Krylov subspace methods for the Dirac equation

Abstract The Lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the Lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The unboundedness of the Dirac Hamiltonian does not hinder the applicability of the Lanczos algorithm. As the Lanczos algorithm requires only matrix–vector products and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations. The excellent parallelization capabilities are demonstrated by a parallel implementation of the Dirac Lanczos propagator utilizing the Message Passing Interface standard. Program summary Program title: Dirac_Laczos Catalogue identifier: AEUY_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEUY_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 526 428 No. of bytes in distributed program, including test data, etc.: 2 181 729 Distribution format: tar.gz Programming language: C++11. Computer: Multi-core systems or cluster computers. Operating system: Any. Has the code been vectorized or parallelized?: Parallelized using MPI. RAM: Typically 10 megabyte to 1 gigabyte depending on the chosen problem size Classification: 2.7. External routines: Boost [1], LAPACK [2] Nature of problem: Solving the time-dependent Dirac equation in two spatial dimensions Solution method: Lanczos propagator Running time: Depending on the problem size and computer hardware typically several minutes to several days References: [1] Boost C++ Libraries, http://www.boost.org [2] LAPACK-Linear Algebra PACKage, http://www.netlib.org/lapack/

[1]  D. Eng,et al.  \computing Eigenvalues of Very Large Symmetric Matrices { an Implementation of a Lanczos Algorithm with No Reorthogonalization," , 1996 .

[2]  C. Paige.  Computational variants of the Lanczos method for the eigenproblem , 1972 .

[3]  John C. Light,et al.  On the Exponential Form of Time‐Displacement Operators in Quantum Mechanics , 1966 .

[4]  J. Light,et al.  Generalized discrete variable approximation in quantum mechanics , 1985 .

[5]  Franz Gross,et al.  Relativistic quantum mechanics and field theory , 1993 .

[6]  Christoph H. Keitel,et al.  A real space split operator method for the Klein-Gordon equation , 2009, J. Comput. Phys..

[7]  C. Paige.  Error Analysis of the Lanczos Algorithm for Tridiagonalizing a Symmetric Matrix , 1976 .

[8]  G. Meurant.  The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations , 2006 .

[9]  V. Szalay.  Discrete variable representations of differential operators , 1993 .

[10]  L. Brey,et al.  Electronic states of graphene nanoribbons studied with the Dirac equation , 2006 .

[11]  K. Z. Hatsagortsyan,et al.  Extremely high-intensity laser interactions with fundamental quantum systems , 2011, 1111.3886.

[12]  Barry I. Schneider,et al.  ALTDSE: An Arnoldi-Lanczos program to solve the time-dependent Schrödinger equation , 2009, Comput. Phys. Commun..

[13]  R. Haydock.  The recursive solution of the Schrödinger equation , 1980 .

[14]  Werner Scheid,et al.  FINITE ELEMENT FORMULATION OF THE DIRAC EQUATION AND THE PROBLEM OF FERMION DOUBLING , 1998 .

[15]  H. Bethe,et al.  Theory of Atomic Collisions , 1951, Nature.

[16]  H. Bauke,et al.  The Kapitza-Dirac effect in the relativistic regime , 2013, 1305.5507.

[17]  T. Park,et al.  Unitary quantum time evolution by iterative Lanczos reduction , 1986 .

[18]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[19]  Suk-Geun Hwang,et al.  Cauchy's Interlace Theorem for Eigenvalues of Hermitian Matrices , 2004, Am. Math. Mon..

[20]  J. Kamiński,et al.  Fundamental processes of quantum electrodynamics in laser fields of relativistic power , 2009 .

[21]  D. Gitman,et al.  Exact solutions of relativistic wave equations , 1990 .

[22]  Guido R. Mocken,et al.  Quantum dynamics of relativistic electrons , 2004 .

[23]  M. Huber,et al.  Relativistic entanglement of two massive particles , 2009, 0912.4863.

[24]  Claude Leforestier,et al.  A comparison of different propagation schemes for the time dependent Schro¨dinger equation , 1991 .

[25]  Christoph H. Keitel,et al.  Accelerating the Fourier split operator method via graphics processing units , 2010, Comput. Phys. Commun..

[26]  Rainer Grobe,et al.  Numerical approach to solve the time-dependent Dirac equation , 1999 .

[27]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[28]  Walter Pötz,et al.  Staggered grid leap-frog scheme for the (2+1) D Dirac equation , 2013, Comput. Phys. Commun..

[29]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[30]  G. L. Payne,et al.  Relativistic Quantum Mechanics , 2007 .

[31]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[32]  M. Ancona,et al.  Cluster computing , 2003, Eleventh Euromicro Conference on Parallel, Distributed and Network-Based Processing, 2003. Proceedings..

[33]  Vlatko Vedral,et al.  Physical interpretation of the Wigner rotations and its implications for relativistic quantum information , 2011, 1111.7145.

[34]  Guido R. Mocken,et al.  FFT-split-operator code for solving the Dirac equation in 2+1 dimensions , 2008, Comput. Phys. Commun..

[35]  R. Blatt,et al.  Quantum simulation of the Dirac equation , 2009, Nature.

[36]  C. Vafa,et al.  Advanced Quantum Mechanics , 2012 .

[37]  André D. Bandrauk,et al.  Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling , 2011, Comput. Phys. Commun..

[38]  Nicholas Wilt,et al.  The CUDA Handbook: A Comprehensive Guide to GPU Programming , 2013 .

[39]  Daniel R. Terno,et al.  Quantum Information and Relativity Theory , 2002, quant-ph/0212023.

[40]  André D. Bandrauk,et al.  A split-step numerical method for the time-dependent Dirac equation in 3-D axisymmetric geometry , 2013, J. Comput. Phys..

[41]  Eva Lindroth,et al.  Solution of the Dirac equation for hydrogenlike systems exposed to intense electromagnetic pulses , 2009 .

[42]  P. Strange.  Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics , 1998 .

[43]  Christoph H. Keitel,et al.  Relativistic high-power laser–matter interactions , 2006 .

[44]  S. Blanes,et al.  The Magnus expansion and some of its applications , 2008, 0810.5488.

[45]  M. Feit,et al.  Solution of the Schrödinger equation by a spectral method , 1982 .

[46]  S. X. Hu,et al.  Parallel solver for the time-dependent linear and nonlinear Schrödinger equation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[48]  H. Simon.  The Lanczos algorithm with partial reorthogonalization , 1984 .

[49]  Y. Saad.  Numerical Methods for Large Eigenvalue Problems , 2011 .

[50]  Sidney D. Drell,et al.  Relativistic Quantum Mechanics , 1965 .

[51]  Y. Saad.  Analysis of some Krylov subspace approximations to the matrix exponential operator , 1992 .