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      Topological Boundary Modes in Isostatic Lattices

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          Abstract

          Frames, or lattices consisting of mass points connected by rigid bonds or central force springs, are important model constructs that have applications in such diverse fields as structural engineering, architecture, and materials science. The difference between the number of bonds and the number of degrees of freedom of these lattices determines the number of their zero-frequency "floppy modes". When these are balanced, the system is on the verge of mechanical instability and is termed isostatic. It has recently been shown that certain extended isostatic lattices exhibit floppy modes localized at their boundary. These boundary modes are insensitive to local perturbations, and appear to have a topological origin, reminiscent of the protected electronic boundary modes that occur in the quantum Hall effect and in topological insulators. In this paper we establish the connection between the topological mechanical modes and the topological band theory of electronic systems, and we predict the existence of new topological bulk mechanical phases with distinct boundary modes. We introduce model systems in one and two dimensions that exemplify this phenomenon.

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          Most cited references 10

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          Topological insulators and superconductors

          Topological insulators are new states of quantum matter which can not be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. These topological materials have been theoretically predicted and experimentally observed in a variety of systems, including HgTe quantum wells, BiSb alloys, and Bi\(_2\)Te\(_3\) and Bi\(_2\)Se\(_3\) crystals. We review theoretical models, materials properties and experimental results on two-dimensional and three-dimensional topological insulators, and discuss both the topological band theory and the topological field theory. Topological superconductors have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions. We review the theory of topological superconductors in close analogy to the theory of topological insulators.
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            Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells

            We show that the Quantum Spin Hall Effect, a state of matter with topological properties distinct from conventional insulators, can be realized in HgTe/CdTe semiconductor quantum wells. By varying the thickness of the quantum well, the electronic state changes from a normal to an "inverted" type at a critical thickness \(d_c\). We show that this transition is a topological quantum phase transition between a conventional insulating phase and a phase exhibiting the QSH effect with a single pair of helical edge states. We also discuss the methods for experimental detection of the QSH effect.
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              Continuous deformations in random networks

               M.F. Thorpe (1983)
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                Author and article information

                Journal
                02 August 2013
                2013-11-12
                Article
                10.1038/nphys2835
                1308.0554

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                Nature Physics 10, 39 (2014)
                8 pages, 5 figures, including supplementary information
                cond-mat.mes-hall cond-mat.soft

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