Highly Scalable Quantum Transfer Matrix Simulations of Molecule-Based Nanomagnets on a Parallel IBM BlueGene/P Architecture
Antkowiak Michał 1*, Kucharski Łukasz 1, Matysiak Ryszard 2, Kamieniarz Grzegorz 1
1 Faculty of Physics, A. Mickiewicz University
Umultowska 85, 61-614 Poznań, Poland2 Institute of Engineering and Computer Education, University of Zielona Góra,
ul. Prof. Szafrana 4a, 65-516 Zielona Góra, Poland∗E-mail: antekm@amu.edu.pl
Received:
Received: 15 June 2015; revised: 04 May 2016; accepted: 18 May 2016; published online: 31 May 2016
DOI: 10.12921/cmst.2016.22.02.003
Abstract:
In this work we present a very efficient scaling of our two applications based on the quantum transfer matrix method which we exploited to simulate the thermodynamic properties of Cr9 and Mn6 molecules as examples of the uniform and non-uniform molecular nanomagnets. The test runs were conducted on the IBM BlueGene/P supercomputer JUGENE of the Tier-0 performance class installed in the Jülich Supercomputing Centre.
Key words:
Heisenberg model, magnetic rings, MPI, numerical simulations, parallelization of processing
References:
[1] D. Gatteschi, R. Sessoli, and J. Villain, Molecular nanomagnets, Oxford University Press, Oxford, 2006.
[2] M. Mannini, F. Pineider, P. Sainctavit, C. Danieli, E. Otero, C. Sciancalepore, A.M. Talarico, M-A. Arrio, A. Cornia, D. Gatteschi, and R. Sessoli, Magnetic memory of a single-molecule quantum magnet wired to a gold surface, Nature Mater. 8, 194-197 (2009).
[3] A. Ardavan, O. Rival, J.J.L. Morton, S.J. Blundell, A.M. Tyryshkin, G.A. Timco, and R.E.P. Winpenny, Will spin-relaxation times in molecular magnets permit quantum information processing?, Phys. Rev. Lett. 98, 057201 (2007).
[4] J. Lehmann, A. Gaita-Ariño, E. Coronado, and D. Loss, Spin qubits with electrically gated polyoxometalate molecules, Nature Nano. 2, 312-317 (2007).
[5] G.A. Timco, S. Carretta, F. Troiani, F. Tuna, R.J. Pritchard, Ch. A. Muryn, E.J.L. McInnes, A. Ghirri, A. Candini, P. Santini, G. Amoretti, M. Affronte, and R.E.P. Winpenny, Engineering the coupling between molecular spin qubits by coordination chemistry, Nature Nano. 4, 173-178 (2009).
[6] B. Georgeot and F. Mila, Chirality of triangular antiferromagnetic clusters as qubit, Phys. Rev. Lett. 104, 200502 (2010).
[7] D. Gatteschi, R. Sessoli, and A. Cornia, Single-molecule magnets based on iron (III) oxo clusters, Chem. Commun., pages 725-732 (2000).
[8] S.G. Louie, Nanoparticles behaving oddly, Nature 384(6610), 612-613 (1996), 10.1038/384612a0.
[9] O. Cador, D. Gatteschi, R. Sessoli, A-L. Barra, G.A. Timco, and R.E.P. Winpenny, Spin frustration effects in an odd-membered antiferromagnetic ring and the magnetic Möbius strip, J. Magn. Magn. Mater. 290-291, 55-60 (2005).
[10] Y. Furukawa, K. Kiuchi, K. Kumagai, Y. Ajiro, Y. Narumi, M. Iwaki, K. Kindo, A. Bianchi, S. Carretta, P. Santini, F. Borsa, G.A. Timco, and R.E.P. Winpenny, Evidence of spin singlet ground state in the frustrated antiferromagnetic ring Cr8Ni, Phys. Rev. B 79,
134416 (2009).
[11] K. Bärwinkel, H.J. Schmidt, and J. Schnack, Ground-state properties of antiferromagnetic Heisenberg spin rings, J. Magn. Magn. Mater. 220, 227-234 (2000).
[12] J. Schnack, Properties of the first excited state of nonbipartite Heisenberg spin rings, Phys. Rev. B 62, 14855-14859 (2000).
