GPU-based acceleration of free energy calculations in solid state physics

Abstract Obtaining a thermodynamically accurate phase diagram through numerical calculations is a computationally expensive problem that is crucially important to understanding the complex phenomena of solid state physics, such as superconductivity. In this work we show how this type of analysis can be significantly accelerated through the use of modern GPUs. We illustrate this with a concrete example of free energy calculation in multi-band iron-based superconductors, known to exhibit a superconducting state with oscillating order parameter (OP). Our approach can also be used for classical BCS-type superconductors. With a customized algorithm and compiler tuning we are able to achieve a 19×speedup compared to the CPU (119×compared to a single CPU core), reducing calculation time from minutes to mere seconds, enabling the analysis of larger systems and the elimination of finite size effects. Program summary Program title: Free_Energy Catalogue identifier: AEVX_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEVX_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU Lesser General Public License, version 3 No. of lines in distributed program, including test data, etc.: 786 No. of bytes in distributed program, including test data, etc.: 6304 Distribution format: tar.gz Programming language: Fortran, CUDA C. Computer: Any with a CUDA-compliant GPU. Operating system: No limits (tested on Linux). RAM: Typically tens of megabytes. Classification: 7, 6.5. Nature of problem: GPU-accelerated free energy calculations in multi-band iron-based superconductor models. Solution method: Parallel parameter space search for a global minimum of free energy. Unusual features: The same core algorithm is implemented in Fortran with OpenMP and OpenACC compiler annotations, as well as in CUDA C. The original Fortran implementation targets the CPU architecture, while the CUDA C version is hand-optimized for modern GPUs. Running time: Problem-dependent, up to several seconds for a single value of momentum and a linear lattice size on the order of 10 3

[1]  D. J. Scalapino,et al.  Near-degeneracy of several pairing channels in multiorbital models for the Fe pnictides , 2008, 0812.0343.

[2]  I. Eremin,et al.  Theory of magnetic excitations in iron-based layered superconductors , 2008, 0804.1793.

[3]  H. Hosono,et al.  To What Extent Iron-Pnictide New Superconductors Have Been Clarified: A Progress Report , 2009, 0906.2045.

[4]  C. Ting,et al.  Impurity effects on the quasiparticle spectrum of the Fulde-Ferrell-Larkin-Ovchinnikov state of a d-wave superconductor , 2006 .

[5]  Manfred Krafczyk,et al.  TeraFLOP computing on a desktop PC with GPUs for 3D CFD , 2008 .

[6]  R. Arita,et al.  Erratum: Unconventional pairing originating from the disconnected fermi surfaces of superconducting LaFeAsO1-xFx (Physical Review Letters (2008) 101 (087004)) , 2009 .

[7]  Joshua A. Anderson,et al.  General purpose molecular dynamics simulations fully implemented on graphics processing units , 2008, J. Comput. Phys..

[8]  M. Du,et al.  Density functional study of LaFeAsO(1-x)F(x): a low carrier density superconductor near itinerant magnetism. , 2008, Physical review letters.

[9]  T. Kopp,et al.  Superconducting state with a finite-momentum pairing mechanism in zero external magnetic field , 2009, 0907.4345.

[10]  Y. Gao,et al.  Mixed-state effect on quasiparticle interference in iron-based superconductors , 2012, 1205.2418.

[11]  M. Johannes,et al.  Unconventional superconductivity with a sign reversal in the order parameter of LaFeAsO1-xFx. , 2008, Physical review letters.

[12]  K. Matsubayashi,et al.  Determination of the Upper Critical Field of a Single Crystal LiFeAs: The Magnetic Torque Study up to 35 Tesla , 2010, 1011.1334.

[13]  Order N Monte Carlo Algorithm for Fermion Systems Coupled with Fluctuating Adiabatical Fields , 2003, cond-mat/0308298.

[14]  L. Boeri,et al.  Is LaFeAsO1-xFx an electron-phonon superconductor? , 2008, Physical review letters.

[15]  P. Hirschfeld,et al.  Spin-Orbit Coupling in Fe-Based Superconductors , 2011, 1608.02230.

[16]  Chaoxing Liu,et al.  Minimal two-band model of the superconducting iron oxypnictides , 2008, 0804.1113.

[17]  A. Ptok.  The Fulde–Ferrell–Larkin–Ovchinnikov State in Quantum Rings , 2012 .

[18]  M. Machida,et al.  Efficient Numerical Self-Consistent Mean-Field Approach for Fermionic Many-Body Systems by Polynomial Expansion on Spectral Density , 2011, 1105.4939.

