Coordinatespace solver for superfluid manyfermion systems with the shifted conjugateorthogonal conjugategradient method
Abstract
Selfconsistent approaches to superfluid manyfermion systems in three dimensions (and their subsequent use in timedependent studies) require a large number of diagonalizations of very large dimension Hermitian matrices, which results in enormous computational costs. We present an approach based on the shifted conjugateorthogonal conjugategradient (COCG) Krylov method for the evaluation of the Green's function, from which we subsequently extract various densities (particle number, spin, current, kinetic energy, anomalous, etc.) of a nuclear system. The approach eschews the determination of the quasiparticle wave functions and their corresponding quasiparticle energies, which never explicitly appear in the construction of a singleparticle Hamiltonian or needed for the calculation of various static nuclear properties, which depend only on densities. As benchmarks this study presents calculations for nuclei with axial symmetry, including the ground state of spherical (magic or semimagic) and axially deformed nuclei, the saddle point in the $$^{240}Pu$$ constrained fission path, and a vortex in the neutron star crust, and demonstrate the superior efficiency of the shifted COCG Krylov method over traditional approaches.
 Authors:

 Univ. of Washington, Seattle, WA (United States). Dept. of Physics
 Univ. of Washington, Seattle, WA (United States). Dept. of Physics; Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
 Warsaw Univ. of Technology (Poland). Faculty of Physics; Univ. of Washington, Seattle, WA (United States). Dept. of Physics
 Publication Date:
 Research Org.:
 Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF); Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC); Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC)
 OSTI Identifier:
 1544399
 Alternate Identifier(s):
 OSTI ID: 1349702
 Grant/Contract Number:
 FG0297ER41014; AC0500OR22725; AC0205CH11231
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review C
 Additional Journal Information:
 Journal Volume: 95; Journal Issue: 4; Journal ID: ISSN 24699985
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 73 NUCLEAR PHYSICS AND RADIATION PHYSICS
Citation Formats
Jin, Shi, Bulgac, Aurel, Roche, Kenneth, and Wlazłowski, Gabriel. Coordinatespace solver for superfluid manyfermion systems with the shifted conjugateorthogonal conjugategradient method. United States: N. p., 2017.
Web. doi:10.1103/PhysRevC.95.044302.
Jin, Shi, Bulgac, Aurel, Roche, Kenneth, & Wlazłowski, Gabriel. Coordinatespace solver for superfluid manyfermion systems with the shifted conjugateorthogonal conjugategradient method. United States. https://doi.org/10.1103/PhysRevC.95.044302
Jin, Shi, Bulgac, Aurel, Roche, Kenneth, and Wlazłowski, Gabriel. Mon .
"Coordinatespace solver for superfluid manyfermion systems with the shifted conjugateorthogonal conjugategradient method". United States. https://doi.org/10.1103/PhysRevC.95.044302. https://www.osti.gov/servlets/purl/1544399.
@article{osti_1544399,
title = {Coordinatespace solver for superfluid manyfermion systems with the shifted conjugateorthogonal conjugategradient method},
author = {Jin, Shi and Bulgac, Aurel and Roche, Kenneth and Wlazłowski, Gabriel},
abstractNote = {Selfconsistent approaches to superfluid manyfermion systems in three dimensions (and their subsequent use in timedependent studies) require a large number of diagonalizations of very large dimension Hermitian matrices, which results in enormous computational costs. We present an approach based on the shifted conjugateorthogonal conjugategradient (COCG) Krylov method for the evaluation of the Green's function, from which we subsequently extract various densities (particle number, spin, current, kinetic energy, anomalous, etc.) of a nuclear system. The approach eschews the determination of the quasiparticle wave functions and their corresponding quasiparticle energies, which never explicitly appear in the construction of a singleparticle Hamiltonian or needed for the calculation of various static nuclear properties, which depend only on densities. As benchmarks this study presents calculations for nuclei with axial symmetry, including the ground state of spherical (magic or semimagic) and axially deformed nuclei, the saddle point in the $^{240}Pu$ constrained fission path, and a vortex in the neutron star crust, and demonstrate the superior efficiency of the shifted COCG Krylov method over traditional approaches.},
doi = {10.1103/PhysRevC.95.044302},
journal = {Physical Review C},
number = 4,
volume = 95,
place = {United States},
year = {2017},
month = {4}
}
Web of Science
Figures / Tables:
Works referenced in this record:
Stochastic estimation of nuclear level density in the nuclear shell model: An application to paritydependent level density in 58 Ni
journal, February 2016
 Shimizu, Noritaka; Utsuno, Yutaka; Futamura, Yasunori
 Physics Letters B, Vol. 753
Adaptive multiresolution 3D HartreeFockBogoliubov solver for nuclear structure
journal, August 2014
 Pei, J. C.; Fann, G. I.; Harrison, R. J.
