Kernel Polynomial Method Research Articles

Semiclassical dynamics of spin density waves

We present a theoretical framework for equilibrium and nonequilibrium dynamical simulation of quantum states with spin-density-wave (SDW) order. Within a semiclassical adiabatic approximation that retains electron degrees of freedom, we demonstrate that the SDW order parameter obeys a generalized Landau-Lifshitz equation. With the aid of an enhanced kernel polynomial method, our linear-scaling quantum Landau-Lifshitz dynamics (QLLD) method enables dynamical SDW simulations with $N \simeq 10^5$ lattice sites. Our real-space formulation can be used to compute dynamical responses, such as dynamical structure factor, of complex and even inhomogeneous SDW configurations at zero or finite temperatures. Applying the QLLD to study the relaxation of a noncoplanar topological SDW under the excitation of a short pulse, we further demonstrate the crucial role of spatial correlations and fluctuations in the SDW dynamics.

Fast Computation of Spectral Densities for Generalized Eigenvalue Problems

The distribution of the eigenvalues of a Hermitian matrix (or of a Hermitian matrix pencil) reveals important features of the underlying problem, whether a Hamiltonian system in physics or a social network in behavioral sciences. However, computing all the eigenvalues explicitly is prohibitively expensive for real-world applications. This paper presents two types of methods to efficiently estimate the spectral density of a matrix pencil (A, B) where both A and B are sparse Hermitian and B is positive definite. The methods are targeted at the situation when the matrix B scaled by its diagonal is very well conditioned, as is the case when the problem arises from some finite element discretizations of certain partial differential equations. The first method is an adaptation of the kernel polynomial method (KPM) and the second is based on Gaussian quadrature by the Lanczos procedure. By employing Chebyshev polynomial approximation techniques, we can avoid direct factorizations in both methods, making the resulting algorithms suitable for large matrices. Under some assumptions, we prove bounds that suggest that the Lanczos method converges twice as fast as the KPM method. Numerical examples further indicate that the Lanczos method can provide more accurate spectral densities when the eigenvalue distribution is highly nonuniform. As an application, we show how to use the computed spectral density to partition the spectrum into intervals that contain roughly the same number of eigenvalues. This procedure, which makes it possible to compute the spectrum by parts, is a key ingredient in the new breed of eigensolvers that exploit “spectrum slicing."

Large-scale calculations of thermoelectric transport coefficients: a case study of γ-graphyne with point defects

Defects such as vacancies and impurities could have profound effects on the transport properties of thermoelectric materials. However, it is usually quite difficult to directly calculate the thermoelectric properties of defect-containing systems via first-principles methods since a very large supercell is required. In this work, based on the linear response theory and the kernel polynomial method, we present an efficient approach that can help to calculate the thermoelectric transport coefficients of a large system containing millions of atoms at arbitrary chemical potential and temperature. As a prototype example, we consider dilute vacancies and hydrogen impurities in a large-scale γ-graphyne sheet and discuss their effects on the thermoelectric transport properties.

Wavelet estimation in time-varying coefficient time series models with measurement errors

ABSTRACTThe article studies a time-varying coefficient time series model in which some of the covariates are measured with additive errors. In order to overcome the bias of estimator of the coefficient functions when measurement errors are ignored, we propose a modified least squares estimator based on wavelet procedures. The advantage of the wavelet method is to avoid the restrictive smoothness requirement for varying-coefficient functions of the traditional smoothing approaches, such as kernel and local polynomial methods. The asymptotic properties of the proposed wavelet estimators are established under the α-mixing conditions and without specifying the error distribution. These results can be used to make asymptotically valid statistical inference.

Finite-temperature phase transition to a Kitaev spin liquid phase on a hyperoctagon lattice: A large-scale quantum Monte Carlo study

The quantum spin liquid is an enigmatic quantum state in insulating magnets, in which conventional long-range order is suppressed by strong quantum fluctuations. Recently, an unconventional phase transition was reported between the low-temperature quantum spin liquid and the high-temperature paramagnet in the Kitaev model on a three-dimensional hyperhoneycomb lattice. Here, we show that a similar ``liquid-gas'' transition takes place in another three-dimensional lattice, the hyperoctagon lattice. We investigate the critical phenomena by adopting the Green-function based Monte Carlo technique with the kernel polynomial method, which enables systematic analysis of up to 2048 sites. The critical temperature is lower than that in the hyperhoneycomb case, reflecting the smaller flux gap. We also discuss the transition on the basis of an effective model in the anisotropic limit.

