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Abstract. -The theory of almost commuting matrices can be used to quantify topological obstructions to the existence of localized Wannier functions with time-reversal symmetry in systems with time-reversal symmetry and strong spin-orbit coupling. We present a numerical procedure that calculates a Z2 invariant using these techniques, and apply it to a model of HgTe. This numerical procedure allows us to access sizes significantly larger than procedures based on studying twisted boundary conditions. Our numerical results indicate the existence of a metallic phase in the presence of scattering between up and down spin components, while there is a sharp transition when the system decouples into two copies of the quantum Hall effect. In addition to the Z2 invariant calculation in the case when up and down components are coupled, we also present a simple method of evaluating the integer invariant in the quantum Hall case where they are decoupled.The study of topological insulators is one of the most active areas of physics today. Experimental and theoretical work has shown physical realizations of time-reversal invariant insulators with strong spin-orbit coupling in both two [1] and three dimensions [2] and a complete classification of different insulating phases has been recently obtained using methods of Anderson localization [3] and, more generally, K-theoretic techniques [4].However, numerically it is difficult to determine the Z 2 invariants that are signatures of topological insulating phases. For systems with translational invariance, one can study the bundle over the momentum torus [5], while for systems without translation invariance, Essin and Moore [6] were able to study the phase diagram of a graphene model by studying the model over a flux torus corresponding to twisted boundary conditions. Unfortunately, the flux torus approach is very computationally intensive: for each disorder realization, the Hamiltonian must be diagonalized once for each point on a discrete grid on the flux torus, and then the connection on the torus must be computed. This limited the study to small systems, with at most 64 sites.In this paper, we present a different approach to calculating a Z 2 invariant, based on ideas in C * -algebras, in particular the K-theory of almost commuting matrices. We present a fast numerical algorithm based on these ideas. Computing the invariant requires a single diagonalization of the Hamiltonian, matrix function calculations on matrices at most half the size of the Hamiltonian, and finally the calculation of the Pfaffian of a real anti-symmetric matrix that is at most the size of the Hamiltonian. The most costly step is a single diagonalization, allowing us to study significantly larger samples, up to 1600 sites.Our invariant serves the same fundamental purpose as the invariant used in [6]-proving that for certain lowenergy bands it is impossible to find well-localized Wannier functions with time-reversal symmetry. These invariants are most likely equivalent, but that is another topic [7].We apply ...

We apply ideas from C * -algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems.We use this approach to calculate the index for time-reversal invariant systems with spinorbit scattering in three dimensions, on sizes up to 12 3 , averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an "order parameter" for the topological insulator) begins to fluctuate from sample too sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the C * -algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.

Let K(A) denote the sum of all the K-theory groups of a C*-algebra A in all degrees and with all cyclic coefficient groups. The Bockstein operations (which generate a category Λ) act on K(A). We establish a universal coefficient exact sequence 0 → Pext(K * (A), K * (B)) δ − → KK(A, B) Γ − → Hom Λ (K(A), K(B)) → 0. that holds in the same generality as the universal coefficient theorem of Rosenberg and Schochet. There are advantages, in some circumstances, to using Hom Λ (K(A), K(B)) in place of KK(A, B). These advantages derive from the fact that K(A) can be equipped with order and scale structures similar to those on K 0 (A). With this additional structure, the "Λ − module" K(A) becomes a powerful invariant of C*algebras. We show that it is a complete invariant for the class of real-rank-zero AD algebras. The AD algebras are a certain kind of approximately subhomogeneous C *-algebras which may have torsion in K 1 [Ell]. In addition to classifying these algebras, we calculate their automorphism groups up to approximately innerautomorphisms.

On the oscillator realization of conformal U(2, 2) quantum particles and their particle-hole coherent states On a classification of irreducible almost commutative geometries, a second helping For models of noninteracting fermions moving within sites arranged on a surface in three-dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are K-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward and does not require translational invariance or introduce a flux torus. We further show that there is a Z 2 index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as time-reversal or particle-hole conjugation. Finally, in the case of the sphere-mathematically speaking, three almost commuting Hermitians whose sum of square is near the identity-we give the first quantitative result, showing that this index is the only obstruction to finding commuting approximations. We review the known nonquantitative results for the torus.Given a list of bounded operators on infinite dimensional Hilbert space, it is often natural to seek a finite-rank projection P that almost commutes with that set. The C ‫ء‬ -algebraist would do so in the study of quasidiagonality. 1-3 In physics, we are interested in a projection onto a band of energy states separated from the rest of the spectrum by an energy gap; assuming the underlying Hamiltonian is local, this projection will itself be local due to the gap, and hence will approximately commute with a list of observables.Whatever exact relations might be known to hold for the original operators ͑H 1 , ... ,H r ͒ will generally hold only approximately for the compressions ͑PX 1 P , ... , PX r P͒. In a lattice model, the projection might be from a finite dimensional space to a space whose dimension is much lower, but still the outcome is finite-rank operators that approximately satisfy some relations. Can these be approximated by finite-rank operators that exactly satisfy those relations?For example, if X 1 and X 2 in B͑H͒ satisfy −I Յ X j Յ I and ͓X 1 , X 2 ͔ = 0, then P almost commuting with the X j implies − I Յ PX j P Յ I, a͒ Electronic We especially want to know if these almost commuting Hermitian operators are close to commuting Hermitian operators in the corner PB͑H͒P Х M k ͑C͒. It is sufficient to answer this question: can two almost commuting Hermitian matrices be approximated by commuting Hermitian matrices? The answer is yes. This is Lin's theorem. 4 The si...

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Our main theorem is a characterization of C*-algebras that have a separating family of finite-dimensional representations. This characterization makes possible a solution to a problem posed by Goodearl and Menaul. Specifically, we prove that the free product of such C*-algebras again has this property.

Weak topological phases are usually described in terms of protection by the lattice translation symmetry. Their characterization explicitly relies on periodicity since weak invariants are expressed in terms of the momentum-space torus. We prove the compatibility of weak topological superconductors with aperiodic systems, such as quasicrystals. We go beyond usual descriptions of weak topological phases and introduce a novel, real-space formulation of the weak invariant, based on the Clifford pseudospectrum. A non-trivial value of this index implies a non-trivial bulk phase, which is robust against disorder and hosts localized zero-energy modes at the edge. Our recipe for determining the weak invariant is directly applicable to any finite-sized system, including disordered lattice models. This direct method enables a quantitative analysis of the level of disorder the topological protection can withstand.Introduction -One of the hallmarks of a topological phase is the presence of quantized macroscopic observables which are insensitive to perturbations or random modulations of the local microscopic environment [1, 2]. This remarkable feature can be understood by expressing the quantized observables as topological invariants of an underlying microscopic theory of the bulk system, such that they are unchanged by arbitrary local deformations which do not close the bulk gap. The most well-known example is the transverse conductivity of the quantum Hall effect [3]: owing to the topological protection [4], it shows a level of quantization which is used to define the metrological standard of resistance [5].