We present an application of quantics tensor trains (QTTs) and tensor cross interpolation (TCI) to the solution of a full set of self-consistent equations for multivariate functions, the so-called parquet equations. We show that the steps needed to evaluate the equations (Bethe-Salpeter equations, parquet equation, and Schwinger-Dyson equation) can be decomposed into basic operations on the QTT-TCI compressed objects. The repeated application of these operations does not lead to a loss of accuracy beyond a specified tolerance and the iterative scheme converges even for numerically demanding parameters. As examples, we take the Hubbard model in the atomic limit and the single impurity Anderson model, where the basic objects in parquet equations, the two-particle vertices, depend on three frequencies, but not on momenta. The results show that this approach is able to overcome major computational bottlenecks of standard numerical methods. The applied methods allow for an exponential increase of the number of grid points included in the calculations, and a corresponding exponential reduction of the computational error, for a linear increase in computational cost.We present an application of quantics tensor trains (QTTs) and tensor cross interpolation (TCI) to the solution of a full set of self-consistent equations for multivariate functions, the so-called parquet equations. We show that the steps needed to evaluate the equations (Bethe-Salpeter equations, parquet equation, and Schwinger-Dyson equation) can be decomposed into basic operations on the QTT-TCI compressed objects. The repeated application of these operations does not lead to a loss of accuracy beyond a specified tolerance and the iterative scheme converges even for numerically demanding parameters. As examples, we take the Hubbard model in the atomic limit and the single impurity Anderson model, where the basic objects in parquet equations, the two-particle vertices, depend on three frequencies, but not on momenta. The results show that this approach is able to overcome major computational bottlenecks of standard numerical methods. The applied methods allow for an exponential increase of the number of grid points included in the calculations, and a corresponding exponential reduction of the computational error, for a linear increase in computational cost.

arXiv:2410.22975Physical Review ResearchApr 24, 202517 citations

Parquetlike equations for the Hedin three-leg vertex

Friedrich Krien, Angelo Valli

Taking the competition and the mutual screening of various bosonic fluctuations in correlated electron systems into account requires an unbiased approach to the many-body problem. One such approach is the self-consistent solution of the parquet equations, whose numerical treatment in lattice systems is however prohibitively expensive. In a recent article it was shown that there exists an alternative to the parquet decomposition of the four-point vertex function, which classifies the vertex diagrams according to the principle of single-boson exchange (SBE) [Krien, Valli, and Capone, Phys. Rev. B 100, 155149 (2019)]. Here we show that the SBE decomposition leads to a closed set of equations for the Hedin three-leg vertex, the polarization, and the electronic self-energy, which sums self-consistently the diagrams of the Maki-Thompson type. This circumvents the calculation of four-point vertex functions and the inversion of the Bethe-Salpeter equations, which are the two major bottlenecks of the parquet equations. The convergence of the calculation scheme starting from a fully irreducible vertex is demonstrated for the Anderson impurity model.Taking the competition and the mutual screening of various bosonic fluctuations in correlated electron systems into account requires an unbiased approach to the many-body problem. One such approach is the self-consistent solution of the parquet equations, whose numerical treatment in lattice systems is however prohibitively expensive. In a recent article it was shown that there exists an alternative to the parquet decomposition of the four-point vertex function, which classifies the vertex diagrams according to the principle of single-boson exchange (SBE) [Krien, Valli, and Capone, Phys. Rev. B 100, 155149 (2019)]. Here we show that the SBE decomposition leads to a closed set of equations for the Hedin three-leg vertex, the polarization, and the electronic self-energy, which sums self-consistently the diagrams of the Maki-Thompson type. This circumvents the calculation of four-point vertex functions and the inversion of the Bethe-Salpeter equations, which are the two major bottlenecks of the parquet equations. The convergence of the calculation scheme starting from a fully irreducible vertex is demonstrated for the Anderson impurity model.

arXiv:1909.02793Physical review. B./Physical review. BDec 26, 201942 citations

Exact and approximate results for the Anderson impurity model

U. Brandt, Andreas Giesekus

The European Physical Journal BDec 1, 19891 citation

Multiorbital simplified parquet equations for strongly correlated electrons

Pavel Augustinský, V. Janiš

We extend an approximation that we developed earlier for the single-impurity Anderson model to a full-size impurity solver for models of interacting electrons with multiple orbitals. The approximation is based on parquet equations simplified by separating small and large energy fluctuations justified in the critical region of a pole in the two-particle vertex. We show that an $l$ -orbital model with the most general interaction is described within this approximation by $4{l}^{2}\ifmmode\times\else\texttimes\fi{}4{l}^{2}$ matrices and is Fermi liquid in the metallic phase. We explicitly calculate properties of a paramagnetic solution of a two-orbital Hubbard model with a Hund exchange and orbital splitting within the dynamical mean-field approximation. We trace the genesis of a metal-insulator transition induced by a crystal field and vanishing of the Kondo quasiparticle peak in strongly correlated orbitally asymmetric systems.We extend an approximation that we developed earlier for the single-impurity Anderson model to a full-size impurity solver for models of interacting electrons with multiple orbitals. The approximation is based on parquet equations simplified by separating small and large energy fluctuations justified in the critical region of a pole in the two-particle vertex. We show that an $l$-orbital model with the most general interaction is described within this approximation by $4{l}^{2}\ifmmode\times\else\texttimes\fi{}4{l}^{2}$ matrices and is Fermi liquid in the metallic phase. We explicitly calculate properties of a paramagnetic solution of a two-orbital Hubbard model with a Hund exchange and orbital splitting within the dynamical mean-field approximation. We trace the genesis of a metal-insulator transition induced by a crystal field and vanishing of the Kondo quasiparticle peak in strongly correlated orbitally asymmetric systems.

arXiv:1008.3498Physical Review BJan 18, 20119 citations

l

-orbital model with the most general interaction is described within this approximation by $4{l}^{2}\ifmmode\times\else\texttimes\fi{}4{l}^{2}$ matrices and is Fermi liquid in the metallic phase. We explicitly calculate properties of a paramagnetic solution of a two-orbital Hubbard model with a Hund exchange and orbital splitting within the dynamical mean-field approximation. We trace the genesis of a metal-insulator transition induced by a crystal field and vanishing of the Kondo quasiparticle peak in strongly correlated orbitally asymmetric systems.