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Computer Physics Communications

Numerical renormalization group calculations for magnetic impurity systems with spin-orbit coupling and crystal-field effects

Aitor Calvo-Fernández, M. Blanco-Rey, Asier Eiguren

Published Apr 8, 2025• 1 citation
Publisher

Abstract

Exploiting symmetries in the numerical renormalization group (NRG) method significantly enhances performance by improving the accuracy, increasing the computational speed, and optimizing the memory efficiency. Published codes focus on continuous rotations and unitary groups, which generally are not applicable to systems with strong crystal-field effects. The PointGroupNRG code implements symmetries related to discrete rotation groups, which are defined by the user in terms of Clebsch-Gordan coefficients, together with particle conservation and spin rotation symmetries. In this paper we present a new version of the code that extends the available finite groups, previously limited to simply reducible point groups, in a way that all point and double groups become accessible. It also includes the full spin-orbital rotation group. Moreover, to improve the code's flexibility for impurities with complex interactions, this new version allows to choose between a standard Anderson Hamiltonian for the impurity or, as another novel feature, an ionic model that requires only the spectrum and the impurity Lehmann amplitudes. Program summary Program Title: PointGroupNRG CPC Library link to program files: https://doi.org/10.17632/hjwmt6cc55.1 Developer's repository link: https://github.com/aitorcf/PointGroupNRG Licensing provisions: GPLv3 Programming language: Julia Journal reference of previous version: Comput. Phys. Commun. 296 (2024), 109032, https://doi.org/10.1016/j.cpc.2023.109032 Does the new version supersede the previous version?: Yes. Reasons for the new version: Extension. Nature of problem: Numerical renormalization group (NRG) calculations for realistic models are computationally expensive, mainly due to their hard scaling with the number of orbital and spin configurations available for the electrons. Symmetry considerations reduce the computational cost of the calculations by exploiting the block structure of the operator matrix elements and by removing the redundancy in the symmetry-related matrix elements. Existing codes implement continuous symmetries, which are not generally and/or straightforwardly applicable to systems where spin-orbit and crystal-field effects need to be taken into account. Solution method: The first version of the code [1] introduced finite point group symmetries together with particle conservation and spin isotropy , useful for systems with strong crystal-field effects but negligible spin-orbit coupling. This new version also includes total angular momentum conservation and double group symmetries, together with particle conservation. This allows to deal with magnetic impurity systems with strong spin-orbit coupling. To make the code more versatile in handling systems of this type, we have added the possibility to use ionic models.

