arXiv subjects

Terry A. Loring

Publications and source records attributed to Terry A. Loring.

At least 37 records · Page 2Linked to original sources

arXiv2022

Local invariants identify topology in metals and gapless systems

Although topological band theory has been used to discover and classify a wide array of novel topological phases in insulating and semi-metal systems, it is not well-suited to identifying topological phenomena in metallic or gapless systems. Here, we develop a theory of topological metals based on the system's spectral localizer and associated Clifford pseudospectrum, which can both determine whether a system exhibits boundary-localized states despite the presence of degenerate bulk bands and provide a measure of these states' topological protection even in the absence of a bulk band gap. We demonstrate the generality of this method across symmetry classes in two lattice systems, a Chern metal and a higher-order topological metal, and prove the topology of these systems is robust to relatively strong perturbations. The ability to define invariants for metallic and gapless systems allows for the possibility of finding topological phenomena in a broad range of natural, photonic, and other artificial materials that could not be previously explored.

cond-mat.mes-hall↗

arXiv2022

Quadratic pseudospectrum for identifying localized states

We examine the utility of the quadratic pseudospectrum in photonics and condensed matter. Specifically, the quadratic pseudospectrum represents a method for approaching systems with incompatible observables, as it both minimizes the "eigen-error" in the joint approximate spectrum of the incompatible observables and does not increase the system's computational complexity. Moreover, we derive an important estimate relating the Clifford and quadratic pseudospectra. Finally, we prove that the quadratic pseudospectrum is local, and derive the bounds on the errors that are incurred by truncating the system in the vicinity of where the pseudospectrum is being calculated.

quant-ph↗

arXiv2021

Locality of the windowed local density of states

We introduce a generalization of local density of states which is "windowed" with respect to position and energy, called the windowed local density of states (wLDOS). This definition generalizes the usual LDOS in the sense that the usual LDOS is recovered in the limit where the position window captures individual sites and the energy window is a delta distribution. We prove that the wLDOS is local in the sense that it can be computed up to arbitrarily small error using spatial truncations of the system Hamiltonian. Using this result we prove that the wLDOS is well-defined and computable for infinite systems satisfying some natural assumptions. We finally present numerical computations of the wLDOS at the edge and in the bulk of a "Fibonacci SSH model", a one-dimensional non-periodic model with topological edge states.

math-ph↗

arXiv2020

Wave-packet propagation in a finite topological insulator and the spectral localizer index

We consider a model of electrons in a finite topological insulator. We numerically study the propagation of electronic wave-packets localized near edges of the structure in the presence of defects and random disorder. We compare the propagation with computations of the \emph{spectral localizer index}: a spatially local topological index. We find that without disorder, wave-packets propagate along boundaries between regions of differing spectral localizer index with minimal loss, even in the presence of strong defects. With disorder, wave-packets still propagate along boundaries between regions of differing localizer index, but lose significant mass as they propagate. We also find that with disorder, the \emph{localizer gap}, a measure of the localizer index "strength", is generally smaller away from the boundary than without disorder. Based on this result, we conjecture that wave-packets propagating along boundaries between regions of differing spectral localizer index do not lose significant mass whenever the localizer gap is sufficiently large on both sides of the boundary.

cond-mat.mes-hall↗

arXiv2019

Surfaces and hypersurfaces as the joint spectrum of matrices

The Clifford spectrum is an elegant way to define the joint spectrum of several Hermitian operators. While it has been know that for examples as small as three $2$-by-$2$ matrices the Clifford spectrum can be a two-dimensional manifold, few concrete examples have been investigated. Our main goal is to generate examples of the Clifford spectrum of three or four matrices where, with the assistance of a computer algebra package, we can calculate the Clifford spectrum.

math.OA↗

arXiv2019

Bulk Spectrum and K-theory for Infinite-Area Topological Quasicrystal

The bulk spectrum of a possible Chern insulator on a quasicrystalline lattice is examined. The effect of being a 2D insulator seems to override any fractal properties in the spectrum. We compute that the spectrum is either two continuous bands, or that any gaps other than the main gap are small. After making estimates on the spectrum, we deduce a finite system size, above which the K-theory must coincide with the K-theory of the infinite system. Knowledge of the spectrum and $K$-theory of the infinite-area system will control the spectrum and K-theory of sufficiently large finite systems. The relation between finite volume $K$-theory and infinite volume Chern numbers is only proven to begin, for the model under investigation here, for systems on Hilbert space of dimension around 17 million. The real-space method based on the Clifford spectrum allows for computing Chern numbers for systems on Hilbert space of dimension around 2.7 million. New techniques in numerical K-theory are used to equate the K-theory of systems of different sizes.

cond-mat.mes-hall↗

arXiv2019

A Guide to the Bott Index and Localizer Index

The Bott index is inherently global. The pseudospectal index is inherently local, and so now the preferred name is the localizer index. We look at these on a rather standard model for a Chern insulator, with an emphasis how to program these effectively. We also discuss how to tune the localizer index so it behaves like a global index.

math-ph↗

arXiv2016

K-Theory for Real C*-algebras via Unitary Elements with Symmetries

We prove that all eight KO groups for a real C*-algebra can be constructed from homotopy classes of unitary matrices that respect a variety of symmetries. In this manifestation of the KO groups, all eight boundary maps in the 24-term exact sequence associated to an ideal in a real C*-algebra can be computed as exponential or index maps with formulas that are nearly identical to the complex case.

