The Kernel Polynomial Method

Abstract

Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed matter physics. In this article we review basic properties and recent developments of Chebyshev expansion based algorithms and the Kernel Polynomial Method. Characterized by a resource consumption that scales linearly with the problem dimension these methods enjoyed growing popularity over the last decade and found broad application not only in physics. Representative examples from the fields of disordered systems, strongly correlated electrons, electron-phonon interaction, and quantum spin systems we discuss in detail. In addition, we illustrate how the Kernel Polynomial Method is successfully embedded into other numerical techniques, such as Cluster Perturbation Theory or Monte Carlo simulation.

32 pages, 17 figs; revised version

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• Dynamical mean-field studies of metal-insulator transitions
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• Implicit Purification for Temperature-Dependent Density Matrices
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• Optical absorption and activated transport in polaronic systems
• Chebyshev expansion approach to the AC conductivity of the Anderson model
• Unusual localisation effects in quantum percolation
• The Truncated Polynomial Expansion Monte Carlo Method for Fermion Systems Coupled to Classical Fields: A Model Independent Implementation
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• Slow dynamics and strong finite-size effects in many-body localization with random and quasi-periodic potential
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• Diverging dc conductivity due to a flat band in disordered pseudospin-1 Dirac-Weyl fermions
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• Many-body localization in presence of cavity mediated long-range interactions
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• Tilted disordered Weyl semimetals
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• Many-Body Non-Hermitian Skin Effect for Multipoles
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• Magic-Angle Semimetals with Chiral Symmetry
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• Topological phase transitions in glassy quantum matter
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• Spectral signatures of the Luttinger liquid to charge-density-wave transition
• RKKY Interactions in Graphene: Dependence on Disorder and Gate Voltage
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• Electron dynamics in graphene with gate-defined quantum dots
• Avoided quantum criticality in exact numerical simulations of a single disordered Weyl cone
• The self-consistent quantum-electrostatic problem in strongly non-linear regime
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• Quantum Hall studies of a Semi-Dirac Nanoribbon
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• Electrically tunable resonant scattering in fluorinated bilayer graphene
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• Dynamic mean-field theory for dense spin systems at infinite temperature
• On the Nature of Localization in Ti doped Si
• Perturbation theory and thermal transport in mass-disordered alloys: Insights from Green's function methods
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• On the absorption of microwaves by the one-dimensional spin-1/2 Heisenberg-Ising magnet
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• Scaling up the lattice dynamics of amorphous materials by orders of magnitude
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• Optical conductivity of polaronic charge carriers
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• Excitons and trions in monolayer semiconductors with correlated electrons
• Tight-binding study of bilayer graphene Josephson junctions
• Chebyshev Filter Diagonalization on Modern Manycore Processors and GPGPUs
• Nuclear Spin Noise in the Central Spin Model
• Probing unconventional superconductivity in proximitized graphene by impurity scattering
• Parallel sparse matrix-vector multiplication as a test case for hybrid MPI+OpenMP programming
• Relation between far-from-equilibrium dynamics and equilibrium correlation functions for binary operators
• Inducing a many-body topological state of matter through Coulomb-engineered local interactions
• Probe-type of superconductivity by impurity in materials with short coherence length: the s-wave and $\eta$-wave phases study
