Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0512484 · Physical Review B

Solving the dynamical mean-field theory at very low temperatures using the Lanczos exact diagonalization

Massimo Capone, Luca de’ Medici, Antoine Georges

Published Dec 17, 2007• 119 citations
arXivHTMLPDF

Abstract

We present an efficient method to solve the impurity Hamiltonians involved in dynamical mean-field theory at low but finite temperature based on the extension of the Lanczos algorithm from ground state properties alone to excited states. We test the approach on the prototypical Hubbard model and find extremely accurate results from $T=0$ up to relatively high temperatures up to the scale of the critical temperature for the Mott transition. The algorithm substantially decreases the computational effort involved in finite temperature calculations.

DOI
10.1103/physrevb.76.245116

Related papers

Augmented hybrid exact-diagonalization solver for dynamical mean field theory

Cédric Weber, A. Amaricci, Massimo Capone +1 more

We present a methodology to solve the Anderson impurity model in the context of dynamical mean field theory based on the exact-diagonalization method. We propose a strategy to effectively refine the exact-diagonalization solver by combining a finite-temperature Lanczos algorithm with an adapted version of the cluster perturbation theory. We show that the augmented diagonalization yields an improved accuracy in the description of the spectral function of the single-band Hubbard model and is a reliable approach for a full $d$-orbital manifold calculation.

arXiv:1207.5610•Physical Review B•Sep 26, 2012•30 citations

Quasi-continuous-time impurity solver for the dynamical mean-field theory with linear scaling in the inverse temperature

D. Rost, Fakher F. Assaad, Nils Blümer

We present an algorithm for solving the self-consistency equations of the dynamical mean-field theory (DMFT) with high precision and efficiency at low temperatures. In each DMFT iteration, the impurity problem is mapped to an auxiliary Hamiltonian, for which the Green function is computed by combining determinantal quantum Monte Carlo (BSS-QMC) calculations with a multigrid extrapolation procedure. The method is numerically exact, i.e., yields results which are free of significant Trotter errors, but retains the BSS advantage, compared to direct QMC impurity solvers, of linear (instead of cubic) scaling with the inverse temperature. The new algorithm is applied to the half-filled Hubbard model close to the Mott transition; detailed comparisons with exact diagonalization, Hirsch-Fye QMC, and continuous-time QMC are provided.

arXiv:1303.2004•Physical Review E•May 29, 2013•16 citations

Exact diagonalization as an impurity solver in dynamical mean field theory

Yi Lu, M. W. Haverkort

The dynamical mean-field theory (DMFT) maps a correlated lattice problem onto an impurity problem of a single correlated site coupled to an uncorrelated bath. Most implementations solve the DMFT equations using quantum Monte-Carlo sampling on the imaginary time and frequency (Matsubara) axis. We will here review alternative methods using exact diagonalization, i.e., representing the many-body ground state of the impurity as a sum over Slater determinants and calculating Green’s functions using iterative Lanczos procedures. The advantage being that these methods have no sign problem, can handle involved multi-orbital Hamiltonians (low crystal symmetry, spin-orbit coupling) and – when working completely on the real axis – do not need a mathematically ill-posed analytical continuation. The disadvantage of traditional implementations of exact diagonalization has been the exponential scaling of the calculation problem as a function of number of bath discretization points. In the last part we will review how recent advances in exact diagonalization can evade the exponential barrier thereby increasing the number of bath discretization points to reach the thermodynamic limit.

The European Physical Journal Special Topics•Jul 1, 2017•17 citations

A Davidson-Lanczos iteration method for computation of continued-fraction expansion of the Green's function at very low temperatures: Applications to the dynamical mean field theory

Medha Sharma, M. A. H. Ahsan

We present a combination method based on orignal version of Davidson algorithm for extracting few of the lowest eigenvalues and eigenvectors of a sparse symmetric Hamiltonian matrix and the simplest version of Lanczos technique for obtaining a tridiagonal representation of the Hamiltonian to compute the continued fraction expansion of the Green's function at a very low temperature. We compare the Davidson$+$Lanczos method with the full diagonalization on a one-band Hubbard model on a Bethe lattice of infinite-coordination using dynamical mean field theory.

arXiv:1409.3374•Sep 11, 2014•0 citations

Efficient variational approach to the impurity problem and its application to the dynamical mean-field theory

Chungwei Lin, Alexander A. Demkov

Within the framework of exact diagonalization (ED), we compute the ground state of Anderson impurity problem using the variational approach based on the configuration interaction (CI) expansion. We demonstrate that an accurate ground state can be obtained by iteratively diagonalizing a matrix with the dimension that is less than 10$%$ of the full Hamiltonian. The efficiency of the CI expansion for different problems is analyzed. By way of example, we apply this method to the single-site dynamical mean-field theory using ED as the impurity solver. Specifically, to demonstrate the usefulness of this approach, we solve the attractive Hubbard model in the grand-canonical ensemble, where the $s$-wave superconducting solution is explicitly obtained.

arXiv:1307.4982•Physical Review B•Jul 18, 2013•38 citations

Natural orbital impurity solver for real-frequency properties at finite temperature