math.OA↗

arXiv2016

Local Matrix Homotopies and Soft Tori

We present solutions to local connectivity problems in matrix representations of the form $C([-1,1]^{N}) \to A_{n,\varepsilon} \leftarrow C_{\varepsilon}(\mathbb{T}^{2})$ for any $\varepsilon\in[0,2]$ and any integer $n\geq 1$, where $A_{n,\varepsilon}\subseteq M_n$ and where $C_{\varepsilon}(\mathbb{T}^{2})$ denotes the {\bf Soft Torus}. We solve the connectivity problems by introducing the so called toroidal matrix links, which can be interpreted as normal contractive matrix analogies of free homotopies in differential algebraic topology. In order to deal with the locality constraints, we have combined some techniques introduced in this document with some techniques from matrix geometry, combinatorial optimization, classification and representation theory of C$^*$-algebras.

math.OA↗

arXiv2015

Pictures of KK-theory for real C*-algebras and almost commuting matrices

We give a systematic account of the various pictures of KK-theory for real C*-algebras, proving natural isomorphisms between the groups that arise from each picture. As part of this project, we develop the universal properties of KK-theory, and we use CRT-structures to prove that a natural transformation from F(A) to G(A) between homotopy equivalent, stable, half-exact functors defined on real C*-algebras is an isomorphism provided it is an isomorphism on the smaller class of C*-algebras. Finally, we develop E-theory for real C*-algebras and use that to obtain new negative results regarding the problem of approximating almost commuting real matrices by exactly commuting real matrices.

math.OA↗

arXiv2015

K-Theory and Pseudospectra for Topological Insulators

We derive formulas and algorithms for Kitaev's invariants in the periodic table for topological insulators and superconductors for finite disordered systems on lattices with boundaries. We find that K-theory arises as an obstruction to perturbing approximately compatible observables into compatible observables. We derive formulas in all symmetry classes up to dimension two, and in one symmetry class in dimension three, that can be computed with sparse matrix algorithms. We present algorithms in two symmetry classes in 2D and one in 3D and provide illustrative studies regarding how these algorithms can detect the scaling properties of phase transitions.

cond-mat.mes-hall↗

arXiv2014

Quantitative K-Theory Related to Spin Chern Numbers

We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the "log method" for commutator norms up to a specific constant.

math.OA↗

arXiv2013

Estimating Norms of Commutators

We find estimates on the norms commutators of the form [f(x), y] in terms of the norm of [x, y] assuming that x and y are contractions in a C*-algebra A, with x normal and with spectrum within the domain of f. In particular we discuss [x^2, y] and [x^(1/2), y] for 0 <=, x <=, 1. For larger values of δ= \|[x; y]\| we can rigorous calculate the best possible upper bound \|[f(x), y]\| for many f. In other cases we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound.

math.OA↗

arXiv2013

Almost Commuting Orthogonal Matrices

We show that almost commuting real orthogonal matrices are uniformly close to exactly commuting real orthogonal matrices. We prove the same for symplectic unitary matrices. This is in contrast to the general complex case, where not all pairs of almost commuting unitaries are close to commuting pairs. Our techniques also yield results about almost normal matrices over the reals and the quaternions.

math.OA↗

arXiv2013

Making Laplacians commute

In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. Our construction naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of applications in dimensionality reduction, shape analysis, and clustering, demonstrating that our method better captures the inherent structure of multi-modal data.

cs.CV↗

arXiv2013

Almost commuting self-adjoint matrices --- the real and self-dual cases

We show that a pair of almost commuting self-adjoint, symmetric matrices is close to a pair of commuting self-adjoint, symmetric matrices (in a uniform way). Moreover we prove that the same holds with self-dual in place of symmetric. The notion of self-dual Hermitian matrices is important in physics when studying fermionic systems that have time reversal symmetry. Since a symmetric, self-adjoint matrix is real, we get a real version of Huaxin Lin's famous theorem on almost commuting matrices. Similarly the self-dual case gives a version for matrices over the quaternions. We prove analogous results for element of real C^*-algebras of "low rank." In particular, these stronger results apply to paths of almost commuting Hermitian matrices that are real or self-dual. Along the way we develop a theory of semiprojectivity for real C^*-algebras.

math.OA↗

arXiv2013

Principal angles and approximation for quaternionic projections

We extend Jordan's notion of principal angles to work for two subspaces of quaternionic space, and so have a method to analyze two orthogonal projections in M_n(A) for A the real, complex or quaternionic field (or skew field). From this we derive an algorithm to turn almost commuting projections into commuting projections that minimizes the sum of the displacements of the two projections. We quickly prove what we need using the universal real C*-algebra generated by two projections.

math.OA↗

arXiv2013

Almost commuting unitary matrices related to time reversal

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 mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either the matrices over A for A the real numbers, A the complex numbers or A the algebra of quaternions. There are potential obstructions keeping k-tuples of almost commuting operators from being close to a commuting k-tuple. We consider two-dimensional geometries and so this obstruction lives in KO_{-2}(A). This obstruction corresponds to either the Chern number or spin Chern number in physics. We show that if this obstruction is the trivial element in K-theory then the approximation by commuting matrices is possible.

math.OA↗