[13] P. Kozłowski, M. Antkowiak, and G. Kamieniarz, Frustration signatures in the anisotropic model of a nine-spin s = 3/2 ring with bond defect, J. Nanopart. Res. (2011).
[14] M.L. Baker, G.A. Timco, S. Piligkos, J.S. Mathieson, H.Mutka, F. Tuna, P.Kozłłowski, M. Antkowiak, T.Guidi,T. Gupta, H.Rath, R.J. Woolfson, G. Kamieniarz, R.G. Pritchard, H. Weihe, L. Cronin, G. Rajaraman, D. Collison, E.J.L. McInnes, and R.E.P. Winpenny, Spin frustration in molecular magnets – a classification: physical studies of large odd-numbered-metal, odd-electron rings, Proc. Natl. Acad. Sci. USA109(47), 19113-19118 (2012).
[15] M. Antkowiak, P. Kozłłowski, G. Kamieniarz, G.A. Timco, F. Tuna, and R.E.P. Winpenny, Detection of ground states in frustrated molecular rings by the in-field local magnetization profiles, Phys. Rev. B 87, 184430 (2013).
[16] A. Caramico D’Auria, U. Esposito, F. Esposito, G. Kamieniarz, and R. Matysiak, Exact simulations of quantum rings and characterization of hexanuclear manganese and dodecanuclear cyclic complexes, J. Phys.: Condens. Matter 13, 2017 (2001).
[17] G. Kamieniarz, R. Matysiak, A. Caramico D’Auria, F. Esposito, and C. Benelli, Finite-temperature characterization and simulations of the molecular assemblies Mn6 and Ni12, Eur. Phys. J. B 23, 183 (2001).
[18] D. Gatteschi, A. Caneschi, L. Pardi, and R. Sessoli, Large clusters of metal ions: The transition from molecular to bulk magnets, Science 265(5175), 1054 (1994).
[19] A. Caneschi, D. Gatteschi, C. Sangregorio, R. Sessoli, L. Sorace, A. Cornia, MA Novak, C. Paulsen, and W. Wernsdorfer, The
molecular approach to nanoscale magnetism, Journal of magnetism and magnetic materials 200(1-3), 182-201, (1999).
[20] G. Kamieniarz and R. Matysiak, Simulations of the low-dimensional magnetic systems by the quantum transfer-matrix technique, Computational Materials Science 28(2), 353-365 (2003), Proceedings of the Symposium on Software Development for Process and
Materials Design.
[21] G.Kamieniarzand, R.Matysiak, Transfer matrix simulation technique: effectiveness and applicability to the low-dimensional magnetic spin systems, J. Comput. Appl. Math. 189, 471-480 (May 2006).
[22] G. Kamieniarz, P. Kozłłowski, M. Antkowiak, P. Sobczak,T. Ślusarski, D.M. Tomecka, A.Barasiński, B.Brzostowski, A.Drzewiński, A. Bieńko, and J. Mroziński, Anisotropy, Geometric Structure and Frustration Effects in Molecule-Based Nanomagnets, Acta Physica
Polonica A121(5-6), 992-998 (2011).
[23] G. Kamieniarz, W. Florek, and M. Antkowiak, Universal sequence of ground states validating the classification of frustration in
antiferromagnetic rings with a single bond defect, Phys. Rev. B 92, 140411(R) (2015).
[24] G. Kamieniarz, P. Kozłowski, G. Musiał, W. Florek, M. Antkowiak, M. Haglauer, A.C. D’Auria, and F. Esposito, Phenomenological modeling of molecular-based rings beyond the strong exchange limit: Bond alternation and single-ion anisotropy effects, Inorg. Chin. Acta 361, 3690-3696 (2008).
[25] P. Kozłowski, G. Kamieniarz, M. Antkowiak, F. Tuna, G.A. Timco, and R.E.P. Winpenny, Phenomenological modeling of the
anisotropic molecular-based ring Cr7Cd, Polyhedron 28, 1852-1855 (2009).
[26] R. Matysiak, G. Kamieniarz, P. Gegenwart, and A. Ochiai, Field-dependent specific heat of Yb4As3: Agreement between a spin-1/2 model and experiment, Phys. Rev. B 79, 224413 (2009).