• Structural spillage: an efficient method to identify non-crystalline topological materials
• Third-order topological insulator induced by disorder
• Characterization and comparison of large directed graphs through the spectra of the magnetic Laplacian
• Orbital Hall Responses in Disordered Topological Materials
• Replica Higher-Order Topology of Hofstadter Butterflies in Twisted Bilayer Graphene
• Topological diffusive metal in amorphous transition metal monosilicides
• Real-space imaging of dispersive triplon excitations in engineered quantum magnets
• Entropy and Temperature in finite isolated quantum systems
• Diffusion in the Anderson model in higher dimensions
• Solitonic in-gap modes in a superconductor-quantum antiferromagnet interface
• Probing Band-center Anomaly with the Kernel Polynomial Method
• Charge and Spin Transport Anisotropy in Nanopatterned Graphene
• Strong disorder in nodal semimetals: Schwinger-Dyson--Ward approach
• Long-time coherence in fourth-order spin correlation functions
• Hot Electrons Regain Coherence in Semiconducting Nanowires
• Topological insulators in random potentials
• The effect of amplitude fluctuations on the BKT transition
• Dynamical spin Hall conductivity in a magnetic disordered system
• Monte Carlo study of thermal fluctuations and Fermi-arc formation in d-wave superconductors
• Pushing the limit of quantum transport simulations
• Optimized Attenuated Interaction: Enabling Stochastic Bethe-Salpeter Spectra for Large Systems
• GPUQT: An efficient linear-scaling quantum transport code fully implemented on graphics processing units
• Chiral spin liquids with crystalline ${\mathbb{Z}}_2$ gauge order in a three-dimensional Kitaev model
• Effective dissipation rate in a Liouvillian graph picture of high-temperature quantum hydrodynamics
• Comment on "Anderson transition in disordered graphene"
• Sudden removal of a static force in a disordered system: Induced dynamics, thermalization, and transport
• Machine learning assisted derivation of effective low energy models for metallic magnets
• Integer Quantum Hall Effect: Disorder, temperature, floating, and plateau width
• Reduced-Shifted Conjugate-Gradient Method for a Green's Function: Efficient Numerical Approach in a Nano-structured Superconductor
• Fingerprints of Composite Fermion Lambda Levels in Scanning Tunneling Microscopy
• Interaction effects in a microscopic quantum wire model with strong spin-orbit interaction
• Superfluid-insulator transition of two-dimensional disordered Bose gases
• Tunable valley filtering in dynamically strained $\alpha$-${\mathcal{T}}_3$ lattices
• Grain boundary-induced variability of charge transport in hydrogenated polycrystalline graphene
• Scaling of the Integrated Quantum Metric in Disordered Topological Phases
• Quasi-Nambu-Goldstone modes in many-body scar models
• Landauer transport as a quasisteady state on finite chains under unitary quantum dynamics
• Energy-filtered random-phase states as microcanonical thermal pure quantum states
• Dynamics of Phase Separated States in the Double Exchange Model
• Carrier induced ferromagnetism in the insulating Mn doped III-V semiconductor InP
• Sublinear Time Spectral Density Estimation
• Photoinduced topological spin texture in a metallic ferromagnet
• Calculating the many-body density of states on a digital quantum computer
• Accuracy of the typicality approach using Chebyshev polynomials
• From Bloch Oscillations to a Steady-State Current in Strongly Biased Mesoscopic Devices
• Spectral Functions of One-Dimensional Systems with Correlated Disorder
• Anisotropic optics and gravitational lensing of tilted Weyl fermions
• Nonequilibrium dynamics of superconductivity in the attractive Hubbard model
• Initial State Dependent Dynamics Across Many-body Localization Transition
• Density of States of the lattice Schwinger model
• Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems
• Dynamical quantum typicality: Simple method for investigating transport properties applied to the Holstein model
• Divide and conquer the Hilbert space of translation-symmetric spin systems
• Towards a Comprehensive Theory of Metal-Insulator Transitions in Doped Semiconductors
• Establishing Coherent Momentum-Space Electronic States in Locally Ordered Materials
• Faster spectral density calculation using energy moments
• Disorder-Induced Delocalization in Magic-Angle Twisted Bilayer Graphene
• Many-polaron problem by cluster perturbation theory
