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Lecture notes in physics

Lattice Systems

Karsten Balzer, M. Bönitz

Published Dec 12, 2012• 0 citations
Publisher
DOI
10.1007/978-3-642-35082-5_5

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Properties of general lattice systems are investigated using the group properties of such systems. The duality transformation is generalized and is expressed in terms of exact short sequence between the groups associated with one lattice and the groups associated with the dual lattice. A general method is given for constructing explicitly a dual lattice for any lattice system and is illustrated with some two- and three-dimensional models. In particular it is shown that the two-dimensional triangular lattice with three-body forces is self-dual.
We study the continuous extension of discrete shift translations on one-dimensional quantum lattice systems. We explore a specific example provided by a quasi-free C ∗ {C}^{\ast } -flow on the one-dimensional fermion lattice system, comparing it with its original discrete shift-translations. Specifically, we demonstrate its explicit dynamical formula, which reveals violations of causality and locality. Furthermore, we prove that this quasi-free C ∗ {C}^{\ast } -flow on the one-dimensional fermion lattice system cannot be extended to the one-dimensional quantum spin lattice system via the Jordan-Wigner transformation.
Abstract. Quantum spin-lattice systems in low dimensions exhibit a variety of interesting zero-temperature phases, some of which show non-classical (i.e., non-magnetic) long-range orders, such as dimer or trimer valence-bond order. These symmetry-breaking systems with localized valence bonds are referred to as valence-bond lattices (VBL) in this article. A review of our systematic microscopic formalism based on a proper set of composite operators for the ground and excited states of the VBL systems is given. The onedimensional (1D) spin-1 2 frustrated model is investigated in detail. Several possible VBL systems on the 1D spin-1 chains, the 2D square and kagomé lattices are also discussed. That our microscopic theory guarantees the rotational symmetry of the VBL systems is emphasized. 1.