Citation: Naaijkens, Pieter. Quantum spin systems on infinite lattices. eScholarship, University of California, 2013.
Web: https://arxiv.org/abs/1311.2717
Tags: Expository, Operator-algebras, Spin-chains, Lieb-Robinson
In this textbook, Pieter Naaijkens gives a lovely introduction to the analysis of quantum spin systems on infinite lattices - the thermodynamic limit of the finite spins systems people most typically consider. In this context, the standard approach of states on a Hilbert space fails because the tensor product of infinitely many finite dimensional systems behaves poorly. Instead, one adopts the perspective of C* algebras. The infinite lattice limit is now the direct limit of the C* algebras over the finite systems. This operator algebraic approach is necessary for thermodynamic limits, and hence is also the foundation of quantum statistical mechanics.
This book introduces the theory of operator algebras, shows how they apply to infinite lattices, and then use Lieb-Robinson bounds to show how to prove rigorous theorems in this context. This seems to be a fantastic place for people in condensed matter and quantum information to learn how C* algebras work.