Pseudo-fermion functional renormalization group for spin models

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.

arXiv commentReview article for Reports on Progress in Physics. Comments are welcome! (43+7) pages, (16+4) figures, 3 tables

References in corpus (159)

• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• Exact evolution equation for the effective potential
• Quantum Spin Liquids
• Machine learning phases of matter
• Renormalization Group Approach to Interacting Fermions
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• The Exact Renormalisation Group and Approximate Solutions
• Aspects of the Functional Renormalisation Group
• Relevance of the Heisenberg-Kitaev model for the honeycomb lattice iridates A_2IrO_3
• Functional renormalization group approach to correlated fermion systems
• The nonperturbative functional renormalization group and its applications
• Computational Studies of Quantum Spin Systems
• Properties of a classical spin liquid: the Heisenberg pyrochlore antiferromagnet
• Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory
• Studying Two Dimensional Systems With the Density Matrix Renormalization Group
• Renormalization group analysis of the 2D Hubbard model
• Spin liquid nature in the Heisenberg $J_1$-$J_2$ triangular antiferromagnet
• Dipolar spin correlations in classical pyrochlore magnets
• Density waves and Cooper pairing on the honeycomb lattice
• Functional Renormalization Group for multi-orbital Fermi Surface Instabilities
• Vaporization of Kitaev spin liquids
• Finite-temperature phase diagram of the Heisenberg-Kitaev model
• Continuous Symmetry Breaking in a Two-dimensional Rydberg Array
• Magnetization plateaus in frustrated antiferromagnetic quantum spin models
• Tracking the Footprints of Spin Fluctuations: A MultiMethod, MultiMessenger Study of the Two-Dimensional Hubbard Model
• The temperature-flow renormalization group and the competition between superconductivity and ferromagnetism
• Physical realization of a quantum spin liquid based on a novel frustration mechanism
• On the fulfillment of Ward identities in the functional renormalization group approach
• J_1-J_2 frustrated two-dimensional Heisenberg model: Random phase approximation and functional renormalization group
• Magnetization plateaus of the Shastry-Sutherland model for SrCu_2(BO_3)_2: SDW, supersolid and bound states
• Renormalization Group Approach towards the QCD Phase Diagram
• Functional renormalization group for the anisotropic triangular antiferromagnet
• Magnetic order and paramagnetic phases in the quantum J1-J2-J3 honeycomb model
• Nonequilibrium functional RG with frequency dependent vertex function: A study of the single impurity Anderson model
• Functional renormalization group for non-equilibrium quantum many-body problems
• Critical temperature and the transition from quantum to classical order parameter fluctuations in the three-dimensional Heisenberg antiferromagnet
• Renormalization group flows into phases with broken symmetry
• A finite-frequency functional RG approach to the single impurity Anderson model
• A novel approach to transport through correlated quantum dots
• High-frequency asymptotics of the vertex function: diagrammatic parametrization and algorithmic implementation
• Evidence for a three-dimensional quantum spin liquid in PbCuTe${}_2$O${}_6$
• Paramagnetism in the kagome compounds (Zn,Mg,Cd)Cu${}_3$(OH)${}_6$Cl${}_2$
• Efficient implementation of the parquet equations -- role of the reducible vertex function and its kernel approximation
• Phase Diagram and Quantum Order by Disorder in the Kitaev $K_1$-$K_2$ Honeycomb Magnet
• Witnessing quantum criticality and entanglement in the triangular antiferromagnet KYbSe${}_2$
• Quantum spin liquids in frustrated spin-1 diamond antiferromagnets
• Multiloop functional renormalization group for general models
• Multiloop functional renormalization group that sums up all parquet diagrams
• Spiral order in the honeycomb iridate Li2IrO3
• New Fermionic Description of Quantum S = 1/2 Antiferromargnet
• Interaction flow method for many-fermion systems
• Spin Liquid State in the 3D Frustrated Antiferromagnet PbCuTe2O6: NMR and muSR Studies
• Continuum limit in matrix models for quantum gravity from the Functional Renormalization Group
• Renormalization group analysis of competing quantum phases in the J1-J2 Heisenberg model on the kagome lattice
• Quantum Phases of the Planar Antiferromagnetic J_1-J_2-J_3 Heisenberg-Model
• Functional renormalization group for three-dimensional quantum magnetism
• Functional renormalization group for Luttinger liquids with impurities
• Functional renormalization group approach to zero-dimensional interacting systems
• Bold Diagrammatic Monte Carlo Method Applied to Fermionized Frustrated Spins
• Magnetic ordering phenomena of interacting quantum spin Hall models
• Superconductivity in the two-dimensional $t$-$t^{\prime }$-Hubbard model
• Breathing chromium spinels: a showcase for a variety of pyrochlore Heisenberg Hamiltonians
• Antiferromagnetic and d-wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group
• Fingerprints of the local moment formation and its Kondo screening in the generalized susceptibilities of many-electron problems
• Functional renormalization group study of the interacting resonant level model in and out of equilibrium
• Electron-doping versus hole-doping in the 2D t-t' Hubbard model
• Self-energy flows in the two-dimensional repulsive Hubbard model
• Numerical treatment of spin systems with unrestricted spin length $S$: A functional renormalization group study
• The pyrochlore S=1/2 Heisenberg antiferromagnet at finite temperature
• Frustrated Quantum Spins at finite Temperature: Pseudo-Majorana functional RG approach
• Possible inversion symmetry breaking in the $S=1/2$ pyrochlore Heisenberg magnet
• Spin-Ice State of the Quantum Heisenberg Antiferromagnet on the Pyrochlore Lattice
• Quantitative functional renormalization-group description of the two-dimensional Hubbard model
• Spin-orbit entangled j=1/2 moments in Ba${}_2$CeIrO${}_6$ -- a frustrated fcc quantum magnet
• Spin-spin correlators in Majorana representation
• Competing magnetic orders and spin liquids in two- and three-dimensional kagome systems: A pseudo-fermion functional renormalization group perspective
• Intertwined nematic orders in a frustrated ferromagnet
• Signatures of a gearwheel quantum spin liquid in a spin-$\frac{1}{2}$ pyrochlore molybdate Heisenberg antiferromagnet
• Phase diagram of a distorted kagome antiferromagnet and application to Y-kapellasite
• Multiloop functional renormalization group for the two-dimensional Hubbard model: Loop convergence of the response functions
• Functional-renormalization-group analysis of Dzyaloshinsky-Moriya and Heisenberg spin interactions on the kagome lattice
• Magnetic field induced quantum spin liquid in the two coupled trillium lattices of K${}_2$Ni${}_2$(SO${}_4$)${}_3$