DOI
10.1016/j.cpc.2025.109613

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Exploiting symmetries in the numerical renormalization group (NRG) method significantly enhances performance by improving the accuracy, increasing the computational speed, and optimizing the memory efficiency. Published codes focus on continuous rotations and unitary groups, which generally are not applicable to systems with strong crystal-field effects. The PointGroupNRG code implements symmetries related to discrete rotation groups, which are defined by the user in terms of Clebsch-Gordan coefficients, together with particle conservation and spin rotation symmetries. In this paper we present a new version of the code that extends the available finite groups, previously limited to simply reducible point groups, in a way that all point and double groups become accessible. It also includes the full spin-orbital rotation group. Moreover, to improve the code's flexibility for impurities with complex interactions, this new version allows to choose between a standard Anderson Hamiltonian for the impurity or, as another novel feature, an ionic model that requires only the spectrum and the impurity Lehmann amplitudes. Program summary Program Title: PointGroupNRG CPC Library link to program files: https://doi.org/10.17632/hjwmt6cc55.1 Developer's repository link: https://github.com/aitorcf/PointGroupNRG Licensing provisions: GPLv3 Programming language: Julia Journal reference of previous version: Comput. Phys. Commun. 296 (2024), 109032, https://doi.org/10.1016/j.cpc.2023.109032 Does the new version supersede the previous version?: Yes. Reasons for the new version: Extension. Nature of problem: Numerical renormalization group (NRG) calculations for realistic models are computationally expensive, mainly due to their hard scaling with the number of orbital and spin configurations available for the electrons. Symmetry considerations reduce the computational cost of the calculations by exploiting the block structure of the operator matrix elements and by removing the redundancy in the symmetry-related matrix elements. Existing codes implement continuous symmetries, which are not generally and/or straightforwardly applicable to systems where spin-orbit and crystal-field effects need to be taken into account. Solution method: The first version of the code [1] introduced finite point group symmetries together with particle conservation and spin isotropy , useful for systems with strong crystal-field effects but negligible spin-orbit coupling. This new version also includes total angular momentum conservation and double group symmetries, together with particle conservation. This allows to deal with magnetic impurity systems with strong spin-orbit coupling. To make the code more versatile in handling systems of this type, we have added the possibility to use ionic models.
Exploiting symmetries in the numerical renormalization group (NRG) method significantly enhances performance by improving accuracy, increasing computational speed, and optimizing memory efficiency. Published codes focus on continuous rotations and unitary groups, which generally are not applicable to systems with strong crystal-field effects. The PointGroupNRG code implements symmetries related to discrete rotation groups, which are defined by the user in terms of Clebsch-Gordan coefficients, together with particle conservation and spin rotation symmetries. In this paper we present a new version of the code that extends the available finite groups, previously limited to simply reducible point groups, in a way that all point and double groups become accessible. It also includes the full spin-orbital rotation group. Moreover, to improve the code's flexibility for impurities with complex interactions, this new version allows to choose between a standard Anderson Hamiltonian for the impurity or, as another novel feature, an ionic model that requires only the spectrum and the impurity Lehmann amplitudes.
The numerical renormalization group (NRG) has been widely used as a magnetic impurity solver since the pioneering works by Wilson. Over the past decades, a significant attention has been focused on the application of symmetries in order to reduce the computational cost of the calculations and to improve their accuracy. In particular, a notable progress has been made in implementing continuous symmetries such as SO(3), useful for studying impurities in an isotropic medium, or SU(N), which is applicable to a wide range of systems. In this work, we focus on the application of discrete point group symmetries, which are particularly relevant for impurity systems in metals where crystal field effects are important. With this aim, we have developed an original NRG code written in the Julia language, PointGroupNRG, where we have implemented crystal point-group symmetries for the Anderson impurity model, as well as the continuous spin and charge symmetries. Among other results, we demonstrate the advantage of our procedure by applying the code to a two-impurity model with RKKY interaction and an impurity system with two orbitals of Eg symmetry and two channels. We also provide benchmarks to show the performance improvements obtained by exploiting the orbital symmetries.
Quantum impurity models describe interactions between some local degrees of freedom and a continuum of non-interacting fermionic or bosonic states. The investigation of quantum impurity models is a starting point towards the understanding of more complex strongly correlated systems, but quantum impurity models also provide the description of various correlated mesoscopic structures, biological and chemical processes, atomic physics and describe phenomena such as dissipation or dephasing. Prototypes of these models are the Anderson impurity model, or the single- and multi-channel Kondo models. The numerical renormalization group method (NRG) proposed by Wilson in mid 70's has been used in its original form for a longtime as one of the most accurate and powerful methods to deal with quatum impurity problems. Recently, a number of new developments took place: First, a spectral sum-conserving density matrix NRG approach (DM-NRG) has been developed, which has also been generalized for non-Abelian symmetries. In this manual we introduce some of the basic concepts of the NRG method and present recently developed Flexible DM-NRG code. This code uses user-defined non-Abelian symmetries dynamically, computes spectral functions, expectation values of local operators for user-defined impurity models. The code can also use a uniform density of states as well as a user-defined density of states. The current version of the code assumes fermionic bath's and it uses any number of U(1), SU(2) charge SU(2) or Z(2) symmetries. The Flexible DM-NRG code can be downloaded from http://www.phy.bme.hu/~dmnrg
Quantum impurity problems can be solved using the numerical renormalization group (NRG), which involves discretizing the free conduction electron system and mapping to a ``Wilson chain.'' It was shown recently that Wilson chains for different electronic species can be interleaved by use of a modified discretization, dramatically increasing the numerical efficiency of the RG scheme [Phys. Rev. B 89, 121105(R) (2014)]. Here we systematically examine the accuracy and efficiency of the ``interleaved'' NRG (iNRG) method in the context of the single impurity Anderson model, the two-channel Kondo model, and a three-channel Anderson-Hund model. The performance of iNRG is explicitly compared with ``standard'' NRG (sNRG): when the average number of states kept per iteration is the same in both calculations, the accuracy of iNRG is equivalent to that of sNRG but the computational costs are significantly lower in iNRG when the same symmetries are exploited. Although iNRG weakly breaks SU(NNN) channel symmetry (if present), both accuracy and numerical cost are entirely competitive with sNRG exploiting full symmetries. iNRG is therefore shown to be a viable and technically simple alternative to sNRG for high-symmetry models. Moreover, iNRG can be used to solve a range of lower-symmetry multiband problems that are inaccessible to sNRG.
We present the SpinParser open-source software [https://github.com/fbuessen/SpinParser]. The software is designed to perform pseudofermion functional renormalization group (pf-FRG) calculations for frustrated quantum magnets in two and three spatial dimensions. It aims to make such calculations readily accessible without the need to write specialized program code; instead, custom lattice graphs and microscopic spin models can be defined as plain-text input files. Underlying symmetries of the model are automatically analyzed and exploited by the numerical core written in C++ in order to optimize the performance across large-scale shared memory and/or distributed memory computing platforms.
The numerical renormalization group (NRG) is rephrased as a variational method with the cost function given by the sum of all the energies of the effective low-energy Hamiltonian. This allows us to systematically improve the spectrum obtained by NRG through sweeping. The ensuing algorithm has a lot of similarities to the density matrix renormalization group (DMRG) when targeting many states, and this synergy of NRG and DMRG combines the best of both worlds and extends their applicability. We illustrate this approach with simulations of a quantum spin chain and a single impurity Anderson model where the accuracy of the effective eigenstates is greatly enhanced as compared to the NRG, especially in the transition to the continuum limit.