Functional renormalization group approach to correlated fermion systems

Abstract

Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems, (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals, (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and non-equilibrium transport through quantum wires and quantum dots. The functional renormalization group is a flexible and unbiased tool for dealing with such scale-dependent behavior. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a continuously decreasing energy scale. Expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes. This review is a comprehensive introduction to the functional renormalization group method for interacting Fermi systems. We present a self-contained derivation of the exact flow equations and describe frequently used truncation schemes. Reviewing selected applications we then show how approximations based on the functional renormalization group can be fruitfully used to improve our understanding of correlated fermion systems.

Review article, final version, 59 pages, 28 figures

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• Functional renormalization group study of an eight-band model for the iron arsenides
• Two-particle-correlations in a functional renormalization group scheme using a dynamical mean-field theory approach
• Thermalization of magnons in yttrium-iron garnet: nonequilibrium functional renormalization group approach
• Breakdown of Fermi liquid behavior near the hot spots in a two-dimensional model: A two-loop renormalization group analysis
• Correlated metallic state in honeycomb lattice: Orthogonal Dirac semimetal
• Scaling theory for Mott-Hubbard transitions
• Quantum phase transition and unusual critical behavior in multi-Weyl semimetals
• Taming pseudo-fermion functional renormalization for quantum spins: Finite-temperatures and the Popov-Fedotov trick
• Dynamic criticality far-from-equilibrium: one-loop flow of Burgers-Kardar-Parisi-Zhang systems with broken Galilean invariance
• Ab-initio description of excited states of a one-dimensional nuclear matter with the Hohenberg-Kohn-theorem-inspired functional-renormalization-group method
• Finite temperature line-shapes of hard-core bosons in quantum magnets: A diagrammatic approach tested in one dimension
• Fermi-liquid Landau parameters for a nondegenerate band: Spin and charge instabilities in the extended Hubbard model
• Charge Density Wave Ordering in NdNiO${}_2$: Effects of Multiorbital Nonlocal Correlations
• Effect of spin-orbit interactions on the 0.7 anomaly in quantum point contacts
• Hierarchical Parallelisation of Functional Renormalisation Group Calculations -- hp-fRG
• Mott transition and pseudogap of the square-lattice Hubbard model: results from center-focused cellular dynamical mean-field theory
• Finite-frequency-dependent noise of a quantum dot in a magnetic field
• Functional Renormalization Group Approach for Inhomogeneous One-Dimensional Fermi Systems with Finite-Ranged Interactions
• Physics in one dimension: theoretical concepts for quantum many-body systems
• Superconductivity from repulsive interactions in Bernal-stacked bilayer graphene
• Interplay of Coulomb interaction and spin-orbit effects in multi-level quantum dots
• Transport in a hybrid normal-topological superconductor Kondo model
• Real-frequency quantum field theory applied to the single-impurity Anderson model
• Ferromagnetic fluctuations in the Rashba-Hubbard model
• The Ohmic two-state system from the perspective of the interacting resonant level model: Thermodynamics and transient dynamics
• Anisotropy of electronic correlations: On the applicability of local theories to layered materials
• The effect of vertex corrections on the possibility of chiral symmetry breaking, induced by long-range Coulomb repulsion in graphene
• Theory of intervalley-coherent AFM order and topological superconductivity in tWSe${}_2$
• A rich man's derivation of scaling laws for the Kondo model
• Mean-field approximation for thermodynamic and spectral functions of correlated electrons: Strong-coupling and arbitrary band filling
• Fermionic Functional Renormalization Group Approach to Superfluid Phase Transition
• Functional renormalization group in Floquet space
• Fluctuation effects at the onset of $2{\mathbf{k}}_{\mathbf{F}}$ density wave order with one pair of hot spots in two-dimensional metals
• Complementary views on electron spectra: From Fluctuation Diagnostics to real space correlations
