LEADER 05801nam 22006735 450 001 9910254581003321 005 20220329235823.0 010 $a3-319-55023-3 024 7 $a10.1007/978-3-319-55023-7 035 $a(CKB)3710000001109676 035 $a(DE-He213)978-3-319-55023-7 035 $a(MiAaPQ)EBC4825726 035 $a(PPN)199765200 035 $a(EXLCZ)993710000001109676 100 $a20170317d2017 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 12$aA computational non-commutative geometry program for disordered topological insulators$b[electronic resource] /$fby Emil Prodan 205 $a1st ed. 2017. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2017. 215 $a1 online resource (X, 118 p. 19 illus. in color.) 225 1 $aSpringerBriefs in Mathematical Physics,$x2197-1757 ;$v23 311 $a3-319-55022-5 320 $aIncludes bibliographical references at the end of each chapters. 327 $aDisordered Topological Insulators: A Brief Introduction -- Homogeneous Materials -- Homogeneous Disordered Crystals -- Classification of Homogenous Disordered Crystals -- Electron Dynamics: Concrete Physical Models -- Notations and Conventions -- Physical Models -- Disorder Regimes -- Topological Invariants -- The Non-Commutative Brillouin Torus -- Disorder Configurations and Associated Dynamical Systems -- The Algebra of Covariant Physical Observables -- Fourier Calculus -- Differential Calculus -- Smooth Sub-Algebra -- Sobolev Spaces -- Magnetic Derivations -- Physics Formulas -- The Auxiliary C*-Algebras -- Periodic Disorder Configurations -- The Periodic Approximating Algebra -- Finite-Volume Disorder Configurations -- The Finite-Volume Approximating Algebra -- Approximate Differential Calculus -- Bloch Algebras -- Canonical Finite-Volume Algorithm -- General Picture -- Explicit Computer Implementation -- Error Bounds for Smooth Correlations -- Assumptions -- First Round of Approximations -- Second Round of Approximations -- Overall Error Bounds -- Applications: Transport Coefficients at Finite Temperature -- The Non-Commutative Kubo Formula -- The Integer Quantum Hall Effect -- Chern Insulators -- Error Bounds for Non-Smooth Correlations -- The Aizenman-Molchanov Bound -- Assumptions -- Derivation of Error Bounds -- Applications II: Topological Invariants -- Class AIII in d = 1 -- Class A in d = 2 -- Class AIII in d = 3 -- References. 330 $aThis work presents a computational program based on the principles of non-commutative geometry and showcases several applications to topological insulators. Noncommutative geometry has been originally proposed by Jean Bellissard as a theoretical framework for the investigation of homogeneous condensed matter systems. Recently, this approach has been successfully applied to topological insulators, where it facilitated many rigorous results concerning the stability of the topological invariants against disorder. In the first part of the book the notion of a homogeneous material is introduced and the class of disordered crystals defined together with the classification table, which conjectures all topological phases from this class. The manuscript continues with a discussion of electrons? dynamics in disordered crystals and the theory of topological invariants in the presence of strong disorder is briefly reviewed. It is shown how all this can be captured in the language of noncommutative geometry using the concept of non-commutative Brillouin torus, and a list of known formulas for various physical response functions is presented. In the second part, auxiliary algebras are introduced and a canonical finite-volume approximation of the non-commutative Brillouin torus is developed. Explicit numerical algorithms for computing generic correlation functions are discussed. In the third part upper bounds on the numerical errors are derived and it is proved that the canonical-finite volume approximation converges extremely fast to the thermodynamic limit. Convergence tests and various applications concludes the presentation. The book is intended for graduate students and researchers in numerical and mathematical physics. 410 0$aSpringerBriefs in Mathematical Physics,$x2197-1757 ;$v23 606 $aPhysics 606 $aMathematical physics 606 $aCondensed matter 606 $aK-theory 606 $aFunctional analysis 606 $aMathematical Methods in Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/P19013 606 $aMathematical Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/M35000 606 $aCondensed Matter Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/P25005 606 $aK-Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M11086 606 $aFunctional Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12066 615 0$aPhysics. 615 0$aMathematical physics. 615 0$aCondensed matter. 615 0$aK-theory. 615 0$aFunctional analysis. 615 14$aMathematical Methods in Physics. 615 24$aMathematical Physics. 615 24$aCondensed Matter Physics. 615 24$aK-Theory. 615 24$aFunctional Analysis. 676 $a512.4 700 $aProdan$b Emil$4aut$4http://id.loc.gov/vocabulary/relators/aut$0803691 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910254581003321 996 $aA Computational Non-commutative Geometry Program for Disordered Topological Insulators$92047115 997 $aUNINA