[27] R. Matysiak, P. Gegenwart, A. Ochiai, M. Antkowiak, G. Kamieniarz, and F. Steglich, Specific heat of segmented Heisenberg quantum spin chains in (Yb1−xLux)4As3, Phys. Rev. B 88, 224414 (2013).
In this work we present a very efficient scaling of our two applications based on the quantum transfer matrix method which we exploited to simulate the thermodynamic properties of Cr9 and Mn6 molecules as examples of the uniform and non-uniform molecular nanomagnets. The test runs were conducted on the IBM BlueGene/P supercomputer JUGENE of the Tier-0 performance class installed in the Jülich Supercomputing Centre.
Key words:
Heisenberg model, magnetic rings, MPI, numerical simulations, parallelization of processing
References:
[1] D. Gatteschi, R. Sessoli, and J. Villain, Molecular nanomagnets, Oxford University Press, Oxford, 2006.
[2] M. Mannini, F. Pineider, P. Sainctavit, C. Danieli, E. Otero, C. Sciancalepore, A.M. Talarico, M-A. Arrio, A. Cornia, D. Gatteschi, and R. Sessoli, Magnetic memory of a single-molecule quantum magnet wired to a gold surface, Nature Mater. 8, 194-197 (2009).
[3] A. Ardavan, O. Rival, J.J.L. Morton, S.J. Blundell, A.M. Tyryshkin, G.A. Timco, and R.E.P. Winpenny, Will spin-relaxation times in molecular magnets permit quantum information processing?, Phys. Rev. Lett. 98, 057201 (2007).
[4] J. Lehmann, A. Gaita-Ariño, E. Coronado, and D. Loss, Spin qubits with electrically gated polyoxometalate molecules, Nature Nano. 2, 312-317 (2007).
[5] G.A. Timco, S. Carretta, F. Troiani, F. Tuna, R.J. Pritchard, Ch. A. Muryn, E.J.L. McInnes, A. Ghirri, A. Candini, P. Santini, G. Amoretti, M. Affronte, and R.E.P. Winpenny, Engineering the coupling between molecular spin qubits by coordination chemistry, Nature Nano. 4, 173-178 (2009).
[6] B. Georgeot and F. Mila, Chirality of triangular antiferromagnetic clusters as qubit, Phys. Rev. Lett. 104, 200502 (2010).
[7] D. Gatteschi, R. Sessoli, and A. Cornia, Single-molecule magnets based on iron (III) oxo clusters, Chem. Commun., pages 725-732 (2000).
[8] S.G. Louie, Nanoparticles behaving oddly, Nature 384(6610), 612-613 (1996), 10.1038/384612a0.
[9] O. Cador, D. Gatteschi, R. Sessoli, A-L. Barra, G.A. Timco, and R.E.P. Winpenny, Spin frustration effects in an odd-membered antiferromagnetic ring and the magnetic Möbius strip, J. Magn. Magn. Mater. 290-291, 55-60 (2005).
[10] Y. Furukawa, K. Kiuchi, K. Kumagai, Y. Ajiro, Y. Narumi, M. Iwaki, K. Kindo, A. Bianchi, S. Carretta, P. Santini, F. Borsa, G.A. Timco, and R.E.P. Winpenny, Evidence of spin singlet ground state in the frustrated antiferromagnetic ring Cr8Ni, Phys. Rev. B 79,
134416 (2009).
[11] K. Bärwinkel, H.J. Schmidt, and J. Schnack, Ground-state properties of antiferromagnetic Heisenberg spin rings, J. Magn. Magn. Mater. 220, 227-234 (2000).
[12] J. Schnack, Properties of the first excited state of nonbipartite Heisenberg spin rings, Phys. Rev. B 62, 14855-14859 (2000).
[13] P. Kozłowski, M. Antkowiak, and G. Kamieniarz, Frustration signatures in the anisotropic model of a nine-spin s = 3/2 ring with bond defect, J. Nanopart. Res. (2011).
