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

Physical review. B./Physical review. B

Pseudo-fermion functional renormalization group with magnetic fields

Vincent Noculak, Johannes Reuther

Published May 7, 2024• 8 citations
Publisher

Abstract

The pseudo-fermion functional renormalization group is generalized to treat spin Hamiltonians with finite magnetic fields, enabling its application to arbitrary spin lattice models with linear and bilinear terms in the spin operators. We discuss in detail an efficient numerical implementation of this approach making use of the system's symmetries. Particularly, we demonstrate that the inclusion of small symmetry-breaking magnetic seed fields regularizes divergences of the susceptibility at magnetic phase transitions. This allows the investigation of spin models within magnetically ordered phases at $T=0$ in the physical limit of vanishing renormalization group parameter $\mathrm{\ensuremath{\Lambda}}$. We explore the capabilities and limitations of this method extension for Heisenberg models on the square, honeycomb, and triangular lattices. While the zero-field magnetizations of these systems are systematically overestimated, the types of magnetic orders are correctly captured, even if the local orientations of the seed field are chosen differently than the spin orientations of the realized magnetic order. Furthermore, the magnetization curve of the square lattice Heisenberg antiferromagnet shows good agreement with error controlled methods. In the future, the inclusion of magnetic fields in the pseudo-fermion functional renormalization group, which is also possible in three-dimensional spin systems, will enable a variety of additional interesting applications such as the investigation of magnetization plateaus.

DOI
10.1103/physrevb.109.174414

Related papers

Pseudo-fermion functional renormalization group with magnetic fields

Vincent Noculak, Johannes Reuther

The pseudo-fermion functional renormalization group is generalized to treat spin Hamiltonians with finite magnetic fields, enabling its application to arbitrary spin lattice models with linear and bilinear terms in the spin operators. We discuss in detail an efficient numerical implementation of this approach making use of the system's symmetries. Particularly, we demonstrate that the inclusion of small symmetry breaking magnetic seed fields regularizes divergences of the susceptibility at magnetic phase transitions. This allows the investigation of spin models within magnetically ordered phases at $T=0$ in the physical limit of vanishing renormalization group parameter $Λ$. We explore the capabilities and limitations of this method extension for Heisenberg models on the square, honeycomb and triangular lattices. While the zero-field magnetizations of these systems are systematically overestimated, the types of magnetic orders are correctly captured, even if the local orientations of the seed field are chosen differently than the spin orientations of the realized magnetic order. Furthermore, the magnetization curve of the square lattice Heisenberg antiferromagnet shows good agreement with error controlled methods. In the future, the inclusion of magnetic fields in the pseudo-fermion functional renormalization group, which is also possible in three dimensional spin systems, will enable a variety of additional interesting applications such as the investigation of magnetization plateaus.

arXiv:2312.12097•Dec 19, 2023•1 citation

Pseudo-Majorana functional renormalization for frustrated XXZ spin- 1 2 models with field or magnetization along the spin- Z direction at finite temperature

Frederic Bippus, Benedikt Schneider, Björn Sbierski

The numerical study of high-dimensional frustrated quantum magnets remains a challenging problem. Here we present an extension of the pseudo-Majorana functional renormalization group to spin-1/2 XXZ-type Hamiltonians with field or magnetization along spin- Z direction at finite temperature. We consider a U(1) symmetry-adapted fermionic spin representation and derive the diagrammatic framework and its renormalization group flow equations. We discuss benchmark results and application to two antiferromagnetic triangular lattice materials recently studied in experiments with applied magnetic fields: First, we numerically reproduce the magnetization data measured for CeMgAl 11 O 19 confirming model parameters previously estimated from inelastic neutron spectrum in high fields. Second, we showcase the accuracy of our method by studying the thermal phase transition into the spin solid up-up-down phase of Na 2 BaCo ( PO 4 ) 2 in good agreement with experiment.