Quantum phases of disordered three-dimensional Majorana-Weyl fermions

The gapless Bogoliubov-de Gennes (BdG) quasiparticles of a clean three dimensional spinless $p_x+ip_y$ superconductor provide an intriguing example of a thermal Hall semimetal (ThSM) phase of Majorana-Weyl fermions in class D of the Altland-Zirnbauer symmetry classification; such a phase can support a large anomalous thermal Hall conductivity and protected surface Majorana-Fermi arcs at zero energy. We study the effect of quenched disorder on such a topological phase with both numerical and analytical methods. Using the kernel polynomial method, we compute the average and typical density of states for the BdG quasiparticles; based on this, we construct the disordered phase diagram. We show for infinitesimal disorder, the ThSM is converted into a diffusive thermal Hall metal (ThDM) due to rare statistical fluctuations. Consequently, the phase diagram of the disordered model only consists of ThDM and thermal insulating phases. Nonetheless, there is a cross-over at finite energies from a ThSM regime to a ThDM regime, and we establish the scaling properties of the avoided quantum critical point which marks this cross-over. Additionally, we show the existence of two types of thermal insulators: (i) a trivial thermal band insulator (ThBI) [or BEC phase] supporting only exponentially localized Lifshitz states (at low energy), and (ii) a thermal Anderson insulator (AI) at large disorder strengths. We determine the nature of the two distinct localization transitions between these two types of insulators and ThDM.We also discuss the experimental relevance of our results for three dimensional, time reversal symmetry breaking, triplet superconducting states.

Kernel polynomial representation for imaginary-time Green’s functions in continuous-time quantum Monte Carlo impurity solver**Project supported by the National Natural Science Foundation of China (Grant No. 11504340).

Inspired by the recently proposed Legendre orthogonal polynomial representation for imaginary-time Green’s functions G(τ), we develop an alternate and superior representation for G(τ) and implement it in the hybridization expansion continuous-time quantum Monte Carlo impurity solver. This representation is based on the kernel polynomial method, which introduces some integral kernel functions to filter the numerical fluctuations caused by the explicit truncations of polynomial expansion series and can improve the computational precision significantly. As an illustration of the new representation, we re-examine the imaginary-time Green’s functions of the single-band Hubbard model in the framework of dynamical mean-field theory. The calculated results suggest that with carefully chosen integral kernel functions, whether the system is metallic or insulating, the Gibbs oscillations found in the previous Legendre orthogonal polynomial representation have been vastly suppressed and remarkable corrections to the measured Green’s functions have been obtained.

Uncovering the hidden quantum critical point in disordered massless Dirac and Weyl semimetals

We study the properties of the avoided or hidden quantum critical point (AQCP) in three dimensional Dirac and Weyl semi-metals in the presence of short range potential disorder. By computing the averaged density of states (along with its second and fourth derivative at zero energy) with the kernel polynomial method (KPM) we systematically tune the effective length scale that eventually rounds out the transition and leads to an AQCP. We show how to determine the strength of the avoidance, establishing that it is not controlled by the long wavelength component of the disorder. Instead, the amount of avoidance can be adjusted via the tails of the probability distribution of the local random potentials. A binary distribution with no tails produces much less avoidance than a Gaussian distribution. We introduce a double Gaussian distribution to interpolate between these two limits. As a result we are able to make the length scale of the avoidance sufficiently large so that we can accurately study the properties of the underlying transition (that is eventually rounded out), unambiguously identify its location, and provide accurate estimates of the critical exponents $\nu=1.01\pm0.06$ and $z=1.50\pm0.04$. We also show that the KPM expansion order introduces an effective length scale that can also round out the transition in the scaling regime near the AQCP.