[14] M.L. Baker, G.A. Timco, S. Piligkos, J.S. Mathieson, H.Mutka, F. Tuna, P.Kozłłowski, M. Antkowiak, T.Guidi,T. Gupta, H.Rath, R.J. Woolfson, G. Kamieniarz, R.G. Pritchard, H. Weihe, L. Cronin, G. Rajaraman, D. Collison, E.J.L. McInnes, and R.E.P. Winpenny, Spin frustration in molecular magnets – a classification: physical studies of large odd-numbered-metal, odd-electron rings, Proc. Natl. Acad. Sci. USA109(47), 19113-19118 (2012).
[15] M. Antkowiak, P. Kozłłowski, G. Kamieniarz, G.A. Timco, F. Tuna, and R.E.P. Winpenny, Detection of ground states in frustrated molecular rings by the in-field local magnetization profiles, Phys. Rev. B 87, 184430 (2013).
[16] A. Caramico D’Auria, U. Esposito, F. Esposito, G. Kamieniarz, and R. Matysiak, Exact simulations of quantum rings and characterization of hexanuclear manganese and dodecanuclear cyclic complexes, J. Phys.: Condens. Matter 13, 2017 (2001).
[17] G. Kamieniarz, R. Matysiak, A. Caramico D’Auria, F. Esposito, and C. Benelli, Finite-temperature characterization and simulations of the molecular assemblies Mn6 and Ni12, Eur. Phys. J. B 23, 183 (2001).
[18] D. Gatteschi, A. Caneschi, L. Pardi, and R. Sessoli, Large clusters of metal ions: The transition from molecular to bulk magnets, Science 265(5175), 1054 (1994).
[19] A. Caneschi, D. Gatteschi, C. Sangregorio, R. Sessoli, L. Sorace, A. Cornia, MA Novak, C. Paulsen, and W. Wernsdorfer, The
molecular approach to nanoscale magnetism, Journal of magnetism and magnetic materials 200(1-3), 182-201, (1999).
[20] G. Kamieniarz and R. Matysiak, Simulations of the low-dimensional magnetic systems by the quantum transfer-matrix technique, Computational Materials Science 28(2), 353-365 (2003), Proceedings of the Symposium on Software Development for Process and
Materials Design.
[21] G.Kamieniarzand, R.Matysiak, Transfer matrix simulation technique: effectiveness and applicability to the low-dimensional magnetic spin systems, J. Comput. Appl. Math. 189, 471-480 (May 2006).
[22] G. Kamieniarz, P. Kozłłowski, M. Antkowiak, P. Sobczak,T. Ślusarski, D.M. Tomecka, A.Barasiński, B.Brzostowski, A.Drzewiński, A. Bieńko, and J. Mroziński, Anisotropy, Geometric Structure and Frustration Effects in Molecule-Based Nanomagnets, Acta Physica
Polonica A121(5-6), 992-998 (2011).
[23] G. Kamieniarz, W. Florek, and M. Antkowiak, Universal sequence of ground states validating the classification of frustration in
antiferromagnetic rings with a single bond defect, Phys. Rev. B 92, 140411(R) (2015).
[24] G. Kamieniarz, P. Kozłowski, G. Musiał, W. Florek, M. Antkowiak, M. Haglauer, A.C. D’Auria, and F. Esposito, Phenomenological modeling of molecular-based rings beyond the strong exchange limit: Bond alternation and single-ion anisotropy effects, Inorg. Chin. Acta 361, 3690-3696 (2008).
[25] P. Kozłowski, G. Kamieniarz, M. Antkowiak, F. Tuna, G.A. Timco, and R.E.P. Winpenny, Phenomenological modeling of the
anisotropic molecular-based ring Cr7Cd, Polyhedron 28, 1852-1855 (2009).
[26] R. Matysiak, G. Kamieniarz, P. Gegenwart, and A. Ochiai, Field-dependent specific heat of Yb4As3: Agreement between a spin-1/2 model and experiment, Phys. Rev. B 79, 224413 (2009).
[27] R. Matysiak, P. Gegenwart, A. Ochiai, M. Antkowiak, G. Kamieniarz, and F. Steglich, Specific heat of segmented Heisenberg quantum spin chains in (Yb1−xLux)4As3, Phys. Rev. B 88, 224414 (2013).