• Superconductivity of repulsive spinless fermions with sublattice potentials
• Nonanalyticities in a Strongly Correlated Fermi Liquid: Corrections to Scaling at the Fermi-Liquid Fixed Point
• Single-boson exchange functional renormalization group application to the two-dimensional Hubbard model at weak coupling
• Universality at work -- the local sine-Gordon model, lattice fermions, and quantum circuits
• Effective dimension reduction with mode transformations: Simulating two-dimensional fermionic condensed matter systems
• Cooling Stochastic Quantization with colored noise
• Spin and Charge Fluctuation Induced Pairing in ABCB Tetralayer Graphene
• Exponential and power-law renormalization in phonon-assisted tunneling
• Effect of exchange interaction on electronic instabilities in the honeycomb lattice: A functional renormalization group study
• Spin and pair density waves in 2D altermagnetic metals
• Summing parquet diagrams using the functional renormalization group: X-ray problem revisited
• Ab initio construction of the energy density functional for electron systems with the functional-renormalization-group-aided density functional theory
• Critical Casimir forces for O(N) models from functional renormalization
• Superfluid ground state phase diagram of the two-dimensional Hubbard model in the emergent Bardeen-Cooper-Schrieffer regime
• $f$-wave superfluidity from repulsive interaction in Rydberg-dressed Fermi gas
• Twisted bilayer graphene at charge neutrality: competing orders of SU(4) Dirac fermions
• Renormalization flow towards gravitational catalysis in the 3d Gross-Neveu model
• Functional-renormalization-group aided density-functional analysis for the correlation energy of the two-dimensional homogeneous electron gas
• Counting Feynman diagrams via many-body relations
• Effect of weak impurities on electronic properties of graphene: functional renormalization-group analysis
• Critical phenomena at the complex tensor ordering phase transition
• Transport properties of a two-lead Luttinger liquid junction out of equilibrium: fermionic representation
• Schwinger-Dyson Renormalization Group
• Enhanced nematicity emerging from higher-order van Hove singularities
• Quasiparticle interference and charge order in a heavily overdoped non-superconducting cuprate
• Thermodynamic electric quadrupole moments of nematic phases from first-principles calculation
• Functional renormalization and mean-field approach to multiband systems with spin-orbit coupling: Application to the Rashba model with attractive interaction
• Competition between d-wave superconductivity and magnetism in uniaxially strained Sr2RuO4
• Existence and Construction of Exact FRG Flows of a UV-Interacting Scalar Field Theory
• Non-equilibrium Properties of Berezinskii-Kosterlitz-Thouless Phase Transitions
• Fermion pairing in mixed-dimensional atomic mixtures
• Critical spin dynamics of Heisenberg ferromagnets revisited
• Transport properties of fully screened Kondo models
• Dominant Superconducting Fluctuations in the One-Dimensional Extended Holstein-Extended Hubbard model
• Hierarchical equations of motion approach to transport through an Anderson impurity coupled to interacting Luttinger liquid leads
• 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
• Fermi-edge singularity and the functional renormalization group
• Holographic unitary renormalization group for correlated electrons -- II: insights on fermionic criticality
• Holographic unitary renormalization group for correlated electrons -- I: a tensor network approach
• Stability of fixed points and generalized critical behavior in multifield models
• Electronic orders on the kagome lattice at the lower Van Hove filling
• Bond-ordered states and $f$-wave pairing of spinless fermions on the honeycomb lattice
• From high-$T_c$ to low-$T_c$: Multi-orbital effects in transition metal oxides
• Field theoretic study of electron-electron interaction effects in Dirac liquids
• Intertwined orders from symmetry projected wavefunctions of repulsively interacting Fermi gases in optical lattices
• Renormalization group analysis of competition between distinct order parameters
• The Crossover from Ordinary to Higher-Order van Hove Singularity in a Honeycomb System: A Parquet Renormalization Group Analysis
• Spectral representation of Matsubara n-point functions: Exact kernel functions and applications
• Error estimates and specification parameters for functional renormalization
