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:1711.02182 · Physical review. B./Physical review. B

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

Finn Lasse Buessen, Dietrich Roscher, Sebastian Diehl, Simon Trebst

Published Feb 20, 2018• 42 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.97.064415

Related papers

Functional renormalization group approach to SU(N) Heisenberg models: Momentum-space renormalization group for the large- N limit

Dietrich Roscher, Finn Lasse Buessen, Michael M. Scherer +2 more

In frustrated magnetism, making a stringent connection between microscopic spin models and macroscopic properties of spin liquids remains an important challenge. A recent step towards this goal has been the development of the pseudofermion functional renormalization group approach (pf-FRG) which, building on a fermionic parton construction, enables the numerical detection of the onset of spin liquid states as temperature is lowered. In this work, focusing on the $\mathrm{SU}(N)$ Heisenberg model at large $N$, we extend this approach in a way that allows us to directly enter the low-temperature spin liquid phase, and to probe its character. Our approach proceeds in momentum space, making it possible to keep the truncation minimalistic, while also avoiding the bias introduced by an explicit decoupling of the fermionic parton interactions into a given channel. We benchmark our findings against exact mean-field results in the large-$N$ limit, and show that even without prior knowledge the pf-FRG approach identifies the correct mean-field decoupling channel. On a technical level, we introduce an alternative finite temperature regularization scheme that is necessitated to access the spin liquid ordered phase. In a companion paper [Buessen et al., Phys. Rev. B 97, 064415 (2018)] we present a different set of modifications of the pf-FRG scheme that allow us to study $\mathrm{SU}(N)$ Heisenberg models (using a real-space RG approach) for arbitrary values of $N$, albeit only up to the phase transition towards spin liquid physics.

arXiv:1711.02183•Physical review. B./Physical review. B•Feb 20, 2018•35 citations

Cluster functional renormalization group and absence of a bilinear spin liquid in the J1−J2 Heisenberg model

Dietrich Roscher, Nico Gneist, Michael M. Scherer +2 more

The pseudofermion functional renormalization group (pf-FRG) has been put forward as a semianalytical scheme that, for a given microscopic spin model, allows to discriminate whether the low-temperature states exhibit magnetic ordering or a tendency toward the formation of quantum spin liquids. However, the precise nature of the putative spin-liquid ground state has remained hard to infer from the original (single-site) pf-FRG scheme. Here, we introduce a cluster pf-FRG approach, which allows for a more stringent connection between a microscopic spin model and its low-temperature spin-liquid ground states. In particular, it allows to calculate spatially structured fermion bilinear expectation values on spatial clusters, which are formed by splitting the original lattice into several sublattices, thereby allowing for the positive identification of a family of bilinear spin-liquid states. As an application of this cluster pf-FRG approach, we consider the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)$ Heisenberg model on a square lattice, which is a paradigmatic example for a frustrated quantum magnet exhibiting quantum spin-liquid behavior for intermediate coupling strengths. In the well-established large-$N$ limit of this model, we show that our approach correctly captures the emergence of the $\ensuremath{\pi}$-flux spin-liquid state at low temperatures. For small $N$, where the precise nature of the ground state remains controversial, we focus on the widely studied case of $N=2$, for which we determine the low-temperature phase diagram near the strongly frustrated regime after implementing the fermion-number constraint by the flowing Popov-Fedotov method. Our results suggest that the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model does not support the formation of a fermion bilinear spin-liquid state.

arXiv:1905.01060•Physical review. B./Physical review. B•Sep 13, 2019•20 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

Pseudo-fermion functional renormalization group for spin models

Tobias Müller, Dominik Kiese, Nils 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.

arXiv:2307.10359•Jul 19, 2023•4 citations

Spin-functional renormalization group for the J1J2J3 quantum Heisenberg model

Dmytro Tarasevych, Andreas Rückriegel, Savio Keupert +2 more

We use our recently developed functional renormalization group (FRG) approach for quantum spin systems to investigate the phase diagram of the frustrated ${J}_{1}{J}_{2}{J}_{3}$ quantum Heisenberg model on a cubic lattice. From a simple truncation of the hierarchy of FRG flow equations for the irreducible spin vertices, which retains only static spin fluctuations and neglects the flow of the four-spin interaction, we can estimate the critical temperature with a similar accuracy as the numerically more expensive pseudofermion FRG. In the regime where the ground state exhibits either ferromagnetic or antiferromagnetic order, a more sophisticated truncation including the renormalization of the four-spin interaction as well as dynamic spin fluctuations reveals the underlying renormalization group fixed point and yields critical temperatures which deviate from the accepted values by at most $4%$.