• Derivation of exact flow equations from the self-consistent parquet relations
• Multiloop functional renormalization group approach to quantum spin systems
• Stability of the spiral spin liquid in MnSc${}_2$S${}_4$
• Characterization of quantum spin liquids and their spinon band structures via functional renormalization
• Renormalization-group analysis of the one-dimensional extended Hubbard model with a single impurity
• Fermionic two-loop functional renormalization group for correlated fermions: Method and application to the attractive Hubbard model
• Bold Diagrammatic Monte Carlo technique for frustrated spin systems
• Absence of long-range order in a triangular spin system with dipolar interactions
• Functional renormalization group analysis of the soft mode at the QCD critical point
• Functional renormalization group approach to SU(N) Heisenberg models: Real-space RG at arbitrary N
• Rise and fall of plaquette order in Shastry-Sutherland magnet revealed by pseudo-fermion functional renormalization group
• Tiling with triangles: parquet and $GW\gamma$ methods unified
• Self-energy feedback and frequency-dependent interactions in the functional renormalization group flow for the two-dimensional Hubbard model
• Pinwheel valence-bond-crystal ground state of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the $shuriken$ lattice
• Properties of multi-particle Green and vertex functions within Keldysh formalism
• Fermionic theory for quantum antiferromagnets with spin S > 1/2
• Classical spiral spin liquids as a possible route to quantum spin liquids
• Dimerization tendencies of the pyrochlore Heisenberg antiferromagnet: A functional renormalization group perspective
• Exact renormalization group for quantum spin systems
• Majorana representation for dissipative spin systems
• Single boson exchange representation of the functional renormalization group for strongly interacting many-electron systems
• Pinch-points to half-moons and up in the stars: the kagome skymap
• Functional renormalization group approach to SU(N) Heisenberg models: Momentum-space RG for the large-N limit
• Majorana representations of spin and an alternative solution of the Kitaev honeycomb model
• Quantum effects on unconventional pinch point singularities
• Dynamical functional renormalization group computation of order parameters and critical temperatures in the two-dimensional Hubbard model
• Functional renormalization group study of the Kitaev-$\Gamma$ model on the honeycomb lattice and emergent incommensurate magnetic correlations
• Adapting Planck's route to investigate the thermodynamics of the spin-half pyrochlore Heisenberg antiferromagnet
• $\mathbf{q}=0$ long-range magnetic order in centennialite CaCu${}_3$(OD)${}_6$Cl${}_2$$\cdot$0.6D${}_2$O: A spin-1/2 perfect kagome antiferromagnet with $J_1$-$J_2$-$J_d$
• Cluster Functional Renormalization Group
• Ground-state phase diagram of spin-$S$ Kitaev-Heisenberg models
• Emergence and stability of spin-valley entangled quantum liquids in moiré heterostructures
• Functional renormalization group for frustrated magnets with nondiagonal spin interactions
• Quantum paramagnetism and helimagnetic orders in the Heisenberg model on the body centered cubic lattice
• Effects of two-loop contributions in the pseudofermion functional renormalization group method for quantum spin systems
• Pseudogap opening in the two-dimensional Hubbard model: A functional renormalization group analysis
• Fulfillment of sum rules and Ward identities in the multiloop functional renormalization group solution of the Anderson impurity model
• Multiloop flow equations for single-boson exchange fRG
• Thermodynamics of the pyrochlore-lattice quantum Heisenberg antiferromagnet
• Continuous spin excitations in the three-dimensional frustrated magnet K2Ni2(SO4)3
• Dynamics of K${}_2$Ni${}_2$(SO${}_4$)${}_3$ governed by proximity to a 3D spin liquid model
• Spin functional renormalization group for quantum Heisenberg ferromagnets: Magnetization and magnon damping in two dimensions
• Supercurrent through a serial quantum dot close to singlet-triplet degeneracy
• Classical and quantum phases of the pyrochlore $S=1/2$ magnet with Heisenberg and Dzyaloshinskii-Moriya interactions
• Critical scales in anisotropic spin systems from functional renormalization
• A Non-Perturbative Renormalization Group approach to quantum XY spin models
• Candidate quantum disordered intermediate phase in the Heisenberg antiferromagnet on the maple-leaf lattice
• Semi-fermionic representation of SU(N) Hamiltonians
• Magnetism in the two-dimensional dipolar XY model
• Quantum disordered insulating phase in the frustrated cubic-lattice Hubbard model
• Cluster functional renormalization group and absence of a bilinear spin liquid in the $J_1$-$J_2$-Heisenberg model
• Dissipative spin dynamics in hot quantum paramagnets
• Zero-magnon sound in quantum Heisenberg ferromagnets
• Solution of the Anderson impurity model via the functional renormalization group
• Equality of bulk wave functions and edge correlations in topological superconductors: A spacetime derivation
• Competing Gauge Fields and Entropically-Driven Spin Liquid to Spin Liquid Transition in non-Kramers Pyrochlores
• Enhanced symmetry-breaking tendencies in the $S=1$ pyrochlore antiferromagnet
• Taming pseudo-fermion functional renormalization for quantum spins: Finite-temperatures and the Popov-Fedotov trick
• Real-frequency quantum field theory applied to the single-impurity Anderson model
• A rich man's derivation of scaling laws for the Kondo model
• $\require{mhchem}$Quantum paramagnetism in the decorated square-kagome antiferromagnet $\ce{Na6Cu7BiO4(PO4)4Cl3}$
• Using Majorana spin-1/2 representation for the spin-boson model
• Majorana fermion description of the Kondo lattice: Variational and Path integral approach
• Bilinear Majorana representations for spin operators with spin magnitudes $S>1/2$
• Critical spin dynamics of Heisenberg ferromagnets revisited
• Feasibility of Kitaev quantum spin liquids in ultracold polar molecules
• Spin functional renormalization group for the $J_1{J}_2{J}_3$ quantum Heisenberg model
• Spectral representation of Matsubara n-point functions: Exact kernel functions and applications
• Fluctuations and the QCD phase diagram
• A pseudofermion functional renormalization group study of dipolar-octupolar pyrochlore magnets
• Ground-State Phase Diagram of the Kitaev-Heisenberg Model on a Three-dimensional Hyperhoneycomb Lattice
• Moments and multiplets in moiré materials: A pseudo-fermion functional renormalization group for spin-valley models
• Spin-valley magnetism on the triangular moiré lattice with SU(4) breaking interactions
• Functional renormalization flow and dynamical chiral symmetry breaking of QCD
• Ground state of the $S$=1/2 pyrochlore Heisenberg antiferromagnet: A quantum spin liquid emergent from dimensional reduction