Motoharu Kitatani, 志朗 酒井, Ryotaro Arita

We extend the natural orbital impurity solver [Y. Lu, M. H\"oppner, O. Gunnarsson, and M. W. Haverkort, Phys. Rev. B 90, 085102 (2014)] to finite temperatures and apply it to calculate spectral and transport properties of correlated electrons within the dynamical mean-field theory. First, we benchmark our method against the exact diagonalization result for small clusters, finding that the natural orbital scheme works well not only for zero temperature but for low finite temperatures. The method yields smooth and sufficiently accurate spectra, which agree well with the results of the numerical renormalization group. Using the smooth spectra, we calculate the electric conductivity and Seebeck coefficient for the two-dimensional Hubbard model at low temperatures which are within the scope of many experiments and practical applications. These results demonstrate the usefulness of the natural orbital framework for obtaining the real frequency information of correlated electron systems.

arXiv:2107.06517•Physical review. B./Physical review. B•Nov 13, 2023•1 citation
We present an efficient method to solve the impurity Hamiltonians involved in dynamical mean-field theory at low but finite temperature based on the extension of the Lanczos algorithm from ground state properties alone to excited states. We test the approach on the prototypical Hubbard model and find extremely accurate results from T=0T=0T=0 up to relatively high temperatures up to the scale of the critical temperature for the Mott transition. The algorithm substantially decreases the computational effort involved in finite temperature calculations.
We present a methodology to solve the Anderson impurity model in the context of dynamical mean field theory based on the exact-diagonalization method. We propose a strategy to effectively refine the exact-diagonalization solver by combining a finite-temperature Lanczos algorithm with an adapted version of the cluster perturbation theory. We show that the augmented diagonalization yields an improved accuracy in the description of the spectral function of the single-band Hubbard model and is a reliable approach for a full ddd-orbital manifold calculation.
We present an algorithm for solving the self-consistency equations of the dynamical mean-field theory (DMFT) with high precision and efficiency at low temperatures. In each DMFT iteration, the impurity problem is mapped to an auxiliary Hamiltonian, for which the Green function is computed by combining determinantal quantum Monte Carlo (BSS-QMC) calculations with a multigrid extrapolation procedure. The method is numerically exact, i.e., yields results which are free of significant Trotter errors, but retains the BSS advantage, compared to direct QMC impurity solvers, of linear (instead of cubic) scaling with the inverse temperature. The new algorithm is applied to the half-filled Hubbard model close to the Mott transition; detailed comparisons with exact diagonalization, Hirsch-Fye QMC, and continuous-time QMC are provided.
The dynamical mean-field theory (DMFT) maps a correlated lattice problem onto an impurity problem of a single correlated site coupled to an uncorrelated bath. Most implementations solve the DMFT equations using quantum Monte-Carlo sampling on the imaginary time and frequency (Matsubara) axis. We will here review alternative methods using exact diagonalization, i.e., representing the many-body ground state of the impurity as a sum over Slater determinants and calculating Green’s functions using iterative Lanczos procedures. The advantage being that these methods have no sign problem, can handle involved multi-orbital Hamiltonians (low crystal symmetry, spin-orbit coupling) and – when working completely on the real axis – do not need a mathematically ill-posed analytical continuation. The disadvantage of traditional implementations of exact diagonalization has been the exponential scaling of the calculation problem as a function of number of bath discretization points. In the last part we will review how recent advances in exact diagonalization can evade the exponential barrier thereby increasing the number of bath discretization points to reach the thermodynamic limit.
We present a combination method based on orignal version of Davidson algorithm for extracting few of the lowest eigenvalues and eigenvectors of a sparse symmetric Hamiltonian matrix and the simplest version of Lanczos technique for obtaining a tridiagonal representation of the Hamiltonian to compute the continued fraction expansion of the Green's function at a very low temperature. We compare the Davidson+++Lanczos method with the full diagonalization on a one-band Hubbard model on a Bethe lattice of infinite-coordination using dynamical mean field theory.
Within the framework of exact diagonalization (ED), we compute the ground state of Anderson impurity problem using the variational approach based on the configuration interaction (CI) expansion. We demonstrate that an accurate ground state can be obtained by iteratively diagonalizing a matrix with the dimension that is less than 10% of the full Hamiltonian. The efficiency of the CI expansion for different problems is analyzed. By way of example, we apply this method to the single-site dynamical mean-field theory using ED as the impurity solver. Specifically, to demonstrate the usefulness of this approach, we solve the attractive Hubbard model in the grand-canonical ensemble, where the sss-wave superconducting solution is explicitly obtained.
We extend the natural orbital impurity solver [Y. Lu, M. H\"oppner, O. Gunnarsson, and M. W. Haverkort, Phys. Rev. B 90, 085102 (2014)] to finite temperatures and apply it to calculate spectral and transport properties of correlated electrons within the dynamical mean-field theory. First, we benchmark our method against the exact diagonalization result for small clusters, finding that the natural orbital scheme works well not only for zero temperature but for low finite temperatures. The method yields smooth and sufficiently accurate spectra, which agree well with the results of the numerical renormalization group. Using the smooth spectra, we calculate the electric conductivity and Seebeck coefficient for the two-dimensional Hubbard model at low temperatures which are within the scope of many experiments and practical applications. These results demonstrate the usefulness of the natural orbital framework for obtaining the real frequency information of correlated electron systems.