• Pymablock: an algorithm and a package for quasi-degenerate perturbation theory
• Cloaking Resonant Scatterers and Tuning Electron Flow in Graphene
• Temporal relaxation of gapped many-body quantum systems
• Dynamical topological excitations in parafermion chains
• Impurities and electronic localization in graphene bilayers
• Disorder effects on triple-point fermions
• Efficient Time-Domain Approach for Linear Response Functions
• Doped Mott insulator on Penrose tiling
• Topological zero modes and correlation pumping in an engineered Kondo lattice
• The Calculation of Two-Particle Quantities in the Typical Medium Dynamical Cluster Approximation
• Signatures of nodeless multiband superconductivity and particle-hole crossover in the vortex cores of FeTe${}_{0.55}$Se${}_{0.45}$
• Nodal vacancy bound states and resonances in three-dimensional Weyl semimetals
• Exactly solvable models for universal operator growth
• Effect of interaction on the Majorana zero modes in the Kitaev chain at half filling
• NanoNET: an extendable Python framework for semi-empirical tight-binding models
• Finite-size scaling of typicality-based estimates
• Variational discrete variable representation for excitons on a lattice
• Absence of quantization in the circular photogalvanic effect in disordered chiral Weyl semimetals
• Analysis of stochastic Lanczos quadrature for spectrum approximation
• Spectral function for ${}^4$He using the Chebyshev expansion in coupled-cluster theory
• Understanding the dynamics of randomly positioned dipolar spin ensembles
• Berezinskii-Kosterlitz-Thouless localization-localization transitions in disordered two-dimensional quantized quadrupole insulators
• Interaction-enhanced many body localization in a generalized Aubry-Andre model
• Ballistic transport in disordered Dirac and Weyl semimetals
• Finite-size prethermalization at the chaos-to-integrable crossover
• Neural network enhanced hybrid quantum many-body dynamical distributions
• Quantum Hall effect and Landau levels without spatial long-range correlations
• Intertwined Orders and Electronic Structure in Superconducting Vortex Halos
• Electronic Properties of $\alpha -{\mathcal{T}}_3$ Quantum Dots in Magnetic Fields
• Failure of the geometric approach prediction of excess work scaling for open and isolated quantum systems
• Spectral features of polaronic excitations in a superconducting analog simulator
• Effect of incommensurate potential on nodal-link semimetals
• Strong gate-tunability of flat bands in bilayer graphene due to moiré encapsulation between hBN monolayers
• Theory of Mixed-State Effect on Quasiparticle Interference in Iron-based Superconductors
• Chebyshev pseudosite matrix product state approach for the spectral functions of electron-phonon coupling systems
• RKKY interaction in heavily vacant graphene
• Sparse-Stochastic Fragmented Exchange for Large-Scale Hybrid TDDFT Calculations
• Approximating spectral densities of large matrices
• Magnetic Doublon Bound States in the Kondo Lattice Model
• Multifractality and the distribution of the Kondo temperature at the Anderson transition
• Disordered graphene Josephson junctions
• Topological Transition of Graphene from Quantum Hall Metal to Quantum Hall Insulator at $\nu =0$
• Fate of Weyl semimetals in the presence of incommensurate potentials
• Linear-scale simulations of quench dynamics
• Adiabatic time evolution of highly excited states
• Kernel Polynomial Method for Linear Spin Wave Theory
• Decoding the DC and optical conductivities of disordered MoS${}_2$ films: an inverse problem
• Determination of the Fermi Level Position in Dilute Magnetic Ga${}_{1-x}$Mn${}_x$N Films
• Chebyshev Polynomial Method to Landauer-Büttiker Formula of Quantum Transport in Nanostructures
• Relaxation of dynamically prepared out-of-equilibrium initial states within and beyond linear response theory
• Scattering from spin-polarized charged impurities in graphene
• Interplay of quantum spin Hall effect and spontaneous time-reversal symmetry breaking in electron-hole bilayers II: Zero-field topological superconductivity
• Ioffe-Regel criterion and viscoelastic properties of amorphous solids
• Optical Hall response of bilayer graphene: the manifestation of chiral hybridised states in broken mirror symmetry lattices
• Spatiotemporal dynamics of classical and quantum density profiles in low-dimensional spin systems
• Strange metal phase of disordered magic-angle twisted bilayer graphene at low temperatures: from flatbands to weakly coupled Sachdev-Ye-Kitaev bundles
