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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