Rare-Region-Induced Avoided Quantum Criticality in Disordered Three-Dimensional Dirac and Weyl Semimetals

We numerically study the effect of short ranged potential disorder on massless noninteracting three-dimensional Dirac and Weyl fermions, with a focus on the question of the proposed quantum critical point separating the semimetal and diffusive metal phases. We determine the properties of the eigenstates of the disordered Dirac Hamiltonian ($H$) and exactly calculate the density of states (DOS) near zero energy, using a combination of Lanczos on $H^2$ and the kernel polynomial method on $H$. We establish the existence of two distinct types of low energy eigenstates contributing to the disordered density of states in the weak disorder semimetal regime. These are (i) typical eigenstates that are well described by linearly dispersing perturbatively dressed Dirac states, and (ii) nonperturbative rare eigenstates that are weakly-dispersive and quasi-localized in the real space regions with the largest (and rarest) local random potential. Using twisted boundary conditions, we are able to systematically find and study these two types of eigenstates. We find that the Dirac states contribute low energy peaks in the finite-size DOS that arise from the clean eigenstates which shift and broaden in the presence of disorder. On the other hand, we establish that the rare quasi-localized eigenstates contribute a nonzero background DOS which is only weakly energy-dependent near zero energy and is exponentially small at weak disorder. We find that the expected semimetal to diffusive metal quantum critical point is converted to an {\it avoided} quantum criticality that is "rounded out" by nonperturbative effects, with no signs of any singular behavior in the DOS near the Dirac energy. We discuss the implications of our results for disordered Dirac and Weyl semimetals, and reconcile the large body of existing numerical work showing quantum criticality with the existence of the rare region effects.

Quantum Hall effect and semiconductor-to-semimetal transition in biased black phosphorus

We study the quantum Hall effect of 2D electron gas in black phosphorus in the presence of perpendicular electric and magnetic fields. In the absence of a bias voltage, the external magnetic field leads to a quantization of the energy spectrum into equidistant Landau levels, with different cyclotron frequencies for the electron and hole bands. The applied voltage reduces the band gap, and eventually a semiconductor to semimetal transition takes place. This nontrivial phase is characterized by the emergence of a pair of Dirac points in the spectrum. As a consequence, the Landau levels are not equidistant anymore, but follow the $\varepsilon_n\propto \sqrt{nB}$ characteristic of Dirac crystals as graphene. By using the Kubo-Bastin formula in the context of the kernel polynomial method, we compute the Hall conductivity of the system. We obtain a $\sigma_{xy}\propto 2n$ quantization of the Hall conductivity in the gapped phase (standard quantum Hall effect regime), and a $\sigma_{xy}\propto 4(n+1/2)$ quantization in the semimetalic phase, characteristic of Dirac systems with non-trivial topology.

Anderson metal-insulator transitions with classical magnetic impurities

We study numerically the effects of classical magnetic impurities on the Anderson metal-insulator transition. We find that a small concentration of Heisenberg impurities enhances the critical disorder amplitude ${W}_{\mathrm{c}}$ with increasing exchange coupling strength $J$. The resulting scaling with $J$ is analyzed which supports an anomalous scaling prediction by Wegner due to the combined breaking of time-reversal and spin-rotational symmetry. Moreover, we find that the presence of magnetic impurities lowers the critical correlation length exponent $\ensuremath{\nu}$ and enhances the multifractality parameter ${\ensuremath{\alpha}}_{0}$. The new value of $\ensuremath{\nu}$ improves the agreement with the value measured in experiments on the metal-insulator transition (MIT) in doped semiconductors like phosphor-doped silicon, where a finite density of magnetic moments is known to exist in the vicinity of the MIT. The results are obtained by a finite-size scaling analysis of the geometric mean of the local density of states which is calculated by means of the kernel polynomial method. We establish this combination of numerical techniques as a method to obtain critical properties of disordered systems quantitatively.