• The low energy excitation spectrum of magic-angle semimetals
• Notes on density matrix perturbation theory
• Valley-dependent transport through graphene quantum dots due to proximity-induced, staggered spin-orbit couplings
• Long-time memory effects in a localizable central spin problem
• Phase transitions in the binary-alloy Hubbard model: insight from strong-coupling perturbation theory
• Impurity coupled to a lattice with disorder
• Strong-coupling perturbative study of the disordered Hubbard model on honeycomb lattice
• Dissipative Edge Transport in Disordered Axion Insulator Films
• Kernel-Function Based Quantum Algorithms for Finite Temperature Quantum Simulation
• A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation
• The Aubry-Andre Anderson model: Magnetic impurities coupled to a fractal spectrum
• Near-term quantum algorithm for computing molecular and materials properties based on recursive variational series methods
• Interplay of resonant states and Landau levels in functionalized graphene
• Randomized semi-quantum matrix processing
• Vacancy-induced tunable Kondo effect in twisted bilayer graphene
• Nonlinear electric transport in graphene with magnetic disorder
• Effective three-band structure in Fe-based superconductors
• Partial preservation of chiral symmetry and colossal magnetoresistance in adatom doped graphene
• Correlated states in super-moiré materials with a kernel polynomial quantics tensor cross interpolation algorithm
• Theory of collective excitations in the quadruple-$Q$ magnetic hedgehog lattices
• Effective Heisenberg exchange integrals of diluted magnetic semiconductors determined within realistic multi-band tight-binding models
• Optical response of many-polaron systems
• Landau levels in a topological insulator
• Kinetics of orbital ordering in cooperative Jahn-Teller models: Machine-learning enabled large-scale simulations
• Many-body Liouvillian dynamics with a non-Hermitian tensor-network kernel polynomial algorithm
• Two-doublon Bloch oscillations in the mass-imbalanced extended Fermi-Hubbard model
• Large scale calculations of thermoelectric transport coefficients: a case study of γ-graphyne with point defects
• Edge non-collinear magnetism in nanoribbons of Fe3GeTe2 and Fe3GaTe2
• Subspace recursive Fermi-operator expansion strategies for large-scale DFT eigenvalue problems on HPC architectures
• Transport and quantum coherence in graphene rings: Aharonov-Bohm oscillations, Klein tunneling and particle localization
• Unambiguous Simulation of Diffusive Charge Transport in Disordered Nanoribbons
• Particularities of polaron formation in the extended Holstein model with next nearest neighbor transfer
• Anderson Metal-Insulator Transitions With Classical Magnetic Impurities
• Interband scattering- and nematicity-induced quantum oscillation frequency in FeSe
• Energy and particle currents in a driven integrable system
• Random matrix approach to plasmon resonances in the random impedance network model of disordered nanocomposites
• Observation of spin-charge separation and boundary bound states via the local density of states
• Reconstruction of chiral edge states in a magnetic Chern insulator
• Kernel polynomial representation of imaginary-time Green's functions
• RKKY interaction in triplet superconductors: Dzyaloshinskii-Moriya-type interaction mediated by spin-polarized Cooper pairs
• A spectrum adaptive kernel polynomial method
• Spectroscopy of hot-electron pair emission from a driven quantum dot
• Selective final state spectroscopy and multifractality in two-component ultracold Bose-Einstein condensates: a numerical study
• Time-domain interferometry of electron weak localization through terahertz nonlinear response
• Half-Quantized Hall Metal and Marginal Metal in Disordered Magnetic Topological Insulators
• Nano-welding of quantum spin-$1/2$ chains at minimal dissipation
• Spectral properties of fractionalized Shiba states
• Emergent quantum matter in graphene nanoribbons
• Phase Diagram of a Spin-ice Kondo Lattice Model in a Breathing Pyrochlore Lattice
• Family of Aperiodic Tilings with Tunable Quantum Geometric Tensor
• Restoring the continuum limit in the time-dependent numerical renormalization group approach
• Dynamical typicality for initial states with a preset measurement statistics of several commuting observables
• The shear Alfvén continuum of quasisymmetric stellarators
• Chiral states around a mass-inverted quantum dot in graphene
