PDF Publication Title:
Text from PDF Page: 048
R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann Then the U(1) symmetry Q and conjugation by the time reversal symmetry T have matrix representations on the cˆj Q=iσ0 ⊗σy, T =−iσy ⊗σz (5.8) where σ0 = 1 0 and σi are the Pauli matrices. Q2 =T2 =−1, {Q,T}=0. A Dirac Hamiltonian H = i Ajkcˆjcˆk represented by a real skew-symmetric 2N×2N matrix A has U(1) symmetry if and only if [Q,A] = 0. Similarly, H being time reversal invariant is equivalent to {T,A} = 0. To simplify things, one can rescale A, e.g., by regarding it as a skew-symmetric quadratic form or iA as a Hermitian form, to have Eigenvalues i, −i. Denote the result by A ̃, then A ̃2 = −1 Finding a compatible matrix A reduces to the problem of finding an extension of a Clifford algebra with n generators by one more generator A ̃. Here n = 0 for nosymmetry,n=1forT onlyandn=2forT andQ.Forexampleforn=2: e1 =T ande2 =QT andwewanttoaddonemoregeneratore3 =A ̃fortheDirac Hamiltonian with both time-reversal symmetry and U(1) symmetry. In general, the Clifford extension problem is given by considering the extensions i : Cl0,n(R) → Cl0,n+1(R) Here Clp,q is the real Clifford algebra for the quadratic form with signature (+,...,+,−,...,−). pq Bott periodicity in Clifford algebra manifests itself as Morita equivalence, Cln+2(C) ≃ M2(Cln(C)) ∼M Cln(C), Clp,q+8(R) ≃ Clp+8,q(R) ≃ M16(Clp,q(R)) ∼M Clp,q(R) where Mn(R) is the n × n matrix of some commutative ring R and ∼M stands for the Morita equivalence in ring theory. Recall that two unital rings are Morita equivalent if they have equivalent categories of left modules. In order to classify the Dirac Hamiltonians with or without the above mentioned Z2 symmetries, we actually deal with the representations of the above Clifford algebras, i.e. Clifford modules, since the ei are represented by matrices. Denote by M(Cl0,n(R)) the free abelian group generated by irreducible Z2-graded Cl0,n(R)- modules. There exists a restriction map induced by i, i∗ : M(Cl0,n+1(R)) → M(Cl0,n(R)). Each Clifford extension i defines the cokernel of i∗, An := coker(i∗) = M(Cl0,n(R))/i∗M(Cl0,n+1(R)). (5.9) For the complex case, we define Mc(Cl0,n(R)) as the Clifford modules of the com- plexification Cln(C) = Cl0,n(R) ⊗R C, Acn can be obtained similarly. It is An (or 1630003-48 01 4 Rev. Math. Phys. 2016.28. Downloaded from www.worldscientific.com by PURDUE UNIVERSITY on 08/11/17. For personal use only.PDF Image | Notes on topological insulators
PDF Search Title:
Notes on topological insulatorsOriginal File Name Searched:
RMP.pdfDIY PDF Search: Google It | Yahoo | Bing
Sulfur Deposition on Carbon Nanofibers using Supercritical CO2 Sulfur Deposition on Carbon Nanofibers using Supercritical CO2. Gamma sulfur also known as mother of pearl sulfur and nacreous sulfur... More Info
CO2 Organic Rankine Cycle Experimenter Platform The supercritical CO2 phase change system is both a heat pump and organic rankine cycle which can be used for those purposes and as a supercritical extractor for advanced subcritical and supercritical extraction technology. Uses include producing nanoparticles, precious metal CO2 extraction, lithium battery recycling, and other applications... More Info
CONTACT TEL: 608-238-6001 Email: greg@infinityturbine.com (Standard Web Page)