Structure and relevant dimension of the Heisenberg model and applications to spin rings
01 Mar 2000-Journal of Magnetism and Magnetic Materials (North-Holland)-Vol. 212, Iss: 1, pp 240-250
TL;DR: In this article, for the diagonalization of the Hamilton matrix in the Heisenberg model relevant dimensions are determined depending on the applicable symmetries, both by general formulae in closed form and by the respective numbers for a variety of special systems.
About: This article is published in Journal of Magnetism and Magnetic Materials.The article was published on 2000-03-01 and is currently open access. It has received 38 citations till now. The article focuses on the topics: Heisenberg model & Magnon.
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TL;DR: In this article, numerical approaches to diagonalize the Heisenberg Hamiltonian that employ symmetries are reviewed, in particular focusing on the spin-rotational symmetry SU(2) in combination with point-group symmetry.
Abstract: The determination of the energy spectra of small spin systems as for instance given by magnetic molecules is a demanding numerical problem. In this work we review numerical approaches to diagonalize the Heisenberg Hamiltonian that employ symmetries; in particular we focus on the spin-rotational symmetry SU(2) in combination with point-group symmetries. With these methods one is able to block-diagonalize the Hamiltonian and thus to treat spin systems of unprecedented size. Thermodynamic observables such as the magnetization are then easily evaluated. In addition it provides a spectroscopic labeling by irreducible representations that can be related to selection rules which can be helpful when interpreting transitions induced by electron paramagnetic resonance, nuclear magnetic resonance or inelastic neutron scattering. It is our aim to provide the reader with detailed knowledge on how to set up such a diagonalization scheme.
58 citations
TL;DR: In this article, the authors 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.
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.
50 citations
Book•
22 Dec 2015
TL;DR: In this paper, the voter model with homogeneous mixing is used for the non-Markovian case, where overlapping versus non-overlapping generations are compared.
Abstract: Introduction.- Background and Concepts.- Agent-based Models as Markov Chains.- The Voter Model with Homogeneous Mixing.- From Network Symmetries to Markov Projections.- Application to the Contrarian Voter Model.- Information-Theoretic Measures for the Non-Markovian Case.- Overlapping Versus Non-Overlapping Generations.- Aggretion and Emergence: A Synthesis.- Conclusion.
42 citations
TL;DR: In this paper, a general rule for the shift quantum numbers k of the relative ground states of antiferromagnetic Heisenberg spin rings is proposed. But this rule is only applicable to systems with a Haldane gap.
Abstract: We suggest a general rule for the shift quantum numbers k of the relative ground states of antiferromagnetic Heisenberg spin rings. This rule generalizes the well-known results of Marshall, Peierls, Lieb, Schultz, and Mattis for even rings. Our rule is confirmed by numerical investigations and rigorous proofs for special cases, including systems with a Haldane gap for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ Implications for the total spin quantum number S of relative ground states are discussed as well as generalizations to the XXZ model.
41 citations
TL;DR: Magnetic molecules are zero-dimensional quantum spin objects as discussed by the authors, and the general scientific knowledge obtained through their study can be found in the paper "Magnetic Molecules: General Scientific Knowledge".
Abstract: Magnetic molecules are zero-dimensional quantum spin objects. This article is devoted to the general scientific knowledge we obtained through their study. In particular the following concepts will ...
41 citations
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16,450 citations
"Structure and relevant dimension of..." refers background in this paper
... and (11), Mmay be replaced by |M| since the dimension of H(M) equals those of H(−M). [ν/µ] in the sum symbolizes the greatest integer less or equal to ν/µ. Eq. (11) is known as a result of De Moivre [18]. Proof: The proof of (10) and (11) may be accomplished by comparing any product state (5) with the completely aligned state |Ωi= |m1 = s(1),m2 = s(2),...,mN = s(N)i , (13) which is also called magnon...
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11,456 citations
"Structure and relevant dimension of..." refers background in this paper
...[ν/� ] in the sum symbolizes the greatest integer less or equal to ν/� . Eq. (11) is known as a result of De Moivre [ 18 ]....
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01 Jan 1950
TL;DR: A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
Abstract: Office hours: MWF, immediately after class or early afternoon (time TBA). We will cover the mathematical foundations of probability theory. The basic terminology and concepts of probability theory include: random experiments, sample or outcome spaces (discrete and continuous case), events and their algebra, probability measures, conditional probability A First Course in Probability (8th ed.) by S. Ross. This is a lively text that covers the basic ideas of probability theory including those needed in statistics. Theoretical concepts are introduced via interesting concrete examples. In 394 I will begin my lectures with the basics of probability theory in Chapter 2. However, your first assignment is to review Chapter 1, which treats elementary counting methods. They are used in applications in Chapter 2. I expect to cover Chapters 2-5 plus portions of 6 and 7. You are encouraged to read ahead. In lectures I will not be able to cover every topic and example in Ross, and conversely, I may cover some topics/examples in lectures that are not treated in Ross. You will be responsible for all material in my lectures, assigned reading, and homework, including supplementary handouts if any.
10,221 citations
"Structure and relevant dimension of..." refers background in this paper
...(11) is known asa resultof deMoivre[18]....
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...(11) is known as a result of De Moivre [18]....
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2,687 citations