 Physical Review C, Vol. 90, Issue 2
BiCGStab(?) for Families of Shifted Linear Systems
journal, April 2003
 Frommer, A.
 Computing, Vol. 70, Issue 2
Energy Density Functional Approach to Superfluid Nuclei
journal, June 2003
 Yu, Yongle; Bulgac, Aurel
 Physical Review Letters, Vol. 90, Issue 22
Nuclear response in the continuum
journal, May 1975
 Shlomo, S.; Bertsch, G.
 Nuclear Physics A, Vol. 243, Issue 3
HartreeFockBogolyubov description of nuclei near the neutrondrip line
journal, June 1984
 Dobaczewski, J.; Flocard, H.; Treiner, J.
 Nuclear Physics A, Vol. 422, Issue 1
Axially symmetric HartreeFockBogoliubov calculations for nuclei near the drip lines
journal, June 2003
 Terán, E.; Oberacker, V. E.; Umar, A. S.
 Physical Review C, Vol. 67, Issue 6
Broyden's method in nuclear structure calculations
journal, July 2008
 Baran, Andrzej; Bulgac, Aurel; Forbes, Michael McNeil
 Physical Review C, Vol. 78, Issue 1
Fission fragment charge and mass distributions in ${}^{239}\mathrm{Pu}\left(n,f\right)$ in the adiabatic nuclear energy density functional theory
journal, May 2016
 Regnier, D.; Dubray, N.; Schunck, N.
 Physical Review C, Vol. 93, Issue 5
An Efficient MAINV Preconditioned COCG Method for FEM Analysis of Millimeter Wave Filters
journal, January 2011
 Li, Yue Hui; Nie, Zai Ping; Meng, Min
 Journal of Infrared, Millimeter, and Terahertz Waves, Vol. 32, Issue 2
Finite amplitude method applied to the giant dipole resonance in heavy rareearth nuclei
journal, March 2016
 Oishi, Tomohiro; Kortelainen, Markus; Hinohara, Nobuo
 Physical Review C, Vol. 93, Issue 3
A parallel method for large sparse generalized eigenvalue problems using a GridRPC system
journal, June 2008
 Sakurai, Tetsuya; Kodaki, Yoshihisa; Tadano, Hiroto
 Future Generation Computer Systems, Vol. 24, Issue 6
Relativistic Coulomb Excitation within the Time Dependent Superfluid Local Density Approximation
journal, January 2015
 Stetcu, I.; Bertulani, C. A.; Bulgac, A.
 Physical Review Letters, Vol. 114, Issue 1
Finite amplitude method for the solution of the randomphase approximation
journal, August 2007
 Nakatsukasa, Takashi; Inakura, Tsunenori; Yabana, Kazuhiro
 Physical Review C, Vol. 76, Issue 2
Finite amplitude method in linear response TDDFT calculations
journal, September 2014
 Nakatsukasa, Takashi
 Journal of Physics: Conference Series, Vol. 533
Preconditioning iterative algorithm for the electromagnetic scattering from a large cavity
journal, May 2009
 Wang, Yingxi; Du, Kui; Sun, Weiwei
 Numerical Linear Algebra with Applications, Vol. 16, Issue 5
Coordinate space HartreeFockBogoliubov calculations for the zirconium isotope chain up to the twoneutron drip line
journal, May 2005
 Blazkiewicz, A.; Oberacker, V. E.; Umar, A. S.
 Physical Review C, Vol. 71, Issue 5
Largeamplitude motion in superfluid fermi droplets
journal, January 1990
 Barranco, F.; Bertsch, G.; Broglia, R.