• Many-body Majorana-like zero modes without gauge symmetry breaking
• Charge dynamics in magnetically disordered Mott insulators
• Direct topological insulator transitions in three dimensions are destabilized by non-perturbative effects of disorder
• Carrier-density effects in many-polaron systems
• Critical Filaments and Superconductivity in Quasiperiodic Twisted Bilayer Graphene
• Diffusion and localization in quantum random resistor networks
• Many Body Density of States of a system of non interacting spinless fermions
• Escaping the Krylov space during finite precision Lanczos
• Dynamical conductivity of disordered quantum chains
• Shubnikov-de Haas oscillations in the anomalous Hall conductivity of Chern insulators
• Phonons in disordered alloys : a multiple scattering approach
• Spectroscopic Feature of Quantum Many-Body Scar States
• Spontaneous edge corner currents in $s+is$ superconductors and time reversal symmetry breaking surface states
• Emergent fractals in dirty topological crystals
• Effect of dimerization on dynamics of spin-charge separation in Pariser-Parr-Pople model: A time-dependent density matrix renormalization group study
• Probing thermalization in quenched (non-)integrable Fermi-Hubbard models
• Gapped-filtering for efficient Chebyshev expansion of the density projection operator
• Atomic collapse of high-order singular potentials in graphene
• Unveiling Stripe-shaped Charge Density Modulations in Doped Mott Insulators
• Low-energy interband Kondo bound states in orbital-selective Mott phases
• A domain-specific language and matrix-free stencil code for investigating electronic properties of Dirac and topological materials
• Fate of Quadratic Band Crossing under quasiperiodic modulation
• Self-consistent tensor network method for correlated super-moiré matter beyond one billion sites
• Advancing Quantum Many-Body GW Calculations on Exascale Supercomputing Platforms
• Numerical investigation of the electrical conductivity of irradiated graphene
• Bogoliubov quasiparticles coupled to the antiferromagnetic spin mode in a vortex core
• Thermometry for Laughlin States of Ultracold Atoms
• Multichannel active-space embedding of atomic multiplets in plane-wave DFT/PAW for core-level spectroscopies
• A Study of Parallelizing O(N) Green-Function-Based Monte Carlo Method for Many Fermions Coupled with Classical Degrees of Freedom
• Angular momentum of vortex-core Majorana zero modes
• Spin Hall Conductivity on the Anisotropic Triangular Lattice
• A stochastic method to compute the $L^2$ localisation landscape
• Upper limit of spin relaxation in suspended graphene
• Superfluid-insulator transition in weakly interacting disordered Bose gases: a kernel polynomial approach
• Structural transitions in dense disordered silicon from quantum-accurate ultra-large-scale simulations
• Magnetic Quantum Phase Diagram of Magnetic Impurities in 2 Dimensional Disordered Electron Systems
• Evaluation of real-space second Chern number using the kernel polynomial method
• Anderson Transition in Disordered Bilayer Graphene
• Hybrid kernel polynomial method
• Enhanced Convergence of Quantum Typicality using a Randomized Low-Rank Approximation
• Review of the tight-binding method applicable to the properties of moiré superlattices
• Genuine topological Anderson insulator from impurity induced chirality reversal
• Fast computation of spectral densities for generalized eigenvalue problems
• Kernel polynomial method to Anderson transition in disordered $\beta$-graphyne
• Quasiperiodic Quadrupole Insulators
• Modeling and Computation of Kubo Conductivity for 2D Incommensurate Bilayers
• Quantum Hall Effect without Chern Bands
• Real-space spectral simulation of quantum spin models: Application to generalized Kitaev models
• Numerical methods for localization
• Memory efficient Fock-space recursion scheme for computing many-fermion resolvents
• Orthogonal layers of parallelism in large-scale eigenvalue computations
• Exact numerical methods for electron-phonon problems
• Tensor network method for real-space topology in quasicrystal Chern mosaics
• Optimized binning for response function reconstruction via Chebyshev expansions
• Aharanov-Bohm oscillations and perfectly transmitted mode in amorphous topological insulator nanowires
• The Ba${}_{0.6}$K${}_{0.4}$Fe${}_2$As${}_2$ superconducting four-gap temperature evolution: a multi-band Chebyshev-BdG approach
• Three-Dimensional Quantum Hall Effect in Topological Amorphous Metals
