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Exact Diagonalization Techniques for Quantum Spin Systems

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Computational Modelling of Molecular Nanomagnets

Part of the book series: Challenges and Advances in Computational Chemistry and Physics ((COCH,volume 34))

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Abstract

The essence of quantum magnetism lies in correlated many-body states. They determine the physical properties beyond paramagnetism. Even constructions as simple as singlets or triplets cannot be understood without the mathematical concept of superpositions. This is even more evident if time evolution of quantum states and phenomena such as quantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalization of the many-body Hamiltonian describing the physical system. Symmetries are of great help when tackling the numerical task of matrix diagonalization since they allow to decompose a problem of large Hilbert space dimension into a number of smaller problems. This approach will be presented first. It rests on the identification of the available symmetries—e.g., U(1), SU(2), as well as point groups—and their application. However, the applicability of exact diagonalization is limited by the exponential growth of the Hilbert space of the spin system with the number of spins: \(\dim \left( {\mathcal{H}} \right) = \mathop \prod \nolimits_{i} \left( {2 s_{i} + 1} \right)\). Nowadays, complex Hermitian matrices of a linear dimension of about 100,000 can be completely diagonalized on supercomputers which limits the number of spins depending on their size \(s_{i}\) to just a few. I will therefore present a very accurate approximation to exact diagonalization that rests on Lanczos procedures and the concept of typicality. The most popular version, the finite-temperature Lanczos method, allows to evaluate low-energy spectroscopic data, thermal averages as well as time evolutions for systems with Hilbert space dimensions of up to about 1011. Hands-on experience and examples will be provided.

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Acknowledgements

I thank the Deutsche Forschungsgemeinschaft DFG for funding (355031190 (FOR 2692); 355031190 (SCHN 615/25-2); 449703145 (SCHN 615/28-1)) and the Leibniz Supercomputing Center in Garching/Germany for supercomputing resources. I would also like to thank all past and present members of my group for a great collaboration and for all the results we achieved. Proofreading by Kilian Irländer and Dennis Westerbeck is gratefully acknowledged.

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  1. Faculty of Physics, Bielefeld University, 33601, Bielefeld, Germany

    Jürgen Schnack

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  1. of Chemistry, Indian Institute of Technology Bombay, Mumbai, India

    Gopalan Rajaraman

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Schnack, J. (2023). Exact Diagonalization Techniques for Quantum Spin Systems. In: Rajaraman, G. (eds) Computational Modelling of Molecular Nanomagnets. Challenges and Advances in Computational Chemistry and Physics, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-031-31038-6_4

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