 Nuclear Physics A, Vol. 512, Issue 2
A projection method for generalized eigenvalue problems using numerical integration
journal, October 2003
 Sakurai, Tetsuya; Sugiura, Hiroshi
 Journal of Computational and Applied Mathematics, Vol. 159, Issue 1
Solution of the Skyrme–HF+BCS equation on a 3D mesh, II: A new version of the Ev8 code
journal, February 2015
 Ryssens, W.; Hellemans, V.; Bender, M.
 Computer Physics Communications, Vol. 187
Finiteamplitude method for chargechanging transitions in axially deformed nuclei
journal, August 2014
 Mustonen, M. T.; Shafer, T.; Zenginerler, Z.
 Physical Review C, Vol. 90, Issue 2
HartreeFockBogoliubov descriptions of deformed weakly bound nuclei in large coordinate spaces
journal, November 2013
 Zhang, Y. N.; Pei, J. C.; Xu, F. R.
 Physical Review C, Vol. 88, Issue 5
Microscopic calculation of ${}^{240}\mathrm{Pu}$ scission with a finiterange effective force
journal, November 2009
 Younes, W.; Gogny, D.
 Physical Review C, Vol. 80, Issue 5
Augmented Lagrangian method for constrained nuclear density functional theory
journal, August 2010
 Staszczak, A.; Stoitsov, M.; Baran, A.
 The European Physical Journal A, Vol. 46, Issue 1
A Preconditioned COCG Method for Solving Complex Symmetric Linear Systems Arising from Scattering Problems
journal, January 2008
 Li, L.; Huang, T. Z.; Ren, Z. G.
 Journal of Electromagnetic Waves and Applications, Vol. 22, Issue 1415
HartreeFockBogoliubov calculations in coordinate space: Neutronrich sulfur, zirconium, cerium, and samarium isotopes
journal, December 2003
 Oberacker, V. E.; Umar, A. S.; Terán, E.
 Physical Review C, Vol. 68, Issue 6
Solution of the Skyrme–Hartree–Fock–Bogolyubov equations in the Cartesian deformed harmonicoscillator basis.
journal, January 2012
 Schunck, N.; Dobaczewski, J.; McDonnell, J.
 Computer Physics Communications, Vol. 183, Issue 1
Meanfield description of groundstate properties of dripline nuclei: Pairing and continuum effects
journal, June 1996
 Dobaczewski, J.; Nazarewicz, W.; Werner, T. R.
 Physical Review C, Vol. 53, Issue 6
Application of the imaginary time step method to the solution of the static HartreeFock problem
journal, June 1980
 Davies, K. T. R.; Flocard, H.; Krieger, S.
 Nuclear Physics A, Vol. 342, Issue 1
Linear algebraic calculation of the Green’s function for largescale electronic structure theory
journal, April 2006
 Takayama, R.; Hoshi, T.; Sogabe, T.
 Physical Review B, Vol. 73, Issue 16
The nuclear density of states in the space of nuclear shapes
journal, September 1980
 Bertsch, G.
 Physics Letters B, Vol. 95, Issue 2
Pairing renormalization and regularization within the local density approximation
journal, April 2006
 Borycki, P. J.; Dobaczewski, J.; Nazarewicz, W.
 Physical Review C, Vol. 73, Issue 4
Pulsar glitches and restlessness as a hard superfluidity phenomenon
journal, July 1975
 Anderson, P. W.; Itoh, N.
 Nature, Vol. 256, Issue 5512
Application of the gradient method to HartreeFockBogoliubov theory
journal, July 2011
 Robledo, L. M.; Bertsch, G. F.
 Physical Review C, Vol. 84, Issue 1
Vortex Pinning and Dynamics in the Neutron Star Crust
journal, November 2016
 Wlazłowski, Gabriel; Sekizawa, Kazuyuki; Magierski, Piotr
 Physical Review Letters, Vol. 117, Issue 23
Nuclear groundstate properties from meanfield calculations
journal, September 2002
 Dobaczewski, J.; Nazarewicz, W.; Stoitsov, M. V.
