Almost commuting orthogonal matrices

Loring, Terry A.; Sørensen, Adam P. W.

doi:10.1016/j.jmaa.2014.06.019

Cited by 7 publications

(4 citation statements)

References 26 publications

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“…Almost commuting matrices and insulator systems (Loring et al) Some of the first mathematical attempts to understand the topological insulator problem came from Loring in collaboration with Hastings and Sørensen [29,30,54,55,56,57,58].…”

Section: 3mentioning

confidence: 99%

A non-commutative framework for topological insulators

2016

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Abstract. We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of noncommutative index theory of operator algebras. In particular we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realised as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample's (possibly noncommutative) Brillouin zone.

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Section: 3mentioning

confidence: 99%

A non-commutative framework for topological insulators

2016

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“…Quite remarkably, the question whether one can find triples of almost commuting matrices has generically a negative answer [30]. A similar story about unitary matrices is more involved [31,32]. There, the existence of almost commuting unitary matrices have some topological obstructions given by the so-called Bott indices.…”

Section: Introductionmentioning

confidence: 99%

Quasiconserved quantities in the perturbed spin- 12 XXX model

2022

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We consider the isotropic spin-1 2 Heisenberg spin chain weakly perturbed by a local translationally and SU(2)-invariant perturbation. Starting from the local integrals of motion of the unperturbed model, we modify them in order to obtain quasiconserved integrals of motion (charges) for the perturbed model. Such quasiconserved quantities are believed to be responsible for the existence of the prethermalization phase at intermediate timescales. We find that for a sufficiently local perturbation the quasiconserved quantities indeed exist, and we construct an explicit form for the first few of them.

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“…This is due to a K-theoretic obstruction [12][13][14], though it is true if this obstruction vanishes [7,8,15], or under the assumption of a spectral gap [16]. Imposing a form of self-duality analogous to time-reversal symmetry the relevant K-theoretic obstruction reduces to the spin Chern number of a fermionic system [17], highlighting a link between the fields of topologically ordered quantum systems [18] and approximately commuting matrices.…”

Section: Introductionmentioning

confidence: 99%

Approximate symmetries of Hamiltonians

Chubb

Flammia

2017

Journal of Mathematical Physics

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We explore the relationship between approximate symmetries of a gapped Hamiltonian and the structure of its ground space. We start by considering approximate symmetry operators, defined as unitary operators whose commutators with the Hamiltonian have norms that are sufficiently small. We show that when approximate symmetry operators can be restricted to the ground space while approximately preserving certain mutual commutation relations. We generalize the Stone-von Neumann theorem to matrices that approximately satisfy the canonical (Heisenberg-Weyl-type) commutation relations, and use this to show that approximate symmetry operators can certify the degeneracy of the ground space even though they only approximately form a group. Importantly, the notions of "approximate" and "small" are all independent of the dimension of the ambient Hilbert space, and depend only on the degeneracy in the ground space. Our analysis additionally holds for any gapped band of sufficiently small width in the excited spectrum of the Hamiltonian, and we discuss applications of these ideas to topological quantum phases of matter and topological quantum error correcting codes. Finally, in our analysis we also provide an exponential improvement upon bounds concerning the existence of shared approximate eigenvectors of approximately commuting operators under an added normality constraint, which may be of independent interest.

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Almost commuting orthogonal matrices

Cited by 7 publications

References 26 publications

A non-commutative framework for topological insulators

A non-commutative framework for topological insulators

Quasiconserved quantities in the perturbed spin- 12 XXX model

Approximate symmetries of Hamiltonians

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