analysis which shows that the cRPA contribution to the screening is to a large extent cancelled by other diagrams in the direct particle-hole channel. We comment on the implications of this finding for the ab-initio estimation of interaction parameters in low-energy descriptions of solids.
