Quantum spin systems on infinite lattices

Abstract

This is an extended and corrected version of lecture notes originally written for a one semester course at Leibniz University Hannover. The main aim of the notes is to give an introduction to the mathematical methods used in describing discrete quantum systems consisting of infinitely many sites. Such systems can be used, for example, to model the materials in condensed matter physics. The notes provide the necessary background material to access recent literature in the field. Some of these recent results are also discussed. The contents are roughly as follows: (1) quick recap of essentials from functional analysis, (2) introduction to operator algebra, (3) algebraic quantum mechanics, (4) infinite systems (quasilocal algebra), (5) KMS and ground states, (6) Lieb-Robinson bounds, (7) algebraic quantum field theory, (8) superselection sectors of the toric code, (9) Haag-Ruelle scattering theory in spin systems, (10) applications to gapped phases. The level is aimed at students who have at least had some exposure to (functional) analysis and have a certain mathematical "maturity".

To be published in Lecture Notes in Physics (Springer). v2: extended and corrected. 157 pages

References in corpus (11)

• Non-Abelian Anyons and Topological Quantum Computation
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Propagation of Correlations in Quantum Lattice Systems
• On thermalization in Kitaev's 2D model
• A statistical mechanics view on Kitaev's proposal for quantum memories
• Localized endomorphisms in Kitaev's toric code on the plane
• Entanglement, Haag-duality and type properties of infinite quantum spin chains
• Haag duality for Kitaev's quantum double model for abelian groups
• The complete set of infinite volume ground states for Kitaev's abelian quantum double models
• Much Ado About Something: Why Lieb-Robinson bounds are useful

Cited by in corpus (26)

• A review of Quantum Cellular Automata
• Unified theory of local quantum many-body dynamics: Eigenoperator thermalization theorems
• Hydrodynamic projections and the emergence of linearised Euler equations in one-dimensional isolated systems
• Pauli topological subsystem codes from Abelian anyon theories
• Fusion rules from entanglement
• Flow of (higher) Berry curvature and bulk-boundary correspondence in parametrized quantum systems
• A converse to Lieb-Robinson bounds in one dimension using index theory
• Enriched string-net models and their excitations
• Quantum cellular automata for quantum error correction and density classification
• Charting the space of ground states with tensor networks
• Long-time dynamics in quantum spin lattices: ergodicity and hydrodynamic projections at all frequencies and wavelengths
• The Wasserstein distance of order 1 for quantum spin systems on infinite lattices
• Relativistic Quantum Fields Are Universal Entanglement Embezzlers
• An entropic invariant for 2D gapped quantum phases
• Generalized Phase-Space Techniques to Explore Quantum Phase Transitions in Critical Quantum Spin Systems
• Almost everywhere ergodicity in quantum lattice models
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Exponential tail estimates for quantum lattice dynamics
• Quantum cellular automata and categorical dualities of spin chains
• Pure state entanglement and von Neumann algebras
• Disentangling anomaly-free symmetries of quantum spin chains
• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• A non-semisimple non-invertible symmetry
• Perturbative criteria for the ergodicity of interacting dissipative quantum lattice systems
• Continuous Dependence on the Initial Data in the Kadison Transitivity Theorem and GNS Construction
• Uniqueness of purifications is equivalent to Haag duality