Sound Topology, Duality, Coherence and Wave-mixing : An Introduction to the Emerging New Science of Sound

374 Pages·2017·10.67 MB·English

by  Deymier

Springer Series in Solid-State Sciences 188
Pierre Deymier
Keith Runge
Sound Topology,
Duality,
Coherence and
Wave-Mixing
An Introduction to the Emerging New
Science of Sound
Springer Series in Solid-State Sciences
Volume 188
Serieseditors
BernhardKeimer,Stuttgart,Germany
RobertoMerlin,AnnArbor,MI,USA
Hans-JoachimQueisser,Stuttgart,Germany
KlausvonKlitzing,Stuttgart,Germany
The Springer Series in Solid-State Sciences consists of fundamental scientific
books prepared by leading researchers in the field. They strive to communicate,
in a systematic and comprehensive way, the basic principles as well as new
developmentsintheoreticalandexperimentalsolid-satephysics.
Moreinformationaboutthisseriesathttp://www.springer.com/series/682
Pierre Deymier • Keith Runge
Sound Topology, Duality,
Coherence and Wave-Mixing
An Introduction to the Emerging
New Science of Sound
PierreDeymier KeithRunge
Dept.ofMaterialsScience Dept.ofMaterialsScienceandEngineering
andEngineering UniversityofArizona
UniversityofArizona Tucson,Arizona
Tucson,Arizona USA
USA
ISSN0171-1873 ISSN2197-4179 (electronic)
SpringerSeriesinSolid-StateSciences
ISBN978-3-319-62379-5 ISBN978-3-319-62380-1 (eBook)
DOI10.1007/978-3-319-62380-1
LibraryofCongressControlNumber:2017946710
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To Le´onard Dobrzynski
Preface
Anewscienceofsoundisemergingbasedonthescientificprinciplesofsymmetry
breaking and interactions. This new science of sound focuses on aspects of wave
phenomenathathavenotbeenemphasized intraditionalinstructionandinpartic-
ular the four paradigm-changing scientific notions of sound topology, duality,
coherence,andwavemixing.
Topology Whensoundwavespropagateinmediaundersymmetrybreakingcon-
ditions,theymayexhibitamplitudesA(k)¼A eiη(k)thatacquireageometricphaseη
0
leading to non-conventional topology. Broken symmetry phenomena lead to the
concept of symmetry-protected topological order. Topological acoustic waves
promise designs and new device functionalities for acoustic systems that are
unique,robust,andavoidthelossofcoherence.
Duality The self-interaction of a wave through its supporting medium creates
acoustic wave states determined by self-interference phenomena. In the phonon
representation of sound waves, these self-interference phenomena uncover the
notionofdualityinthequantumstatistics(i.e.,bosonversusfermioncharacterized
by the symmetry of multiple particle states). It also enables the development of
analogieswithquantummechanicsandQuantumFieldTheory.
Coherence Interactions in nonlinear elastic media cause multiple scattering and
resonances of sound waves. Individually, these resonances may lead to sound
waveswithnon-conventionaltopologies.However,inthecaseofmultiplechannels
for phonon scattering, nonlinearity leads to the loss of phase coherence while
retaining a broken time-reversalsymmetry demonstrated byacousticwaveampli-
tudedegradation.Mechanicalenergycanbedirectedinaone-way,irreversible,and
targeted fashion from a linear system to a nonlinear system in coupled nonlinear
andlinearvibrationalsystems.Targetedenergytransferoffersstrategiesforvibra-
tionandsoundmanagement.
Wave Mixing Media supporting different types of waves and their sources can
coherentlyconvertenergybetweensoundandotherphysicalandbiologicalwaves.
vii
viii Preface
Topological effects have been shown for biological waves (calcium signals)
resulting fromacoustic wave-drivenspatiotemporalmodulation ofcell membrane
conductance in biological tissue. Coherent phonons can bestow non-conventional
topologicalcharacteristicstoelectronicwavefunctionsandtopologicalspinwaves
can arise from the spatiotemporal modulation of the spin coupling constant in
ferromagneticmediaduetophonons.
These notions are taken further and related to the development of acoustic
analogues of other physical phenomena ranging from quantum mechanics to
generalrelativity.Theseanaloguesofferperspectivesforapplicationsandtechno-
logicaldevelopmentsofthenewscienceofsound.
This book gives an introduction to these scientific notions and analogues by
attempting to present a number of “simple” models. The authors have developed
these models in the spirit of Jacques Friedel’s approach to theoretical research.
Friedel’s research utilized extensively approximate but simple models that can be
understoodbynon-specialists:“J’aiessaye´dede´velopperdesmode`lesapproximatifs
maissimples,compre´hensiblesetmeˆmeutilisablespardesnon-spe´cialistes”[1].
This book is not an exhaustive review of models relative to the four notions
introduced above. These models serve as demonstrations of these notions and
provideonlyapartialfoundationfortheirunificationintoanewscienceofsound.
Mostofthe simple models presented hererepresent contributions ofthe authors
andmanyoftheircollaborators,studentsandpostdoctoralfellows,whomwewould
liketoacknowledgeatthistime.Inparticular,Prof.JeromeVasseur,whoproofread
portionsofseveralchapters.Thisbookpaintsapictureoftheemergingnewscience
ofsoundthatreflectsthe biasedperception andunderstandingoftheauthorsalone.
We would also like to acknowledge partial support from the US National Science
Foundation(NSF)aswellastheFrenchCentreNationaldelaRechercheScientifique
