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Numerical solution of large scale Hartree-Fock-Bogoliubov equations. (English) Zbl 07405582

Summary: The Hartree-Fock-Bogoliubov (HFB) theory is the starting point for treating superconducting systems. However, the computational cost for solving large scale HFB equations can be much larger than that of the Hartree-Fock equations, particularly when the Hamiltonian matrix is sparse, and the number of electrons \(N\) is relatively small compared to the matrix size \(N_b\). We first provide a concise and relatively self-contained review of the HFB theory for general finite sized quantum systems, with special focus on the treatment of spin symmetries from a linear algebra perspective. We then demonstrate that the pole expansion and selected inversion (PEXSI) method can be particularly well suited for solving large scale HFB equations. For a Hubbard-type Hamiltonian, the cost of PEXSI is at most \(\mathcal{O} (N_b^2)\) for both gapped and gapless systems, which can be significantly faster than the standard cubic scaling diagonalization methods. We show that PEXSI can solve a two-dimensional Hubbard-Hofstadter model with \(N_b\) up to \(2.88 \times 10^6\), and the wall clock time is less than 100 s using 17 280 CPU cores. This enables the simulation of physical systems under experimentally realizable magnetic fields, which cannot be otherwise simulated with smaller systems.

MSC:

65-XX Numerical analysis
81-XX Quantum theory
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[1] M. Aidelsburger, M. Atala, M. Lohse, J.T. Barreiro, B. Paredes and I. Bloch, Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett. 111 (2013) 185301.
[2] V. Bach, E.H. Lieb and J.P. Solovej, Generalized Hartree-Fock theory and the Hubbard model. J. Stat. Phys. 76 (1994) 3-89. · Zbl 0839.60095
[3] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Microscopic theory of superconductivity. Phys. Rev. 106 (1957) 162-164. · Zbl 0090.45401
[4] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Theory of superconductivity. Phys. Rev. 108 (1957) 1175-1204. · Zbl 0090.45401
[5] L.S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker and R.C. Whaley, ScaLAPACK Users’ Guide. SIAM, Philadelphia, PA (1997). · Zbl 0886.65022
[6] J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems. MIT Press (1986).
[7] G. Bräunlich, C. Hainzl and R. Seiringer, Translation-invariant quasi-free states for fermionic systems and the BCS approximation. Rev. Math. Phys. 26 (2014) 1450012. · Zbl 1327.82099
[8] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices. Nature 556 (2018) 43.
[9] S. Chiesa and S. Zhang, Phases of attractive spin-imbalanced fermions in square lattices. Phys. Rev. A 88 (2013) 043624.
[10] D. Cocks, P.P. Orth, S. Rachel, M. Buchhold, K. Le Hur and W. Hofstetter, Time-reversal-invariant Hofstadter-Hubbard model with ultracold fermions. Phys. Rev. Lett. 109 (2012) 205303.
[11] L. Covaci, F.M. Peeters and M. Berciu, Efficient numerical approach to inhomogeneous superconductivity: the Chebyshev-Bogoliubov-de Gennes method. Phys. Rev. Lett. 105 (2010) 167006.
[12] S.M. Cronenwett, T.H. Oosterkamp and L.P. Kouwenhoven, A tunable kondo effect in quantum dots. Science 281 (1998) 540-544.
[13] A. Erisman and W. Tinney, On computing certain elements of the inverse of a sparse matrix. Comm. ACM 18 (1975) 177. · Zbl 0296.65012
[14] A. Georges, G. Kotliar, W. Krauth and M.J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68 (1996) 13.
[15] C. Hainzl, M. Lewin and E. Séré, Self-consistent solution for the polarized vacuum in a no-photon QED model. J. Phys. A: Math. General 38 (2005) 4483-4499. · Zbl 1073.81677
[16] C. Hainzl, M. Lewin and J.P. Solovej, The mean-field approximation in quantum electrodynamics: the no-photon case. Commun. Pure Appl. Math. 60 (2007) 546-596. · Zbl 1113.81126
[17] D.R. Hofstadter, Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14 (1976) 2239-2249.