 The European Physical Journal A, Vol. 15, Issue 12
Comment on “Spontaneous Fission: A Kinetic Approach”
journal, November 1997
 Bertsch, George F.; Bulgac, Aurel
 Physical Review Letters, Vol. 79, Issue 18
Renormalization of the HartreeFockBogoliubov Equations in the Case of a Zero Range Pairing Interaction
journal, January 2002
 Bulgac, Aurel; Yu, Yongle
 Physical Review Letters, Vol. 88, Issue 4
Axially deformed solution of the SkyrmeHartree–Fock–Bogoliubov equations using the transformed harmonic oscillator basis (II) hfbtho v2.00d: A new version of the program
journal, June 2013
 Stoitsov, M. V.; Schunck, N.; Kortelainen, M.
 Computer Physics Communications, Vol. 184, Issue 6
Description of induced nuclear fission with Skyrme energy functionals: Static potential energy surfaces and fission fragment properties
journal, November 2014
 Schunck, N.; Duke, D.; Carr, H.
 Physical Review C, Vol. 90, Issue 5
Gamow Shell Model Description of NeutronRich Nuclei
journal, July 2002
 Michel, N.; Nazarewicz, W.; Płoszajczak, M.
 Physical Review Letters, Vol. 89, Issue 4
Use of the discrete variable representation basis in nuclear physics
journal, May 2013
 Bulgac, Aurel; Forbes, Michael McNeil
 Physical Review C, Vol. 87, Issue 5
Oddeven mass differences from selfconsistent mean field theory
journal, March 2009
 Bertsch, G. F.; Bertulani, C. A.; Nazarewicz, W.
 Physical Review C, Vol. 79, Issue 3
Finite amplitude method for the quasiparticle randomphase approximation
journal, July 2011
 Avogadro, Paolo; Nakatsukasa, Takashi
 Physical Review C, Vol. 84, Issue 1
Towards a universal nuclear density functional
journal, August 1998
 Fayans, S. A.
 Journal of Experimental and Theoretical Physics Letters, Vol. 68, Issue 3
Eigenvalue computation in the 20th century
journal, November 2000
 Golub, Gene H.; van der Vorst, Henk A.
 Journal of Computational and Applied Mathematics, Vol. 123, Issue 12
QuasiMinimal Residual Variants of the COCG and COCR Methods for Complex Symmetric Linear Systems in Electromagnetic Simulations
journal, December 2014
 Gu, XianMing; Huang, TingZhu; Li, Liang
 IEEE Transactions on Microwave Theory and Techniques, Vol. 62, Issue 12
Filter diagonalization of shellmodel calculations
journal, August 2010
 Mizusaki, Takahiro; Kaneko, Kazunari; Honma, Michio
 Physical Review C, Vol. 82, Issue 2
Numerical accuracy of meanfield calculations in coordinate space
journal, December 2015
 Ryssens, W.; Heenen, P. H.; Bender, M.
 Physical Review C, Vol. 92, Issue 6
Shifted ConjugateOrthogonal–ConjugateGradient Method and Its Application to Double Orbital Extended Hubbard Model
journal, November 2008
 Yamamoto, Susumu; Sogabe, Tomohiro; Hoshi, Takeo
 Journal of the Physical Society of Japan, Vol. 77, Issue 11
Selfconsistent meanfield models for nuclear structure
journal, January 2003
 Bender, Michael; Heenen, PaulHenri; Reinhard, PaulGerhard
 Reviews of Modern Physics, Vol. 75, Issue 1
On the solution of the HartreeFockBogoliubov equations by the conjugate gradient method
journal, November 1995
 Egido, J. L.; Lessing, J.; Martin, V.
 Nuclear Physics A, Vol. 594, Issue 1
Deformed coordinatespace HartreeFockBogoliubov approach to weakly bound nuclei and large deformations
journal, December 2008
 Pei, J. C.; Stoitsov, M. V.; Fann, G. I.
 Physical Review C, Vol. 78, Issue 6
Nuclear isotope shifts within the local energydensity functional approach
journal, August 2000
 Fayans, S. A.; Tolokonnikov, S. V.; Trykov, E. L.
 Nuclear Physics A, Vol. 676, Issue 14
Skyrme interaction and nuclear matter constraints
journal, March 2012
 Dutra, M.; Lourenço, O.; Sá Martins, J. S.
