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Interplay between Point-Group Symmetries and the Choice of the Bloch Basis in Multiband Models

Stefan Alexander Maier, Carsten Honerkamp A, Qiang-Hua Wang

Published Jan 1, 2013• 6 citations
OpenAlex

Abstract

We analyze the point-group symmetries of generic multiband tight-binding models with respect to the transformation properties of the effective interactions. While the vertex functions in the orbital language may transform non-trivially under point-group operations, their point-group behavior in the band language can be simplified by choosing a suitable Bloch basis. We first give two analytically accessible examples. Then, we show that, for a large class of models, a natural Bloch basis exists, in which the vertex functions in the band language transform trivially under all point-group operations. As a consequence, the point-group symmetries can be used to reduce the computational effort in perturbative many-particle approaches, such as the functional renormalization group.

Related papers

Interplay between Point-Group Symmetries and the Choice of the Bloch Basis in Multiband Models

Stefan Maier, Carsten Honerkamp A, Qiang-Hua Wang

We analyze the point-group symmetries of generic multiband tight-binding models with respect to the transformation properties of the effective interactions. While the vertex functions in the orbital language may transform non-trivially under point-group operations, their point-group behavior in the band language can be simplified by choosing a suitable Bloch basis. We first give two analytically accessible examples. Then, we show that, for a large class of models, a natural Bloch basis exists, in which the vertex functions in the band language transform trivially under all point-group operations. As a consequence, the point-group symmetries can be used to reduce the computational effort in perturbative many-particle approaches, such as the functional renormalization group.

arXiv:1310.0278•Symmetry•Nov 11, 2013•6 citations

Functional renormalization group approaches to multiband models and antiferromagnetically ordered phases

S. Maier, Carsten Honerkamp A

This thesis is devoted to purely fermionic functional renormalization-group (fRG) studies of multiband effects and of antiferromagnetically ordered phases. The formal framework of the one-particle irreducible fRG scheme is recapitulated and extended for charge-conserving theories with a collinear spin symmetry. For multiband models, point-group symmetries and their interplay with the phases of Bloch basis states are discussed. Further, an improved truncation of the fRG flow equations is proposed that keeps track of three-particle interactions generated by virtual excitations of bands away from the Fermi surface. The feedback of this three-particle term on the two-particle interaction is first investigated in a g-ology approach to a simple two-band model for cuprate superconductors at weak coupling. The results suggest a potentially strong impact of the three-particle term. Subsequently, the three-band Emery model is analyzed within a channel-decomposition, again at weak coupling. In contrast to earlier work, the approach pursued here does not rely on a form-factor expansion, which allows for the observation of a deformed d-wave pairing gap for increasingly incommensurate ordering tendencies in the magnetic channel. In this more refined study, the three-particle feedback plays a minor role. Despite considerable qualitative agreement with results for the one-band Hubbard model with effective parameters, multi-orbital effects are found to play an important role on a quantitative level. Phases of antiferromagnetic ordering are first studied for a reduced mean-field model. In order to go beyond the mean-field picture, a hierarchy of approximations is devised, which starts from the corresponding channel-decomposed one-loop flow equations. Interaction terms breaking discrete symmetries are thereby successively neglected. In the course of these approximations, also an exchange parametrization is employed. For the spin-density wave phase of a two-pocket model initially proposed by Chubukov et al. in Phys. Rev. B 78:134512, the flow equations are then integrated numerically at the most approximate level of the hierarchy. This yields a gap that is significantly reduced compared to mean-field theory. The resulting violation of the SU(2) Ward identity appears acceptable on a qualitative level, suggesting that the approximations causing this violation are physically meaningful.

•Jan 1, 2014•1 citation

Efficient vertex parametrization for the constrained functional renormalization group for effective low-energy interactions in multiband systems

Carsten Honerkamp A

We describe an efficient approximation for the electron-electron interaction in the determination of the low-energy effective interaction in multiband lattice systems. By using ideas for channel decomposition, form-factor expansion, and the truncated-unity technique we describe the interaction as arising from the nonlocal and orbital-dependent coupling of particle-hole and particle-particle bilinears formed by fields residing in the same one or two orbitals. This allows us to employ the constrained functional renormalization group (cfRG) with a suitable momentum and frequency discretization. The approach gives insights into the nonlocal screening of spin and charge interactions when bands away from the Fermi level are integrated out. Specifically, we compute the effective low-energy interactions in the low-energy target band of a three-band model with on-site and nonlocal bare interactions. We show that the cfRG adds important features to the effective target-band interaction that cannot be found using the constrained random phase approximation (cRPA).