[18] S. Goedecker, Linear scaling electronic structure methods. Rev. Mod. Phys. 71 (1999) 1085-1123.
[19] D. Goldhaber-Gordon, J. Göres, M.A. Kastner, H. Shtrikman, D. Mahalu and U. Meirav, From the kondo regime to the mixed-valence regime in a single-electron transistor. Phys. Rev. Lett. 81 (1998) 5225.
[20] M. Iskin, Attractive Hofstadter-Hubbard model with imbalanced chemical and vector potentials. Phys. Rev. A - At. Mol. Opt. Phy. 91 (2015) 1-12.
[21] M. Iskin, Hofstadter-Hubbard model with opposite magnetic fields: Bardeen-Cooper-Schrieffer pairing and superfluidity in the nearly flat butterfly bands. Phys. Rev. A 96 (2017).
[22] M. Jacquelin, L. Lin and C. Yang, PSelInv - A distributed memory parallel algorithm for selected inversion: the symmetric case. ACM Trans. Math. Softw. 43 (2016) 21.
[23] M. Jacquelin, L. Lin and C. Yang, PSelInv - A distributed memory parallel algorithm for selected inversion: the non-symmetric case. Parallel Comput. 74 (2018) 84.
[24] G. Karypis and V. Kumar, A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20 (1998) 359-392. · Zbl 0915.68129
[25] G. Karypis and V. Kumar, A parallel algorithm for multilevel graph partitioning and sparse matrix ordering. J. Parallel Distrib. Comput. 48 (1998) 71-85.
[26] C.J. Kennedy, G.A. Siviloglou, H. Miyake, W.C. Burton and W. Ketterle, Spin-orbit coupling and quantum spin hall effect for neutral atoms without spin flips. Phys. Rev. Lett. 111 (2013) 225301.
[27] G. Knizia and G. Chan, Density matrix embedding: a simple alternative to dynamical mean-field theory. Phys. Rev. Lett. 109 (2012) 186404.
[28] D.S. Kosov, Nonequilibrium fock space for the electron transport problem. J. Chem. Phys. 131 (2009) 171102.
[29] C.V. Kraus and J.I. Cirac, Generalized Hartree-Fock theory for interacting fermions in lattices: numerical methods. New J. Phys. 12 (2010) 113004.
[30] A.L. Kuzemsky, Variational principle of Bogoliubov and generalized mean fields in many-particle interacting systems. Int. J. Mod. Phys. B 29 (2015) 1530010. · Zbl 1341.82054
[31] A. Kuzmin, M. Luisier and O. Schenk, Fast methods for computing selected elements of the Green’s function in massively parallel nanoelectronic device simulations. In: Euro-Par 2013 Parallel Proc. Springer (2013) 533-544.
[32] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Butterworth-Heinemann (1991). · Zbl 0178.57901
[33] E. Lenzmann and M. Lewin, Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs. Duke Math. J. 152 (2010) 257-315. · Zbl 1202.49013
[34] M. Lewin and S. Paul, A numerical perspective on Hartree-Fock-Bogoliubov theory. ESAIM: M2AN 48 (2014) 53-86. · Zbl 1301.82069
[35] X.S. Li and J.W. Demmel, SuperLU_DIST: a scalable distributed-memory sparse direct solver for unsymmetric linear systems. ACM Trans. Math. Softw. 29 (2003) 110. · Zbl 1068.90591
[36] E.H. Lieb, Variational principle for many-fermion systems. Phys. Rev. Lett. 46 (1981) 457-459.
[37] L. Lin, J. Lu, L. Ying and E. Weinan, Pole-based approximation of the Fermi-Dirac function. Chin. Ann. Math. 30B (2009) 729. · Zbl 1188.41007
[38] L. Lin, J. Lu, L. Ying, R. Car and E. Weinan, Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems. Comm. Math. Sci. 7 (2009) 755. · Zbl 1182.65072
[39] L. Lin, C. Yang, J. Meza, J. Lu, L. Ying and E. Weinan, SelInv - An algorithm for selected inversion of a sparse symmetric matrix. ACM. Trans. Math. Softw. 37 (2011) 40. · Zbl 1365.65069
[40] L. Lin, M. Chen, C. Yang and L. He, Accelerating atomic orbital-based electronic structure calculation via pole expansion and selected inversion. J. Phys.: Condens. Matter 25 (2013) 295501.