21pBA-5 局在長の多項式展開 : エルミート系と非エルミート系

21pBA-5 局在長の多項式展開 : エルミート系と非エルミート系

Many-body effects in the optical absorption of lithium azide (LiN3)

Until recently most of the understanding achieved for solid explosives has been obtained using various semi-empirical approaches due to a major role of excitonic effects in the mechanisms of decomposition. Nevertheless, during the last two decades, thanks to the ongoing progress in iterative computational methods, the inclusion of the electron-hole interaction in ab initio calculations has become a standard approach in solid-state theory. In this paper, the electronic structure and optical properties of bulk lithium azide are investigated, taking into account the electron-hole interaction via the Bethe–Salpeter equation (BSE). Here, we employ the kernel polynomial method (KPM), which significantly reduces the computational cost compared to direct diagonalization methods. The calculations of the imaginary part of the polarization dependent dielectric function including excitonic effects are reported for the first time. Then, we show a density map of the two-particle wave function and propose an alternative interpretation of the initial stages of the externally triggered chemical decomposition, based on the analysis of two-particle states near the absorption edge.

Boson peak and Ioffe-Regel criterion in amorphous siliconlike materials: The effect of bond directionality.

The vibrational properties of model amorphous materials are studied by combining complete analysis of the vibration modes, dynamical structure factor, and energy diffusivity with exact diagonalization of the dynamical matrix and the kernel polynomial method, which allows a study of very large system sizes. Different materials are studied that differ only by the bending rigidity of the interactions in a Stillinger-Weber modelization used to describe amorphous silicon. The local bending rigidity can thus be used as a control parameter, to tune the sound velocity together with local bonds directionality. It is shown that for all the systems studied, the upper limit of the Boson peak corresponds to the Ioffe-Regel criterion for transverse waves, as well as to a minimum of the diffusivity. The Boson peak is followed by a diffusivity's increase supported by longitudinal phonons. The Ioffe-Regel criterion for transverse waves corresponds to a common characteristic mean-free path of 5-7 Å (which is slightly bigger for longitudinal phonons), while the fine structure of the vibrational density of states is shown to be sensitive to the local bending rigidity.

Disorder-driven itinerant quantum criticality of three-dimensional massless Dirac fermions

Progress in the understanding of quantum critical properties of itinerant electrons has been hindered by the lack of effective models which are amenable to controlled analytical and numerically exact calculations. Here we establish that the disorder driven semimetal to metal quantum phase transition of three dimensional massless Dirac fermions could serve as a paradigmatic toy model for studying itinerant quantum criticality, which is solved in this work by exact numerical and approximate field theoretic calculations. As a result, we establish the robust existence of a non-Gaussian universality class, and also construct the relevant low energy effective field theory that could guide the understanding of quantum critical scaling for many strange metals. Using the kernel polynomial method (KPM), we provide numerical results for the calculated dynamical exponent ($z$) and correlation length exponent ($\nu$) for the disorder-driven semimetal (SM) to diffusive metal (DM) quantum phase transition at the Dirac point for several types of disorder, establishing its universal nature and obtaining the numerical scaling functions in agreement with our field theoretical analysis.