arXiv:1805.01669•Physical review. B./Physical review. B•Oct 17, 2018•25 citations

Effective three-particle interactions in low-energy models for multiband systems

Stefan Alexander Maier, Carsten Honerkamp A

We discuss different approximations for effective low-energy interactions in multiband models for weakly correlated electrons. In the study of Fermi-surface instabilities of the conduction band(s), the standard approximation consists only keeping those terms in the bare interactions that couple only to the conduction band(s), while corrections due to virtual excitations into bands away from the Fermi surface are typically neglected. Here, using a functional renormalization group approach, we present an improved truncation for the treatment of the effective interactions in the conduction band that keeps track of the generated three-particle interactions (six-point term) and hence allows one to include important aspects of these virtual interband excitations. Within a simplified two-patch treatment of the conduction band, we demonstrate that these corrections can have a rather strong effect in parts of the phase diagram by changing the critical scales for various orderings and the phase boundaries.

arXiv:1112.0229•Physical Review B•Feb 27, 2012•12 citations

Point-group symmetry analysis of many-electron wavefunctions on a quantum computer

Rei Sakuma, Kenji Sugisaki, Shu Kanno +2 more

A point group is a set of spatial symmetry operations in molecular systems and is an indispensable tool for analyzing molecular orbitals and spectroscopy experiments in chemistry. Several quantum algorithms to exploit this symmetry have been proposed, but practical implementations of point-group symmetry operations and the detailed symmetry analysis of realistic many-electron wavefunctions are still missing. In this work, we propose an ancilla-free hybrid method to analyze point-group symmetries of many-electron states, which works for both abelian and non-abelian groups. For a given wavefunction, our method calculates the projection weights of point-group irreducible representations by applying orbital rotations derived from the eigenvectors of the representation matrices, making it applicable to arbitrary basis functions. The usefulness of our approach is demonstrated through numerical simulations of benzene and ferrocene molecules. Furthermore, we perform a hardware demonstration of the weight calculation of the ground state and the first excited state of benzene in $D_{2h}$ symmetry, using up to 32 qubits of IBM's ibm_kawasaki device. By combining a tensor-network based encoding scheme and error mitigation techniques, we find the weights of irreducible representations for both states are faithfully reproduced within a few percent error. Our results suggest that the proposed method serves as a practical tool for analyzing symmetry properties of many-electron wavefunctions in realistic material simulations on near-term and early fault-tolerant quantum computers.

quant-pharXiv:2605.24824•May 26, 2026•0 citations

The PointGroupNRG code for numerical renormalization group calculations with discrete point-group symmetries

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

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.