[41] L. Lin, A. Garca, G. Huhs and C. Yang, Massively parallel method for efficient and accurate ab initio materials simulation without matrix diagonalization. J. Phys.: Condens. Matter 26 (2014) 305503.
[42] A. Marek, V. Blum, R. Johanni, V. Havu, B. Lang, T. Auckenthaler, A. Heinecke, H.-J. Bungartz and H. Lederer, The {ELPA} library: scalable parallel eigenvalue solutions for electronic structure theory and computational science. J. Phys.: Condens. Matter. 26 (2014) 213201.
[43] J.E. Moussa, Minimax rational approximation of the fermi-dirac distribution. J. Chem. Phys. 145 (2016) 164108.
[44] Y. Nagai, Y. Ota and M. Machida, Efficient numerical self-consistent mean-field approach for fermionic many-body systems by polynomial expansion on spectral density. J. Phys. Soc. Jpn. 81 (2012).
[45] D. Nakamura, A. Ikeda, H. Sawabe, Y.H. Matsuda and S. Takeyama, Record indoor magnetic field of 1200 T generated by electromagnetic flux-compression. Rev. Sci. Instrum. 89 (2018) 95106.
[46] J.W. Negele and H. Orland, Quantum Many-Particle Systems. Westview (1988). · Zbl 0984.82503
[47] P. Pulay, Convergence acceleration of iterative sequences: the case of SCF iteration. Chem. Phys. Lett. 73 (1980) 393-398.
[48] P. Pulay, Improved SCF convergence acceleration. J. Comput. Chem. 3 (1982) 54-69.
[49] P. Rosenberg, S. Chiesa and S. Zhang, FFLO order in ultra-cold atoms in three-dimensional optical lattices. J. Phys. Condens. Matter 27 (2015) 225601.
[50] H. Shi and S. Zhang, Many-body computations by stochastic sampling in Hartree-Fock-Bogoliubov space. Phys. Rev. B 95 (2017) 045144.
[51] A. Szabo and N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory. McGraw-Hill, New York (1989).
[52] K. Takahashi, J. Fagan and M. Chin, Formation of a sparse bus impedance matrix and its application to short circuit study. In: 8th PICA Conf. Proc. (1973).
[53] R.O. Umucalllar and M. Iskin, BCS theory of time-reversal-symmetric Hofstadter-Hubbard Model. Phys. Rev. Lett. 119 (2017).
[54] L. Wang, H.-H. Hung and M. Troyer, Topological phase transition in the Hofstadter-Hubbard model. Phys. Rev. B 90 (2014) 205111.
[55] C. Zeng, T.D. Stanescu, C. Zhang, V.W. Scarola and S. Tewari, Majorana corner modes with solitons in an attractive Hubbard-Hofstadter model of cold atom optical lattices. Phys. Rev. Lett. 123 (2019) 060402.
[56] G.-Q. Zha, L. Covaci, S.-P. Zhou and F.M. Peeters, Proximity-induced pseudogap in mesoscopic superconductor/normal-metal bilayers. Phys. Rev. B 82 (2010) 140502.
[57] B.X. Zheng and G.K.L. Chan, Ground-state phase diagram of the square lattice Hubbard model from density matrix embedding theory. Phys. Rev. B 93 (2016) 1-17.
[58] B.X. Zheng, C.M. Chung, P. Corboz, G. Ehlers, M.P. Qin, R.M. Noack, H. Shi, S.R. White, S. Zhang and G.K.L. Chan, Stripe order in the underdoped region of the two-dimensional Hubbard model. Science 358 (2017) 1155-1160. · Zbl 1404.82093
[59] J.-X. Zhu, Bogoliubov-de Gennes Method and Its Applications. Springer 924 (2016). · Zbl 1361.82007
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