Two types of non-Hermitian flat bands

Daniel Leykam, Sergej Flach, Yidong Chong

Phys. Rev. B 96, 064305 (2017)

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

Geometrical frustration in spin networks

Moessner & Ramirez, Physics Today 59, 24 (2006)

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).

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

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

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)

Non-Hermitian flat bands

• Photonic systems do not need to be Hermitian

• Gain or loss => non-Hermitian Hamiltonians

• Can gain/loss induce flat bands?

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)

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

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)

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

Frustration: competing interactions

Moessner & Ramirez, Physics Today 59, 24 (2006)

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

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)

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

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

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

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

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

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

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

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

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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