arXiv:2411.18198•Physical review. B./Physical review. B•Feb 13, 2025•6 citations

Pseudo-fermion functional renormalization group for spin models

Tobias Müller, Dominik Kiese, Nils Niggemann +5 more

Abstract For decades, frustrated quantum magnets have been a seed for scientific progress and innovation in condensed matter. As much as the numerical tools for low-dimensional quantum magnetism have thrived and improved in recent years due to breakthroughs inspired by quantum information and quantum computation, higher-dimensional quantum magnetism can be considered as the final frontier, where strong quantum entanglement, multiple ordering channels, and manifold ways of paramagnetism culminate. At the same time, efforts in crystal synthesis have induced a significant increase in the number of tangible frustrated magnets which are generically three-dimensional in nature, creating an urgent need for quantitative theoretical modeling. We review the pseudo-fermion (PF) and pseudo-Majorana (PM) functional renormalization group (FRG) and their specific ability to address higher-dimensional frustrated quantum magnetism. First developed more than a decade ago, the PFFRG interprets a Heisenberg model Hamiltonian in terms of Abrikosov pseudofermions, which is then treated in a diagrammatic resummation scheme formulated as a renormalization group flow of m-particle pseudofermion vertices. The article reviews the state of the art of PFFRG and PMFRG and discusses their application to exemplary domains of frustrated magnetism, but most importantly, it makes the algorithmic and implementation details of these methods accessible to everyone. By thus lowering the entry barrier to their application, we hope that this review will contribute towards establishing PFFRG and PMFRG as the numerical methods for addressing frustrated quantum magnetism in higher spatial dimensions.

Reports on Progress in Physics•Jan 19, 2024•40 citations

Quantitative functional renormalization for three-dimensional quantum Heisenberg models

Nils Niggemann

We employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by $\sim5\%$ and we further confirm the established values for the critical exponent $\nu$ and the anomalous dimension $\eta$. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit $T\rightarrow0$. While extensions towards higher loop orders $\ell$ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing $\ell$ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.

•May 1, 2022•33 citations

Pseudo fermion functional renormalization group for spin models

T. Mueller, D. Kiese, N. Niggemann +5 more

For decades, frustrated quantum magnets have been a seed for scientific progress and innovation in condensed matter. As much as the numerical tools for low dimensional quantum magnetism have thrived and improved in recent years due to breakthroughs inspired by quantum information and quantum computation, higher dimensional quantum magnetism can be considered as the final frontier, where strong quantum entanglement, multiple ordering channels, and manifold ways of paramagnetism culminate. At the same time, efforts in crystal synthesis have induced a significant increase in the number of tangible frustrated magnets which are generically three dimensional in nature, creating an urgent need for quantitative theoretical modeling. We review the pseudo fermion PF and pseudo Majorana PM functional renormalization group FRG and their specific ability to address higher dimensional frustrated quantum magnetism. First developed more than a decade ago, the PFFRG interprets a Heisenberg model Hamiltonian in terms of Abrikosov pseudofermions, which is then treated in a diagrammatic resummation scheme formulated as a renormalization group flow of m particle pseudofermion vertices. The article reviews the state of the art of PFFRG and PMFRG and discusses their application to exemplary domains of frustrated magnetism, but most importantly, it makes the algorithmic and implementation details of these methods accessible to everyone. By thus lowering the entry barrier to their application, we hope that this review will contribute towards establishing PFFRG and PMFRG as the numerical methods for addressing frustrated quantum magnetism in higher spatial dimensions

•Jan 1, 2024•0 citations

Functional renormalization group approach to SU(N) Heisenberg models: Real-space renormalization group at arbitrary N

Finn Lasse Buessen, Dietrich Roscher, Sebastian Diehl +1 more

The pseudofermion functional renormalization group (pf-FRG) is one of the few numerical approaches that has been demonstrated to quantitatively determine the ordering tendencies of frustrated quantum magnets in two and three spatial dimensions. The approach, however, relies on a number of presumptions and approximations, in particular the choice of pseudofermion decomposition and the truncation of an infinite number of flow equations to a finite set. Here we generalize the pf-FRG approach to $\mathrm{SU}(N)$-spin systems with arbitrary $N$ and demonstrate that the scheme becomes exact in the large-$N$ limit. Numerically solving the generalized real-space renormalization group equations for arbitrary $N$, we can make a stringent connection between the physically most significant case of SU(2) spins and more accessible $\mathrm{SU}(N)$ models. In a case study of the square-lattice $\mathrm{SU}(N)$ Heisenberg antiferromagnet, we explicitly demonstrate that the generalized pf-FRG approach is capable of identifying the instability indicating the transition into a staggered flux spin liquid ground state in these models for large, but finite, values of $N$. In a companion paper [Roscher et al., Phys. Rev. B 97, 064416 (2018)] we formulate a momentum-space pf-FRG approach for $\mathrm{SU}(N)$ spin models that allows us to explicitly study the large-$N$ limit and access the low-temperature spin liquid phase.