arXiv:2208.05741•Physical review. B./Physical review. B•Nov 14, 2022•9 citations

Keldysh pseudofermion functional renormalization group for quantum magnetism

Janik Potten, Yasir Iqbal, Ronny Thomale +1 more

The functional renormalization group (FRG) approach for spin models relying on a pseudofermionic description has proven to be a powerful technique in simulating ground-state properties of strongly frustrated magnetic lattices. A drawback of the FRG framework is that it is formulated in the imaginary-time Matsubara formalism and thus only able to access static correlations, a limitation shared with most other many-body approaches. A description of the dynamical properties of magnetic systems is the key to bridging the gap between theory and neutron-scattering spectra. We take the decisive step of expanding the scope of pseudofermion FRG to the Keldysh formalism, which, while originally developed to address nonequilibrium phenomena, enables a direct calculation of the equilibrium dynamical spin structure factors on generic lattices in arbitrary dimensions. We identify the principal features characterizing the low-energy spectra of exemplary zero-, one- and two-dimensional spin-$1/2$ Heisenberg models as well as the Kitaev honeycomb model while identifying current limitations of the method that have to be improved upon.

arXiv:2503.11596•Physical review. B./Physical review. B•Nov 17, 2025•1 citation
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.
In frustrated magnetism, making a stringent connection between microscopic spin models and macroscopic properties of spin liquids remains an important challenge. A recent step towards this goal has been the development of the pseudofermion functional renormalization group approach (pf-FRG) which, building on a fermionic parton construction, enables the numerical detection of the onset of spin liquid states as temperature is lowered. In this work, focusing on the SU(N)\mathrm{SU}(N)SU(N) Heisenberg model at large NNN, we extend this approach in a way that allows us to directly enter the low-temperature spin liquid phase, and to probe its character. Our approach proceeds in momentum space, making it possible to keep the truncation minimalistic, while also avoiding the bias introduced by an explicit decoupling of the fermionic parton interactions into a given channel. We benchmark our findings against exact mean-field results in the large-NNN limit, and show that even without prior knowledge the pf-FRG approach identifies the correct mean-field decoupling channel. On a technical level, we introduce an alternative finite temperature regularization scheme that is necessitated to access the spin liquid ordered phase. In a companion paper [Buessen et al., Phys. Rev. B 97, 064415 (2018)] we present a different set of modifications of the pf-FRG scheme that allow us to study SU(N)\mathrm{SU}(N)SU(N) Heisenberg models (using a real-space RG approach) for arbitrary values of NNN, albeit only up to the phase transition towards spin liquid physics.
The pseudofermion functional renormalization group (pf-FRG) has been put forward as a semianalytical scheme that, for a given microscopic spin model, allows to discriminate whether the low-temperature states exhibit magnetic ordering or a tendency toward the formation of quantum spin liquids. However, the precise nature of the putative spin-liquid ground state has remained hard to infer from the original (single-site) pf-FRG scheme. Here, we introduce a cluster pf-FRG approach, which allows for a more stringent connection between a microscopic spin model and its low-temperature spin-liquid ground states. In particular, it allows to calculate spatially structured fermion bilinear expectation values on spatial clusters, which are formed by splitting the original lattice into several sublattices, thereby allowing for the positive identification of a family of bilinear spin-liquid states. As an application of this cluster pf-FRG approach, we consider the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)$ Heisenberg model on a square lattice, which is a paradigmatic example for a frustrated quantum magnet exhibiting quantum spin-liquid behavior for intermediate coupling strengths. In the well-established large-NNN limit of this model, we show that our approach correctly captures the emergence of the $\ensuremath{\pi}$-flux spin-liquid state at low temperatures. For small NNN, where the precise nature of the ground state remains controversial, we focus on the widely studied case of N=2N=2N=2, for which we determine the low-temperature phase diagram near the strongly frustrated regime after implementing the fermion-number constraint by the flowing Popov-Fedotov method. Our results suggest that the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model does not support the formation of a fermion bilinear spin-liquid state.
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.
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 mmm-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 use our recently developed functional renormalization group (FRG) approach for quantum spin systems to investigate the phase diagram of the frustrated J1J2J3{J}_{1}{J}_{2}{J}_{3}J1​J2​J3​ quantum Heisenberg model on a cubic lattice. From a simple truncation of the hierarchy of FRG flow equations for the irreducible spin vertices, which retains only static spin fluctuations and neglects the flow of the four-spin interaction, we can estimate the critical temperature with a similar accuracy as the numerically more expensive pseudofermion FRG. In the regime where the ground state exhibits either ferromagnetic or antiferromagnetic order, a more sophisticated truncation including the renormalization of the four-spin interaction as well as dynamic spin fluctuations reveals the underlying renormalization group fixed point and yields critical temperatures which deviate from the accepted values by at most 44%4.
The functional renormalization group (FRG) approach for spin models relying on a pseudofermionic description has proven to be a powerful technique in simulating ground-state properties of strongly frustrated magnetic lattices. A drawback of the FRG framework is that it is formulated in the imaginary-time Matsubara formalism and thus only able to access static correlations, a limitation shared with most other many-body approaches. A description of the dynamical properties of magnetic systems is the key to bridging the gap between theory and neutron-scattering spectra. We take the decisive step of expanding the scope of pseudofermion FRG to the Keldysh formalism, which, while originally developed to address nonequilibrium phenomena, enables a direct calculation of the equilibrium dynamical spin structure factors on generic lattices in arbitrary dimensions. We identify the principal features characterizing the low-energy spectra of exemplary zero-, one- and two-dimensional spin-1/21/21/2 Heisenberg models as well as the Kitaev honeycomb model while identifying current limitations of the method that have to be improved upon.