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Two types of non-Hermitian flat bands
Daniel Leykam, Sergej Flach, Yidong Chong
Phys. Rev. B 96, 064305 (2017)
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Outline
1. Introduction: flat band photonic lattices
2. Non-Hermitian flat bands with parity-time symmetry
3. Flat bands from non-Hermitian coupling
4. Robustness of non-Hermitian flat bands to disorder
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Geometrical frustration in spin networks
Moessner & Ramirez, Physics Today 59, 24 (2006)
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Flat band lattices
Tight binding / coupled mode Hamiltonians with a band En
(k) = constant
for all k (c.f. slow light, group velocity vanishes locally)
Macroscopic degeneracy & geometrical frustration
Sensitive to perturbations (nonlinearity, disorder, interactions)
1D, 2D, 3D, fractal networks...
Photonic lattices, BECs, Hubbard models
Sawtooth Stub
Derzhko, Richter, & Maksymenko, Int.
J. Mod. Phys. B 29, 153007 (2015)
Diamond
M Hyrkas, V. Apaja & M Manninen, PRA 87, 023614 (2013).
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Flat bands for scalar waves
• Frustration leads to compactly localized, nondiffracting flat band eigenstates
• Destructive interference preventing diffraction, vanishing group velocity
Vicencio et al, Phys. Rev. Lett. 114, 245503 (2015)Xia et al, Opt. Lett. 41, 1435 (2016)
Mukherjee et al, Phys. Rev. Lett. 114,
245504 (2015)
Flat band
output
Dispersive
band outputFlat band eigenstate
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Aharonov-Bohm caging
Extreme case: all bands flat
Diamond lattice + π effective magnetic flux
Complete destructive interference prevents diffraction
Oscillatory dynamics (beating between bands)
Hasan, Iorsh, Kibis, & Shelykh, Phys. Rev. B 93, 125401 (2016)Longhi, Opt. Lett. 39, 5892 (2014)
Vidal, Mosseri, Doucot, Phys. Rev. Lett. 81, 5888 (1998)
z
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Sensitivity to perturbations & disorder
• As group velocity -> 0, effective perturbation strength diverges
• Dispersive band states strongly interact with localized flat band states
• Compact localized states destabilized by weak disorder
• Fano-resonance-like enhancement of perturbations
• Amplification of effective disorder strength
Band structure LatticeFano-Anderson model
Disorder strength
• Anderson localisation length
• Dispersive bands: ν = 2
• Flat band: ν = 1
Leykam, Flach, Bahat-Treidel, & A. S. Desyatnikov, Phys. Rev. B, 88, 224203 (2013)
Flach, Leykam, Bodyfelt, Matthies, & A. S. Desyatnikov, Europhys. Lett. 105, 30001 (2014)
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Non-Hermitian flat bands
• Photonic systems do not need to be Hermitian
• Gain or loss => non-Hermitian Hamiltonians
• Can gain/loss induce flat bands?
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Non-Hermitian Hamiltonians & parity-time (PT) symmetry
xnR(x)
nI(x)
• Gain balanced by loss: nR(x) = nR(-x), nI(x) = -nI(-x)
• Below threshold: propagating modes
• Threshold: exceptional point (non-Hermitian degeneracy)
• Above threshold: spontaneous amplification
Gain
Loss
Linear PT-symmetric coupler
Ruter et al., Nature Phys. 6, 192 (2010)
Zhang, Yong, Zhang, & He, Scientific
Reports 6, 24487 (2016)
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PT symmetric lattices
Each wavevector has different PT-breaking threshold
EPs separate real & complex eigenvalue k space regions
Propagation: asymmetric and non-reciprocal
Power not conserved: oscillates below threshold
Exponential amplification above threshold
Spectrum above PT-breaking threshold
Makris, El-Ganainy, Christodoulides, Musslimani,
Phys. Rev. Lett. 100, 103904 (2008)
Propagation below threshold
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Flat bands + PT symmetry
Add PT potential to Hermitian FB lattice
If PT potential breaks FB symmetry: thresholdless PT breaking
If PT potential preserves FB symmetry: FB unaffected
“PT dimer” flat band lattices -> nearly flat bands
No evidence of non-Hermitian frustration
Chern & Saxena, Opt. Lett. 40, 5806 (2015)Ge, Phys. Rev. A 92, 052103 (2015)
Molina, Phys. Rev. A 92, 063813 (2015)
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Non-Hermiticity induced flat band in trimer lattice
Trimer lattices: gain/loss can induce flat bands
Flat band occurs at critical gain/loss, embedded in dispersive band
Increasing gain/loss further, spectrum becomes complex
Can this example be generalized?