arXiv:1711.02182•Physical review. B./Physical review. B•Feb 20, 2018•42 citations
The pseudo-fermion functional renormalization group is generalized to treat spin Hamiltonians with finite magnetic fields, enabling its application to arbitrary spin lattice models with linear and bilinear terms in the spin operators. We discuss in detail an efficient numerical implementation of this approach making use of the system's symmetries. Particularly, we demonstrate that the inclusion of small symmetry-breaking magnetic seed fields regularizes divergences of the susceptibility at magnetic phase transitions. This allows the investigation of spin models within magnetically ordered phases at T=0T=0T=0 in the physical limit of vanishing renormalization group parameter \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ. We explore the capabilities and limitations of this method extension for Heisenberg models on the square, honeycomb, and triangular lattices. While the zero-field magnetizations of these systems are systematically overestimated, the types of magnetic orders are correctly captured, even if the local orientations of the seed field are chosen differently than the spin orientations of the realized magnetic order. Furthermore, the magnetization curve of the square lattice Heisenberg antiferromagnet shows good agreement with error controlled methods. In the future, the inclusion of magnetic fields in the pseudo-fermion functional renormalization group, which is also possible in three-dimensional spin systems, will enable a variety of additional interesting applications such as the investigation of magnetization plateaus.
The pseudo-fermion functional renormalization group is generalized to treat spin Hamiltonians with finite magnetic fields, enabling its application to arbitrary spin lattice models with linear and bilinear terms in the spin operators. We discuss in detail an efficient numerical implementation of this approach making use of the system's symmetries. Particularly, we demonstrate that the inclusion of small symmetry breaking magnetic seed fields regularizes divergences of the susceptibility at magnetic phase transitions. This allows the investigation of spin models within magnetically ordered phases at T=0T=0T=0 in the physical limit of vanishing renormalization group parameter ΛΛΛ. We explore the capabilities and limitations of this method extension for Heisenberg models on the square, honeycomb and triangular lattices. While the zero-field magnetizations of these systems are systematically overestimated, the types of magnetic orders are correctly captured, even if the local orientations of the seed field are chosen differently than the spin orientations of the realized magnetic order. Furthermore, the magnetization curve of the square lattice Heisenberg antiferromagnet shows good agreement with error controlled methods. In the future, the inclusion of magnetic fields in the pseudo-fermion functional renormalization group, which is also possible in three dimensional spin systems, will enable a variety of additional interesting applications such as the investigation of magnetization plateaus.
The numerical study of high-dimensional frustrated quantum magnets remains a challenging problem. Here we present an extension of the pseudo-Majorana functional renormalization group to spin-1/2 XXZ-type Hamiltonians with field or magnetization along spin- Z direction at finite temperature. We consider a U(1) symmetry-adapted fermionic spin representation and derive the diagrammatic framework and its renormalization group flow equations. We discuss benchmark results and application to two antiferromagnetic triangular lattice materials recently studied in experiments with applied magnetic fields: First, we numerically reproduce the magnetization data measured for CeMgAl 11 O 19 confirming model parameters previously estimated from inelastic neutron spectrum in high fields. Second, we showcase the accuracy of our method by studying the thermal phase transition into the spin solid up-up-down phase of Na 2 BaCo ( PO 4 ) 2 in good agreement with experiment.