• Phase diagram of the $J_1$-$J_2$ quantum Heisenberg model for arbitrary spin
• Expansion of the effective action around non-Gaussian theories
• Y-junction of Luttinger-liquid wires out of equilibrium
• Time-dependent spectral functions of the Anderson impurity model in response to a quench with application to time-resolved photoemission spectroscopy
• Topological superconductivity on the honeycomb lattice: Effect of normal state topology
• Feshbach hypothesis of high-Tc superconductivity in cuprates
• Dynamical regimes of dissipative quantum systems
• ${\mathbb{Z}}_4$-symmetric perturbations to the XY model from functional renormalization
• Electronic orders near the type-II van Hove singularity in BC${}_3$
• Functional-renormalization-group approach to classical liquids with short-range repulsion: a scheme without repulsive reference system
• Unified theory of quantum phase transitions in quantum dots with gapped host bands
• Effective average action based approach to correlation functions at finite momenta
• Fermionic criticality is shaped by Fermi surface topology: a case study of the Tomonaga-Luttinger liquid
• Interaction-driven first-order and higher-order topological superconductivity
• Correlated Phases of Population Imbalanced Fermi-Fermi Mixtures on an Optical Lattice
• $d$- and $p$-wave quantum liquid crystal orders in cuprate superconductors, $\kappa$-(BEDT-TTF)${}_2$X, and coupled chain Hubbard models: functional-renormalization-group analysis
• Renormalization group and Fermi liquid theory for many-nucleon systems
• Ground-State Phase Diagram of the Kitaev-Heisenberg Model on a Three-dimensional Hyperhoneycomb Lattice
• Approximating the frequency dependence of the effective interaction in the functional renormalization group for many-fermion systems
• Electronic nematic phase transition in the presence of anisotropy
• Low temperature thermodynamics of multi-flavored hardcore bosons by the Brückner approach
• Stability of the Fulde-Ferrell-Larkin-Ovchinnikov states in anisotropic systems and critical behavior at thermal $m$-axial Lifshitz points
• A pseudofermion functional renormalization group study of dipolar-octupolar pyrochlore magnets
• Suppression of Bogoliubov momentum pairing and emergence of non-Gaussian correlations in ultracold interacting Bose gases
• A stochastic root finding approach: The Homotopy Analysis Method applied to Dyson-Schwinger Equations
• Spontaneous ferromagnetism in the spinor Bose gas with Rashba spin-orbit coupling
• Pseudo-fermion functional renormalization group with magnetic fields
• Space of non-Fermi liquids
• KeldyshQFT: A C++ codebase for real-frequency multiloop functional renormalization group and parquet computations of the single-impurity Anderson model
• The SpinParser software for pseudofermion functional renormalization group calculations on quantum magnets
• Out-of-equilibrium transport in the interacting resonant level model: the surprising relevance of the boundary sine-Gordon model
• Keldysh functional renormalization group for electronic properties of graphene
• Renormalization group flow for fermions into antiferromagnetically ordered phases: Method and mean-field models
• Accessing topological superconductivity via a combined STM and renormalization group analysis
• Quasi-particle functional Renormalisation Group calculations in the two-dimensional half-filled Hubbard model at finite temperatures
• Phonon Induced Instabilities in Correlated Electron Hamiltonians
• Weakly-interacting Bose-Bose mixtures from the functional renormalisation group
• Supercell Wannier functions and a faithful low-energy model for Bernal bilayer graphene
• Dimensional crossover in ultracold Fermi gases from Functional Renormalisation
• Parquet approximation and one-loop renormalization group: Equivalence on the leading-logarithmic level
• A Renormalization-Group Study of Interacting Bose-Einstein condensates: Absence of the Bogoliubov Mode below Four ($T>0$) and Three ($T=0$) Dimensions
• Better Integrators for Functional Renormalization Group Calculations
• Magic angle of Sr${}_2$RuO${}_4$: Optimizing correlation-driven superconductivity
• Compressing the two-particle Green's function using wavelets: Theory and application to the Hubbard atom
• Prediction of pseudogap formation due to $d$-wave bond-order in organic superconductor $\kappa$-(BEDT-TTF)${}_2$X
• Response Theory of the Electron-Phonon Coupling
