Non-Hermitian Topological Insulators

By Eber Nolasco-Martinez

Topological Insulators: A summary

Graph from Asboth, J.K. https://arxiv.org/pdf/1509.02295.pdf

PT SymmetryParity Symmetry (Unitary Operator): Spatial Inversion

Time Symmetry (Antiunitary Operator):

So that

Systems where PT symmetry appearIncorporating system interaction with the environment can lead to PT symmetry

ex) laser systems with gain/loss regions

Real Spectrum of PT symmetry possibleIf PT symmetry is preserved, real eigenspectrum is possible.

Requires that [H,PT]=0 and that eigenvectors are the same for both H and PT.

SSH Model

B

A

m=7m=5 m=6

t1 t2 t1 t2

SSH Bulk Hamiltonian

Zero Modes of SSH model

Graph from Asboth, J.K. https://arxiv.org/pdf/1509.02295.pdf

SSH Topological Number Graph from Asboth, J.K. https://arxiv.org/pdf/1509.02295.pdf

Non-Hermitian SSH Model

B

A

m=7m=5 m=6

t1-𝝲/2t1+𝝲/2

t2

Spectrum of Non-Hermitian SSH

Invariant of the non-Hermitian SSH model

Graph from Yao S. and Wang Z. https://arxiv.org/pdf/1803.01876.pdf

Current Research● Studies exists to classify

topological insulators● Topological photonic

insulator systems can be made PT-invariant with gain loss regions

Conclusions● Studying the SSH model, we can study the existence of zero modes with

topological numbers● The non-Hermitian model shows the existence of a real spectrum of

eigenenergies due to PT symmetry● Existence of the zero mode can be quantified with a topological number as

well in analogous way to the hermitian model

ReferencesYao, Shunyu and Wang, Zhong. Edge states and topological invariants of non-Hermitian systems https://arxiv.org/pdf/1803.01876.pdf

Asboth, J.K. et al. A Short Course on Topological Insulators https://arxiv.org/pdf/1509.02295.pdf

Bender, Carl. Introduction to PT -Symmetric Quantum Theory https://arxiv.org/pdf/quant-ph/0501052.pdf

Feng, Liang et al. Single-mode laser by parity-time symmetry breaking http://science.sciencemag.org/content/346/6212/972

Viewpoint: Non-Hermitian Topological Systems https://physics.aps.org/articles/v11/96

https://arxiv.org/pdf/1803.01876.pdfhttps://arxiv.org/pdf/1509.02295.pdfhttps://arxiv.org/pdf/quant-ph/0501052.pdfhttp://science.sciencemag.org/content/346/6212/972https://physics.aps.org/articles/v11/96