 Physical Review C, Vol. 85, Issue 3
Induced Fission of ${}^{240}\mathrm{Pu}$ within a RealTime Microscopic Framework
journal, March 2016
 Bulgac, Aurel; Magierski, Piotr; Roche, Kenneth J.
 Physical Review Letters, Vol. 116, Issue 12
Isovector giant dipole resonance from the 3D timedependent density functional theory for superfluid nuclei
journal, November 2011
 Stetcu, I.; Bulgac, A.; Magierski, P.
 Physical Review C, Vol. 84, Issue 5
A Skyrme parametrization from subnuclear to neutron star densities Part II. Nuclei far from stabilities
journal, May 1998
 Chabanat, E.; Bonche, P.; Haensel, P.
 Nuclear Physics A, Vol. 635, Issue 12
A study of the nuclear response function
journal, May 1975
 Bertsch, G. F.; Tsai, S. F.
 Physics Reports, Vol. 18, Issue 2
A fast configuration space method for solving local Kohn–Sham equations
journal, September 2005
 Aichinger, M.; Krotscheck, E.
 Computational Materials Science, Vol. 34, Issue 2
A filter diagonalization for generalized eigenvalue problems based on the Sakurai–Sugiura projection method
journal, February 2010
 Ikegami, Tsutomu; Sakurai, Tetsuya; Nagashima, Umpei
 Journal of Computational and Applied Mathematics, Vol. 233, Issue 8
A PetrovGalerkin type method for solving Axk=b, where A is symmetric complex
journal, March 1990
 van der Vorst, H. A.; Melissen, J. B. M.
 IEEE Transactions on Magnetics, Vol. 26, Issue 2
Coordinatespace solution of the Skyrme–Hartree–Fock– Bogolyubov equations within spherical symmetry. The program HFBRAD (v1.00)
journal, June 2005
 Bennaceur, K.; Dobaczewski, J.
 Computer Physics Communications, Vol. 168, Issue 2
Restarted GMRES for Shifted Linear Systems
journal, January 1998
 Frommer, Andreas; Glässner, Uwe
 SIAM Journal on Scientific Computing, Vol. 19, Issue 1
Lowenergy collective modes of deformed superfluid nuclei within the finiteamplitude method
journal, June 2013
 Hinohara, Nobuo; Kortelainen, Markus; Nazarewicz, Witold
 Physical Review C, Vol. 87, Issue 6
Largeamplitude motion in superfluid fermi droplets
journal, January 1990
 Barranco, F.; Bertsch, G. F.; Broglia, R. A.
 Nuclear Physics A, Vol. 512, Issue 2
Eigenvalue computation in the 20th century
book, January 2001
 Golub, Gene H.; van der Vorst, Henk A.
 Numerical Analysis: Historical Developments in the 20th Century
Lowenergy collective modes of deformed superfluid nuclei within the finiteamplitude method
text, January 2013
 Markus, Kortelainen,; Witold, Nazarewicz,; Nobuo, Hinohara,
 The University of North Carolina at Chapel Hill University Libraries
Nuclear groundstate properties from meanfield calculations
book, January 2003
 Dobaczewski, J.; Nazarewicz, W.; Stoitsov, M. V.
 Exotic Nuclei and Atomic Masses
Finiteamplitude method for chargechanging transitions in axially deformed nuclei
text, January 2014
 T., Shafer,; J., Engel,; T., Mustonen, M.
 The University of North Carolina at Chapel Hill University Libraries
Axially deformed solution of the SkyrmeHartree–Fock–Bogoliubov equations using the transformed harmonic oscillator basis (II) hfbtho v2.00d: A new version of the program
dataset, January 2019
 Stoitsov, M. V.
 Mendeley
Solution of the Skyrme–HF+BCS equation on a 3D mesh, II: A new version of the Ev8 code
dataset, January 2019
 Ryssens, W.