• Quantum discontinuity fixed point and renormalization group flow of the SYK model
• Interplay between Point-Group Symmetries and the Choice of the Bloch Basis in Multiband Models
• Transport through a magnetic impurity: a slave-spin approach
• Criticality of the $O(2)$ model with cubic anisotropies from nonperturbative renormalization
• Local magnetic moments and electronic transport in closed loop quantum dot systems: a case of quadruple quantum dot ring at and away from equilibrium
• Domes of $T_c$ in single-band and multiband superconductors with finite-range attractive interactions
• Scattering theory of mesons in doped antiferromagnetic Mott insulators: Multichannel perspective and Feshbach resonance
• Magnetoelectric Response in Electric Octupole State: Possible Hidden Order in Cuprate Superconductors
• Possible high temperature odd parity superconductivity in LaNi${}_{7/8}$Co${}_{1/8}$O${}_3$ (111) bilayers
• Analytic continuation of multipoint correlation functions
• Superconductivity from repulsion: Variational results for the 2D Hubbard model in the limit of weak interaction
• Competing many-body instabilities in two-dimensional dipolar Fermi gases
• Review of recent developments of the functional renormalization group for systems out of equilibrium
• Pseudo-Majorana functional renormalization for frustrated XXZ spin-1/2 models with field or magnetization along the spin-Z direction at finite temperature
• Second order functional renormalization group approach to one-dimensional systems in real and momentum space
• Fluctuation effects at the onset of $2{\mathbf{k}}_{\mathbf{F}}$ density wave order with two pairs of hot spots in two-dimensional metals
• Low-Energy Effective Theory at a Quantum Critical Point of the Two-Dimensional Hubbard Model: Mean-Field Analysis
• Spin-valley magnetism on the triangular moiré lattice with SU(4) breaking interactions
• Loop Current Order on the Kagome Lattice
• Structure of Exact Renormalization Group Equations for field theory
• Non-Fermi liquid fixed points and anomalous Landau damping in a quantum critical metal
• Magnetic, charge, and transport properties of graphene nanoflakes
• Functional renormalization-group approach to the Pokrovsky-Talapov model via modified massive Thirring fermion model
• The {\it victory} project v1.0: an efficient parquet equations solver
• Combinatorial summation of Feynman diagrams: Equation of state of the 2D SU(N) Hubbard model
• Functional renormalization group approach to interacting three-dimensional Weyl semimetals
• Observation of spin-charge separation and boundary bound states via the local density of states
• Spectral properties and observables in ultracold Fermi gases
• Chern number landscape of spin-orbit coupled chiral superconductors
• Consistent combination of truncated-unity functional renormalization group and mean-field theory
• Possible Nematic Order Driven by Magnetic Fluctuations in Iron Pnictides
• Momentum-dependent quasiparticle properties of the Fermi polaron from the functional renormalization group
• Finite-bias transport through the interacting resonant level model coupled to a phonon mode -- a functional renormalization group study
• Quantum interface unbinding transitions
• Quantum spin Hall insulator on the honeycomb lattice induced by ferromagnetic exchange interaction
• MatsubaraFunctions.jl: An equilibrium Green's function library in the Julia programming language
• Bilayer honeycomb lattice with ultracold atoms: Multiple Fermi surfaces and incommensurate spin density wave instability
• Angle evolution of the superconducting phase diagram in twisted bilayer WSe2
• Testing the parquet equations and the U(1) Ward identity for real-frequency correlation functions from the multipoint numerical renormalization group
• Phonon renormalization and Pomeranchuk instability in the Holstein model
• Functional renormalization of QCD in $1+1$ dimensions: four-fermion interactions from quark-gluon dynamics
• Theoretical studies of superconductivity in doped BaCoSO
• Robust d-wave pairing symmetry in multi-orbital cobalt high temperature superconductors
• Spin susceptibilities in magnetic type-I and type-II Weyl semimetals
• Quantitative analysis of interaction effects in generalized Aubry-Andŕe-Harper models
• Second-order functional renormalization group approach to quantum wires out of equilibrium
• Robust topological superconductivity in spin-orbit coupled systems at higher-order van Hove filling