arXiv:2307.03658•Computer Physics Communications•Nov 29, 2023•3 citations
We analyze the point-group symmetries of generic multiband tight-binding models with respect to the transformation properties of the effective interactions. While the vertex functions in the orbital language may transform non-trivially under point-group operations, their point-group behavior in the band language can be simplified by choosing a suitable Bloch basis. We first give two analytically accessible examples. Then, we show that, for a large class of models, a natural Bloch basis exists, in which the vertex functions in the band language transform trivially under all point-group operations. As a consequence, the point-group symmetries can be used to reduce the computational effort in perturbative many-particle approaches, such as the functional renormalization group.
We analyze the point-group symmetries of generic multiband tight-binding models with respect to the transformation properties of the effective interactions. While the vertex functions in the orbital language may transform non-trivially under point-group operations, their point-group behavior in the band language can be simplified by choosing a suitable Bloch basis. We first give two analytically accessible examples. Then, we show that, for a large class of models, a natural Bloch basis exists, in which the vertex functions in the band language transform trivially under all point-group operations. As a consequence, the point-group symmetries can be used to reduce the computational effort in perturbative many-particle approaches, such as the functional renormalization group.
This thesis is devoted to purely fermionic functional renormalization-group (fRG) studies of multiband effects and of antiferromagnetically ordered phases. The formal framework of the one-particle irreducible fRG scheme is recapitulated and extended for charge-conserving theories with a collinear spin symmetry. For multiband models, point-group symmetries and their interplay with the phases of Bloch basis states are discussed. Further, an improved truncation of the fRG flow equations is proposed that keeps track of three-particle interactions generated by virtual excitations of bands away from the Fermi surface. The feedback of this three-particle term on the two-particle interaction is first investigated in a g-ology approach to a simple two-band model for cuprate superconductors at weak coupling. The results suggest a potentially strong impact of the three-particle term. Subsequently, the three-band Emery model is analyzed within a channel-decomposition, again at weak coupling. In contrast to earlier work, the approach pursued here does not rely on a form-factor expansion, which allows for the observation of a deformed d-wave pairing gap for increasingly incommensurate ordering tendencies in the magnetic channel. In this more refined study, the three-particle feedback plays a minor role. Despite considerable qualitative agreement with results for the one-band Hubbard model with effective parameters, multi-orbital effects are found to play an important role on a quantitative level. Phases of antiferromagnetic ordering are first studied for a reduced mean-field model. In order to go beyond the mean-field picture, a hierarchy of approximations is devised, which starts from the corresponding channel-decomposed one-loop flow equations. Interaction terms breaking discrete symmetries are thereby successively neglected. In the course of these approximations, also an exchange parametrization is employed. For the spin-density wave phase of a two-pocket model initially proposed by Chubukov et al. in Phys. Rev. B 78:134512, the flow equations are then integrated numerically at the most approximate level of the hierarchy. This yields a gap that is significantly reduced compared to mean-field theory. The resulting violation of the SU(2) Ward identity appears acceptable on a qualitative level, suggesting that the approximations causing this violation are physically meaningful.
We describe an efficient approximation for the electron-electron interaction in the determination of the low-energy effective interaction in multiband lattice systems. By using ideas for channel decomposition, form-factor expansion, and the truncated-unity technique we describe the interaction as arising from the nonlocal and orbital-dependent coupling of particle-hole and particle-particle bilinears formed by fields residing in the same one or two orbitals. This allows us to employ the constrained functional renormalization group (cfRG) with a suitable momentum and frequency discretization. The approach gives insights into the nonlocal screening of spin and charge interactions when bands away from the Fermi level are integrated out. Specifically, we compute the effective low-energy interactions in the low-energy target band of a three-band model with on-site and nonlocal bare interactions. We show that the cfRG adds important features to the effective target-band interaction that cannot be found using the constrained random phase approximation (cRPA).
We discuss different approximations for effective low-energy interactions in multiband models for weakly correlated electrons. In the study of Fermi-surface instabilities of the conduction band(s), the standard approximation consists only keeping those terms in the bare interactions that couple only to the conduction band(s), while corrections due to virtual excitations into bands away from the Fermi surface are typically neglected. Here, using a functional renormalization group approach, we present an improved truncation for the treatment of the effective interactions in the conduction band that keeps track of the generated three-particle interactions (six-point term) and hence allows one to include important aspects of these virtual interband excitations. Within a simplified two-patch treatment of the conduction band, we demonstrate that these corrections can have a rather strong effect in parts of the phase diagram by changing the critical scales for various orderings and the phase boundaries.
A point group is a set of spatial symmetry operations in molecular systems and is an indispensable tool for analyzing molecular orbitals and spectroscopy experiments in chemistry. Several quantum algorithms to exploit this symmetry have been proposed, but practical implementations of point-group symmetry operations and the detailed symmetry analysis of realistic many-electron wavefunctions are still missing. In this work, we propose an ancilla-free hybrid method to analyze point-group symmetries of many-electron states, which works for both abelian and non-abelian groups. For a given wavefunction, our method calculates the projection weights of point-group irreducible representations by applying orbital rotations derived from the eigenvectors of the representation matrices, making it applicable to arbitrary basis functions. The usefulness of our approach is demonstrated through numerical simulations of benzene and ferrocene molecules. Furthermore, we perform a hardware demonstration of the weight calculation of the ground state and the first excited state of benzene in D2hD_{2h}D2h​ symmetry, using up to 32 qubits of IBM's ibm_kawasaki device. By combining a tensor-network based encoding scheme and error mitigation techniques, we find the weights of irreducible representations for both states are faithfully reproduced within a few percent error. Our results suggest that the proposed method serves as a practical tool for analyzing symmetry properties of many-electron wavefunctions in realistic material simulations on near-term and early fault-tolerant quantum computers.
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.