[19]  E. Dagotto,et al.  Three orbital model for the iron-based superconductors , 2009, 0910.1573.

[20]  Yoo Jang Song,et al.  Nodeless two-gap superconducting state in single crystals of the stoichiometric iron pnictide LiFeAs , 2011 .

[21]  M. Januszewski,et al.  Sailfish: A flexible multi-GPU implementation of the lattice Boltzmann method , 2013, Comput. Phys. Commun..

[22]  V. Cvetkovic,et al.  Valley density-wave and multiband superconductivity in iron-based pnictide superconductors , 2008, 0808.3742.

[23]  A. Ptok.  Influence of s± symmetry on unconventional superconductivity in pnictides above the Pauli limit – two-band model study , 2013, 1311.5413.

[24]  Peter Schröder,et al.  Quantum Monte Carlo on graphical processing units , 2007, Comput. Phys. Commun..

[25]  C. Ting,et al.  Evolution of the Fermi surface topology in doped 122 iron pnictides , 2013, 1311.7047.

[26]  W. Marsden.  I and J , 2012 .

[27]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[28]  T. Kondo,et al.  Momentum dependence of the superconducting gap in NdFeAsO0.9F0.1 single crystals measured by angle resolved photoemission spectroscopy. , 2008, Physical review letters.

[29]  H. Löhneysen,et al.  Pauli-limited multiband superconductivity in KFe2As2. , 2013, Physical review letters.

[30]  Hideo Hosono,et al.  Iron-based layered superconductor La[O(1-x)F(x)]FeAs (x = 0.05-0.12) with T(c) = 26 K. , 2008, Journal of the American Chemical Society.

[31]  Tetsuya Sakurai,et al.  Numerical construction of a low-energy effective Hamiltonian in a self-consistent Bogoliubov-de Gennes approach of superconductivity , 2013, 1303.3683.

[32]  X. Dai,et al.  Observation of Fermi-surface–dependent nodeless superconducting gaps in Ba0.6K0.4Fe2As2 , 2008, 0807.0419.

[33]  M. Januszewski,et al.  Accelerating numerical solution of stochastic differential equations with CUDA , 2009, Comput. Phys. Commun..

[34]  M. Mierzejewski,et al.  Mutual enhancement of magnetism and Fulde-Ferrell-Larkin-Ovchinnikov superconductivity in CeCoIn 5 , 2009 .

[35]  Ari Harju,et al.  Exact diagonalization of the Hubbard model on graphics processing units , 2012, Comput. Phys. Commun..

[36]  M. Mierzejewski,et al.  The Fulde–Ferrell–Larkin–Ovchinnikov phase in the presence of pair hopping interaction , 2009, Journal of physics. Condensed matter : an Institute of Physics journal.

[37]  F. Balakirev,et al.  Upper critical field and its anisotropy in LiFeAs , 2011, 1101.3159.

[38]  Paul Rigge,et al.  Performance of FORTRAN and C GPU Extensions for a Benchmark Suite of Fourier Pseudospectral Algorithms , 2012, 2012 Symposium on Application Accelerators in High Performance Computing.

[39]  M. Maśka,et al.  Vortex structure in the d-density-wave scenario , 2003 .

[40]  I. Eremin,et al.  Three-orbital Model for Fe-Pnictides , 2013 .

[41]  R. Arita,et al.  Unconventional pairing originating from the disconnected Fermi surfaces of superconducting LaFeAsO1-xFx. , 2008, Physical review letters.

[42]  J. Kaczmarczyk,et al.  Superconducting Bardeen-Cooper-Schrieffer versus Fulde-Ferrell-Larkin-Ovchinnikov states of heavy quasiparticles with spin-dependent masses and Kondo-type pairing , 2010 .

[43]  H. Eisaki,et al.  Hysteretic superconducting resistive transition in Ba0.07K0.93Fe2As2 , 2013, 1303.7281.

[44]  J. Kaczmarczyk,et al.  Unconventional superconducting phases in a correlated two-dimensional Fermi gas of nonstandard quasiparticles: a simple model , 2009, Journal of physics. Condensed matter : an Institute of Physics journal.

[45]  T. Kopp,et al.  Crossover from hc=e to hc=2e current oscillations in rings of s-wave superconductors , 2008, 0808.2891.

[46]  A. Gurevich.  Upper critical field and the Fulde-Ferrel-Larkin-Ovchinnikov transition in multiband superconductors , 2010 .

