Related papers: Quantitative K-Theory Related to Spin Chern Number…
The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the…
This article reviews the foundations of the theory of the Bott index of a pair of unitary matrices in the context of condensed matter theory, as developed by Hastings and Loring in J. Math. Phys. (51), 015214, (2010) and Annals of Physics…
A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to exactly commuting matrices of the same kind. It has long been known that there is a winding number obstruction for approximately…
To easily calculate statistical properties of pairs correlated through Schmidt decomposition, as commonly used in Quantum Information, we propose a "commutator formalism" for these single-index pairs, somewhat simpler than the one we…
In this paper we show that the `quantization commutes with reduction' principle of Guillemin-Sternberg holds for the coadjoint orbits that parametrize the discrete series of a real connected semi-simple Lie group.
We detect the topological properties of Chern insulators with strong Coulomb interactions by use of cluster perturbation theory and variational cluster approach. The common scheme in previous studies only involves the calculation of the…
Identification of a non-trivial $\mathbb{Z}_{2}$ index in a spinful two dimensional insulator indicates the presence of an odd, quantized (pseudo)spin-resolved Chern number, $C_{s}=(C_{\uparrow}-C_{\downarrow})/2$. However, the statement is…
We define commutants mod normed ideals associated with compact smooth manifolds with boundary. The results about the K-theory of these operator algebras include an exact sequence for the connected sum of manifolds, derived from the…
Suppose two Hermitian matrices $A,B$ almost commute ($\Vert [A,B] \Vert \leq \delta$). Are they close to a commuting pair of Hermitian matrices, $A',B'$, with $\Vert A-A' \Vert,\Vert B-B'\Vert \leq \epsilon$? A theorem of H. Lin shows that…
Quantum walks have attracted attention as a promising platform realizing topological phenomena and many physicists have introduced various types of indices to characterize topologically protected bound states that are robust against…
We consider quantum mechanical systems of spin chain type, with finite-dimensional Hilbert spaces and $\mathcal{N}=2$ or $\mathcal{N}=4$ supersymmetry, described in $\mathcal{N}=2$ superspace in terms of nonlinear chiral multiplets. We…
We define and study the properties of observables associated to any link in $\Sigma\times {\bf R}$ (where $\Sigma$ is a compact surface) using the combinatorial quantization of hamiltonian Chern-Simons theory. These observables are traces…
We extend to the case of a threshold ideal our result with J. Bourgain about the essential centre of the commutant mod a diagonalization ideal for a n-tuple of commuting Hermitian operators . We also compute the $K_0$-group of the commutant…
Employing the currently discussed notion of pseudo-Hermiticity, we define a pseudo-unitary group. Further, we develop a random matrix theory which is invariant under such a group and call this ensemble of pseudo-Hermitian random matrices as…
Whether an almost-commuting pair of operators must be close to a commuting pair is a central question in operator and matrix theory. We investigate this problem for pairs of $C^*$-subalgebras $\mathcal{A}$ and $\mathcal{B}$ of…
Quantum Hamiltonians containing nonseparable products of non-commuting operators, such as $\hat{\bf x}^m \hat{\bf p}^n$, are problematic for numerical studies using split-operator techniques since such products cannot be represented as a…
Real index pairings of projections and unitaries on a separable Hilbert space with a real structure are defined when the projections and unitaries fulfill symmetry relations invoking the real structure, namely projections can be real,…
We propose an alternative formulation of the $Z_2$ topological index for quantum spin Hall systems and band insulators when time reversal invariance is not broken. The index is expressed in terms of the Chern numbers of the bands of the…
This work considers the asymptotic behavior of the distance between two sample covariance matrices (SCM). A general result is provided for a class of functionals that can be expressed as sums of traces of functions that are separately…
Building from work by Cedzich et al. and Suzuki et al., we consider topological and index-theoretic properties of chiral unitaries, which are an abstraction of the time evolution of a chiral-symmetric self-adjoint operator. Split-step…