• Isotropic 3D topological phases with broken time reversal symmetry
• Quantum transport in an environment parametrized by dispersive bosons
• Randomized estimation of spectral densities of large matrices made accurate
• Application of the random matrix theory to the boson peak in glasses
• Real-Time Out-of-Equilibrium Quantum Dynamics in Disordered Materials
• Fragile magnetic order in metallic quasicrystals
• Regularized universal topological markers for Dirac systems
• Machine-learning force-field models for dynamical simulations of metallic magnets
• Analysis of Kohn-Sham Eigenfunctions Using a Convolutional Neural Network in Simulations of the Metal-insulator Transition in Doped Semiconductors
• Temperature dependent energy gap for Yu-Shiba-Rusinov states at the quantum phase transition
• Machine learning dynamics of phase separation in correlated electron magnets
• Microscopic understanding of NMR signals by dynamic mean-field theory for spins
• Disorder-induced instability of a Weyl nodal loop semimetal towards a diffusive topological metal with protected multifractal surface states
• Metal-insulator transition of spinless fermions coupled to dispersive optical bosons
• Disorder-induced delocalization and reentrance in a Chern-Hopf insulator
• Increasing the number of topological nodal lines in semimetals via uniaxial pressure
• Reconciling Kubo and Keldysh Approaches to Fermi-Sea-Dependent Nonequilibrium Observables: Application to Spin Hall Current and Spin-Orbit Torque in Spintronics
• Two-dimensional plasmons in the random impedance network model of disordered thin-film nanocomposites
• A polynomial time algorithm for studying physical observables in chaotic eigenstates
• Bodge: Python package for efficient tight-binding modeling of superconducting nanostructures
• Effects of dilution in a 2D topological magnon insulator
• Linear-scaling electronic structure theory: Electronic temperature in the Kernel Polynomial Method
• Locality of the windowed local density of states
• Tangent equations of motion for nonlinear response functions
• Simulated floating zone method
• Proximity Effects Between the Graphene Quasicrystal and Magic-Angle Twisted Bilayer Graphene
• Machine-learning modeling of magnetization dynamics in quasi-equilibrium and driven metallic spin systems
• Creating two-qudit maximally entangled quantum link through bulk
• Performance Acceleration of Kernel Polynomial Method Applying Graphics Processing Units
• Markov Inequality as a Tool for Linear-Scaling Estimation of Local Observables
• Faster randomized partial trace estimation
• Robust Zero Modes in Disordered Two-Dimensional Honeycomb Lattice with Kekulé Bond Ordering
• An entropic method for discrete systems with Gibbs entropy
• Multifractal correlations of the local density of states in dirty superconducting films
• Quantum typicality approach to energy flow between two spin-chain domains at different temperatures
• Impact of clustering of substitutional impurities on quasiparticle lifetimes and localization
• Matrix-product state approach to the generalized nuclear pairing Hamiltonian
• Rashba Spin-Orbit Coupling and Nonlocal Correlations in Disordered 2D Systems
• Disordered Parity Anomalous Semimetal
• Effects of critical correlations on quantum percolation in two dimensions
• Electron-hole asymmetry in electrical conductivity of low-fluorinated graphene: Numerical study
• Analytic Formulas for the Anomalous Hall Effect in Itinerant Magnets
• Impurity states and Localization in Bilayer Graphene: the Low Impurity Concentration Regime
• Critical States of Fermions with ${\mathbb{Z}}_2$ Flux Disorder
• Robust quantum anomalous Hall effect with spatially uncorrelated disorder
• A $C^{\ast }$-algebraic view on the interaction of real- and reciprocal space topology in skyrmion crystals
• paces: Parallelized Application of Co-Evolving Subspaces, a method for computing quantum dynamics on GPUs
• Identification of the Majorana edge modes in tight-binding systems based on the Krylov method
• Resistivity size effect due to surface steps on ruthenium thin films computed with a realistic tight-binding model
• Supercurrent effect in a charge density wave intertwined superconductor
• Chiral Graviton Modes in Fermionic Fractional Chern Insulators
• The Spectral Coupled Cluster Method with $k$ dependency for strongly correlated lattices
• Applicability of the cumulant expansion method for the calculation of transport properties in electron-phonon systems