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• Dynamics of K${}_2$Ni${}_2$(SO${}_4$)${}_3$ governed by proximity to a 3D spin liquid model
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• Competing Gauge Fields and Entropically-Driven Spin Liquid to Spin Liquid Transition in non-Kramers Pyrochlores
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• High temperature series expansions of S = 1/2 Heisenberg spin models: algorithm to include the magnetic field with optimized complexity
• A pseudofermion functional renormalization group study of dipolar-octupolar pyrochlore magnets
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• Modified large-$N$ approach to gapless spin liquids, magnetic orders, and dynamics: Application to triangular lattice antiferromagnets
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• Pseudo-Majorana functional renormalization for frustrated XXZ spin-1/2 models with field or magnetization along the spin-Z direction at finite temperature
• Taming spin susceptibilities in frustrated quantum magnets: Mean-field form and approximate nature of the quantum-to-classical correspondence
• Entanglement in the pseudogap regime of cuprate superconductors
• High-temperature series expansion of the dynamic Matsubara spin correlator
• Phase transitions in the spin-1/2 Heisenberg antiferromagnet on the dimerized diamond lattice
• Phase diagram of the antiferromagnetic $J_1$-$J_2$ spin-$1$ pyrochlore Heisenberg model
• Unconventional orders in the maple-leaf ferro-antiferromagnetic Heisenberg model
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• Quantum Coulomb Liquids of Different Rank in the Breathing Pyrochlore Antiferromagnet
• Crystallography-driven molecularization of a two-dimensional spin-$3/2$ magnet
• Majorana Diagrammatics for Quantum Spin-1/2 Models
• Mapping the Phase Diagram of a Frustrated Magnet: Degeneracies, Flat Bands, and Canting Cycles on the Pyrochlore Lattice