 Mendeley
Deformed CoordinateSpace HartreeFockBogoliubov Approach to Weakly Bound Nuclei and Large Deformations
text, January 2008
 Pei, J. C.; Stoitsov, M. V.; Fann, G. I.
 arXiv
Augmented Lagrangian Method for Constrained Nuclear Density Functional Theory
text, January 2010
 Staszczak, A.; Stoitsov, M.; Baran, A.
 arXiv
Application of the gradient method to HartreeFockBogoliubov theory
text, January 2011
 Robledo, L. M.; Bertsch, G. F.
 arXiv
Isovector Giant Dipole Resonance from the 3D TimeDependent Density Functional Theory for Superfluid Nuclei
text, January 2011
 Stetcu, I.; Bulgac, A.; Magierski, P.
 arXiv
Skyrme Interaction and Nuclear Matter Constraints
text, January 2012
 Dutra, M.; Lourenço, O.; Martins, J. S. Sá
 arXiv
Axially deformed solution of the SkyrmeHartreeFockBogolyubov equations using the transformed harmonic oscillator basis (II) HFBTHO v2.00d: a new version of the program
text, January 2012
 Stoitsov, M. V.; Schunck, N.; Kortelainen, M.
 arXiv
Low energy collective modes of deformed superfluid nuclei within the finite amplitude method
text, January 2013
 Hinohara, Nobuo; Kortelainen, Markus; Nazarewicz, Witold
 arXiv
Description of Induced Nuclear Fission with Skyrme Energy Functionals: I. Static Potential Energy Surfaces and Fission Fragment Properties
text, January 2013
 Schunck, N.; Duke, D.; Carr, H.
 arXiv
Finite Amplitude Method for ChargeChanging Transitions in AxiallyDeformed Nuclei
text, January 2014
 Mustonen, M. T.; Shafer, T.; Zenginerler, Z.
 arXiv
Solution of the Skyrme HF+BCS equation on a 3D mesh. II. A new version of the Ev8 code
text, January 2014
 Ryssens, W.; Hellemans, V.; Bender, M.
 arXiv
Adaptive Multiresolution 3D HartreeFockBogoliubov Solver for Nuclear Structure
text, January 2014
 Pei, Junchen; Fann, George; Harrison, Robert
 arXiv
Stochastic Estimation of Nuclear Level Density in the Nuclear Shell Model: An Application to ParityDependent Level Density in $^{58}$Ni
text, January 2015
 Shimizu, Noritaka; Utsuno, Yutaka; Futamura, Yasunori
 arXiv
Vortex pinning and dynamics in the neutron star crust
text, January 2016
 Wlazłowski, Gabriel; Sekizawa, Kazuyuki; Magierski, Piotr
 arXiv
Linear Algebraic Calculation of Green's function for LargeScale Electronic Structure Theory
text, January 2005
 Takayama, R.; Hoshi, T.; Sogabe, T.
 arXiv
Gamow Shell Model Description of NeutronRich Nuclei
text, January 2002
 Michel, N.; Nazarewicz, W.; Ploszajczak, M.
 arXiv
Nuclear groundstate properties from meanfield calculations
text, January 2002
 Dobaczewski, J.; Nazarewicz, W.; Stoitsov, M. V.
 arXiv
Axially symmetric HartreeFockBogoliubov Calculations for Nuclei Near the DripLines
text, January 2002
 Teran, E.; Oberacker, V. E.; Umar, A. S.
 arXiv
HartreeFockBogoliubov Calculations in Coordinate Space: NeutronRich Sulfur, Zirconium, Cerium, and Samarium Isotopes
text, January 2003
 Oberacker, V. E.; Umar, A. S.; Teran, E.
 arXiv
Pairing renormalization and regularization within the local density approximation
text, January 2006
 Borycki, P. J.; Dobaczewski, J.; Nazarewicz, W.
 arXiv
Meanfield description of groundstate properties of dripline nuclei: Pairing and continuum effects
text, January 1995
 Dobaczewski, J.; Nazarewicz, W.; Werner, T. R.
 arXiv
Works referencing / citing this record:
A new statistical method for the structure of the inner crust of neutron stars
journal, August 2017
 Pastore, A.; Shelley, M.; Baroni, S.
 Journal of Physics G: Nuclear and Particle Physics, Vol. 44, Issue 9
A Minimal Nuclear Energy Density Functional
text, January 2017
 Bulgac, Aurel; Forbes, Michael McNeil; Jin, Shi
 arXiv