• Approximate expression for the ground-state energy of the two- and three-dimensional Hubbard model at arbitrary filling obtained from dimensional scaling
• Comment on "Unconventional Fermi Surface Instabilities in the Kagome Hubbard Model" by M. Kiesel, C. Platt, and R. Thomale, Phys. Rev. Lett. {\bf 110}, 126405 (2013)
• Physics-informed neural networks for solving functional renormalization group on a lattice
• Nonperturbative Renormalization Group for the Landau-de Gennes Model
• Anomalous isotope Effect in d-wave superconductors on the square lattice
• Multi-orbital two-particle self-consistent approach -- strengths and limitations
• Solving the Transient Dyson Equation with Quasilinear Complexity via Matrix Compression
• Functional renormalization group study of a four-fermion model with $CP$ violation
• Single-boson exchange formulation of the Schwinger-Dyson equation and its application to the functional renormalization group
• Keldysh Derivation of Oguri's Linear Conductance Formula for Interacting Fermions
• Entanglement in the pseudogap regime of cuprate superconductors
• Intertwined fluctuations and isotope effects in the Hubbard-Holstein model on the square lattice from functional renormalization
• Functional renormalization group study of parallel double quantum dots: Effects of asymmetric dot-lead couplings
• $\mathcal{N}=4$ chiral superconductivity in moiré transition metal dichalcogenides
• A Renormalization-Group Study of Interacting Bose-Einstein Condensates: II. Anomalous Dimension $\eta$ for $d\lesssim 4$ at Finite Temperatures
• Renormalization group flows in one-dimensional lattice models: impurity scaling, umklapp scattering and the orthogonality catastrophe
• Matrix element interference in $N$-patch functional renormalization group
• Fermionic Functional Renormalization Group Approach to Bose-Einstein Condensation of Dimers
• Interaction induced local moments in parallel quantum dots within the functional renormalization group approach
• Conductance scaling of junctions of Luttinger-liquid wires out of equilibrium
• Evolution of superconducting gap anisotropy in hole-doped 122 iron pnictides
• Current noise of the interacting resonant level model
• Functional renormalization-group approaches, one-particle (ir)reducible with respect to local Green functions, using the dynamical mean-field theory as a starting point
• Phases of translation-invariant systems out of equilibrium: Iterative Green's function techniques and renormalization group approaches
• Fractionalized Superconductivity Mediated by Majorana Fermions in the Kitaev-Kondo Lattice
• Efficiency of the hidden fermion determinant states Ansatz in the light of different complexity measures
• Current noise in quantum dot thermoelectric engines
• Addressing energy density functionals in the language of path-integrals II: Comparative study of functional renormalization group techniques applied to the (0+0)-D $O(N)$-symmetric ${\phi }^4$-theory
• Breaking a "poor man" RG approach in the Luttinger liquid with one impurity
• Density of states of interacting quantum wires with impurities: a Dyson equation approach
• Quasi-particle functional Renormalisation Group calculations in the two-dimensional t-t'-Hubbard model
• Functional renormalization group without functional integrals: implementing Hilbert space projections for strongly correlated electrons via Hubbard X-operators
• Effects of Interactions on Dynamic Correlations of Hard-Core Bosons at Finite Temperatures
• Pairing from repulsion in a two-dimensional Fermi gas with soft-core interactions
• Non-Fermi liquid fixed point of the dissipative Yukawa-Sachdev-Ye-Kitaev model
• A functional renormalization group approach to electronic structure calculations for systems without translational symmetry
• Functional renormalization group for extremely correlated electrons
• Functional renormalization group approach to conventional theory of superfluidity and beyond
• Renormalization group study of systems with quadratic band touching
• Full and unbiased solution of the Dyson-Schwinger equation in the functional integro-differential representation
• Weak three-dimensional mediators of two-dimensional triplet pairing
• Exact effective action for the O(N) vector model in the large N limit
• Incommensurate magnetic order: A fingerprint for electronic correlations in hole-doped cuprates
• A functional-analysis derivation of the parquet equation
• Strong Boundary and Trap Potential Effects on Emergent Physics in Ultra-Cold Fermionic Gases