Survey and Review

The first Survey and Review article in this issue is “Symplectic Runge--Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More,” by J. M. Sanz-Serna. It looks at an aspect of geometric integration, a term first coined by the article's author in 1997 to describe an area that revitalized research in the numerical solution of ODEs. A classic 1987 paper of Cooper characterized those Runge--Kutta methods that preserve quadratic invariants---if the ODE has solutions that live on a sphere, then these Runge--Kutta methods produce discrete approximations that also live on a sphere. Soon after, Lasagni, Sanz-Serna, and Suris independently proved that the same set of Runge--Kutta methods are symplectic; as a consequence, when applied to a Hamiltonian problem, they (essentially) produce the exact solution to a nearby problem that is also of Hamiltonian form. The study of symplecticness from the perspective of backward errors and long time integration has since become a core activity in geometric integration. This review article, however, surveys and unifies a collection of results that hark back to Cooper's original work, showing that, notwithstanding symplecticness, many positive consequences arise from a method's ability to preserve quadratic invariants. Strong messages from this article are that (a) adjoint systems are a powerful tool for analysis, and (b) preservation of quadratic invariants (whether explicitly or under the hood) is a key to successful computation in this area. For those unfamiliar with adjoint equations, Figure 1 captures the main ideas in one concrete example. After introducing Lagrange multipliers in this context, the article describes results in automatic differentiation, optimal control, and classical mechanics. Overall, this work makes available and clarifies a range of contributions that have been scattered across the numerical analysis and applications literature. Our second article, “Approximating Spectral Densities of Large Matrices,” by Lin Lin, Yousef Saad, and Chao Yang, looks at a challenging linear algebra problem from physics. Computing one or more eigenvalues of a real, symmetric matrix $A$ is a classical problem in numerical analysis. The related density of states, or spectral density, problem surveyed here requires us to compute a continuous, nonnegative function, $\tilde \phi(t)$, that takes large values when $t$ is close to an eigenvalue of $A$. Computing the full spectrum of $A$ will immediately solve this problem, but in many realistic cases, that approach is not feasible. Before discussing numerical methods, the authors deal with the issue of judging the accuracy of $\tilde \phi(t)$. This is not a clearly defined task, since the target “function” is a linear combination of Dirac distributions, and the authors adopt a particular error measure that allows for different levels of resolution. Two classes of numerical method are then reviewed. The kernel polynomial method (KPM) expands $\tilde \phi(t)$ in terms of Chebyshev polynomials, with coefficients defined via moment matching. This approach is shown to be closely related to the “spectroscopic” method originally proposed by Lanczos. The second main class is built on the classical Lanczos tridiagonalization method. Throughout this article the authors adopt a numerical linear algebra perspective, comparing a number of variations within the two key classes and covering a range of practical issues, including blurring, damping, and preserving nonnegativity. The error measure and numerical methods are then illustrated on a range of large-scale examples. In addition to those directly involved in spectral density computation, this article will appeal more widely to readers interested in practical matrix computation.

Disorder driven superconductor-insulator transition in inhomogenous d-wave superconductor

We study the effect of disorder on the superconductor-insulator transition in an inhomogeneous d -wave superconductor using the kernel polynomial method. As the Bogoliubov-de Gennes equations of the two-dimensional square lattice are solved self-consistently for the cases with more than 100000 unit cells, it is possible to observe the spatial fluctuations of the superconducting order parameters at the nanoscale. We find that strong spatial fluctuation of the superconducting order parameters can be introduced by disorder, and some superconducting specific order parameters are even enhanced. Moreover, we find that some isolated superconducting “islands” can survive the strong disorder, giving a boson insulator with some localized Cooper pairs. Our numerical calculations predict the existence of two sequential transitions with the increasing disorder strength: a d -wave to s -wave superconductor transition, and then an s -wave superconductor to insulator transition. The possibility of the appearance of a metallic phase between the superconducting and insulating phases is excluded by performing the lattice-size scaling of the generalized inverse participation ratio. In addition, we also discuss the effect of disorder on the optical conductivity of the d -wave superconductors.

Study of multiband disordered systems using the typical medium dynamical cluster approximation

We generalize the typical medium dynamical cluster approximation to multiband disordered systems. Using our extended formalism, we perform a systematic study of the non-local correlation effects induced by disorder on the density of states and the mobility edge of the three-dimensional two-band Anderson model. We include inter-band and intra-band hopping and an intra-band disorder potential. Our results are consistent with the ones obtained by the transfer matrix and the kernel polynomial methods. We apply the method to K$_x$Fe$_{2-y}$Se$_2$ with Fe vacancies. Despite the strong vacancy disorder and anisotropy, we find the material is not an Anderson insulator. Our results demonstrate the application of the typical medium dynamical cluster approximation method to study Anderson localization in real materials.

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT.

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires half of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but the algorithm may require more matrix-vector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.

Calculation of characteristics of the absorbance of fluorides with regard to many-particle effects

A new approach to the solution of high dimension spectral problems, which is based on the use of the moment method (kernel polynomial method), is applied to study many-particle effects in a series of alkali fluorides. The electronic structure, absorption spectra, the features of the electronic charge distribution caused by the formation of bound electron-hole states and changes in these distributions in a series of fluorides due to distinctions in their electronic structure are analyzed.