[47]  M. Mierzejewski,et al.  Coexistence of superconductivity and incommensurate magnetic order , 2011, 1109.3534.

[48]  E. Rotenberg,et al.  Electronic properties of iron arsenic high temperature superconductors revealed by angle resolved photoemission spectroscopy (ARPES) , 2009 .

[49]  Z. Tes̆anović,et al.  Multiband magnetism and superconductivity in Fe-based compounds , 2008, 0804.4678.

[50]  Y.,et al.  Fulde – Ferrell – Larkin – Ovchinnikov State in Heavy Fermion Superconductors , 2007 .

[51]  L. Covaci,et al.  Efficient numerical approach to inhomogeneous superconductivity: the Chebyshev-Bogoliubov-de Gennes method. , 2010, Physical review letters.

[52]  D. Crivelli,et al.  Unconventional Superconductivity in Iron-Based Superconductors in a Three-Band Model , 2014, 1401.3066.

[53]  C. Ting,et al.  Calculated phase diagram of doped BaFe2As2 superconductor in a C4-symmetry breaking model , 2013 .

[54]  J. Kaczmarczyk,et al.  Superconductivity in an almost localized Fermi liquid of quasiparticles with spin-dependent masses and effective-field induced by electron correlations , 2008, 0812.2351.

[55]  A. Gurevich.  Iron-based superconductors at high magnetic fields , 2011 .

[56]  Self-consistent interface properties of d- and s-wave superconductors , 1997, cond-mat/9708086.

[57]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[58]  Hiroaki Ikeda,et al.  Wien2wannier: From linearized augmented plane waves to maximally localized Wannier functions , 2010, Comput. Phys. Commun..

[59]  D. Crivelli,et al.  The Fulde–Ferrell–Larkin–Ovchinnikov State in Pnictides , 2013, 1304.2389.

[60]  B. L. Gyorffy,et al.  A recursion method for solving the Bogoliubov equations for inhomogeneous superconductors , 1995 .

[61]  R. Prozorov,et al.  Anisotropic upper critical field and possible Fulde-Ferrel-Larkin-Ovchinnikov state in the stoichiometric pnictide superconductor LiFeAs , 2010, 1011.5126.

[62]  G. Wellein,et al.  The kernel polynomial method , 2005, cond-mat/0504627.

[63]  Wolfgang Paul,et al.  GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model , 2009, J. Comput. Phys..

[64]  T. Wolf,et al.  Strong Pauli-limiting behavior of $H_{c2}$ and uniaxial pressure dependencies in KFe2As2 , 2013, 1303.6822.

[65]  Y. Matsuda,et al.  Nodal structure of unconventional superconductors probed by angle resolved thermal transport measurements , 2006, cond-mat/0611371.

[66]  Lukasz Machura,et al.  GPU accelerated Monte Carlo simulation of Brownian motors dynamics with CUDA , 2014, Comput. Phys. Commun..

[67]  A. Ptok.  The Fulde-Ferrell-Larkin-Ovchinnikov Superconductivity in Disordered Systems , 2010 .

[68]  C. Ting,et al.  Phase diagram of doped BaFe$_2$As$_2$ superconductor under broken $C_4$ symmetry , 2013, 1303.1446.

[69]  Peter Fulde,et al.  Superconductivity in a Strong Spin-Exchange Field , 1964 .

[70]  X. Dai,et al.  Observation of Fermi-surface – dependent nodeless superconducting gaps in Ba 0 . 6 K 0 . 4 Fe 2 As 2 , 2008 .

[71]  A. Moreo,et al.  Spectral density in a nematic state of iron pnictides , 2012, 1202.3656.

[73]  Mark J. Harris,et al.  Parallel Prefix Sum (Scan) with CUDA , 2011 .

[74]  J. Kaczmarczyk,et al.  Conductance spectroscopy of a correlated superconductor in a magnetic field in the Pauli limit: Evidence for strong correlations , 2011, 1106.1574.

[75]  K. Gray,et al.  Inhomogeneous state of superconductors by intense tunnel injection of quasiparticles , 1978 .

[76]  Yun Song,et al.  Self-consistent calculations of the effects of disorder in d-wave and s-wave superconductors , 2013 .

[77]  C. Ting,et al.  Spin-density wave and asymmetry of coherence peaks in iron pnictide superconductors from a two-orbital model , 2009, 0911.0385.

[78]  M. Machida,et al.  Direct numerical demonstration of sign-preserving quasiparticle interference via an impurity inside a vortex core in an unconventional superconductor , 2011, 1103.5842.