• Quantitative functional renormalization for three-dimensional quantum Heisenberg models
• Conventional and practical metallic superconductivity arising from repulsive Coulomb coupling
• Bad Metal Behavior and Lifshitz Transition of a Nagaoka Ferromagnet
• Spin susceptibility in a pseudogap state with fluctuating spiral magnetic order
• Diagrammatic bosonization, aspects of criticality, and the Hohenberg-Mermin-Wagner theorem in parquet approaches
• Compressing local vertex functions from the multipoint numerical renormalization group using quantics tensor cross interpolation
• Towards combined transport and optical studies of the 0.7-anomaly in a quantum point contact
• Kekulé order from diffuse nesting near higher-order Van Hove points
• Critical pairing fluctuations in the normal state of a superconductor: pseudogap and quasi-particle damping
• d_{xy}-Density wave in fermion-fermion cold atom mixtures
• Exploring excited eigenstates of many-body systems using the functional renormalization group
• Leading-logarithmic approximation by one-loop renormalization group within Matsubara formalism
• On the competition between the Kondo effect and the exchange interaction in a parallel double quantum dot system
• Effect of vacancies on magnetic correlations and conductance in graphene nanoflakes with realistic Coulomb interaction
• Emergent chirality in multi-lead Luttinger-liquid junctions out of equilibrium
• Broadening and sharpening of the Drude peak through antiferromagnetic fluctuations
• Functional-renormalization-group approach to strongly coupled Bose-Fermi mixtures in two dimensions
• Development of spin fluctuations under the presence of $d$-wave bond order in cuprate superconductors
• Quantum Adiabatic Doping for Atomic Fermi-Hubbard Quantum Simulations
• Electrical Conductivity in Quantum Materials
• Fluctuation conductivity due to the preformed local pairs
• Surface Functional Renormalization Group for Layered Quantum Materials
• Functional renormalization group for non-Hermitian and $\mathcal{P}\mathcal{T}$-symmetric systems
• Functional Renormalization Group for fermions on a one dimensional lattice at arbitrary filling
• Many-body correlations in Floquet steady-states: Frequency-resolved renormalization group of the driven Anderson impurity
• Non-Fermi Liquid Behavior of the $t$-$J$ Model in the Strange Metal Phase: $U(1)$ Gauge Theory Consistent with Local Constraints
• The plain and simple parquet approximation: single- and multi-boson exchange in the two-dimensional Hubbard model
• Charge density wave breakdown in a heterostructure with electron-phonon coupling
• Effect of mediated interactions on a Hubbard chain in mixed-dimensional fermionic cold atoms
• Some improved nonperturbative bounds for Fermionic expansions
• Weak coupling theory of nickel-based 327 superconductors
• Beyond on-site Hubbard interaction in charge dynamics of cuprate superconductors
• Median-point approximation and its application for the study of fermionic systems
• Multi-Regulator Functional Renormalization Group for Many-Fermion Systems
• Functional renormalization group approach to the singlet-triplet transition in quantum dots
• Quantum criticality and non-Fermi liquids: the Wilsonian renormalization group perspective
• Functional renormalization of spinless triangular-lattice fermions: $N$-patch vs. truncated-unity scheme
• Eliminating Orbital Selectivity from the Metal-Insulator Transition by Strong Magnetic Fluctuations
• Spin bond order driven by extended repulsive interactions in doped graphene
• Spin functional renormalization group for dimerized quantum spin systems
• Magnetically Mediated Cross-Layer Pairing in Pressurized Trilayer Nickelate La${}_4$Ni${}_3$O${}_{10}$
• Theory of unconventional magnetism in a Cu-based kagome metal
• Superconductivity and the quasiparticle mass enhancement near the CDW critical point using Bethe-Salpeter method: Application to cuprates
• Competing instabilities of the extended Hubbard model on the triangular lattice: Truncated-unity functional renormalization group and application to moiré materials
• Chemical-potential flow equations for graphene with Coulomb interactions
• Frequency Dependent Functional Renormalization Group for Interacting Fermionic Systems
• The phases of QCD reached in terrestrial and cosmic colliders
• Functional renormalization with interaction flows: A single-boson exchange perspective and application to electron-phonon systems