(CNRS)throughtheLaboratoireInternationalAssocie´ “MaterialsandOptics.”
The book is divided into six chapters with their own set of references, each of
whichisintendedtobeself-containedandabletobereadindependently.Incasethere
areoverlappingconcepts,werefertotheappropriatesubsectionofotherchapters.
We hope that this book will stimulate future interest in the emerging field of
soundandwillinitiatenewdevelopmentsinthefourscientificnotionsoftopology,
duality,coherence,andwavemixing.
Tucson,AZ PierreA.Deymier
KeithRunge
Reference
[1] Entrevue avec Jacques Friedel, Paris, 17 octobre 2001, Herve´ Arribart et
Bernadette Bensaude-Vincent. http://authors.library.caltech.edu/5456/1/hrst.
mit.edu/hrs/materials/public/index.html
Contents
1 IntroductiontoSpringSystems. . . . .. . . . .. . . . . .. . . . . .. . . . . .. 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 One-DimensionalMonatomicHarmonicCrystal. . . . . . . . . . . . . 1
1.3 PhaseandGroupVelocity. . . .. . . . . .. . . . . .. . . . . . .. . . . . .. 3
1.4 One-DimensionalDiatomicHarmonicCrystal. . . . . . . . . . . . . . . 4
1.5 One-DimensionalMonatomicCrystalwithSpatiallyVarying
Stiffness.. . . .. . . .. . . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. 6
1.6 Green’sFunctionApproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 MonatomicCrystalwithaMassDefect. . . . . . . . . . . . . . . . . . . . 10
1.7.1 MonatomicHarmonicCrystalwithaGeneralPerturbation.... 12
1.7.2 LocallyResonantStructure. . . . . . . . . . . . . . . . . . . . . . . 13
1.8 InterfaceResponseTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8.1 FundamentalEquationsoftheInterfaceResponse
Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8.2 Green’sFunctionoftheCleaved1-DMonatomic
Crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8.3 FiniteMonatomicCrystal. . . . . . . . . . . . . . . . . . . . . . . . 20
1.8.4 1-DMonatomicCrystalwithOneSideBranch. . . . . . . . . 22
1.8.5 1-DMonatomicCrystalwithMultipleSideBranches. . . . 24
1.9 Conclusion. . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . 28
Appendix1:CodeBasedonGreen’sFunctionApproach. . . . . . . . . . . 28
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 PhaseandTopology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 HarmonicOscillatorModelSystems. . . . . . . . . . . . . . . . . . . . . . 40
2.3.1 GeometricPhaseandDynamicalPhaseoftheDamped
HarmonicOscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.2 GeometricPhaseoftheDrivenHarmonicOscillator. . . . . 42
ix
x Contents
2.3.3 TopologicalInterpretationoftheGeometricPhase. . . . . . 44
2.4 ElasticSuperlatticeModelSystem. . . .. . .. . . .. . . .. . . .. . . .. 48
2.4.1 GeometricPhaseofaOne-DimensionalElastic
Superlattice:ZakPhase. . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.2 TopologicalInterpretationoftheZakPhase. . . . . . . . . . . 52
2.5 Green’sFunctionApproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.5.1 Green’sFunctionsandBerryConnection. . . . . . . . . . . . . 53
2.5.2 TheOne-DimensionalHarmonicCrystal. .. . . . . .. . . . .. 55
2.5.3 TheOne-DimensionalHarmonicCrystalwithSide
Branches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.6 TopologicalModesatInterfacesBetweenMediawithDifferent
ZakPhases. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . 63
2.7 Conclusion. . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . 65
Appendix1:EigenValuesandEigenVectorsinOne-Dimensional
ElasticSuperlattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Appendix2:DiscreteOne-DimensionalMonatomicCrystalwithSpatially
VaryingStiffness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Appendix3:IntroductiontoGreen’sFunctionFormalism. . . . . . . . . . . 74
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3 TopologyandDualityofSoundandElasticWaves. . . . . . . . . . . . . . 81
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3 IntrinsicTopologicalPhononicStructures. . . . . . . . . . . . . . . . . . 83
3.3.1 TwoCoupledMass-SpringHarmonicCrystalsinthe
Long-WavelengthLimit. . . .. . . . . . . . . .. . . . . . . . . .. . 83
3.3.2 SingleHarmonicCrystalGroundedtoaRigidSubstrate
intheLong-WavelengthLimit. . . . . . . . . . . . . . . . . . . . . 88
3.3.3 SingleDiscreteHarmonicCrystalGroundedtoaRigid
SubstrateBeyondtheLong-WavelengthLimit. . . . . . . . . 92
3.4 ExtrinsicTopologicalPhononicStructure. . . . . . . . . . . . . . . . . . 109
3.4.1 Time-DependentElasticSuperlattice. . . . . . . . . . . . . . . . 109
3.4.2 MultipletimeScalePerturbationTheoryofthe
Time-DependentSuperlattice. . . . . . . . . . . . . . . . . . . . . . 112
3.4.3 TopologyofElasticWaveFunctions. . . . . . . . . . . . . . . . 115
3.5 MixedIntrinsicandExtrinsicTopologicalPhononicStructure. . . 120
3.5.1 One-DimensionalMass-SpringHarmonicCrystalGrounded
toaSubstratewithSpatio-TemporalModulation. . . . . . . . 120
3.6 SeparableandNon-separableStatesinElasticStructures. . . . . . . 127
3.6.1 SeparabilityofSystemsComposedofHarmonic
Oscillators. . . . .. . . .. . . .. . . . .. . . .. . . .. . . . .. . . .. 128
3.6.2 PartitioningofaHarmonicCrystalintoNormalModes. . . 129
3.6.3 Uncoupled1-DHarmonicCrystals. . . . . . . . . . . . . . . . . . 130
3.6.4 SeparabilityandNon-separabilityoftheStatesofCoupled
HarmonicCrystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133