Titlebook: Quantum Spin Systems on Infinite Lattices; A Concise Introducti Pieter Naaijkens Book 2017 Springer International Publishing AG 2017 Algebr

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Introduction,in physics, also many fields in mathematics have their origin in or progressed significantly alongside the development of quantum mechanics, and vice versa (Wigner, Commun Pure Appl Math 13(1):1–14, 1960). Of particular interest also from a mathematical point of view are quantum systems with infinit

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Operator Algebras,matrix .., which measures the spin in the . direction, and the position and momentum operators . and . of a single particle on the line. The first acts on the Hilbert space ., while the latter are defined on dense subspaces of .. An . is an algebra of such operators, usually with additional conditio

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Infinite Systems,any sites. We will first consider an explicit example, showing that inequivalent representations appear in a natural way. Then we discuss the appropriate mathematical framework to discuss spin systems with infinitely many sites. After that we discuss how to describe dynamics in this setting, and how

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Lieb-Robinson Bounds,m one has .: information cannot travel faster than the speed of light. In lattice systems there is no such natural bound. Correlations can, in principle, spread arbitrarily fast. Nevertheless, under suitable conditions, most notably on the locality of the dynamics, there . a maximum velocity in the

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Local Quantum Physics,i space-time. It turns out that many of the same techniques can be used in this setting. This resemblance is particularly clear in what is called . (AQFT). This is an attempt to formulate quantum field theory in a mathematically rigorous way, using ..-algebraic techniques. One of the first formulati

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Introduction,ely many degrees of freedom. This includes quantum field theory, but one can also take infinitely many copies of a . system, for example a spin-1/2 system. This is the type of systems that we will study in these lecture notes. That is, we will almost exclusively defined on a discrete space (a lattice), rather than with a continuum theory.

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0075-8450 udy guide.Course-based primer which allows for entering the .This course-based primer offers readers a concise introduction to the description of quantum mechanical systems with infinitely many degrees of freedom – and quantum spin systems in particular – using the operator algebraic approach. Here,

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Infinite Systems,ate mathematical framework to discuss spin systems with infinitely many sites. After that we discuss how to describe dynamics in this setting, and how to define equilibrium and ground states. At the end we consider another example: Kitaev’s toric code in the thermodynamic limit.