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arXiv:0812.4126 · Physical Review B

Numerically exact and approximate determination of energy eigenvalues for antiferromagnetic molecules using irreducible tensor operators and general point-group symmetries

Roman Schnalle, Jürgen Schnack

Published Mar 19, 2009• 56 citations
arXivHTMLPDF

Abstract

For small-enough quantum systems numerical exact and complete diagonalization of the Hamiltonian enables one to evaluate and discuss all static, dynamic, and thermodynamic properties. In this article we consider Heisenberg spin systems and extend the range of applicability of the exact diagonalization method by showing how the irreducible tensor operator technique can be combined with an unrestricted use of general point-group symmetries. We also present ideas on how to use spin-rotational and point-group symmetries in order to obtain approximate spectra.

DOI
10.1103/physrevb.79.104419

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For small-enough quantum systems numerical exact and complete diagonalization of the Hamiltonian enables one to evaluate and discuss all static, dynamic, and thermodynamic properties. In this article we consider Heisenberg spin systems and extend the range of applicability of the exact diagonalization method by showing how the irreducible tensor operator technique can be combined with an unrestricted use of general point-group symmetries. We also present ideas on how to use spin-rotational and point-group symmetries in order to obtain approximate spectra.
Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry CN{C}_{N}CN​ in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry SU(2)SU(2)SU(2) can be used, where the Hamiltonian matrix is block structured according to the total spin and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both CN{C}_{N}CN​ and SU(2)SU(2)SU(2), mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry-reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.
A general treatment of the symmetry for quantum spin systems is presented with the use of permutational operators. Symmetry operations which are commutable with a Hamiltonian form the symmetry group. Complete sets of spin functions which are specified by the total spin ( S tot ) are projected onto the irreducible representations of the symmetry group, using the character table. We show from symmetry consideration that a minimum dimension exists for the ground state of the Heisenberg Hamiltonian with finite N spins. The diagonalization of the Hamiltonian with the use of symmetry consideration can be effectively performed for the complete sets of S tot =0 by the method proposed in this paper. Numerical results of the ground state and all the excited states of S tot =0 are presented for spin systems up to N =20.
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 coherenceQuantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalizationDiagonalization 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 spaceHilbert 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 groupsPoint group—and their application. However, the applicability of exact diagonalizationDiagonalization is limited by the exponential growth of the Hilbert spaceHilbert 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 matricesHermitian 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 sis_{i}si​ 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 methodFinite-Temperature Lanczos Method (FTLM), 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.
One of the long standing problems in quantum chemistry had been the inability to exploit full spatial and spin symmetry of an electronic Hamiltonian belonging to a non-Abelian point group. Here, we present a general technique which can utilize all the symmetries of an electronic (magnetic) Hamiltonian to obtain its full eigenvalue spectrum. This is a hybrid method based on Valence Bond basis and the basis of constant z-component of the total spin. This technique is applicable to systems with any point group symmetry and is easy to implement on a computer. We illustrate the power of the method by applying it to a model icosahedral half-filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and in the largest non-Abelian point group. The C60 molecule has this symmetry and hence our calculation throw light on the higher energy excited states of the bucky ball. This method can also be utilized to study finite temperature properties of strongly correlated systems within an exact diagonalization approach. (C) 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012
Abstract One of the long standing problems in quantum chemistry had been the inability to exploit full spatial and spin symmetry of an electronic Hamiltonian belonging to a non‐Abelian point group. Here, we present a general technique which can utilize all the symmetries of an electronic (magnetic) Hamiltonian to obtain its full eigenvalue spectrum. This is a hybrid method based on Valence Bond basis and the basis of constant z‐component of the total spin. This technique is applicable to systems with any point group symmetry and is easy to implement on a computer. We illustrate the power of the method by applying it to a model icosahedral half‐filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and in the largest non‐Abelian point group. The C60 molecule has this symmetry and hence our calculation throw light on the higher energy excited states of the bucky ball. This method can also be utilized to study finite temperature properties of strongly correlated systems within an exact diagonalization approach. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012
While either a spin or point-group adaptation is straightforward when considered independently, the standard technique for factoring isotropic spin Hamiltonians by the total spin S and the irreducible representation Γ of the point group is limited by the complexity of the transformations between different coupling schemes that are related in terms of their site permutations. To overcome these challenges, we apply projection operators directly to uncoupled basis states, enabling the simultaneous treatment of spin and point-group symmetry without the need for recoupling transformations. This provides a simple and efficient approach for the exact diagonalization of isotropic spin models, which we illustrate, with applications in Heisenberg spin rings and polyhedra, including systems that are computationally inaccessible with conventional coupling techniques.