Sensitive to disorder?
Ramezani, Phys. Rev. A 96, 011802(R) (2017)
k k k k
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Frustration: competing interactions
Moessner & Ramirez, Physics Today 59, 24 (2006)
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PT trimers: two routes to PT-breaking
Dimer: single universal route to PT-breaking (coalescence of 2 eigenmodes)
Trimers: non-Hermitian coupling can also be PT symmetric
Two distinct routes to PT breaking: “ordinary” & “frustrated”
“Ordinary” trimer & PT breaking “Frustrated” trimer & PT breaking
“Dark” mode
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Non-Hermitian coupling
Coupling via medium with gain/loss
Microring resonators + auxiliary rings
Waveguides embedded in active medium
Exciton-polariton condensates
Parametric amplification: signal/idler modesAlexeeva, Barashenkov, Rayanov, & Flach,
Phys. Rev. A 89, 013848 (2014)
Longhi, Gatti, & Della Valle, Phys. Rev.
B 92, 094204 (2015)
Gentry and Popovic, Opt. Lett. 39, 4136 (2014)
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Non-Hermitian diamond ladder
General case: arbitrary coupling strengths κj
Assume a-c sites symmetric, Δa = Δc =0, Δb =Δ
Bipartite symmetry: flat band persists even for complex κj
How are other dispersive bands affected?
Effective
detuning
Effective
hopping
Compact, zero energy flat band eigenmodes
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Non-Hermitian Aharonov-Bohm cage
Interference between gain & loss legs
Effective magnetic flux 2θ,
Non-Hermitan coupling flattens dispersion
Critical coupling: completely flat spectrum, band of EPs
No PT breaking until dispersive bands coalesce
EP1
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Non-Hermitian Aharonov-Bohm cage: dynamics
Complete suppression of diffraction
Band of EPs: H(k) defective for all k
Quadratic power growth if EPs excited
Power oscillations if eigenstates excited
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Embedded non-Hermitian flat band
Gain & loss in a single leg
Non-Hermiticity broadens dispersion
Flat, dispersive bands cross at
Petermann factor Γ: crossings form embedded EPs
Despite EPs, Bloch wave spectrum can be purely real!
EP1 EP1
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Embedded non-Hermitian flat band: dynamics
Propagation sensitive to excitation of isolated EPs
Localized input: discrete diffraction + linear power growth
Broad input @ EP: conventional quadratic power growth
Flat band state input: trivial dynamics
Non-Hermiticity -> asymmetric propagation
Input wavevector k/π
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Disorder: Anderson localization?
Introduce random potential (waveguide detunings)
PT symmetry broken: non-real energy eigenvalues
Anderson localization or non-Hermitian delocalization?
http://lpmmc.grenoble.cnrs.fr/ Longhi, Gatti, & Della Valle, Scientific Reports 5, 13376 (2015)
Hatano & Nelson, Phys. Rev. Lett. 77, 570 (1996)
Diffusive transport Ballistic transport
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Non-Hermitian Aharonov-Bohm cage + disorder
Hermitian-like flat band at E=2C: linear eigenvalue shifts,
Non-Hermitian flat band at E=0: square root shifts,
Stronger sensitivity of non-Hermitian degeneracies to perturbations
Eigenfunctions well-approximated by CLS, insensitive to W
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Embedded non-Hermitian flat band + disorder
Mean participation number P: FB eigenmodes localized & W independent!
Most FB eigenvalues have linear shifts,
1 pair of EP-like eigenvalues with
The EP-like eigenmodes delocalize as W -> 0
EP modes protect remaining FB modes from delocalization
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Summary
Non-Hermitian flat bands from non-Hermitian coupling + sublattice symmetry
Exceptional points (EPs) & flat band may be “embedded” in dispersive band
Power law amplification when EPs excited
Disorder: flat band states protected from delocalization by gap or EPs
Phys. Rev. B 96, 064305 (2017)
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Positions available!
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