Topological States of Matter: Classification of Modular Tensor Categories, Study Guides, Projects, Research of Cryptography and System Security

Download Topological States of Matter: Classification of Modular Tensor Categories and more Study Guides, Projects, Research Cryptography and System Security in PDF only on Docsity! Topological States of Matter Zhenghan Wang Indiana University Microsoft Project Q (visiting KITP/CNSI) Classification of MTC Let N0=I, N1, N2, …, Nn-1 be n× n symmetric matrices with non-negative integer entries and commute with each other pairwise. Let S=(sij)0 i,j n-1 be a real, symmetric orthogonal matrix such that SNiS=Di, where Di=(sia/s0aδa,b)0 a,b n-1 is a diagonal matrix whose entries are all algebraic integers. Fix n, find all such data Quantum Hall Effect 1980 von Klitzing ---IQHE (1985 Nobel) 1982 Stormer, Tsui, Gossard---FQHE Laughlin (1998 Nobel) quasi-particle with 1/3 electron charge and braiding statistics (abelian anyons) Fig. 1. A voltage V drives a current I in the positive x direction. Normal Ohmic resistance is V / I. A magnetic field in the positive z direction shifts positive charge carriers in the negative y direction. This generates a Hall potential ( VH) and a Hall resistance (VH/ I ) in the y direction. (Nobel Press Release 1998) Hall resistance Rxy =ν-1 h/e2, ν an integer or a fraction with precision 10-10 (ν is the Landau filling fraction) Atiyah’s Axioms of (2+1)-TQFT: (i.e. a TQFT without excitations and central charge=0 or anomaly free) Surface Σ2
C-vector space V(Σ) 3-manifold M3
a vector Z(M3)∈ V(∂M3) V(∅) ≅ C V(Σ1 ⊔ Σ2) ≅ V(Σ1) ⊗ V(Σ2) V(Σ*) ≅ V*(Σ) Z(Σ × I)=IdV(Σ) Z(M1∪ M2)=Z(M1)·Z(M2) Elementary Excitations Elementary excitations are particle-like in electron liquids, and carry topological charges---particle types. They are modeled by a ribbon tensor category. A ribbon tensor category with non-singular S-matrix is called a modular tensor category (MTC). MTCs=TQFTs (Turaev) TQFT can be extended to include surfaces with boundaries whose boundaries are labeled by particles types=simple object classes. Walker, Segal, Moore-Seiberg,… TQFT with corners Let L={a,b,c,…} be the particle types (labels), a a*, and a**=a, 0=trivial type Disk Axiom: V(D2, a)=0 if a≠ 0, C if a=0 Annulus Axiom: V(A,a,b)=0 if a≠ b*, C if a=b* a a b Simulation of TQFTs by QCM Given a TQFT, set X=⊕a,b,cVa,b,c≅ Cp, By the gluing axiom, we have: V(Σ)→ (Cp)⊗ m V(Σ)→ (Cp)⊗ m Studying pants decompositions by Hatcher-Thurston theory leads to an efficient simulation. f: Σ→ Σ ρ(f) UL (?) Model Hamiltonian Given a TQFT, and a triangulation of a surface Σ, can V(Σ) be constructed as the ground states of a local Hamiltonian on (C2)⊗ n, where n is the number of edges (or vertices or faces)? Yes if the TQFT is a Drinfeld double. Should be Yes for all ??? Fault tolerance of TQFTs A pair (V, (C2)⊗ n) is a (k, n)-code if for every k-local operator Ok, the following composition is a scalar multiple of idV: V→ (C2)⊗ n (C2)⊗ n→ V Ok Axioms for Ribbon Category Underlying categories of C will always be isomorphic to a category of a finite-set indexed vector spaces: Objects: 2-vectors (V0,V1,…,Vm) Morphisms: f.d. vector spaces ⊕i Hom(Vi, Wi) An object X is simple if Mor(X,X)=C Tensor Category Given a C, a tensor product on C is a bi- functor ⊗:C×C→ C such that 1. For any U,V,W, there exists a natural isomorphism αU,V,W: (U⊗V)⊗W→ U⊗(V⊗W) 2. There exists a unit, denoted by 1, satisfying the pentagon and the triangle equations. Rigidity Given X in C, X* is a right dual of X if there exist bX: 1→ X⊗ X* and dX: X*⊗ X→ 1 such that = = Ribbon Category A ribbon category is a rigid, braided tensor category with twists. We have X**=X, a trace on Mor(X,X): f∈ Mor(X,X), Tr(f)∈ Mor(1,1)=C fTr(f)= Link Invariant Each ribbon category defines invariant for links, tangles, and they are quantum amplitudes for certain physical processes. Modularity Let {Xi} be representatives of simple types di= Define D2=∑i di2 Then sij=1/D S=(sij) is the S-matrix and is symmetric, if it is non-singular then it is unitary. Xi Xi Xj Ocneanu Rigidity Fix the fusion rule of a rigid tensor category (fusion category), then there are only finitely many isomorphism classes of such categories. So the finiteness conjecture is reduced to the finiteness of fusion rules when the number of simple types is fixed. Verlinder Formulae Let Ni=(Nijk), then Ni can be simultaneously diagonalized by the S-matrix. Furthermore, the eigenvalues of Ni are given by sia/s0a, i.e. if Di=(sia/s0aδab) Then Ni=SDi S Write out the (jk)-entry of Ni, Verlinder formulas: Nijk=∑l=0n-1 silsjlskl/s0l. Hence finiteness conjecture is reduced to the finiteness of S-matrices. Rep of SL(2, Z) “Theorem“ The image of a rep of SL(2, Z) from a MTC is always a finite group, and the kernel is a always congruence subgroup of SL(2, Z). Fix the number of simple types, there are only finitely many reps of SL(2, Z) from MTCs. Low Rank Classification: Fusion Rules Consider self-dual (a=a*), indecomposable (not C⊗ C) modular tensor categories. If C is generated by a single object X, the principle graph is a connected graph G such that: 1. Each vertex is a simple type 2. For each vertex Y, X⊗ Y=⊕ Z, where Z is connected to Y by a single edge. Ranks=1,2,3,4 Rowell, Stong, W. Rank=1: Rank=2: 1 x 1 x Rank=3: Rank=4 1 x y 1 x y 1 x y z Z2 SO(3)2 SU(2)2 SO(3)5 SU(2)3 I consider it quite possible that physics cannot be based on the field concept, that is, on continuous structures. Then nothing remains of my entire castle in the air, including the theory of gravitation, but also nothing of the rest of modern physics Einstein, 1954 MATH Physics CS