Abstract For decades, frustrated quantum magnets have been a seed for scientific progress and innovation in condensed matter. As much as the numerical tools for low-dimensional quantum magnetism have thrived and improved in recent years due to breakthroughs inspired by quantum information and quantum computation, higher-dimensional quantum magnetism can be considered as the final frontier, where strong quantum entanglement, multiple ordering channels, and manifold ways of paramagnetism culminate. At the same time, efforts in crystal synthesis have induced a significant increase in the number of tangible frustrated magnets which are generically three-dimensional in nature, creating an urgent need for quantitative theoretical modeling. We review the pseudo-fermion (PF) and pseudo-Majorana (PM) functional renormalization group (FRG) and their specific ability to address higher-dimensional frustrated quantum magnetism. First developed more than a decade ago, the PFFRG interprets a Heisenberg model Hamiltonian in terms of Abrikosov pseudofermions, which is then treated in a diagrammatic resummation scheme formulated as a renormalization group flow of m-particle pseudofermion vertices. The article reviews the state of the art of PFFRG and PMFRG and discusses their application to exemplary domains of frustrated magnetism, but most importantly, it makes the algorithmic and implementation details of these methods accessible to everyone. By thus lowering the entry barrier to their application, we hope that this review will contribute towards establishing PFFRG and PMFRG as the numerical methods for addressing frustrated quantum magnetism in higher spatial dimensions.
We employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by ∼5%\sim5\%∼5% and we further confirm the established values for the critical exponent ν\nuν and the anomalous dimension η\etaη. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit T→0T\rightarrow0T→0. While extensions towards higher loop orders ℓ\ellℓ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing ℓ\ellℓ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.
For decades, frustrated quantum magnets have been a seed for scientific progress and innovation in condensed matter. As much as the numerical tools for low dimensional quantum magnetism have thrived and improved in recent years due to breakthroughs inspired by quantum information and quantum computation, higher dimensional quantum magnetism can be considered as the final frontier, where strong quantum entanglement, multiple ordering channels, and manifold ways of paramagnetism culminate. At the same time, efforts in crystal synthesis have induced a significant increase in the number of tangible frustrated magnets which are generically three dimensional in nature, creating an urgent need for quantitative theoretical modeling. We review the pseudo fermion PF and pseudo Majorana PM functional renormalization group FRG and their specific ability to address higher dimensional frustrated quantum magnetism. First developed more than a decade ago, the PFFRG interprets a Heisenberg model Hamiltonian in terms of Abrikosov pseudofermions, which is then treated in a diagrammatic resummation scheme formulated as a renormalization group flow of m particle pseudofermion vertices. The article reviews the state of the art of PFFRG and PMFRG and discusses their application to exemplary domains of frustrated magnetism, but most importantly, it makes the algorithmic and implementation details of these methods accessible to everyone. By thus lowering the entry barrier to their application, we hope that this review will contribute towards establishing PFFRG and PMFRG as the numerical methods for addressing frustrated quantum magnetism in higher spatial dimensions
The pseudofermion functional renormalization group (pf-FRG) is one of the few numerical approaches that has been demonstrated to quantitatively determine the ordering tendencies of frustrated quantum magnets in two and three spatial dimensions. The approach, however, relies on a number of presumptions and approximations, in particular the choice of pseudofermion decomposition and the truncation of an infinite number of flow equations to a finite set. Here we generalize the pf-FRG approach to SU(N)\mathrm{SU}(N)SU(N)-spin systems with arbitrary NNN and demonstrate that the scheme becomes exact in the large-NNN limit. Numerically solving the generalized real-space renormalization group equations for arbitrary NNN, we can make a stringent connection between the physically most significant case of SU(2) spins and more accessible SU(N)\mathrm{SU}(N)SU(N) models. In a case study of the square-lattice SU(N)\mathrm{SU}(N)SU(N) Heisenberg antiferromagnet, we explicitly demonstrate that the generalized pf-FRG approach is capable of identifying the instability indicating the transition into a staggered flux spin liquid ground state in these models for large, but finite, values of NNN. In a companion paper [Roscher et al., Phys. Rev. B 97, 064416 (2018)] we formulate a momentum-space pf-FRG approach for SU(N)\mathrm{SU}(N)SU(N) spin models that allows us to explicitly study the large-NNN limit and access the low-temperature spin liquid phase.